[r.r. huilgol n. thien fluid mechanics of org

FLUID MECHANICS OF VISCOELASTICITY

RHEOLOGY SERIES

Advisory Editor: K. Waiters FRS, Professor of Applied Mathematics, University of Wales, Aberystwyth, U.K.

Vol. 1 Numerical Simulation of Non-Newtonian Flow (M.J. Crochet, A.R. Davies and K. Waiters)

Vol. 2 Rheology of Materials and Engineering Structures (Z. Sobotka)

Vol. 3 An Introduction to Rheology (H.A. Barnes, J.F. Hutton and K. Waiters)

Vol. 4 Rheological Phenomena in Focus (D.V. Boger and K. Waiters)

Vol. 5 Rheology for Polymer Melt Processing (Edited by J-M. Piau and J-F. Agassant)

Vol. 6 Fluid Mechanics of Viscoelasticity (R.R. Huilgol and N. Phan-Thien)

FLUID MECHANICS OF VISCOELASTICITY General Principles, Constitutive Modelling, Analytical and Numerical Techniques

R.R. Huilgol The Flinders University of South Australia, Adelaide, Australia

and

N. Phan-Thien The University of Sydney, Australia

1997

E l s e v i e r

A m s t e r d a m - L a u s a n n e - N e w Y o r k - O x f o r d - S h a n n o n - T o k y o

ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands

The illustration used for the cover was published in Rheologica Acta 9 (1970) 30 and is reproduced with permission from the publishers, Steinkopf Verlag, Darmstadt, Germany. It illustrates how die-injection techniques can be employed to investigate experimentally the secondary flow generated by a rotating sphere. The figure shows the secondary flow for a 1.3% solution of polyisobutylene in decalin.

ISBN: 0 444 82661 0

�9 1997 ELSEVIER SCIENCE B.V. All rights reserved.

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This.book is printed on acid-free paper.

Printed in The Netherlands.

To

Our Parents

Ramarao and Subhadrabai Huilgol Cuc Bourdeau and Canh Phan-Thien

and

The Women in Our Lives

Dimitra Beroukas and Lai-Kuen

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Foreword

It is a pleasure to see the appearance of a new, comprehensive book on rheology. It has been around 10 years since the last new work was published, and in tha t t ime there have been significant changes in the focus and the subject. This new volume addresses these new areas without losing track of classical results.

Specifically, the areas of suspension mechanics, stability and computational the- ology have simply exploded in scope and substance in the last decade, and this is one of the first, if not the first, book of a comprehensive nature to t reat these topics in depth.

On a personal note, the amicable collaboration of two of my former students in this excellent venture gives me great satisfaction, and I commend this new venture to the reader.

Roger I. Tanner Sydney 24 December 1996

Preface

Fifty years ago, theoretical and experimental investigations into rubber elasticity and normal stress effects in non-Newtonian fluids, as well as studies of visco-plastic substances ushered in the modern era of non-linear continuum mechanics. Over the next two decades, the foundations had well and truly been laid with the discovery of the principles underpinning the development of constitutive relations in nonlinear viscolelasticity, and the deep and fundamental results in the kinematics of continuous media. So much so that around 1965, theoretical nonlinear continuum mechanics was far ahead of the experimental techniques and devices necessary to measure the material properties. Since then, of course, experiments have inter- acted with the relevant theoretical concepts to explain the existence of pressure hole errors, the rod climbing effect, and the die swell phenomenon, to name a few.

The influence of microstructural theories on the formulation of constitutive relations began to take effect from the beginning of the 1970s, and it has now reached a level of sophistication comparable to that of the continuum approach. By the end of that decade, initial researches into the numerical modelling of complex flows of viscoelastic fluids had begun. Subsequently, this field of research grew exponentially and, at present, it has led to an excellent agreement with the theoretical predictions, or experimental observations, or a combination of both in many cases. Nevertheless, there are important problems which remain unsolved.

In writing this treatise, our goal has been to highlight the major discoveries in the forementioned areas, and to present a number of them in sufficient breadth and depth so that the novice can learn from the examples chosen, and the expert can use them as a reference when necessary. These aims have forced us to dissociate ourselves from penning a survey of viscoelastic fluid mechanics. On the other hand, to cover in a single volume all the important areas of research at the level of breadth and depth demanded by them is, by now, impossible. Hence, we have been selective in order to be descriptive and instructive, and perhaps prescriptive as well.

Of course, given that a number of topics of interest have not yet been developed to a theoretical level from which applications can be made in a routine manner, we have included them to draw the attention of the reader to the state of affairs so that research into these matters can take place. For example, the sections which deal with domains bounded by fractal curves or surfaces show that the existence of a stress tensor in such regions is still open to question. Similarly, the constitutive modelling of suspensions, especially at high volume concentrations, with the corresponding particle migration from high to low shear regions is still very sketchy.

There are seven chapters in this treatise, and since each chapter begins with its own extensive summary, we shah not reproduce them here. Rather, we shall make some pertinent remarks only. The first two, grouped under the category of General Principles, deal with the kinematics of continuous media and the balance laws of mechanics, including the existence of the stress tensor and extensions of the laws of vector analysis to domains bounded by fractal curves or surfaces. The third and fourth chapters, under the banner of Constitutive Modelling, present the tools necessary to formulate constitutive equations from the continuum or the microstructural approach. The last three chapters, under the rubric of Analytical and Numerical Techniques, contain most of the important results in the domain of the fluid mechanics of viscoelasticity, and form the core of the book. The fifth chapter contains the complete list of steady and unsteady flows which are possible in all incompressible viscoelastic fluids, including viscometric and non-viscometric flows. In the next chapter, to mention a few, we have chosen simple models to examine the roles of the normal stress differences in a variety of situations; to study the bifurcation and stability of solutions affected by elastic and inertial effects; to examine the impact of pre-shearing on an extensional flow. The final chapter is a compendium of numerical methods available to solve fluid flow problems in viscoelasticity. Our desire, naturally, is to convince the reader that we have read widely and produced a book which is rich in its coverage and exhaustive in its contents.

Of course, this coverage has forced us to confront the problem of dealing with different notations for the same concept or entity. We have tried to overcome this difficulty by making each chapter as self-contained as possible. Certainly, if one wishes to combine one of continuum or microstructural modelling with analytical and/or numerical techniques, it may become necessary to learn to use a new set of notations or symbols. We believe that this is a small price to pay for the rewards that will be obtained from exploring new ideas.

The task of writing this treatise has taken us nearly four years and during this time, our colleagues and our respective Universities have supported us handsomely. Specifically, the book began to take shape when one of us (R.R. H) was awarded a period of study leave by Flinders University in 1993, and completed by us over the next two years, and during the second period of leave from Flinders University in 1996-97, time was spent in revising the manuscript in which we both took our share of the load.

A small part of this period was spent in Wales by one of us (R.R.H) with Pro- fessor K. Waiters FRS, University of Wales, Aberystwyth, and we wish to thank him for his hospitality. Over a larger period of nearly four months, CNRS (France) and the Laboratoire de Rh~ologie, Grenoble, were instrumental in helping us to

xi

put the finishing touches to the manuscript through the award of a Visiting Re- search Fellowship to one of the authors (R.R.H). We wish to thank Professor J.-M. Piau, Directeur, Laboratoire de Rhbologie, for organising the visit and making the necessary resources available, and his staff for their assistance.

The editing of the manuscript was made easier by the enormous help given prior to this period of activity by Professor K. Walters FRS. He read the initial draft and made many suggestions for improvement from the grammatical to the stylistic parts. We are deeply indebted to him for the investment of his time and energy in our endeavour. Naturally, we alone are responsible for all the errors and infelicitous use of language which may remain.

We are also appreciative of the Foreword written by Professor R.I. Tanner FAA FTS. It is a pleasure to thank him for introducing us to the subject of viscoelastic fluids many years ago, and being a colleague, friend and collaborator since that time.

This book is dedicated to our parents for setting us along the scientific path, and to our spouses for their support. Without the encouragement and affection of the women in our lives, this tome would have been just a dream.

Grenoble & Sydney January 1997

R.R. Huilgol & N. Phan-Thien

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Contents

G e n e r a l P r i n c i p l e s

I K i n e m a t i c s o f F l u i d F l o w 1

1 Kinemat ica l Prel iminaries . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1 Relat ion between the Velocity and Deformation Gradients . . 5

1.2 Connection with Dynamical System Theory . . . . . . . . . . 5

1.3 Polar Decomposit ion . . . . . . . . . . . . . . . . . . . . . . . 9

1.4 Relat ive Cauchy-Green Strain Tensor . . . . . . . . . . . . . 11

2 P a t h Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1 Linear Autonomous Systems . . . . . . . . . . . . . . . . . . 12

2.2 Calculat ing the Exponent ial Function of a Matr ix . . . . . . 13

2.3 Linear Non-Autonomous Systems . . . . . . . . . . . . . . . . 16

2.4 Rigid Body Motion . . . . . . . . . . . . . . . . . . . . . . . . 18

2.5 Per tu rba t ion Problems . . . . . . . . . . . . . . . . . . . . . . 19

2.6 Oldroyd's Method . . . . . . . . . . . . . . . . . . . . . . . . 20

3 The Relat ive Deformation Gradient and Strain Tensors . . . . . . . 23

4 Rivlin-Ericksen Tensors . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.1 Oldroyd's Formulae . . . . . . . . . . . . . . . . . . . . . . . 26

5 Approximat ions to the Relative Strain Tensor . . . . . . . . . . . . . 29 5.1 Infinitesimal Strain . . . . . . . . . . . . . . . . . . . . . . . . 29

5.2 Infinitesimal Velocity . . . . . . . . . . . . . . . . . . . . . . . 30 5.3 Small Displacements added to a Large Motion . . . . . . . . 31

5.4 Small Velocity added to Large . . . . . . . . . . . . . . . . . 33 6 Flows such t h a t An - 0 for any Odd Integer and Higher . . . . . . . 35

6.1 S ta te of Rest . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

xiv Contents

6.2 Flows with Deformation Gradients Linear in t . . . . . . . . . 35 6.3 Flows with Deformation Gradients Quadrat ic in t . . . . . . . 37

6.4 Flows where F is a polynomial in t of Order Three or More . 37 6.5 Non-Homogeneous Deformation Gradients . . . . . . . . . . . 38

6.6 Motions with Zero Acceleration . . . . . . . . . . . . . . . . . 38 6.7 Higher Order Non-Homogeneous Deformation Gradients . . . 38

6.8 Deformat ion Gradients which are Infinite Series in t . . . . . 39

7 Viscometric Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 7.1 Flows with a Nonuniform Shear Ra te . . . . . . . . . . . . . 42

7.2 Flows with a Uniform Shear Rate . . . . . . . . . . . . . . . . 45 8 Deformat ion Gradients Equivalent to Exponential Functions . . . . . 47

8.1 Doubly Superposed Viscometric Flows . . . . . . . . . . . . . 47

8.2 EXtensional Flows . . . . . . . . . . . . . . . . . . . . . . . . 48 8.3 Non-Extensional Flows . . . . . . . . . . . . . . . . . . . . . . 50

9 Motions with Constant Stretch History . . . . . . . . . . . . . . . . . 51 9.1 Relat ive Strain Tensor: Proper t ies . . . . . . . . . . . . . . . 51

9.2 Necessary Conditions . . . . . . . . . . . . . . . . . . . . . . 54 9.3 The Impor tance of A1, A2 and A3 . . . . . . . . . . . . . . . 55 9.4 Sufficient Conditions . . . . . . . . . . . . . . . . . . . . . . . 56 9.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 9.6 Strong and Weak Flows . . . . . . . . . . . . . . . . . . . . . 59

10 Effects on Local Kinematics from Translat ion and Rota t ion . . . . . 60 11 Zorawski Velocity Fields - Global Effects of Translat ion and Rota t ion 64

11.1 Uns teady Flows with A~(t) = 0, n ~_ 3 . . . . . . . . . . . . 66 11.2 Compatibi l i ty Conditions . . . . . . . . . . . . . . . . . . . . 68

11.3 Mater ia l Description of Zorawski Velocity Fields . . . . . . . 69

12 Local Change of Reference Configuration . . . . . . . . . . . . . . . 70 Appendix to Chapter 1" Basic Results from Tensor and Dyadic Analysis 74

Tensor Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 74 Covariant Derivative . . . . . . . . . . . . . . . . . . . . . . . 76 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 Dyadic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 81 Gradients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

Matr ix Multiplication . . . . . . . . . . . . . . . . . . . . . . 83 Divergence and Curl . . . . . . . . . . . . . . . . . . . . . . . 83

2 B a l a n c e E q u a t i o n s for S m o o t h a n d N o n - S m o o t h R e g i o n s 85 13 Reynolds ' Transpor t Theorem . . . . . . . . . . . . . . . . . . . . . . 85

14 Conservation of Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 15 Balance of Linear Momen tum . . . . . . . . . . . . . . . . . . . . . . 89

16 Balance of Angular Momen tum . . . . . . . . . . . . . . . . . . . . . 95 17 Balance of Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

18 Other Forms of Equat ions of Motion . . . . . . . . . . . . . . . . . . 98 19 The Length of a Curve . . . . . . . . . . . . . . . . . . . . . . . . . . 99 20 Fractal Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

21 The Box or Minkowski Dimension . . . . . . . . . . . . . . . . . . . 107 22 Unit Tangent and Unit External Normal . . . . . . . . . . . . . . . . 109

Contents xv

23 Flux across a Fractal Curve: The Divergence Theorem . . . . . . . . 113 24 Stokes' Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 25 A Coordinate Free Proof of the Existence of the Stress Tensor . . . . 117 26 Stress Singularities and Shape of a Body . . . . . . . . . . . . . . . . 120

26.1 Continuity of the Stress Vector . . . . . . . . . . . . . . . . . 120 26.2 Sets of Fini te Per imeter . . . . . . . . . . . . . . . . . . . . . 122 26.3 Shape of a Body and its Boundary . . . . . . . . . . . . . . . 123

Constitutive Modelling

3 Formulat ion o f C o n s t i t u t i v e Equat ions- -The Simple Fluid 125 27 Const i tu t ive Relat ions and General Principles of Formulat ion . . . . 126

27.1 Const i tut ive Relations - Are They Necessary? . . . . . . . . . 126 27.2 General Principles . . . . . . . . . . . . . . . . . . . . . . . . 128 27.3 Rota t ions or Rotat ions and Reflections . . . . . . . . . . . . . 129 27.4 Simple Materials and Simple Fluids . . . . . . . . . . . . . . 130

28 Symmet ry Restr ict ions on Const i tut ive Equations of Elastic Materials130 28.1 Isotropic Solids . . . . . . . . . . . . . . . . . . . . . . . . . . 131 28.2 Elastic Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 28.3 Invariance Restrict ions for Singular F . . . . . . . . . . . . . 133

29 Object ivi ty Restr ict ions on Const i tut ive Equat ions of Elastic Materials 134 29.1 Isotropic Solids . . . . . . . . . . . . . . . . . . . . . . . . . . 136

30 Integri ty Basis for an Elastic Fluid . . . . . . . . . . . . . . . . . . . 137

31 Restr ict ions due to Symmetry: Simple Fluids . . . . . . . . . . . . . 138 32 Object ivi ty Restr ict ions on Const i tut ive Equations of Simple Mater ia l s l40

32.1 Simple Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 33 The Incompressible Simple Fluid . . . . . . . . . . . . . . . . . . . . 142

Appendix A to Chapter 3" Exploiting Integri ty Bases . . . . . . . . . 144 Vector Valued Functions . . . . . . . . . . . . . . . . . . . . . 145 Symmetr ic Tensor Valued Functions . . . . . . . . . . . . . . 146 Integri ty Basis and Function Basis . . . . . . . . . . . . . . . 147

Appendix B to Chapter 3" Const i tut ive Approximations . . . . . . . 148

4 C o n s t i t u t i v e Equat ions Der ived From Micros tructures 155 34 Dilute Polymer Solutions . . . . . . . . . . . . . . . . . . . . . . . . 157

34.1 General Physical Characterist ics . . . . . . . . . . . . . . . . 157 34.2 Random-Walk Model . . . . . . . . . . . . . . . . . . . . . . . 159 34.3 Forces on a Chain . . . . . . . . . . . . . . . . . . . . . . . . 166 34.4 Fluctuat ion-Dissipat ion Theorem . . . . . . . . . . . . . . . . 169 34.5 Fokker-Planck Equat ion . . . . . . . . . . . . . . . . . . . . . 172 34.6 Smoluchowski Equat ion . . . . . . . . . . . . . . . . . . . . . 174 34.7 Smoothed-Out Brownian Force . . . . . . . . . . . . . . . . . 177 34.8 The Stress Tensor . . . . . . . . . . . . . . . . . . . . . . . . 177 34.9 Elastic Dumbbell Model . . . . . . . . . . . . . . . . . . . . . 180 34.10 Rigid Dumbbel l . . . . . . . . . . . . . . . . . . . . . . . . . . 199 34.11 Rouse Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

34.12 T ime-Tempera tu re Superposit ion Principle . . . . . . . . . . 214

xvi Contents

35 Network Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 35.1 Affine Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

35.2 Const i tut ive Equat ion . . . . . . . . . . . . . . . . . . . . . . 222

35.3 Some Special Cases . . . . . . . . . . . . . . . . . . . . . . . . 225 36 Repta t ion Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

36.1 Doi-Edwards Model . . . . . . . . . . . . . . . . . . . . . . . 231

36.2 Approximat ions . . . . . . . . . . . . . . . . . . . . . . . . . . 234 36.3 Differential Models . . . . . . . . . . . . . . . . . . . . . . . . 235 36.4 Curtiss-Bird Model . . . . . . . . . . . . . . . . . . . . . . . . 238

37 Suspension Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 37.1 Int roduct ion . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

37.2 Bulk Suspension Proper t ies . . . . . . . . . . . . . . . . . . . 242

37.3 Dilute Suspension of Spheroids . . . . . . . . . . . . . . . . . 245

37.4 Lubricat ion Theories . . . . . . . . . . . . . . . . . . . . . . . 249 37.5 Fibre Suspensions . . . . . . . . . . . . . . . . . . . . . . . . 256

37.6 Flow-Induced Migrat ion . . . . . . . . . . . . . . . . . . . . . 261

A n a l y t i c a l a n d N u m e r i c a l T e c h n i q u e s

5 T h e S h a p e a n d N a t u r e o f G e n e r a l So lu t ions 271 38 Some Consequences of the Isotropy of the Const i tut ive Functional . 272

38.1 P a t h Line and the Const i tut ive Functional . . . . . . . . . . 272 38.2 Isotropy of the Const i tut ive Functional . . . . . . . . . . . . 273

38.3 Motions with Constant Stretch History . . . . . . . . . . . . 273 38.4 Steady Simple Shearing Flow . . . . . . . . . . . . . . . . . . 274 38.5 Const i tut ive Relation in Terms of A1 and A2 . . . . . . . . . 276 38.6 Extensional Flows . . . . . . . . . . . . . . . . . . . . . . . . 277

38.7 Uns teady Shear Flow . . . . . . . . . . . . . . . . . . . . . . 278 38.8 Non-Viscometric Flows . . . . . . . . . . . . . . . . . . . . . 278

39 Equat ions of Motion in Curvilinear Coordinates . . . . . . . . . . . . 280 39.1 Body Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 39.2 Iner t ia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282

39.3 Homogeneous Velocity Fields . . . . . . . . . . . . . . . . . . 282

39.4 General Procedure for Solutions of Problems . . . . . . . . . 282 40 Viscometric Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

40.1 Couet te Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 40.2 Par t ia l Controllabili ty . . . . . . . . . . . . . . . . . . . . . . 284

40.3 Roles of the Divergence of the Shear Stress Tensor and the Acceleration Field . . . . . . . . . . . . . . . . . . . . . . . . 286

40.4 Divergence of the Shear Stress Tensor is I r rota t ional . . . . . 286

40.5 Viscosity Determines the Velocity Field . . . . . . . . . . . . 287 40.6 Flow between Rota t ing Conical Surfaces . . . . . . . . . . . . 287 40.7 Couet te F l o w - The Velocity Field . . . . . . . . . . . . . . . 288 40.8 Poiseuille Flow . . . . . . . . . . . . . . . . . . . . . . . . . . 289 40.9 Helical Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 40.10 Channel Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 294

Contents xvii

40.11 Torsional Flow and the Normal Stress Differences N1 and N2 295

40.12 Cone-and-Pla te Flow and N1 . . . . . . . . . . . . . . . . . . 297

40.13 Generalised Torsional Flow . . . . . . . . . . . . . . . . . . . 298

40.14 Measurement of N2 . . . . . . . . . . . . . . . . . . . . . . . . 300

41 Rect i l inear Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

41.1 Normal Stress on a P ipe Wall and a Free Surface . . . . . . . 305 41.2 Addi t ional Examples and Counter Examples to Ericksen's

Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

41.3 Secondary Flows . . . . . . . . . . . . . . . . . . . . . . . . . 309

42 Non-Viscometr ic Flows . . . . . . . . . . . . . . . . . . . . . . . . . . 309

42.1 Hel ica l - Torsional Flow . . . . . . . . . . . . . . . . . . . . . 310

42.2 Helical Flow - Axial Motion of Fanned Planes . . . . . . . . . 311

42.3 Flow in t he Ecccentric Disk Rheomete r . . . . . . . . . . . . 311 42.4 Simple Extensional Flows . . . . . . . . . . . . . . . . . . . . 313

43 Dynamical ly Compat ib le Uns teady Flows . . . . . . . . . . . . . . . 315 43.1 Uns teady Helical Flow . . . . . . . . . . . . . . . . . . . . . 316

43.2 Uns teady Channel Flow . . . . . . . . . . . . . . . . . . . . . 316

43.3 Uns teady Torsional and Cone-and-Pla te Flows . . . . . . . . 317 43.4 Uns teady Extensional Flow . . . . . . . . . . . . . . . . . . . 317

43.5 Flow Genera ted by Squeezing a Wedge with Extension . . . . 317

43.6 Flow Genera ted by Squeezing a Cone with Extension . . . . . 318 Append ix to Chap te r 5: Strain Jumps and Stress Relaxat ion . . . . 320

Stress Response to Strain Jumps . . . . . . . . . . . . . . . . 320

Stress Relaxat ion . . . . . . . . . . . . . . . . . . . . . . . . . 322

6 Simple Models and Complex Phenomena 323 44 Condit ions for Identical Velocity Fields in Newtonian and Some Non-

Newtonain Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 44.1 Second Order Fluids . . . . . . . . . . . . . . . . . . . . . . . 324

44.2 Maxwell and Oldroyd-B Fluids . . . . . . . . . . . . . . . . . 328

45 The Role of t he Second Normal Stress Difference in Recti l inear Flows329

45.1 Flow in a Tube of Rec tangular Cross-Section . . . . . . . . . 329

45.2 Flow Along a Slot - An Unbounded Domain . . . . . . . . . . 331 45.3 Edge Frac ture in Rheome t ry . . . . . . . . . . . . . . . . . . 334

46 P lane Creeping Flows and the Relevance of the Firs t Normal Stress Difference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338

46.1 Congruence of t he Two Velocity Fields . . . . . . . . . . . . . 339 46.2 Flow across a Slot . . . . . . . . . . . . . . . . . . . . . . . . 340

46.3 Radia l Flow in and out of an Annulus . . . . . . . . . . . . . 343

47 Exper iments and Theoret ical Results to Delineate a Simple Fluid . . 346 47.1 Exper iments with Small Oscillations Superposed on Simple

Shearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346

47.2 Infinitesimal Strain Superposed on a Large Strain History . . 348

47.3 Pe r tu rba t ion due to Small Displacements . . . . . . . . . . . 349

47.4 Pe r tu rba t ion abou t a S ta te of Rest . . . . . . . . . . . . . . . 350

47.5 Pe r tu rba t ion abou t Simple Shear and Extensional Motions . 350 47.6 The Number of Independent Linear Functionals . . . . . . . . 351

xviii Contents

47.7 S o m e Universal Relat ions . . . . . . . . . . . . . . . . . . . . 352 47.8 Ultrasonic Features . . . . . . . . . . . . . . . . . . . . . . . . 352

47.9 Predic t ions and Per formance . . . . . . . . . . . . . . . . . . 353

48 Cessat ion of One History and Cont inua t ion with Another His tory . . 354

48.1 A New Viscometer . . . . . . . . . . . . . . . . . . . . . . . . 354

48.2 Smoo th Transi t ion from Cessat ion of Shear Flow to t he Ini- t i a t ion of Extens ional Flow . . . . . . . . . . . . . . . . . . . 356

48.3 T h e Oldroyd-B Fluid . . . . . . . . . . . . . . . . . . . . . . . 357

48.4 Predic t ions of a F lu id of the K-BKZ Type . . . . . . . . . . . 358 49 Linearised Stabi l i ty and Bifurcat ion . . . . . . . . . . . . . . . . . . 359

49.1 Stabi l i ty of the Rest S ta te . . . . . . . . . . . . . . . . . . . . 360

49.2 Linearised Stabi l i ty of Fully Developed Flows . . . . . . . . . 363

49.3 Torsional Flow of the Oldroyd-B Fluid . . . . . . . . . . . . . 364

49.4 Non-Axisymmet r ic Flows . . . . . . . . . . . . . . . . . . . . 368

49.5 Non-Axisymmet r ic Spiral Instabil i t ies . . . . . . . . . . . . . 371

49.6 Exper imen ta l Resul ts . . . . . . . . . . . . . . . . . . . . . . 371

49.7 Fin i te Domain . . . . . . . . . . . . . . . . . . . . . . . . . . 373

49.8 Cone-and-Pla te , Coue t t e and Ext rus ion Flows . . . . . . . . 381

49.9 Energy Methods and Squire 's Theo rem . . . . . . . . . . . . 383 50 Qual i t a t ive Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 385

50.1 Classification of a Par t ia l Differential Equa t ion . . . . . . . . 385 50.2 Mixed Equa t ions . . . . . . . . . . . . . . . . . . . . . . . . . 387

50.3 Fi rs t Order Sys tems . . . . . . . . . . . . . . . . . . . . . . . 387

50.4 An Uns teady Shear Flow . . . . . . . . . . . . . . . . . . . . 388

50.5 Eigenvalues and Classification . . . . . . . . . . . . . . . . . . 389

50.6 P l ane Flows of a JRS Fluid . . . . . . . . . . . . . . . . . . . 390

50.7 Accelerat ion Waves . . . . . . . . . . . . . . . . . . . . . . . . 393

50.8 Growth of Accelerat ion Waves . . . . . . . . . . . . . . . . . 394 50.9 P ropaga t ing Vortex Sheets . . . . . . . . . . . . . . . . . . . 394

50.10 Nonlinear Hyperbol ic Equat ions . . . . . . . . . . . . . . . . 395

7 Computat ional Viscoelastic Fluid Dynamics 397 51 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398

51.1 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . 399

51.2 Fourier Me thod . . . . . . . . . . . . . . . . . . . . . . . . . . 402

51.3 Bounda ry Condi t ions . . . . . . . . . . . . . . . . . . . . . . 404 51.4 Na tu re of Solutions . . . . . . . . . . . . . . . . . . . . . . . . 412

52 F in i te Difference Me thod . . . . . . . . . . . . . . . . . . . . . . . . 413

52.1 Pa th -Track ing . . . . . . . . . . . . . . . . . . . . . . . . . . 415

52.2 Two-Dimensional Problems . . . . . . . . . . . . . . . . . . . 418

52.3 T i m e - D e p e n d e n t Problems . . . . . . . . . . . . . . . . . . . 420 53 Fin i te Volume Me thod . . . . . . . . . . . . . . . . . . . . . . . . . . 422

53.1 Chor in -Type Methods . . . . . . . . . . . . . . . . . . . . . . 422 53.2 S I M P L E R - T y p e Methods . . . . . . . . . . . . . . . . . . . . 425

53.3 Secondary Flow in Pipes of Rec tangular Cross-Section . . . . 445

54 F in i te E lement M e t h o d . . . . . . . . . . . . . . . . . . . . . . . . . 447 54.1 F in i te E lement Formula t ion . . . . . . . . . . . . . . . . . . . 448

Contents xix

L

54.2 Viscoelast ic F lu ids . . . . . . . . . . . . . . . . . . . . . . . . 451 55 O the r M e t h o d s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456 56 Epi logue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461

A p p e n d i x t o C h a p t e r 7 . . . . . . . . . . . . . . . . . . . . . . . . . . 463 Linear Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . 463

Numer ica l Q u a d r a t u r e . . . . . . . . . . . . . . . . . . . . . 468 Linear and Quad ra t i c Shape Funct ions . . . . . . . . . . . . 471

I n d e x 473

1

Kinematics of Fluid Flow

Ever since Thomson and Taitl followed Amp6re and decided to consider kinematics as the purely geometrical science of motion in the abstract, the study of kinematics has flourished as a subject where one may consider displacements and motions without imposing any restrictions on them; that is, there is no need to ask whether they are dynamically feasible in the physical world. Of course, the dynamical nature of a flow is impossible to ascertain a priori, because a flow which is possible in one fluid need not be in another. Thus, in this Chapter, we concentrate exclusively on kinematics of fluid motions, using the last forty years of development of the subject of viscoelasticity as our guide.

In w we introduce the concepts of a motion of a particle and relate it to the velocity and acceleration of the particle and extend these to a field description over the whole body. Defining the deformation and velocity gradients, we relate these through a simple matrix differential equation. Next, a motion arising from a steady or an unsteady velocity field is shown to have direct relations with the theory of dynamical systems, and use of the latter is made to obtain the description of the motion in a history format for steady flows. The inverse is shown to hold so that the motion in this history format gives rise to a steady velocity field. In addition, the problem of determining when a velocity field in the Lagrangian description is steady in the Eulerian sense is solved.

Next, use of the polar decomposition theorem is made to derive some algebraic properties of the deformation gradient and recent results on determining the stretch part of the polar decompostion by the use of Cayley-Hamilton theorem are illustrated. Finally, we introduce the right and left Cauchy-Green strain tensors for later use.

1THOMSON, W. and TAIT, P.G., Principles of Mechanics and Dynamics, Part I, Dover, New York, 1962. Reprint of the 1912 edition.

2 1. Kinematics of Fluid Flow

In w the calculation of the path lines from a given velocity field is described and, in the case of homogeneous steady velocity fields, it is shown that the problem is reduced to the calculation of the fundamental matrix, which is the exponential function of the matrix of the velocity gradient. For unsteady homogeneous velocity fields, the fundamental matrix again plays a significant role, although there is no general formula for its determination. Leaving aside the homogeneous velocity fields, at the end of this section we illustrate how regular peturbation techniques may be used to determine the path lines in some complex flows.

In w the path lines are used to determine the relative deformation gradient of the motion and the relative strain tensor, for these two tensors play a significant role in calculating the stresses in a given motion. The intimate relation between the right relative strain tensor and the Rivlin-Ericksen tensors are described in detail in w and the recursive relations to find these tensors from a given velocity field are obtained. In w the problem of calculating approximations to the strain tensor when the displacements or velocities are infinitesimal, or when small displacements are added to a large strain or when a small velocity is added to a large velocity field is solved.

In w in order to prepare the groundwork for a systematic study of viscometric and non-viscometric flows, deformation gradients are devised to produce velocity fields in which Rivlin-Ericksen tensors of any desired odd integer order and higher vanish. In particular, this process delivers a homogeneous, unsteady velocity field, where all Rivlin-Ericksen tensors of order three and higher vanish.

In w viscometric flows are discussed in detail. Apart from the rheologically important flows, attention is also paid to a flow that is unsteady and viscometric and possible for a finite time only. In w three types of non-viscometric flows are studied. These are: doubly superposed viscometric flows, extensional flows and non-extensional flows. In w it is shown that the flows discussed in w fall into the category called, motions with constant stretch history. It is also shown that this unification leads to unsteady flows which have constant stretch history - even viscometric flows may be uns te~y. Additionally, the insight gained from flows with constant stretch history leads to a classification of flows into the weak and strong categories.

In w we examine the local kinematics from the vantage point of a second oberver in motion with respect to the first one. The relative motions of the observers affect significantly the various gradients and strains involved. Some are shown to transform as objective quantities, while others do not. The importance of objective quantities will become apparent in Chapter 3, where constitutive relations are discussed.

In w 11, we seek conditions under which an unsteady velocity field is steady in a second frame of reference. This is a global consideration, for it affects the whole body rather than a single particle and its neighbourhood. The discussion here shows that there is an unsteady flow which is equivalent to the simple shearing flow and also that the unsteady viscometric flow from w is unsteady everywhere. In addition, it is shown that the homogeneous, unsteady velocity field where all Rivlin- Ericksen tensors of order three and higher vanish (cf. w is unsteady everywhere. The relevance of the last result to the objectivity of the relative strain tensor is also made obvious here.

1 Kinematical Preliminaries 3

In w we return to the local kinematics when the reference configuration is altered and here the concept of the symmetry preserving changes of the reference configuration is introduced. This, again, has much relevance to Chapter 3.

Finally, the Appendix lists some basic results from tensor and dyadic analysis and these are referred to throughout this treatise.

1 Kinematical Preliminaries

Let X be a particle of a body (a continuous medium) B, and let X occupy a point in three-dimensional Euclidean space E a at a fixed instant t = 0. We shall call the configuration occupied by the body B, the reference configuration BR, and call the coordinates X a, of the point where X is at tha t fixed instant, the material coordinates of the particle. The position vector of X at t -- 0 will be denoted by X.

Let X trace out a path in E 3. We shall denote this path by the curve

x = M ( X , t ) , x' = Mi(X~, t ) (1.1)

in the Euclidean space. Here t, the time coordinate, acts as a parameter and as t varies over a given time interval Z, the function M(- , t ) describes the path. Tra- ditionally, the curve (1.1) is called the Lagrangian or material description of the motion of the particle. Note tha t X -- M ( X , 0), because X is the initial value for the motion.

Now, we are interested in the motion of the whole body B rather than a single particle. So, the domain of M is the Cartesian product BR • Z, where Z is the t ime interval, which may or may not be finite. In other words, changing X in M ( X , - ) gives us the path of another particle of B so that , at t ime t, M ( X , t), X E BR, gives us the spatial configuration of/~.

We usually demand tha t M be continuously differentiable twice with respect to X and t, or M E C 2'2, though on the boundary M may be C 1'1. Hence the motion M is required to have a continuous gradient with respect to X. We shall demand a stronger condition on this gradient: VXX has a positive determinant everywhere. One notes tha t this guarantees tha t the inverse

X = M -1 (x, t) (1.2)

exists and is unique, locally at least. The stronger assumption of global inversion is made in continuum mechanics, on a piecewise basis if necessary; for without this assumption, many results of continuum mechanics which depend on the possibility of recasting a function, originally given in material coordinates, into spatial coordinates, become very difficult to establish. Indeed, this assumption of global invertibility is needed next in connection with the velocity field and later with the Reynolds' t ransport theorem, for example.

The velocity v and the acceleration a of a particle are defined in the material description through the functions:

cO (X, t ) Oi-- OM' (1.3) �9 ~ = ~ M ' - - " ~ - X '


02 02M ~ f i=~-si(X,t) , &~-- Or2 X" (1.4)

Hence, if the velocity v is defined over the body by the function ~, (X, t ) , the acceleration a is given by

0 ^ a : ~ v ( x , t ) . (1.~)

However, we usually find it more convenient to express the velocity of X at time t in terms of the coordinates it occupies at time t. Such a description leads to an Eulerian or a spatial field v (x, t).

Let us now give the velocity a Lagrangian as well as an Eulerian representation:

v = +~ ( x " , t) G~ - ~' (~,t) g, , (1.6)

where Ga and gi are the base vectors at X a and x i, respectively, and the summation convention on repeated indices is employed. Then the spatial form of (1.5) becomes

0v i a ~ v ~ v~ (1 7) -- " ~ ' + ;3 ~

where ; denotes the covariant derivative. In (1.7), we have come across the velocity gradient in dyadic notation. It is also identified by the symbol L in this treatise. Thus, the mixed and covariant components of X7v T = L are given by

�9

Ltj = vt0 , Lij = v~ 0. (1.8)

Hence the acceleration a has the following spatial representation:

Ov = - - + Lv. ( 1 9 ) &

Using the format of (1.7), we call the derivative

-~(.) = + (.);j v j, (1.10)

the material derivative of a spatial field (-); occasionally, we shall use the superposed dot:

~ ( - ) =-- ~') (i.ii)

to indicate the material derivative. When (1.11) is used, the context will make it clear whether one is employing a material or a spatial field.

A velocity field is said to be steady if v = v(x), or it is independent of t when expressed according to (1.6)2; it is unsteady if the velocity field depends explicitly on time, i.e., v = v(x , t ) . For unsteady flows, the term Or/Or represents the local acceleration, which is the acceleration measured at a fixed point in space; of course, it is zero in steady flows. For both steady and unsteady flows, Lv represents the convected terms which arise because the fluid particle is being convected from a point with one velocity vector to a second point where the velocity vector is different. Thus, convected terms may arise either because the magnitude of the velocity is changing along the path of a particle or because the base vectors suffer a change of magnitude or direction.


1.1 Relation between the Velocity and Deformation Gradients

The gradient of x with respect to X is called the d e f o r m a t i o n gradient . We shall denote it by F and write:

ax i F = V X X ; F ~ - : x ~ (1.12)

~ - O X . ,~"

Note that the above definition means that

F(O) = 1 , (1.13) where 1 is the identity matrix. Next, due to the fact that det F (det denotes the determinant) is the Jacobian of the mapping and hence the measure of the ratio of the volumes in the X and x spaces, we demand that det F > 0. This ensures that the mapping (1.1) is not degenerate, i.e., the conservation of mass is assured, and that the inversion in (1.2) is locally possible, as already remarked.

Now, there is a very simple relation between the two tensors F and L. This arises from the equality of the mixed partial derivatives:

Equivalently,

0 O M ~ 0 0 M ~ O X ~ Ot - Ot O X ~ (1.14)

00 i c3 i OX----- X -- - ~ F s (1.15)

Expressing the velocity as a spatial field, we have

O0 ~ Ov ~ OxJ _ i j

O X " - OxJ O X ~ - L j Fs (1.16)

Using the convention of the superposed dot as the material derivative, (1.15) and (1.16) may be combined and rewritten as

----- LF. (1.17)

This identity due to NOLL 2 has numerous applications throughout this treatise.

P r o b l e m 1.A By using the identity F F - 1 = 1 , prove that

~ -1 _ _ F -1L . (1.A1)

1.2 Connection with Dynamical System Theory

In the theory of ordinary differential equations, a an a u t o n o m o u s s y s t e m is described by

: k - f(x), (1.18)

, |

2NOLL, W., J. Rational Mech. Anal., 4, 3-81 (1955). 3See, for example, BRAUER, F. and NOHEL, J.A., Qualitatsve Theory of Ordinary Differen-

tsal Equations, Benjamin, New York, 1969; Dover, New York, 1989.


whereas a non-autonomous system has the form

• = f (x , t ) . (1.19)

By the very nature of these definitions, there is a simple connection between these systems and the Eulerian velocity fields - steady velocity fields are autonomous while unsteady ones are non-autonomous. Thus, a deep understanding of the kinematics of a continuous medium may be obtained by a study of systems of differential equations. In fact, we shall derive next the necessary and sufficient condition for a velocity field in the Lagrangian description to be steady in the Eulerian sense by appealing to the theory of systems of differential equations.

Suppose now tha t (1.18) has a unique solution; it is known that if f(-) is smooth, this is indeed so. Let this solution be expressed as

~ (X, t), (1.20)

with X being the position of the particle at t = 0. Now, consider the following two solution curves:

(i) a particle whose trajectory begins at X at t ime t = 0 and finds itself at (~(X, tl + t2) at t ime tl + t2;

(ii) a particle which starts from ~b(X, t l) at t ime tl and its position is sought at a t ime interval t2 later. Tha t is, we seek ~ ( ~ ( X , t i ) , t2) .

Uniqueness of the solution to (1.18) must mean that these two positions must be identical. See Figure 1.1 for emphasis.

t = O t = t l t = t l +t2

x / / j +

FIGURE 1.1. Motion of a particle used to illustrate the group property of the function

That is, we have the following identity

~ ( x , t, + t2) = ~ ( V ( x , t~), t2) (1.21)

for all t l and t2. We may thus consider 4) to be the map which acts on X, i.e., the initial position at t = 0 and all t ime lapses tl + t2 and produces the sequential


maps on the right side. Such a map is said to consti tute a dynamical system and (1.21) is said to embody a group property. 4

Let us now return to (1.21) and put t l = T, and t2 = t - T there. Then we have the following:

r t) -- ~(~b(X, T), t -- ~-). (1.22)

Using a notat ion t ha t is more common in fluid mechanics, let

, ( x , ~) - ~(~); , ( x , t ) - x(t) . (1.23)

Then (1.22) says t ha t the position x at t ime t of a particle is identical to tha t particle which begins at ~(T) at t ime T and whose location is sought t - T units of t ime later. Using the format of (1.1), we find tha t in a smooth s teady velocity field:

M (X, t) - M ( M (X, T), t -- T). (1.24)

Or

x ( t ) - M ( ~ , ( T ) , t - - T ) . (1.25)

W h a t has been demons t ra ted is t ha t the Lagrangian description of the motion associated with a smooth steady velocity field must obey (1.24) or (1.25). The above description has been termed a s teady flow in a history format. 5

Is the converse t rue? Tha t is, suppose the Lagrangian description of a motion obeys (1.25). Is it then t rue tha t the velocity field is s teady in the Eulerian sense? It is easy to prove tha t this is so. 6 We simply differentiate both sides with respect to T and derive

O M ~ O~ ~ O M ~ O - -

0~ ~ Or

where we have put s -- t - T. However,

O M ~

Os

(1.26)

O M ~ Ot " (1.27)

We can now write (1.26)-(1.27) in a more suggestive notat ion as

Ox ~ O~ ~ Ox ~ -- O. (1.28)

O~ ~ OT Ot

Appealing to the chain rule, we get

Ox ~ Ox ~ OX/3 = , F ~ ( t ) - F ( t ) F ( ~ - ) - I . (1.29)

oC ox~ oC

Because X has been kept fixed in (1.26)-(1.28), we see immediately tha t the two t ime derivatives are simply the Lagrangian velocities"

O~ a _ ~a Oxi _ Oi (1.30) Or ' Ot "

4BHATIA, N.P. and SZEGO, G.P., Stability Theory of Dynamical Systems, Springer-Verlag, Berlin, 1970; HUILGOL, R.R., Proc. IXth Int. Cong. Rheol., 1,285-296 (1984).

SBERNSTEIN, B., Ill. Inst. Tech. Math. Dept. Report, Chicago, 1971. 6HUILGOL, R.R., Zeit. angew. Math. Phys., 37, 270-273 (1986).


Employing (1.29)-(1.30) in (1.28) leads to

F(t)F(T)-I"~(~ -) -- ";'(t). (1.31)

Hence it follows that the Lagrangian description of the motion given by (1.24), or (1.25) implies that

F(T)- lv(T) = F(t)-l"~(t) - ,~(0) (1.32)

for all t and T because F(0) -- 1 from (1.13).

P r o b l e m 1.B Consider the non-autonomous system (1.19), x = f(x, t). Let the initial value

be X at time t = to and the corresponding unique solution be ~5(X, to, t). Define the map

H((X, to ) , t - to ) - ( ~ ( X , t o , t ) , t ) - (x,t) . (1.B1)

Show that the map H has the group property

II((X, to), t l + t2 ) = II(II((X, to),tl),t2) (1.B2)

for all time lapses t - to -- t l and t - to - t2. Examine why a simplification similar to (1.24) or (1.25) does not occur with unsteady velocity fields.

Indeed, (1.32) tells us that

~'(X, t) -- F ( X , t ) # ( X , 0) (1.33)

for all t. Differentiating both sides with respect to t, and using F - LF(cf.(1.17)), we find that

~(X, t) -- F(X, t),~(X, t). (1.34)

In turn, we may express the above in Eulerian coordinates and discover that

a - Lv, (1.35)

which can occur if and only if the flow is steady. Hence, the relation (1.25) is both necessary and sufficient that the velocity field be steady in the Eulerian sense. To put it another way, if a steady velocity field is given, then the material description of the motion must obey (1.24) or (1.33). Conversely, if the material description of a motion obeys (1.24) or (1.33), then the associated velocity field is steady in an Eulerian sense.

P r o b l e m 1.C Suppose that in one dimension the velocity field in the material description

satisfies

O ( X , t ) - F(X,t)O(X,O). (1.C1)

Prove that the motion is given by

x(X, t) = f (g(X) + t), (1.C2)

where g(X) is such tha t its derivative obeys

g'(X)@(X, 0) = 1. (1.C3)

Simplify the expression in (1.C2) to the case 7 when @(X, 0) = U, where U is a constant, and - o o < X < oo.

1.3 PolarDecomposition

We shall now turn to some algebraic properties of the deformation gradient tensor F. Since it is non-singular, it may be expressed as the product of a positive-definite and symmetric tensor and an isometric tensor. By the polar decomposition theorem s

F------ R U = V R , (1.36)

where U and V are positive-definite and symmetric, and R is the orthogonal tensor representing the isometry. Note tha t det R - i because det F > 0. The two Cauchy- Green strain tensors B and (3 are defined through

B = V 2 -- F F T , BiJ-Ga/3 ~ z 5 x , a ,~, (1.37)

C -- U 2 : F T F Caf~ :g{ j x ~ x j (1.38) ~O~ ~*

In (1.37), G aB are the contravariant components of the metric tensor in the material coordinate system X a, while in (1.38), gij are the covariant components of the metric tensor in the coordinate frame x ~. Also, the superscript T denotes the transpose in (1.37)-(1.38) and elsewhere in this treatise.

Now, given U, it is trivial to find C because the latter is just the square of the former matrix. Suppose we are given C. Can we find its square root? The traditional method uses the following technique. Because C is positive definite, its eigenvalues are positive and with respect to the eigenvector basis, the matrix of C is diagonal. Hence, if the columns of the matrix Q consist of the orthonormal eigenvectors of C, then the matrix [U] of U is given by

[ U ] - [Q][A][QT], (1.39)

where [A] is the diagonal matrix with entries equal to the square roots of the eigenvalues of C.

Clearly, this is very cumbersome as the following example shows. Let the deformation gradient have the matrix form

Then we obtain

1 K ) (1.40) [F] = 0 1 "

(1 [C] - [FTF] = K I + K 2 " (1.41)

7GREENBERG, J.M., Arch. Ratzonal Mech. Anal., 24, 1-21 (1967). SFor a proof, see MARTIN, A.D. and MIZEL, V.J.: Introduction to Linear Algebra, McGraw-

Hill, New York, 291-293, 1966.


The eigenvalues of this matrix are

2 + K 2 + Kx/4 + K 2 . ( 1 . 4 2 )

2

It is clear therefore tha t the process of finding the square root of the matr ix in (1.41), which is associated with a simple shear deformation, is not a trivial exercise in algebra, because there still remains the task of finding the orthonormal eigenvector basis.

We now turn to a method 9 which is easy to use in two dimensions at least, for it depends on a simple application of the Cayley-Hamilton theorem. We shall now illustrate this in detail. T w o - d l m e n s i o n a l Case : We begin with the identity

U 2 - IuU + llul = O, (1 .43)

where the two invariants are given by

I u - t r U , I I u = det U. (1.44)

Here, t r denotes the trace operator or t r U is the sum of the diagonal elements of the matrix [U]. A simple rewriting of (1.43) leads to

U = I u - l ( ( 2 + I l v l ) (1.45)

because C -- U 2. Of course,

IU-- A1 + A2, IIu = AIA2, (1.46)

where A 1 and A2 are the eigenvalues of U. Clearly, the corresponding invariants of (3 are

2 2 = + 1 I o = a a2. (1.47)

Hence it follows tha t

llv - x / I I c , (1.48)

= v Io + 2 ,v77- .

Using (1.48) in (1.45), we see tha t U is expressible in terms of (2 and its invariants quite readliy.

For example, turning to the matr ix in (1.41), we find tha t

I I v = 1, It; = X/'4 + K 2, (1.49)

which leads to 1

[U] = x/4 + g 2 g 2 + g 2 "

9HOGER, A. and CARLSON, D.E., Quart. Appl. Math., 42,113-117 (1984).

2 Path Lines 11

T h r e e - d i m e n s i o n a l Case: The formula relating C and U is more complicated and one obtains

U -- ((3 + I l v l ) - l ( I v C + I I I v l ) , (1.51)

where

1 Iv - tr U, I Iu - -~ [I~ - tr U2] , I I I v - det U.

Here the relations between the invariants of U and C are as follows:

I I I v - - v / I I I c ,

= I c + 2IIu ,

(1.52)

(1.53)

The last two equations can be used to eliminate I I v and one obtains

I~ - 2 I c i 2 - 8 V / I I I c I v + 1 2 - 4 I I c = 0. (1 .54)

This equation has but one positive root for Iu, perhaps repeated. Once this is found, one can find I I v from (1.53) and U from (1.51).

1.~ Rela t ive C a u c h y - G r e e n S t ra in Tensor

We shall now turn to the useful strain measure in viscoelasticity, namely the relative Cauchy-Green strain tensor, which is obtained from (1.38)2 when F is the relative deformation gradient. To define this, let us assign the position coordinates of the particle at t ime T to be ~a and x ~ to be the coordinates at t ime t. Then the relative deformation gradient Ft(~-) is defined through

a 0~ a Ft(T) = V~, (Ft(T)){- - Ox'" (1.55)

It should be noted tha t the tensors F~(t) of (1.29) and Ft(T) of (1.55) are inverses of each other. They give rise to different strain measures used in continuum mechanics.

Now, the right relative Cauchy-Green strain tensor Ct(T) is defined through

-

C, (1.56)

Note tha t the components ga/3 are now measured in the ~a coordinate system at the point ~ in E 3, while the covariant components (Ct(T))~j are calculated in the x ~ frame.

2 P a t h L i n e s

In this section we shall explore the analytical techniques tha t are available to integrate an autonomous or non-autonomous system of ordinary differential equations.


That is, if a velocity field is steady and has the autonomous form

= f(x), (2.1)

or is unsteady and has the non-autonomous form

-- f(x, t), (2.2)

and the initial condition x(0) = X is prescribed, what techniques are available to solve these systems of equations?

To put the above in a format used in viscoelastic fluid mechanics, in (2.1) we wish to solve

---~ = f(~) (2.3) d~

subject to the initial condition

{(T)[~=t = x. (2.4)

Corresponding to (2.2) is the form

= f (r (2.5) dT

with the initial condition (2.4). The theory of integration is complete as far as linear systems are concerned and

we shall begin with these first.

2.1 Linear A u tonomous Sys tems

Let us assume tha t the velocity field is steady and is described by

= Lx, (2.6)

where L is a constant matrix. Of course, in this form, L is the velocity gradient. The theory of differential equations l~ tells us tha t the solution to (2.6), with x(0) -- X, is given by

x(t ) = (2.7)

where ~( t ) is called a fundamental matrix of the system (2.6). This fundamental matrix obeys

d ~ = L ~ , ~ ( 0 ) = 1. (2.8)

From (1.17), we note tha t the deformation gradient F(t) is indeed the fundamental matrix and tha t it depends on t only. Appealing to differential equation theory again, we find tha t

r ( t ) - e tL, (2.9)

where we have an exponential function of a matrix. This function has the following properties:

I~ F. and NOHEL, J.A., Qualitative Theory of Ordinary Differential Equations, Benjamin, New York, 1969; Dover, New York, 1989.

1. For every matr ix A, constant or not,

n-----O

where we have put

2 Path Lines 13

(2.10)

A~ = 1, (2.11)

the identity matrix. That is, the exponential function of a matrix is defined in the same way as the exponential function of a scalar. As well, e ~ : 1.

2. For every pair A and B which commute, i.e., A B = B A , we have

eAe B -- e A+B. (2.12)

Because A commutes with itself, it follows tha t

eAe -A = 1. (2.13)

3. If A is a constant, then

d e t A : A e tA = e tAA. (2.14) dt

It follows from (2.12) and (2.14) tha t the solution (2.7) is well defined, however painful the actual process of calculating the infinite series in (2.9) may be. Next, from (2.7), (2 .9 )and (2.13), it follows tha t

X = e- tLx, (2.15)

and thus we have the following relations between the positions of a particle at times T and t:

= e(T-t)Lx. (2.16)

2 .2 C a l c u l a t i n g the E x p o n e n t i a l F u n c t i o n o f a M a t r i x

We shall now seek ways of calculating the exponential function of a matrix. If we restrict our at tent ion to a 3 x 3 matrix L, which is not trivially zero, then it will obey one of the following:

(i) L ~ 0 , L 2 = 0;

(ii) 5 2 ~ 0 , L 3 = 0;

(iii) L n ~ 0 for all n = 1, 2, a, ....

Wha t the above three categories imply for the powers of L is this: it is really a case one, two or infinity! The cases (i) and (ii) are said to fall into the category of nilpotent matrices and for them, the infinite series expansion of the exponential function has two or three non-zero terms only. In fact, when L 2 = 0, we obtain

e tL = 1 + tL. (2.17)


An example of a velocity field for which the velocity gradient obeys the above condition is the steady, simple shear flow:

= 2y, y = 0, $ = 0, (2.18)

where ~ is a constant. Turning to the other nilpotent case, when L 2 # 0, but L 3 = 0, we have

t2L 2 e tL = 1 + t L + . (2.19) 27

The following velocity field which arises by the superposition of two simple shear flows affords an example of a flow of this kind:

-- ay + bz, y = cz, s = 0, (2.20)

where a, b and c are constants.

Problem 2.A If L is a nilpotent matrix, show that its three invariants (cf. (1.52)) IL, I I L and

I I I L are all zero.

Hence, we are left with the difficult case of calculating the infinite series when the velocity gradient matrix is not nilpotent. As an example, consider the extensional flow 11

x = a x , y = b y , s (2.21)

where a, b and c are constants. The velocity gradient matrix is diagonal and is given by (o00)

[L l -- 0 b 0 , (2.22) 0 0 c

and thus the exponential function has the form

/ e at 0 0 ) [e tL] -- 0 e bt 0 �9

0 0 e ct (2.23)

That is to say, when the velocity gradient matrix is diagonal, there is no difficulty in determining the exponential function of the matrix; if the matrix is symmetric, it can be diagonalised. For instance, if

[ L ] - [Q][A][QT], (2.24)

where Q is orthogonal and A is diagonal, we get

[e tL] - - [ q l [ e t A ] [ q T] (2.25)

and this too can be found easily.

. , . , . , , , ,

11COLEMAN, B.D. and NOLL, W., Phys. Fluids, 5,840-843 (1962).

2 Path Lines 15

We are thus left with a matr ix which is not nilpotent and not symmetric. To deal with this, we shall now describe P U T Z E R ' s method 12 which works for all constant matrices, nilpotent or not. Of course, one should use (2.17) or (2.19) when L is nilpotent and (2.23)-(2.24) when L is symmetric. Putzer ' s general procedure is the following:

(i) Given a constant matr ix L, find its eigenvalues hi, i = 1,2, 3;

(ii) Solve the following system of initial value problems for the first order differential equations:

Then

?~1 -- )~lrl, r l (0) = 1;

};2 = ~2r2 -b r l , r2(0) = 0;

r3 : )~3r3 -+- r2, r3(0) = 0.

2

etL -- Z r k + l ( t ) P k ( L ) ' k--O

where we have three matr ix polynomials defined by:

Po(L)

P2(L)

= 1, P I ( L ) = L -- )~11,

--" ( L - ~ l l ) ( L - A21).

(2.26)

(2.27)

(2.28)

The advantages of Putzer ' s method are obvious; we simply find the three eigenvalues, solve three first order differential equations by using integrating factors if necessary, calculate three easy products of matrices and add; the result is the exponential function in (2.27).

There is a hidden problem here: just because L is a real matr ix does not mean tha t its eigenvalues will be real in all situations. For example, consider the flow in the eccentric disk rheometer. 13 The velocity field has the form

= - f l y + flCz,

:y = ~x , (2.29)

, ~ - - " O,

12pUTZER, E.J., Amer. Math. Monthly, 73, 2-7 (1966). While Putzer's method is easy to use as an analytical tool in three dimensions, it is not totally satisfactory when computational stability and efficiency are considered. See, MOLLER, C. and VAN LOAN, C., SIAM Revzew, 20, S01-S36 (197S).

13GENT, A.N., Brit. J. Appl. Phys., 11, 165-167 (1960); MAXWELL, B. and CHARTOFF, R.P., Trans. Soc. Rheol., 9, 41-52 (1965); HUILGOL, R.R., Trans. Soc. Rheol., 13, 513-526 (1969).


where fl is the angular velocity of the two parallel disks whose axes of rotation lie along the z-direction but are not coincident; the parameter r is a measure of this eccentricity. The velocity gradient matrix has the form

The eigenvalues are t4

0 -~ fie) [L l - - ~ o o .

0 0 0

)k 1 -- 0, ,~2 = i~, )k3 = --i~.

The associated functions r i ( t ) , i = 1,2, 3, are given by

r , ( t ) = 1,

r2(t) = - i ( e ' a t - 1)/~,

ra(t) = (1 - ~ n t ) / ~ 2.

(2.30)

(2.31)

(2.32)

Given a real matrix L, assume tha t the eigenvalue )~1 is real, and tha t A2 and A3 are complex conjugates. Determine the functions r i ( t ) , i = 1, 2, 3, and show that

e tL - - r l ( t ) l + (L - ~11)[r2(t)1 + ra(t)(L - ~21)1 (2.B1)

is real.

2.3 Linear Non-Autonomous Systems

Let us now consider the case when the velocity field has the form

= L(t)x, (2.34)

where the velocity gradient depends explicitly on time. For example, the following velocity field:

= ~y, = 0, (2.35)

2, = 7 o w y c o s w t ,

t4HUILGOL, R.R., Rheol. Acta, 27, 111-112 (1988).

( c o Sno~t - sin 12t r sin ~ t [e *L] -- si t cosflt r - cosflt) ) . (2.33)

0 1

P r o b l e m 2 .B

Even though the eigenvalues are not all real and the associated scalar-valued functions are not real either, it is easy to show that e tg is real and, in fact,

2 Path Lines 17

where 70, ~m, w are constants, arises in connection with a transverse oscillation of amplitude 70 and frequency w superposed on a steady simple shear flow with a mean rate of shear ~m" Its velocity gradient matrix depends explicitly on time and is given by (0 0)

[ L ( t ) ] - - - 0 0 0 . (2.36) 0 70wcoswt 0

Although the theory of differential equations tells us tha t the solution to (2.34) is given, as in the autonomous case, by

x ( t ) = r ( t ) x , (2 .37)

and tha t the fundamental matrix, or the deformation gradient F, satisfies

d -:-F- L(t)F, F(0)- 1 (2.38) dt

there is no general formula to calculate this fundamental matrix in a manner similar to tha t in the case of linear, autonomous systems.

Nevertheless, this does not mean that all non-autonomous problems are in- tractable. In the case of (2.35), it is easy to see tha t because the coordinate Y does not change, we can solve (2.35) and obtain

�9 (t) = x + , ~ y t ,

y( t ) = Y, (2.39)

z ( t ) - Z + 7 o Y sinwt,

where we have set (X, Y, Z) and (x, y, z) as the components of X and x respectively. Once (2.39) is derived, to obtain the components (~, r/, (:) of the position vector t~ at t ime T in terms of x is trivial, and we have

= ~ + , ~ ( ~ - - t )y ,

= y, (2 .40)

(: = z +V0Y[sinwT -- sinwt].

P r o b l e m 2.C If the velocity field is given by

= ,~(u - t~) ,

--" Z , (zcx)

- - 0 ,


show that the path lines are:

�9 (t) = x + t # r ,

y ( t ) = Y + t Z ,

z ( t ) = z .

( 2 . c 2 )

2.4 Rigid Body Motion An important type of motion, called the rigid body motion, can be cast as a non- autonomus problem in the following manner. As is well known, a motion is rigid if and only if

x(t) = q ( t ) X , q(O) = 1, (2.41)

where Q(t) is orthogonal, which means that

q ( t ) q T ( t ) - 1 (2.42)

for all t. Hence we have the following:

d..QQ QT + Q d(q T) = O. (2.43) dt dt

A simple exercise in matrix algebra now confirms that

d q ) T d(qT) (2.44) -~ - d t '

which says that the transpose of the derivative is the derivative of the transpose. The above two equations now lead to the conclusion that

QQT_ z, (2.45)

where Z is a skew-symmetric tensor. Hence the velocity field in a rigid body motion has the following form:

= Z(t)x, (2.46)

where Z(t) is skew-symmetric.

Problem 2.D If a matrix �9 obeys the equation

d O o T = Z(t) O(0) = 1 (2.D1) dt ' '

where Z(t) is skew-symmetric, prove that O(t) is orthogonal for all t.

v r - 12 - a f l & (1 - ral/r 3) cot ~ cos r

2.5 Perturbation Problems

So far, the path lines have been obtained by an exact integration procedure. How- ever, there are instances when a solution is sought in terms of a certain parameter (say, a) and the solution is to be linear in this parameter a; in other words, it is assumed that a is so small that terms involving a2 and higher powers of a are to be ignored. Such a problem occurs with the flow in the Kepes apparatus where the following velocity field, 15 in spherical coordinates (See Figure 2.1), is assumed to exist:

v r = 0 ,

v ~ - - a l i a (1 - r3/r 3) sin 0, (2.47)

z

2 Path Lines 19

FIGURE 2.1. Kepes apparatus. The two spheres rotate with the same angular velocity, while the axes of rotation are inclined to one another.

In (2.47), (r, 0, r are the spherical coordinates with r representing the "longi- tudinal" angle and 0 the angle from the "north pole"; the velocity components are the contravariant ones; r l is the radius of the inner sphere which l'otates about the axis 0 -- 0 (or OZ) with an angular velocity ft. The outer sphere of radius r2

_

rotates with the same angular velocity about the axis OZ, inclined to OZat an _

angle ~. The axis OZ lies in the OXZ plane. Finally, ~ is the parameter defined through

= r23/(r23 - r13). (2.48)

15jONES, T.E.R. and WALTERS, K., Brit. J. Appl. Phys. (J. Phys. D) Set. 2, 2, 815-819 (1969). For a recent discussion of the use of this instrument, see MARIN, G., in: Rheological Measurements, Ed. COLLYER, A.A, and CLEGG, D.W., Elsevier Appl. Sci., London, 243-297 (1988).


Letting (~, v/, ~) be the coordinates at time T of a particle which is at (r, 0, r at t ime t, we have

~/ -- 0 - af~A (\1_ ~.q~.3~r sin{ do, (2.49)

--4) + t { " - a f D ~ ( 1 - r~-h~)cot r/cosff} do.

Here (2.49)2 and (2.49)3 are coupled together in a complicated way. In order to solve (2.49)2 first, we use (2.49)3 and express r at t ime T as follows:

r = r + f~(T -- t) + O(a) , (2.50)

where O(a) is a term of order a. Hence at time a,

sin r ~ sin (r + 12(a - t)) + O(a) . (2.51)

On substituting (2.51), (2.49)2 results in

r/ - - 0 + a A ( 1 - ~ ) • (2 .52)

x [cos r (cos 12(T -- t) -- 1) -- sin r sin ~2(T -- t)].

Because of the way r/depends on 0 and a, we appeal to (2.49)3 to obtain that at t ime a:

cot r / ~ cot O + O(a); cosr ~ cos (0 + gt(a - t)) + O(a) . (2.53)

From (2.53) and (2.49)3, it now follows that

= r + - t)

+ a ~ 1 1 - z~. cot 0 cos (r + ~ (a - t)) do t

-- r ~(T - t ) - a,~ ( 1 - " ~ ) c o t O x

x [sinr -- t ) - 1) +cosCsinl2(T -- t)].

(z 4)

Thus, to the first order in a, the equations (2.49)1, (2.52) and (2.54) represent the path lines of the velocity field (2.47). t6

2.6 Oldroyd 's Method

Instead of using a system of ordinary differential equations, one may use an equivalent method for computing the path lines, based on a system of partial differential equations. We shall examine this next. Since

= ( x , (2 .55)

ii ,

tsjONES, T.E.R. and WALTERS, K., Brit. J. Appl. Phys. (J. Phys. D) Ser. 2, ~ 2, 815-819 (1969). See equation (20).

2 Path Lines 21

we have

d ~ ( X , T ) ] X -- 0 t # T . (2.56) dt

]:Now consider X as a function (el. (1.2)) of x and t in (2.56). Then (2.56) may be written as the system of partial differential equations:

0~ ~ 0~ ~ = 0, - r t ) . (2.57)

Ox~

The solution of this set of equations furnishes the path lines in (2.55), 17 and we shall illustrate the use of (2.57) by an example.

Consider the flow of fluid in an eccentric cylinder rheometer which consists of a pair of cylinders is whose axes of rotation do not coincide (see Figure 2.2).

Y f

FIGURE 2.2. Eccentric cylinder rheometer. Each cylinder rotates with an angular velocity in the same direction.

Both of the cylinders rotate with an angular velocity ~ and the eccentricity is reflected by a, the distance between their centres. In cylindrical coordinates, the boundary conditions on the physical components of the velocity field (u, v, w) are

u - - 0, v = ~ r l , w -- 0, o n r - - r l ;

u -- f l acos0 , v -- f l ( r - a s i n 0 ) , ~ o n r _ r 2 + a s i n 0 .

w -- 0. J We shall assume that , in the region between the cylinders,

(2.58)

u - - n ~ f ( r ) e '~ v - - n r + i ~ ( r F ( r ) ) e '~ , w - O. (2.59)

17OLDROYD, J. G., Proc. Roy. Soc. Lond., A200, 523-541 (1950). See equation (21). lSABBOTT, T.N.G. and WALTERS, K., J. Fluid Mech., 43,257-267 (1970).


Here f ( r ) and F(r) are unknown functions, to be determined from the equations of motion. However, (2.58) implies that

F( rs ) -- 0, F ' ( r l ) - F ' ( r2) - 0, F(r2) - 1. (2.60)

We also emphasise tha t only the real parts in (2.54) have any physical meaning. Let (~, r/, (~) be the coordinates at time T of the particle which is at (r, 0, z) at

t ime t. Then (2.57) yields (~(r) = Z and

0~r 0~ 1 0~ ~.+~. ~+7~ ~ - o ,

o'~Or/ ~_. 10. ~ ~ 7 . + u + - v = O. r

(2.61)

We shall now assume tha t there exist three functions, f ( r ) , g(O), h ( t - T) such that

~(~, o, 7-) = ~ + ,~I(~)g(O)~(t - ~) , h(o) = o. (2.62)

The constant a appears in (2.62) because as a --, 0 the motion becomes a rigid one; for a verification of this statement, see the boundary conditions (2.58) above. Now, neglecting all terms of order a2 and substituting (2.62) into (2.61) yields

f gh + ~2Fe ~~ + f g'hfl = O. (2.63)

Hence g(O) -- e '~ and f ( r ) = F(r). Then, with h = dh(a)/da, the differential equation

h + ~ + ih~ = 0 (2.64)

leads to

h(a) = i (1 - e - ' n ~ ) . (2.65)

The form taken by (2.57), correct to O(a) , is

= r + iaF(r)e 'O [ 1 - ei~O-O] . (2.66)

Similarly, by assuming

,7(,', o, ~-) = o + a O - - t) + ~ ( , . ) v ( o ) ~ , ( t - ~-), (2.67)

one obtains

d (rF(r)) . ~/-- O + a ( T - t) -- --~e'0r (1 -- e '•('-t)) ~rr ( 2 . 6 8 )

Thus (2.66), (2.68) and r = z are the path lines for the velocity field (2.59), correct to O (~).

3 The Relative Deformation Gradient and Strain Tensors 23

T h e R e l a t i v e D e f o r m a t i o n G r a d i e n t a n d S t r a i n

T e n s o r s

Suppose that the path lines have been determined. Then, the relative deformation gradient tensor Ft(T) may be calculated through (1.55). That is, we express ~ as a function of (x, t, T) and use

a 0~ ~ (3.1) Ft(T)- V~, (Ft(T)) , : Ox'"

In those cases when the velocity gradient L is a constant tensor, we note from (2.16) tha t

Hence, in these cases, we have

= e(r -0Lx. (3.2)

Ft(T) : e i r -0L. (3.3)

Having determined Ft(T) either by (3.1) or (3.3) when the latter is applicable, we may employ (1.56), viz.,

Ct(T) -- Ft(T)TFt(T), (3.4)

to determine the relative strain tensor Ct(~). When the simple formula (3.3) applies, we get

C t ( T ) - - e (~ ' - t )LT e ( T - t ) L . ( 3 . 5 )

However, quite often we have to deal with more difficult cases and we shall exhibit an example next to illustrate the general procedure.

Suppose that the velocity field at t ime t is steady and is given in contravariant component form in a curvilinear orthogonal coordinate system by

x l __ V 1 __ 0 , X2 __ V 2 __ v ( x l ) , X3 __ V 3 __ W(X 1), ( 3 . 6 )

where v(-) and w(-) are smooth functions of the coordinate x 1. Expressing (3.6) in terms of T, one obtains

d 1 d 2 ~T ~TT~ - - 0 , ~TT~ - - V (~1) , ~3 _. W ( ~ 1 ) . (3.7)

Integration of (3.7) leads to the path lines:

/- d - ~ + ~ (~1) ~ - ~ + O-- t)~ (~1), t /- d = ~ + w (~1) ~ = ~ + ( ~ _ t)w (~1). t

(3.8)


Corresponding to these path lines, the relative deformation gradient has the com- poa nt by

I 1 0 O) (T - t)v' 1 0 , v' =

[F t (T) ]~= (T--t)w' 0 1 dv dw

w' -- (3.9) d x 1 ' ' - d x "

We may now use (3.4) to determine the relative strain tensor. Its covariant components are given by 19

=

g l l -}- g22(T -- t)2V 12 -}- g33(T -- t)2W '2 g22(T -- t)V' g33(T -- t)W'~ �9 g22 0 ) �9 �9 g33

where, in checking the calculations, the following points need attention:

(3.10)

1. Since the coordinate system is orthogonal,

g a f f = 0 ( a r (3.11)

2. g l l , g22 and ga3 are functions of ~ at time (~). If we demand tha t the motion (3.6) be such that the components of the metric tensor do not change along the path line of each particle, then

g,, (~) = g~, (x), (i = 1,2, 3; no sum). (3.12)

In conformity with tensor analysis, the physical components of Ct(T ) are obtained from the covariant components by using the definition that the physical components are defined with respect to an orthonormal basis - see (Al.18) in the Appendix. On using (3.12), (3.10) now yields:

1 + (T- t)2# (T- t)./q22V' (T- t)~/ga~ W'~ vg11

[Ct(T)] = 1 0 ) , �9 . 1

(3.13)

where ~ 2 _ r Vl2 [g22 + g33 w a ] /gll. (3.14)

The above example establishes the principles by which the relative strain tensor may be calculated. These are:

(i) Find the path lines corresponding to the given velocity or displacement field so that one knows ~ in terms of x, T and t;

(ii) Calculate the components of the metric tensor at ~ in terms of x, ~- and t;

(iii) Use (3.4) to determine Ct(~-).

19The dots indicate the symmetry of the matrix.

4 Rivlin-Ericksen Tensors 25

Of course, in many problems it is not possible to determine the pa th lines exactly as we have seen in w already. We shall postpone, until w a discussion of the procedure to be followed in such situations as well as others requiring different techniques.

. . . . i .... ,H . . ,

P r o b l e m 3 . A Show tha t in a rigid motion

for all T and t.

Ct(T) = 1 (3.A1)

4 Rivlin-Ericksen Tensors

Suppose tha t the pa th lines have been determined and tha t the relative strain tensor Ct(T) has been calculated. If this is considered as a function of T, differentiable at T - t, then one may define the n th Rivlin-Ericksen tensor An through 2~

d ~ d T n C t ( T ) l r = t = An(t) , n = 1,2,3 . . . . (4.1)

Note tha t Ct(T)I~= , = 1, (4.2)

because the relative deformation gradient Ft(T) --, 1 as T --, t. Hence we may put A o - 1 and write (4.1) as

1 %

d T n C t ( T ) [ ~ = t = An(t) , n = 0 ,1 ,2 . . . . (4.3)

We shall now show tha t one may define the Rivlin-Ericksen tensors directly from the velocity field by using (1.17), which is the formula F -- LF . We begin with

d -1] ~T Ft(T) -- ~TT [F(T)F( t ) -- L ( T ) F ( T ) F ( t ) -1

-- L(T)Ft(T). (4.4)

It follows therefore tha t

~TFt(~-)T = F t ( T ) T L ( T ) T

Using the definition of Ct(T) in (1.56), we obtain

d "~TCt(T) l~.=t -- L(t) + L(t) T, (4.6)

and hence A l ( t ) = L ( t ) + L(t) T (4.7)

2~ R.S. and ERICKSEN, J.L., J. Ratwnal Mech. Anal., 4, 323-425 (1955).


which establishes that the first Rivlin-Ericksen tensor is twice the symmetric part of the velocity gradient. If we define Ln through

am L ~ - - ~ T F~(~)[~=~, n - - 1, 2 , 3 , . . . . ( 4 . 8 )

then

: -(:) dT n Ct(T)],ffit -- Z LnT,.L,., n - 1,2, 3, ..., (4.9)

r--0

where we have put

(;) n~ L1 = L,

rt(n - r)!' L o = 1. (4.10)

Thus, for example,

A2 - L2 + L T + 2LTL,

where by appealing to (4.4), we have

(4.11)

L 2 - d ~ F t ( T ) l r = t d

= "~T [L(T)Ft (T) I"=t = L(t) + L 2(t). (4.12)

We note tha t

i a ;j �9 Vi V k]

d i v k = ~-~ (v';j) + v ;k ;j,

(4.13)

and hence we have that L2 = Va. Thus L2 is the gradient of the acceleration field. Similarly, if we consider the n th acceleration field, then Ln+l is the gradient of that field. So (4.9) yields An in terms of the velocity gradient and the first n - 1 acceleration gradients. We now turn to OLDROYD's recursive formula 21 for An+l in terms of An and L.

~ . 1 O l d r o y d ' s F o r m u l a e

Let d L 2 be the square of the distance between the points X and X + dX in the material coordinate system. Let the square of the distance between these very same particles be dl 2 at t ime t. Then

dl2(t) = g~j dx ~ dx j 0x ~ 0xJ

- - - d X a d X ~ = g~j O X ~ OX/3

= [C( t ) ]~ d X ~ d X ~. (4.14)

Similarly, az2(~) = [c(~)] .~ aX"dX~, (4.15)

2IOLDROYD, J.G., Proc. Roy. Soc. Lond., A200, 523-541 (1950).

4 Rivlin-Ericksen Tensors 27

and since

we have that

c,(~) = F T (T)F T (T)

= [F(t) - l ] T FT(T)F(r)F(t) -1

= [F(t)-I]Tc(r)F(t) -1,

(4.16)

d~-~ C,(~-) = [F(t) C(r) r(t)-1. (4.17)

Thus the n ta derivative of dl2(T) with respect to T at T -- t is related to An (t). We shall now examine this in more detail.

We can invert (4.17) and obtain

] F(t) T d--"~Ct(r) F ( t ) = ~ - ~ - C ( r ) . (4.18)

Multiply the above equation by dX T from the left and dX from the right. Then we have

d n

Since

dx(t ) - r(t)dX, 0 ~ =

(4.19) may be expressed, in indicial notation, as

(~x i dX a, (4.20)

aX,~

d ~ am dl2(t)" (4.21) (An)q dx'dx j = ~dlU(T)[ .=t = dtn

Thus, for example,

(Ai)ij d x i d x j = ~dl2(t), (4.22)

which is the well known statement relating the rate of stretching to the symmetric part of the velocity gradient. Although we have done nothing new, for we have expressed the Rivlin-Ericksen tensors as time derivatives of the strain, (4.21) will yield a recursive formula for An+l in terms of An as follows. Since

(An+l )ij dxidx j am+l d ( d n dl2(t) ) = dtn+ 1 (dl2(t)) = "~ " ~

= ~(it [(An),j dx'dxJ]

_ [d ( ( A n ) ) + (A~)i, vk;j+ (A~)k j vk;i] dx'dxJ "~ q ,

we have

n - 1 ,2 , . . .

d An+~ = ~ A n + AnL + LTAn, n = 1,2 , . . .

(4.23)

(4.24)


Or

0 (An+l),j - "~ ( (A , ) ) ( ) v k. v k. (4.25) 0 + (a")O ;k vk + ( a . l i k ;.7 + ( a ' l ~ j ;,,

with n = 1,2, 3 . . . In deriving (4.23), we have used (1.17) and (4.20) as follows:

~(dx)d _ ~d (FdX) -- L F d X - - Ldx. (4.26)

The formulae (4.24) and (4.25) show clearly that the Rivlin-Ericksen tensors are der ivable f rom the velocity field in a recursive manner .

We shall now demonstrate the calculations of the first two Rivlin-Ericksen tensors for the Couette flow, as an example of the recursive relations. We begin with the velocity field

=0, O = w(r), $ = 0 (4.27)

in cylindrical coordinates. From (4.7) it follows that

(A1)ij - vi;j + vj;i, (4.28)

and since the components of the velocity field are given in contravariant form, we find the covariant components:

Vl = O, v2 = r2w(r), v3 = 0 . (4.29)

Next, we determine vi;j and compute the matrix of its physical components to be

0 -w 0 ) [L]= w + r w ' 0 0 , w' = dw

0 0 0 dr

Thus, 0 rw ~

[ A i ] = . o

and A1 has the physical components

i)

(4.30)

(4.31)

-2rww t 0 O) [ ]_hi --- . 2rww' 0 .

�9 0

(4.32)

Hence the physical components of A2 are given by

( 2r2w a 0 i ) - . o . (4.33)

Equally, we could have obtained A1 (t) and A2(t) from Ct(7) as follows. Since

gll -- 1, g 2 2 - r2, gs3 = 1, (4.34)

5 Approximations to the Relative Strain Tensor 29

along the pa th lines of the velocity field (4.27), i.e. the components of the metric tensor at t ime T are the same as those at t ime t for each particle, one sees tha t

l + (T-- t)2r2w '2 (T-- t)rw' O) [Ct(T)] -" 1 0

�9 . 1

(4.35)

I (T -- t)2[A2] = [1] + ( T - t)[A1] + ~

Thus we recover (4.31) and (4.33).

(4.36)

P r o b l e m 4 .A (i) Show tha t in a rigid motion the Rivlin-Ericksen tensors obey

A ~ ( t ) - 0, n - 1 , 2 , 3 , , .... (4.A1)

for all t. (ii) If tr A l ( t ) -- 0, i.e., the velocity field is isochoric, show tha t

tr A12(t) - tr Au(t) (4.A2)

for all t. Hence, deduce tha t an isochoric velocity field corresponds to a rigid motion if and only if A2(t) - 0 for all t.

5 Approximations to the Relative Strain Tensor

In this section, we shall be interested in finding the relative strain tensor in a number of si tuations when the pa th lines are not exactly known. We begin with a fairly simple deformation.

5.1 Inf in i tes imal Strain

W h a t we are concerned with may be formulated thus: suppose tha t the motion is such tha t the fluid particle, at all times, is very "close" to its position in the reference configuration. Then, can we express Ct(T) in terms of the displacement gradient from the reference configuration? Let u (X, t) be the displacement at t ime t, i.e.,

x (X , t) - X -- u (X , t) (5.1)

and let this displacement be small in a sense to be made precise below. Defining the displacement gradient through J - V X U , we have

F - ~ T x x - 1 + J.

Let J be "small", i.e., the Schur norm [J[ of J defined by

(5.2)

I J I - x/tr j j T (5.3)

is such tha t its supremum e: e - Supla(t) l (5.4)

t

is very small, or e < < 1. Then we say tha t the deformation (5.1) is infinitesimal, 22 and thus,

F - 1 _ 1 - a + o(e),

Ft(T) -- F(T)F( t ) -1 -- 1 + J ( r ) - J ( t ) + o(e),

Ct(T) -- Ft(T)TFt('r) -- 1 + J(7) T + J(T) -- J ( t ) T - J( t ) + o(e).

Note tha t the infinitesimal strain E(t) is defined by

2E(t) - a( t) + J ( t ) T,

(5.5)

(5.6)

(5.7)

(5.8)

so tha t Ct(T) - 1 + 2 E ( T ) - 2 E ( t ) + o(e).

Since U(t) is the unique square root of C(t) , we obtain

U(t) = 1 + E(t) + o(e).

(5.9)

(5.10)

Thus, R( t ) = F ( t )U( t ) -1 = 1 + W ( t ) + o(e), (5.11)

where 1 (J( t) - J ( t ) T) (5.12) w ( t ) -

is the infinitesimal rotation tensor. We have thus formulated the relative strain (5.9) in terms of the infinitesimal

strain when J ( t ) is small for all t and obtained expressions for other kinematical tensors in such a motion.

5.2 I n f i n i t e s i m a l Velocity

We shall now assume tha t the velocity field has a small norm, i.e., v(x , t) is O(e), where e < < 1. Then, it is clear from the definition of the first Rivlin-Ericksen tensor (4.7) tha t A1 is also O(e). However, from (4.25) we see clearly tha t in the expression for A2, the following is true:

(i) ~ is 0(6);

(ii) Each one of the following terms: v - V A 1 , A l L and LTA1 is 0(62).

It follows therefore tha t each A~, n >_ 2, contains terms of order O(e) and higher. Hence, an approximation to the relative strain expressed as a Taylor series in A~ and correct to O(e) is given by

- 1 + t ) A a ( t ) + - ,~-1 (n + 1)! ~-gA1 + O(e2). (5.13)

22COLEMAN, B.D. and NOLL, W., Rev. Mod. Phys., 33,239-249 (1961).


Interestingly enough, if the velocity field is steady and of order 0(6), then the above reduces to

Ct(T) = 1 + ( T - - t ) A l ( t ) + 0(62). (5.14)

The last one may be compared with its infinitesimal counterpart in (5.9) above.

5.3 S m a l l D i s p l a c e m e n t s added to a Large M o t i o n

We shall suppose tha t we are given a velocity field v ~ (x, t) and we calculate the relative strain tensor ~ ) corresponding to this. If a small perturbation be superposed on this velocity field, the new relative strain C,t(T) can be viewed as

= + (5.15)

where Et(T) is the perturbation term. For example, corresponding to the velocity field (2.35) and the path lines (2.40),

on assuming that 70 is small so that 702 and higher powers can be neglected, we obtain

Ct(T) = ~ + Et(T) + o(A), (5.16)

where

[ ~ . l + ' ~ ( T - - t ) 2 0 , (5.17) �9 1

(oo o ) [Et(T)]-- . 0 f (70 , t , T) , (5.18)

�9 0

and

f(7o, t, T) = 70 [sin WT -- sin wt]. (5.19)

We call such relative strain tensors as those derived by "superposing small on large", i.e., by superposing a small displacement or a disturbance on a given large strain�9

We now reformulate this concept in terms of small displacements superposed on large. Following PIPKIN, 23 let x ~ be the position occupied by the particle X at t ime t and let ~o be the position, at time T, of the particle which is at x ~ at time t. Let us express this as

= (x o, t, (5.20)

Let u(x ~ t) be the small displacement at t ime t, superposed on the particle which would have been at x ~ but for this perturbation. Then, by definition,

(x ~ + u(x ~ , t), t, T) -- ~o (X o ' t, T) + U(~ ~ (X ~ t, T), T). (5.21)

Stated in words, the left side of (5.21) is the position vector at time T of the particle which is at x ~ + u(x ~ t) at time t; the right side is the position vector at t ime T

2apIPKIN, A.C., Trans. Soc. Rheol., 12,397-408 (1968). Though P IPKIN treated a viscometric flow as the base motion, his argument applies to all cases.


of the particle which is at x ~ at t ime t plus the displacement vector of the particle which would have been at t~ ~ at t ime T, had the disturbance not occurred; refer to Figure 5.1 for an illustration of this description. The equality of the two sides in (5.21) becomes quite s traightforward to see.

. (x ~ t)l / (r176

~ O

FIGURE 5.1. Path lines corresponding to a small displacement superposed on a large motion.

In (5.21), replace x ~ by x ~ - u (x ~ t). Then one obtains

(x ~ , t, ~) = ~o (x o _ ~ ( x o ' t), t, ~) + u (~o (x o _ u ( x o ' t)) , t, ~ ) , ~ ) . (~.22)

Linearization with respect to u ( x ~ t) in (5.22) yields

(~o, t, ~-) = r (x o, t, ~) - u ( x ~ t ) . v ~ ~ (x ~ t, ~)

+ u (~o (x o, t, T), T), (5.23)

where we have wri t ten

Vt~ ~ --Vxot~ ~ - - ~ ) = ~ ~ = 0 ~ (5.24) 0 x ~

Hence, by calculating Vxot~, one determines Ct(T) for the per turbed motion in Cartesian coordinates as

(c~(~-))~, = ~ - ~ j - - ~ ~

+~ F~i~ F~j [u~,~ + u~,~] - u,n~ Cij,m. (5.25)

In (5.25), note tha t the derivatives ui5 and ua,~ are defined through

Ou, (,,o , t) ~',~ = o ~ ' (5.26)

o ~ (~o, T) (5.27)


and o C ---- o Ct('r). For example, if we superpose, in a Cartesian coordinate system, a displacement

field in the z-direction: u (x ~ t) = 70y ~ sin wt k (5.28)

on a basic viscometric flow v (x ~ = ~myO i, (5.29)

we can perform simple calculations to obtain

= x ~ + ~ i,

~ : 1 + ( T - t)~/m j | i,

V ~ = 0 .

Using (5.28)-(5.32)in (5.25)leads to

0 0 0 ) [ C t ( T ) - ~ . 0 70 [ s inw~-s inwt ] .

. 0

These results are identical to those in (5.18)-(5.19).

(5.30)

(5.31)

(5.32)

(5.33)

5.~ Small Velocity added to Large

Suppose tha t we can express the velocity field as a sum of two fields:

v ( x , t ) = v ~ ( x , t) + t ) , ( 5 . 3 4 )

where, as usual, the perturbation parameter r is small. Examples of such fields can be found in (2.47) and (2.59) in w Unless one can cast the problem of determining the relative strain tensor as one involving a small displacement on a large strain, i.e., we can integrate vp(x, t), there is no simple formula, similar to (5.25), to determine the new relative strain tensor. This is because the addition from the pertubation Vp to the relative strain ~ obtained from v ~ depends on all the old Rivlin-Ericksen tensors A~, n >_ 1, and the new ones depending on Vp. To emphasise this point, note tha t

A1 - A~ +~A~,

where the latter term depends on vp. Hence, the next Rivlin-Ericksen tensor A2 has the form

A2 O q p

-- A~ + r 1 + v ~ + Vp-VA~

+ {A~L p + A~L ~ + {A~L p + A~L~ T] + O(62), (5.35)

where the terms are all self-explanatory. It seems therefore tha t it is not possible to derive a general formula for the relative strain tensor in terms of v ~ and Vp. Hence, each case has to be handled on an individual basis. Nevertheless, the principles laid down in w are valid. We shall illustrate their uses through two examples.


First, consider the velocity field (2.47) and the path lines corresponding to that flow. The strain tensor can thus be computed to the first order in a only. Because the path lines are known, we need to determine gab (~) in terms of gij (x) first: 24

911

g22 ({,'7, r

933

-- gll (r, O, r -- 1,

= ~2 = r2,

: ~2 s i n 2 r / ~ r 2 s i n 2 (0 q- O(c~ ) )

r 2 [l + 2 cot 0 0 ( a ) ] sin 2 0

r=[1 + 2cA ( 1 - ~ ) c o t 8{cos r (cos 1 2 ( r - t ) - 1)

- s i n Csin f~(r - t)}] sin ~ O. (5.36)

In computing (5.36)3, we have used (2.52). Using (2.49)1, (2.52), (2.54) and (5.36) in (1.56), and employing (3.8), one obtains the physical components of C t ( r ) , correct to O (a): (1

=

where

3~A (R31r a) f(r T) 3c~A (r13/r3) cosO h(r 1 0 , (5.37)

1

f ( r n, T)

h(r f~, T)

-- cos r (cos f~(r - t) - 1) - sin r sin 12(r - t),

= sin r (1 - cos f~(r - t)) - cos r sin f~(r - t). ( 5 . a 8 )

Again, we emphasise that in verifying these calculations, the reader should note that all terms are correct to O(a) only.

As the last example, consider the flow field (2.59). Using the path lines (2.66), (2.68) and (~(r) = z, the relative strain tensor Ct(T) is given in physical component form by 2s

[Ct(T)] -- (5.39)

1 + 2iaF'eW(1 - e i f z C r - t ) ) - a ( r f ' + r2F")ei~ - e i~(r-t)) O~ ei~(~-t)) �9 1 - 2iar2F'e i~ ( 1 - 0 , ) �9 . 1

where F ' = dF/dr and we have used the following formulae for the metric tensor, derived from (2.66):

g~ -- 1, gun -- ~2 _ r 2 + 2iarF(r)e,O ( 1 - eia(~-t)) , gr162 ---- 1. (5.40)

These are, again, computed to O(a).

24jONES, T.E.R. and WALTERS, K., Brit. J. Appl. Phys. (J. Phys. D) Ser. 2, 2, 815-819 (1969). See eq. (21).

25ABBOTT, T.N.G. and WALTERS, K., J. Fluid Mech., 43,257-267 (1970).

6 Flows such that A . - -0 for any Odd Integer and Higher 35

6 Flows such that An Higher

- 0 for any Odd Integer and

In this section, we are interested in finding velocity fields and their relative strain tensors such that As( t ) - 0, for n _ 1, or 3, or 5, etc. These velocity fields are constructed by a study of their deformation gradients, beginning with those deformation gradients leading to homogeneous velocity fields and later examining situations generating non-homogeneous velocity fields. 26

6.1 State of Rest

In the situation when the whole body lies in a state of rest, we know that the motion is described through

x(t ) - x, t e (-oo, oo). (6.1)

Hence, the deformation gradient obeys

F ( t ) - 1 (6.2)

for all t. Since a state of rest is a special case of a rigid body motion, we recall from Problems 3.A and 4.A that A~(t) = 0, n _> 1, in a state of rest. Nothing more need be said on this matter.

6.2 Flows with Deformation Gradients Linear in t

Let us now consider the next place in this hierarchy. Clearly, the deformation gradient will be linear in t, i.e.,

F ( t ) - 1 + tM, (6.3)

where M is a constant tensor such that M ~ 0. The relative deformation gradient

Ft(T) : [1 + TM]F(t) -1 (6.4)

is linear in T and thus

Ct(T) = (F( t ) - I )T[1 + TMT][1 + TM]F(t) -1 (6.5)

is quadratic in T. We can now draw the following important conclusions:

�9 If the tensor M is such that M 2 -- 0, then a comparison with (2.17-18) shows that the deformation gradient (6.3) is the exponential function of M. Hence, the deformation gradient is equivalent to that in steady, simple shearing; moreover, the tensor M is also the velocity gradient L of the flow. Finally, because L 2 -- 0, it is known from Problem 2.A that the first invariant of L is zero. Consequently, such flows are isochoric or det F(t) = 1 for all t.

26The material in this section is based on a seminar delivered by HUILGOL, R.R., at the University of Sydney, Aug. 1995.

�9 If the tensor M is such tha t M 2 ~ 0, we have a motion which has no connection with steady, simple shear. Indeed, consider the velocity field in Problem 2.C, viz.,

= -

~t - - z, (6 .6 )

where ~ is a constant. The motion associated with this velocity field is given by

x( t ) = X + t~/Y,

y( t) = Y + tZ, (6.7)

z ( t ) = z .

Hence, the deformation gradient associated with this unsteady flow has the form (6.3) and the corresponding tensor M is given by

o ~ o] [ M ] - 0 0 1 .

0 0 0 ( 6 . 8 )

Clearly, M z ~ 0. Moreover, this tensor is not the gradient L of the velocity field (6.6), which is divergence free.

�9 The importance of (6.3) lies in the fact tha t Ct(~) is quadratic in T and hence its derivatives of third and higher order with respect to ~ are zero for all - c o < T < CO, and, in particular at T -- t. This has the following interesting consequence, for it follows from (4.3) that , in all flows leading to (6.3),

ar~ dT ~ Ct(T)I,-=t = A,~(t) -- 0, n >_ 3. (6.9)

Thus, there exist flows in which An(t) - 0, n :> 3, and these are separate from steady, simple shearing velocity fields. Indeed, these flows may be unsteady as well; also, they need not be isochoric.

If one demands tha t in (6.3), det F(t) = 1 for all t, it can be shown tha t 27 the tensor M in (6.3) must satisfy M 3 -- 0. This means that , in incompressible materials, linearity of the deformation gradient in t imposes a restriction on the tensor M; hence, it is trivial to show tha t F ( t ) -1 _ _ 1 - t M + t2M 2, and thus, the tensor Ct(T) is a smooth function of both T and t. This point is of relevance when one wishes to s tudy the dynamics of such motions. See w below.

27See HUILGOL, R.R., and TIVER, C., J. Non-Newt. Fluid Mech., 65, 299-306 (1996).

6 Flows such that AN - -0 for any Odd Integer and Higher 37

6.3 Flows with Deformation Gradients Quadratic in t

An argument similar to the one above shows tha t the next position in this nested sequence of deformation gradients is occupied by (cf. (2.19))"

F(t) = 1 + t M l + t2M2, (6.10)

where M1 and M2 are constant tensors; at least the tensor M2 should not be zero. Now, if F( t ) -1 exists, it is trivial to show that in such motions, the relative strain tensor is a quartic polynomial, i.e., of order four, in T. Hence, in such flows, A~(t) - - 0 , n >_ 5.

We can now make the following statements:

* If the tensors M1 and M2 are such tha t M2 - M12/2 and M13 - 0, then a comparison with (2.19) shows tha t the deformation gradient (6.10) is the exponential function of M1. Hence, the deformation gradient is associated with the superposition of two steady, simple shearing flows; moreover, the tensor M1 is also the velocity gradient L of the flow. Finally, because L 3 -- 0, it is known from Problem 2.A tha t the first invariant of L is zero. Consequently, such flows are isochoric or det F( t ) -- 1 for all t.

�9 If the tensor M2 is such tha t M2 ~ M12/2, then we have a motion which has no connection with superposed steady, simple shearing flows. Indeed, consider the following tensors [000] [000]

[M1]-- c~ 0 0 , [ M 2 ] - 0 0 0 , 2"y ~ c~fl. (6.11)

Using these two tensors to calculate the deformation gradient F( t ) in (6.10), one can find the corresponding velocity gradient from the relation (1.17), i.e.,

-- LF. Next, this gradient leads to the velocity field:

~t = ax, (6.12)

- fly + t(27 - afl)x.

So, this velocity field is such tha t An (t) - 0, n >__ 5. Note tha t if 2"y - aft, then a doubly superposed shearing flow results, which is obvious from (6.12).

6.4 Flows where F is a polynomial in t of Order Three or More

The above examples show how one may proceed to generate flows such tha t An (t) - 0, for all n _ 7, or any other higher, odd integer. The procedure consists in choosing a non-singular deformation gradient through

m

F(t) -- 1 + Z tJMj' (6.13) j - - 1


where the tensors Mj are all constants and m > 3. The relative strain tensor is a polynomial of order 2m in T and hence it follows that in such motions, An (t) -- 0, for all n > 2m + 1.

6.5 Non-Homogeneous Deformation Gradients

To discuss the case of non-homogeneous deformation gradients which lead to flows in which Rivlin-Ericksen tensors of any desired odd order and higher vanish, consider the following argument. In order to find motions such that An (t) - 0, n >_ 3, what we really need is that Ct(T) be quadratic in T; it does not matter how it depends on t, as long as this dependence is smooth. So, let us consider non- homogeneous motions with a non-singular deformation gradient of the form

F(X, t ) - Q ( X , t ) P ( X , t ) , (6.14)

where Q(X, t) is orthogonal, Q(X, 0) = 1, and P ( X , t ) - 1 + t M ( X ) , and M ( X ) is a non-zero tensor. Then, suppressing the dependence on X for convenience, it is easily seen that

Ct(T) -- Q(t)(P(t)-I)T[I + TMT]Q(T)Tx

XQ(T)[1 + TM]P(t)-IQ(t) T (6.15)

= + vMT][1 + rMlP(t)- Q(t) T.

So, in these non-homogeneous motions, An(t) - 0, n >_ 3. These flows include viscometric flows, which will be examined extensively in w as a special case.

6.6 Motions with Zero Acceleration

Included in the above class of non-homogeneous motions are those described through

x(t) = x + (6.16)

These have deformation gradients of the form (6.14), with Q(t) = 1, of course. The motions described by (6.16) have zero acceleration, 2s and they generate velocity fields in which An(t) - 0, for all n > 3. Clearly, not all velocity fields derived in this manner will be equivalent to steady, simple shearing; an example is already furnished by (6.6) above.

In addition, in incompressible materials, one has that det P (X , t ) - 1 in (6.14). Consequently, it again transpires that M 3 - 0, when we let P ( X , t) -- 1 + tM(X) . This restriction is identical to that on the tensor M in (6.3), as expected.

6. 7 Higher Order Non-Homogeneous Deformation Gradients

Instead of (6.14), if one lets

F (X, t) = Q(X, t )P (X, t), (6.17)

28For some recent work on these motions, see HUILGOL, R.R. and TIVER, C., J. Non-Newt. Fluid Mech., 65, 299-306 (1996).

6 Flows such that A,~ - -0 for any Odd Integer and Higher 39

where P ( X , t) is a polynomial in t, i.e.,

m

P(X, t) = 1 + E t JMj (X), (6.18) j=l

the tensors M i , - - - , Mm all depend on X only, we will obtain a motion such that A n ( t ) - 0 , n > _ 2 m + l .

As an example of the foregoing, let the velocity field have the form 29

- - u o , 9 - czx2, 2 , - ~y, (6.19)

where u0, a and j3 are constants. The path lines associated with the above velocity field are obtained very easily and they are

y(t)

- - X + t u 0 ,

1 ~ 2+3 -- y + a (X2 t + Xuo t 2 + ~ o o ),

uot4~l Xu~ + 12 - , ,

(6.20)

Although the path lines include a term involving t 4, the deformation gradient is a cubic in t only and so we have a motion such that An (t) - 0, n > 7.

The above example suggests that if we take

x - u 0 , 9 = a x ~, 2 = ~ y , (6.21)

where m > 3, one ends up with a flow such that An (t) - 0, n > 2m + 3.

6.8 D e f o r m a t i o n G r a d i e n t s w h i c h are I n f i n i t e Se r i e s in t

The obvious last step is a deformation gradient which is an infinite series in T and there are at least two well known examples here, although any non-nilpotent, constant velocity gradient matrix L will generate such a series. This is because in such a velocity field, the positions of a particle at times T and t are related through (cf. (2.16))

~,(T) -- e(~-t)Lx(t). (6.22)

This motion will also give rise to an infinite series for the relative strain tensor, because L n # 0 for all n > 1.

We shall now mention two well known flows which give rise to infinite histories. In Cartesian coordinates, they have the following forms:

(i) Simple Extensional Flow, described through

"Jc- ax, 9 - by, 2 , - cz, (6.23)

where a, b and c are constants.

29HUILGOL, R.R., Quart. Appl. Math., 29, 1-15 (1971).


(ii) The flow in the eccentric disk rheometer discussed in (2.29) above and repeated here for convenience"

= - f l y + ~2r

9 -- f/x, (6.24)

---" O~'

where f / a n d r are constants. Finally, in the class of flows listed in this section, two sets are dominant. These

are the flows which arise in rheometers and they possess strain histories with a quadratic dependence in T or an infinite series in T. A special class of the former are called viscometric flows; the latter may be extensional flows or flows which occur in the eccentric disk rheometer, Kepes apparatus, the eccentric cylinder rheometer, etc.

The next few sections will discuss some of these flows in detail and will also provide a unified way of looking at them.

7 V i s c o m e t r i c F l o w s

Consider the steady, simple shear flow described in a Cartesian coordinate system through

= ~y, ~) = ~ = 0, (7.1)

where the shear rate ~/is a constant. This flow has a velocity gradient matrix (0,0) I L l - o o o (7.2)

0 0 0

and this matrix clearly obeys L 2 ---- 0. Hence, the first invariant IL is zero and the velocity field must be isochoric, i.e., d iv v -- O.

Clearly, the deformation gradient F(t) is linear in t and is given by (2.17). Con- sequently, the relative strain history is quadratic in the time lapse s and has the form

ls2 [0, oo) (7.3) C t ( t - - s ) -- 1 -- sA1 + ~ A2, s E ,

where A1 and A2 are the first two Rivlin-Ericksen tensors associated with the velocity field (7.1). These are given by

(0 0 i) [ A l l = 0 0 , [ A 2 ] = �9 2 ~ 2 . ( 7 . 4 )

�9 0

The first derivation of the result that in steady simple shear, A~ = 0, n >_ 3, is due to RIVLIN. 3~ He showed that the above result held in a torsional flow as well

3~ R.S., J. Rational Mech. Anal., 5, 179-188 (1956).

7 Viscometric Flows 41

as in a helical flow which has, as its special cases, Couette flow, Poiseuille flow and annular flow. In 1958, ERICKSEN 31 called flows obeying (7.3), laminar shear

flows, and observed tha t (7.3) held in the case of the azimuthal flow of a fluid held between rotating cones as well. The current term used in desribing flows obeying (7.3) is viscometric f lows and this is due to COLEMAN. 32

Although a great deal of experimental work was done to determine the material functions in viscometric flows from around 1959 onwards, a search for the totali ty of such motions was undertaken by YIN and P IPKIN 33 much later and a thorough knowledge of the kinematical properties of such flows was acquired. Below we summarise these conclusions.

We shall call a flow a viscometric 9flow if the deformation gradient at t ime t is given by (cf. (6 .14)- (6 .15))

F (X , t) - Q(X, t)[1 + tM(X)] , t e 2-, (7.5)

where Q(X, t) is orthogonal, with Q(X, 0) - 1, and the time interval over which the flow is defined may be finite or infinite. Moreover,

M ~ 0 , M 2 = O. (7.6)

These two conditions imply tha t M has the following matrix form

o ,~ o) [M] = 0 0 0 , (7.7)

0 0 0

similar to (7.2), with respect to an orthonormal basis a ~ ~ and c o = a ~ x b ~ That is,

0 (7 .8) Ma~(X) -- ~ (X)a a

The basis vectors are chosen at the fixed instant t = 0, and the function ~(X) is called the shear rate. Because it is a function of X, the shear rate experienced by any particle remains constant in t ime or the material derivative d~//dt -~ O. The crucial point to be understood here is that the direction a ~ need not lie along the streamline at t ime t -- 0, although it does in steady simple shear.

We shall now define three orthonormal vectors (a, b, c) through

a ( t ) - Q( t ) a ~ b ( t ) - Q( t )b ~ c(t) = Q(t )c ~ (7.9)

where the dependence on X has been suppressed. Then

o Fka -- aka a + bkb ~ + + (7.10)

A direct consequence of (7.10) is tha t a material element, which at t ime t = 0 lies in the a~ lies in the a-direction at t ime t and its length is unchanged.

31ERICKSEN, J.L., in: Viscoelasticity - Phenomenological A spects, Ed. BERGEN, J.T., Aca- demic Press, New York, 77-91 (1960).

32COLEMAN, B.D., Arch. Rational Mech. Anal., 9, 273-300 (1962). 33yIN, W.-L. and PIPKIN, A.C., Arch. Rational Mech. Anal., 37, 111-135 (1970).


The same conclusion is valid for an element initially in the direction of c ~ for it will lie in the c-direction and its length is unaltered.

However, the element of original infinitesimal length dL along b ~ suffers an extension, with the new length dl given by

d l - ~/1 + t2~ 2 dL. (7.11)

The position of this element is subsequently given by [b + t~(X)a]dL at t ime t. Thus, this element in the original b~ is sheared toward the a-direction at a constant rate ~. Therefore, a viscometric flow is locally a simple shearing flow, with a being the direction of motion, b being orthogonal to the shearing and c being mutually orthogonal to these two.

Since the a- and c-lines are inextensible, these lines always mesh to form layers of material surfaces, called slip surfaces because they shear or slip past one another. These slip surfaces do not stretch, but may move rigidly or bend. When they are rigid, they need not be coaxial, though in most of the practical flow problems, they are so. The totali ty of such flows are known, but not the totality of flows with flexible slip surfaces which bend.

Before we provide a fairly comprehensive list of viscometric flows, it is necessary to define the shear rate in terms of the velocity field. This may be done easily by calculating the first Rivlin-Ericksen tensor A1 from the velocity field. Then

@2 1 2 ---- ~ t r A 1. (7.12)

The veracity of this formula is obvious from the velocity field (7.1) and the first Rivlin-Ericksen tensor (7.4). For a proof based on the concept of a motion with constant stretch histoky, see Problem 9.B below.

7.1 Flows with a N o n u n i f o r m Shear Rate

1. Tangential sliding of parallel plane slip surfaces:

+ (7.13)

in a Cartesian coordinate system. The simple shear flow is a special case of the above obtained by putt ing w -- 0 and u(y) -- ~/y, while the channel flow occurs with w -- 0. For the flow (7.13), the direction of the gradient of shearing is given by b - - j and the vector a is given by

4/a-- du l + j. ( 7 . 1 4 )

Clearly, a need not be parallel to the streamline or v.

2. Rectilinear flows:

v =

(~176 = "~x + (7.15)


in a Cartesian coordinate system. The channel flow is included here if w is a function of one coordinate only. Although the axial motion of fanned planes

- see (7.20) below, must be included in the class of rectilinear flows, we shall follow YIN and P I P K I N and list it separately.

3. Axial t ranslat ion, rotat ion and screw motions of coaxial circular cylindrical slip surfaces (e.g., helical flow). In cylindrical coordinates,

v--r~(r)ee+u(r)ez , ~ / 2 - r 2 ( d w ) 2 ( d u ) 2 -d"rr + ~ . (7.16)

The above velocity field may be visualised in three separate ways:

(a) When the angular velocity w - 0, one obtains either a pipe flow which occurs in a circular pipe, also known as Poiseuille flow, or the flow may occur between two concentric circular pipes, when it is called annular flow.

(b) When the axial velocity u = 0, the flow is called Couet te flow and this occurs between two concentric circular cylinders, with one or both in rotat ion about the common axis.

(c) When both the axial and rotat ional flows are present, the fluid particles move in helices. Since the angular velocity is w, a particle covers 2~ radians in 2~/w seconds, while every second, the particle suffers a t ranslat ion of amount u in the axial direction. Thus, for every rotat ion about the axis, the rise along the helix is 27ru/w. This rise is constant over each cylinder because w and u are functions of r; in general, the rise varies from one cylinder to another.

Whereas the vector b = er, the vector a is defined through

~ / a - r e0 + e~. (7.17)

4. Screw motions of general helicoidal 34 slip surfaces. In cylindrical coordinates,

v = (ree + cez)w(r, z - c9), (7.18)

where c is a constant and the angular velocity w is a function of its arguments in two groups. In this case, we note tha t the helices have the same rise per turn, viz., 27r/c and the slip surfaces are general helicoids. Interestingly enough, the shear direction a is along the streamline, whereas the direction b is parallel to XTw or it is orthogonal to the helicoids ~ = constant. The shear rate for the flow (7.18) is given by

~2 = (r 2 + c2)Vw. Vw. (7.19)

We note for later use tha t the case c = 0 corresponds to tha t of a torsional flOW.

34For the definition of a helicoid, see WILLMORE, T.J., An Introduction to Differential Geom- etry, Oxford University Press, Oxford, 1959.

5. Axial motion of fanned planes. In cylindrical coordinates,

v= cOez, 0 <_ 0 < 2~, ~/ = c /r , (7.20)

where c is a constant. As observed earlier, this is indeed a rectilinear motion, since one may replace O by arctan ( y / x ) .

In Figure 7.1, 3s we have collected together some of the above viscometric flow fields to give meaning to terms such as shear axes and slip surfaces, used in describing these flows. The reader should refer to (A1.44) - (A1.46) in the Appendix to find the physical components of the first Rivlin-Ericksen tensor A1 and check that the shear rates in (7.13)-(7.20) are indeed as asserted.

b ty ---'t b.x

-,aa

Slip surfaces

v b

a

(a) (b)

Slip.surfaces t b ~ S lip surfaces [ a c~~.~ v

(c) (d) b

(e)

FIGURE 7.1. Viscometric flows in various configurations and their associated slip surfaces. The vector a is tangential to the slip surface, representing the flow direction, b represents the shear direction, and c represents the vorticity axis.

abAfter PIPKIN, A.C. and TANNER, R.I., in Mechanics Today, Ed. S. Nemat-Nasser, Perga- mon, New York, Vol. 1, 262-321 (1972).

7.2 F lows wi th a ' U n i f o r m Shear Ra t e

1. Ro ta t ion of conical slip surfaces abou t a common axis. In spherical coordi-

nates:

v -- ~ r sin 8[ln sin 0 - ln(1 + cos 8)]er ~ 7r - - - a < 8 < -- (7 .21) 2 - - 2 '

where ~ is the cons tant shear rate. Note t ha t v -- 0 when 8 - 7r/2.

2. In order to define a mot ion wi th a flexible slip surface, i.e., a slip surface which bends, we s ta r t with a special o r thonormal t r i ad defined in te rms of {i , j , k}:

a ( a ) = c o s a l + s i n a j ,

b (~ ) - - sin ~ i + cos a J, (7.22)

c(~) = k,

= 00 + t~ In r0 + t2~200.

T h a t is, the t r i ad {a, b, c} is a function of a, which is defined in t e rms of the posi t ion at t ime t = 0 of a part icle in cylindrical coordinates, viz., (r0, 00, z0) and the current t ime t. Thus, a t t ime t - 0, the vector a ~ is radial in direction, b ~ is az imutha l and c o is along the z-axis. In (7.22), ~ is the constant shear rate.

We shall now define the mot ion as follows:

x ( X , t) - ro(1 + ' ~ 2 t 2 ) - 1 [a(c0 -- t~b (a ) ] + zok. (7.23)

P r o b l e m 7 . A

Restr ic t the mot ion (7.23) to two dimensions and rewrite it in (r, 8) coordinates. Show t h a t

x = r e r , (7.A1)

leads to r 2 = r2

1 + t 2 ~ 2 "

Use x = r cos 0 = x - i, y = r sin 0 -- x - J , calculate t an 0 and show t h a t

(7.A2)

8 = a + / 3 , (7.A3)

where t a n / 3 = t~. Finally, establish t h a t 36

rt,~ 2 : _

1 + t2~ 2 '

2t~2(~ + ~) + -~(1 -- t2~ 2) ln[rv/1 + t272] - "y =

1 + t 2 ~ 2

(7.A4)

36HUILGOL, R.R., Arch. Rational Mech. Anal., 76, 183-191 (1981).

Use the above velocity components and (7.12) to verify tha t the shear rate is indeed the constant value ~.

. . . . ,,

To illustrate the above motion in detail, consider a line on the slip surface defined by 0 = 0 at t ime t -- 0. Fix a t tent ion to 1 _< r _< 2 and z = 0, and refer to Figure 7.2.

1 2

0 - 0

FIGURE 7.2. A path line of a viscometric flow in which the slip surface bends.

The point at ( r 0 , 0 0 ) - (1,0) moves to

x ( t ) - 1

1 + t2"~ 2 [i - t'~j], (7.24)

whereas t ha t at (ro, 00) = (2, O) moves to

1 x(t) -- [a(a) - t~b(~)] (7.25)

1 + t2~/2

where

= t# In 2. (7 .26)

Because the line on the slip surface cannot increase in length, it is clear tha t the above motion means t ha t the line has been forced to bend in the course of the motion as shown in Figure 7.2.

However, consider an arc of the circle at r0 = 1 in the initial configuration. As the motion proceeds, the distance from the origin of a point is given by (7.A2) at t ime t. Hence,

1 r( t ) -- V/1 + t2;y ~, (7.27)

which means the segment moves closer to the origin. Concurrently, its length increases according to the formula (7.11) causing the segment to grow and overlap itself in a finite amount of t ime. Hence, the above motion (7.23) is possible for a finite t ime only.

The problem 7.A shows tha t the velocity field associated with the motion (7.23) is not steady, a l though the velocity fields in (7.13)-(7.20) are obviously so. How can an unsteady flow have a relative strain history which looks like (7.3)? If a viscometric flow is not s teady in a given frame of reference, is it possible tha t it is s teady in a second frame which is in relative motion with respect to the first one? We shall re turn to these points in w 11.

8 Deformation Gradients Equivalent to Exponential Functions 47

Deformation Gradients Equivalent to Exponential Functions Non-Viscometric Flows

Any flow that is not viscometric, by definition, is non-viscometric. Thus, the vast number of examples in w above are relevant here. Having said this, we now wish to consider just two classes of steady flows here:

(i) Flows with relative strain histories which are quartic in the time lapse s;

(ii) Flows with strain histories which have an infinite series expansion in s and which arise from constant velocity gradients.

The main reason behind the above is simple: just as every viscometric flow has a deformation gradient (7.5) which is equivalent, to within an orthogonal transformation, to an exponential function of a tensor M such that M 2 - 0, we seek those flows which possess deformation gradients which are, again, exponential functions of tensors; these gradients will be equivalent either to the quadratic expression (2.19) in t or to the infinite series in t. Again, it is to be noted that the above jump from a quadratic dependence on t to an infinite series arises from the way the exponential function of a matrix depends on the matrix in 3-dimensions. (See w above as well as the discussion in w

8.1 D o u b l y S u p e r p o s e d V i s c o m e t r i c F l o w s

If we examine the flow described in (2.20), which is repeated here for convenience:

x. = ay + bz, ~) -- cz, ~ = 0, (8.1)

where a, b and c are constants, we see that it consists of two viscometric flows, viz.,

-- ay + bz, y -- ~ = O, (s.2) and

x= =0 (s.3) superposed on one another. Because the combined velocity field (8.1) has a velocity gradient matrix given by (0a )

[L l - - 0 0 c , ( 8 . 4 )

0 0 0

we see that L 2 ~ 0, whereas L 3 - 0. Hence, the first invariant IL is zero and the velocity field must obey div v - O .

Clearly, the deformation gradient F(t) is quadratic in t - see(2.19); and the relative strain history is quartic in s and has the form �9

1 2 1 1 4 C t ( t - s ) -- 1 - sA1 + ~s A 2 - ~.s3A3 + ~.Is Ad, s E [0,c~), (8.5)

or we have an expansion in terms of the first four Rivlin-Ericksen tensors. Conse- quently, we call strain histories obeying (8.5) doubly superposed viscometric flows.

Using this idea of doubly superposed viscometric flows, HUILGOL a7 extended the examples given by NOLL 38 and OLDROYD 39 and showed that two doubly superposed viscometric flows with a non-constant velocity gradient are of interest. These are

1. Helical - torsional flow described in cylindrical coordinates through

v = + +

2. Helical flow - axial motion of fanned planes. In cylindrical coordinates,

v -- rw(r)e0 + (u(r) + ce)e~, 0 < O < 2~. (8.7)

In (8 .6 )and (8.7), c is a constant.

While the above flows have interesting kinematical properties, they are difficult to produce in the laboratory. We are not aware of any experiments which have looked at the differences, if any, between viscometric material functions and those arising from (8.6). However, the flows (8.6)-(8.7) may be treated as nearly viscometric flows. See w below for a discussion of the theory relevant to this class of flows.

P r o b l e m 8 .A Find the path lines associated with the flows (8.6) and (8.7) and show that the

relative strain histories have the form (8.5).

8.2 Extensional Flows

We have already mentioned an example of extensional flow in (2.21) and in (6.23). We shall repeat it here:

= ax , y = by, cz , (8 .8 )

where a, b and c are constants. Because the deformation gradient F( t ) is given by e tL, it follows tha t the relative strain history is given by

C t ( t - s) = e-sLTe-sL = e -SA', (8.9)

where tha t fact tha t L is symmetric and commutes with its transpose has been used. Hence (2.12) applies and we obtain (8.9) because the first Rivlin-Ericksen tensor is the sum of L and its transpose - see (4.7).

Using (8.9) as a guide, we shall define a flow to be an extensional flow if it has a constant velocity gradient L and

LL T = LTL. (8.10)

37HUILGOL, R.R., Quart. Appl. Math., 29, 1-15 (1971). 3SNOLL, W., Arch. Rational Mech. Anal., 11, 97-105 (1962). 39OLDROYD, J. G., Proc. Roy. Soc. Lond., A283, 115-133 (1965).

8 Deformation Gradients Equivalent to Exponential Functions 49

It then follows that in all extensional flows,

c (t- - e [0, go), (8.11)

and that the nth Rivlin-Ericksen tensor is the nth power of the first, i.e.,

A ~ ( t ) - A~(t), n - 1 ,2 , . . . (8.12)

for all t. Conversely, it can be shown that the much simpler requirement A12(t) - A2(t) for all t is sufficient for a flow to be an extensional flow. See Problem 9.C below.

We shall now give a physical interpretation of the flow (8.8). It can be viewed as the extension, say in the z-direction, of a box of rectangular cross-section which suffers a compression in the other two directions. Converting to cylindrical coordinates, we can visualise the extension of a cylindrical rod through

i ' - - a r , ~ -- O, ~ -- 2az , (8.13)

where a is a constant. The path lines associated with (8.13) are

r - r o e - a t , ~ - ~o, z = zo e 2at. (8.14)

There are only three more examples of extensional flows available in the literature. These are:

1. Extension of a cylindrical rod with a rotation 4~ defined by the motion

r : r0 e - a t , 0 -- Oo e at, z = z o e at. (8.15)

2. Motion of a spinning and contracting sphere: 41

r = ro e - a t , 0 = 0 o , r 1 6 2 e3at. (8.16)

3. The velocity field 42

v = Qx, (8.17)

where Q is a constant, orthogonal matrix.

The important point about an extensional flow is this: unlike viscometric and doubly superposed viscometric flows, the velocity field need not have zero divergence, although in viscoelastic fluids incompressibility demands that the velocity fields be isochoric.

4~ A.C., Brown Univ., Dw. Appl. Math. Report, 1975; PIPKIN, A.C. and TANNER, R.I., Ann. Rev. Fluid Mech., 9, 13-32 (1977).

41HUILGOL, R.R., Unpublished Lecture Notes, Flinders Unwerszty, 1982; HUILGOL, R.R. and PHAN-THIEN, N., Int. J. Engng. Sc~., 24, 161-261 (1986).

42HUILGOL, R.R., Rheol. Acta, 17, 560 (1978).


8.3 Non-ExtensionalFlows

Here we shall be concerned with any velocity field which has a constant, non- symmetric velocity gradient matrix which is not nilpotent. Because they are not easy to produce in the laboratory, not many examples can be given. We are aware of only two sets:

1. Flows with stretching and rotation 43 with a velocity gradient matr ix of the form (a

ILl -- w3 b Wl , (8.18) --~d 2 r 1 C

where a, b, c represent stretching and the wi, i = 1,2, 3 denote the rotation. Giesekus classified the motions arising from the above into a variety of categories depending on the relative magnitudes of stretching and the rotations. In particular, he constructed experimental devices to study two-dimensional flows varying from pure stretching to pure rotation, a4

2. The flow in the eccentric disk rheometer 45 which has been listed in (2.29) and (6.24). This has the velocity components

9 -- Ftx, (8.19)

-- 0,

where Ft and r are constants. The relative strain history corresponding to this flow can be calculated from (3.5) quite easily by replacing T by ( t - s) because the velocity gradient is a constant. We find tha t

C t ( t - 8) : e - sLT e - sL . ( 8 . 2 0 )

Now, in (2.33), we replace t by - s and obtain

cos ~ts sin ~ts - r sin ~ts [e -sL] -- - sin ~ts cosgts r - cos gts) ) . (8.21)

0 0 1

Hence, it follows tha t

1 -

0 - r sin fts 1 r 1) ) . �9 1 + 2 r 2 ( 1 - c o s t s )

( 8 . 2 2 )

43GIESEKUS, H., Rheol. Acta, 2, 101-112; 112-122 (1962). 44See Figs. 4-8 in GIESEKUS, H., Rheol. Acta, 2, 112-122 (1962). 45HUILGOL, R.R., Trans. Soc. Rheol., 13,513-526 (1969).

9 Motions with Constant Stretch History 51

9 M o t i o n s w i t h C o n s t a n t Stretch His tory

We wish to explore here a new way of defining a strain history computed with respect to the t ime t in terms of the strain history computed with respect to the fixed configuration at t ime 0 and explore its consequences. Two important aspects emerge from this: a unification of the flows in w and w under a category called motions with constant stretch history and a simple way of testing whether a given velocity field gives rise to these special histories. The latter procedure will also indicate why unsteady velocity fields may give rise to these special histories.

9.1 R e l a t i v e S t r a i n Tensor : P r o p e r t i e s

We shall call a given motion of a particle, a motion with constant stretch history, if it obeys the following relation between strain histories, computed relative to times t and 0 :

C t ( t - s) = Q ( t ) C 0 ( 0 - s)QT(t) , Q(0) = 1, (9.1)

where Q(t) is again orthogonal. Immediately after this definition due to COLE- MAN 46 appeared, NOLL 47 showed tha t (9.1) followed if and only if the deformation gradient at t ime t is given by

F(t) - Q(t)r M, (9.2)

where M is a constant tensor. As an example of (9.2), see (7.5), which describes a viscometric flow.

As we have already observed in w there are only three, non-trivial, possibilities for the tensor M:

1. M 2 = 0;

2. M 2 # O, M s = O;

3. M ~ # 0, n = 1 , 2 , 3 .....

Clearly, (9.2) implies tha t

c0 (0 - = (9.3)

Corresponding to (1)-(3) above, C 0 ( 0 - s) will be, respectively, a quadratic polynomial, or a quartic polynomial, or an infinite series in the t ime lapse s. Turning to (9.1), we see tha t the dependence of C t ( t - s) on s is not affected by the rotation tensor Q(t) , because it is well known tha t

A e M A -1 - - e AMA-1, (9.4)

46COLEMAN, B.D., Arch. Rational Mech. Anal., 9, 273-300 (1962). 47NOLL, W., Arch. Rational Mech. Anal., 11, 97-105 (1962). See also the proof by JAMEUX,

A., C. R. Acad. Sc. Paris, Set. A 271, 1188-1189 (1970).


for all nonsingular A. Thus, (9.1) and (9.3) lead to

C t ( t - s) - - e - s L T ( t ) e -SL( t ) , (9.5)

where Ll(t) = q ( t ) M q r( t) , LI(0) = M. (9.6)

It is obvious that L1 obeys the same conditions (1)-(3) as M and thus C t ( t - s) depends on s in a polynomial or series fashion in the identical way to that of c0(0-

We shall now make clear the connection between the velocity gradient L and the two new tensors L1 and M. The tensor L is the velocity gradient of a given motion and, as such, it follows that [cf. (1.17), (2.14), (9.2) and (9.6)]

~"F - 1 : L : Z + LI . (9.7)

Here, Z -- QQT _ _ Z T (9.8)

is skew-symmetric (ef. (2.45)). The equations (9.7) and (9.8) imply that in motions obeying (9.1), there exists a time dependent orthogonal tensor Q(t) such that if an observer rotates with the angular velocity Z( t ) in (9.8), he will witness that the velocity gradient L(t) is changed, in his flame, to Ll(t). Of course L1 (t) is related to LI(0) -- M through (9.6). This has the following interesting consequence. Let a rotating orthonormal basis e,(t) be attached to each particle such that

e,(t) = Q(t)e,(0), i = 1,2,3. (9.9)

Then, L~ ( t ) e , ( t ) - Q(t)Me,(0) .

Thus, using (9.6) and (9.10),

(9.10)

(t). - e j ( t ) T L l ( t ) e , ( t )

-- e j ( o ) T Q ( t ) T Q ( t ) M e , ( O )

-- ej(0)- Me,(0).

(9.11)

The interpretation of (9.11) is that the physical components of the tensor L1 at times 0 and t are the same, when computed with respect to this rotating basis. In w we shall see that this rotation is extremely useful in calculating the stress tensor in these motions.

Here, in order to make the rotation of the attached vector basis meaningful, consider Couette flow:

+ - - 0 , 0 = w ( r ) , ~ - - 0 . (9.12)

The velocity gradient matrix is given by (cf.(4.30))

0 - w O) [L] = w + rw ' 0 0 .

0 0 0 (9.13)


This can be written as the sum of a matrix

0 [L1]-- rcz'

0

and the skew-symmetric matrix

0) 0 0

0 - w O ) [ z ] = o o .

0 0 0

(9.14)

(9.15)

The physical interpretation is tha t when an observer rotates with the angular velocity vector wez, where w pertains to a specific particle, the velocity gradient at the particle appears to the observer to have the form (9.14), which is equivalent to that of a steady simple shearing motion.

Having presented the physical meaning behind (9.7), let us now suppose tha t the velocity gradient L is such tha t its material derivative is zero. From (3.5) we obtain the relative strain tensor as

Ct(T) -- e (~-t)Lr e (~-t)L. (9.16)

Thus we see tha t C t ( t - s) -- C 0 ( 0 - s), (9.17)

for all t, or we have a translation invariance of the strain history along the path of the panicle. Hence, if L - 0, we simply take Q(t) - 1 in (9.1). Consequently, all flows with a velocity gradient whose material derivative vanishes are motions with constant stretch history.4S For instance, steady simple shear flow and simple extensional flow fall into this category.

P r o b l e m 9 . A Prove tha t the following velocity field, which is a generalisation of the flow in

the eccentric.disk rheometer (2.29),

u = -~2(y - g(z ) ) , v = ~ 2 ( x - f ( z ) ) , w = 0, (9A.1)

has a velocity gradient with a material derivative equal to zero and hence has constant stretch history. 49

If we now turn to the case of a quadratic dependence of C t ( t - s) on s, it can be seen from (9.5) tha t each exponential term must be linear in s. Hence, the deformation gradient F( t) must be linear in t through (9.2). We have already seen tha t these flows are called viscometric flows and a fairly complete study of them has been provided in w Similarly, flows where the tensor C t ( t - s) is a polynomial in s of degree four or has an infinite series expansion in s have been examined in w Hence, as we remarked at the beginning of this section, the flows in w and w fall under the rubric of motions with constant stretch history. The importance of these flows lies in the fact tha t many of them are possible in the laboratory and we shall discuss their dynamics later in Chapter 5.

4sHUILGOL, R.R., Quart. Appl. Math., 29, 1-15 (1971). 49RAJAGOPAL, K.R., Arch. Ratwnal Mech. Anal., 79, 39-47 (1982).


9.2 Necessary Conditions

We shall now derive some additional properties of the motions which obey (9.5) - (9.6). First, if we differentiate the strain history in (9.5) with respect to s and set s = 0, we find that the first Rivlin-Ericksen tensor obeys

A1 -- L1 q- LI T. (9.18)

Of course, (4.7) applies, i.e.,

A1 -- L + LT; (9.19)

this is because, as we have already seen in (9.7), the two gradients L and L1 differ by a skew-symmetric tensor Z.

P r o b l e m 9.B Consider a viscometric flow and show that (9.11) leads to

t r A 2 = 2 t r L 1 L T = 2 t r M M T. (9.B1)

Assume that the matrix of M has the form

0 # 0) [ M ] - 0 0 0

0 0 0

and show that the shear rate ~ is given by

(9.B2)

2# 2 = t r A 2. (9.B3)

Higher order derivatives of the strain tensor in (9.5) lead to

A , + I -- A , L1 + L T A n , n = 1 , 2 , . . . , (9.20)

which is valid in motions with constant stretch history only. The above should be compared with the formulae valid in all motions (cf. (4.24)):

d An+l = ~ A , + A , L + LTAn, n = 1 ,2 , . . . (9.21)

The formulae (9.18)-(9.21) show that flows with constant stretch history differ from other flows quite remarkedly.

By examining (9.18)-(9.21) and using (9.7), or directly from the definition (9.1), it also follows that , in motions with constant stretch history,

An - ZA,~ - A,~Z, n = 1 , 2 , . . . , (9.22)

and C t ( t - s) = Z ( t ) C t ( t - s) - C t ( t - s)Z(t). (9.23)

As an example of the use of the formulae (9.18), (9.20) and (9.22), let us reconsider the Couette flow discussed previously. The matrix of the velocity gradient L is


given in (9.13) and and those of the first two Rivlin-Ericksen tensors are given by (4.31) and (4.33), respectively. These are repeated here for convenience.

0 rw' O) [ A 1 ] = �9 0 0 ,

�9 0

2r w a 0 O) [A2]= . 0 0 .

�9 0

If we now consider the following system of linear equations

(9.24)

(9.25)

AI - L1 + L T,

A2 = A I L I + L T A I , (9.26)

and solve these for the matrix L1, the unique result is L1 of (9.14) and the skew- symmetric tensor Z = L - L1 is given by (9.15).

Hence, we may use (9.22) in the form

h i = ZA1 - A I Z (9.27)

and obtain the matrix of/kl. It is easy to check that this agrees with that in (4.32), viz.,

( - 2 r w ~ ' 0 O) . [ A l l - - 2rr 0 . (9.28) t A

�9 0

9.3 The Importance of A 1 , A 2 and A3

In the preceding example, it has been shown that the tensor L1, which determines the viscometric flow strain history through (9.5), can be found from a solution of a system of linear equations (9.26) which involved the first two Rivlin-Ericksen tensors only. The situation with respect t9 the non-viscometric flows is slightly more complicated in that one may need the first three Rivlin-Ericksen tensors. This result is due to WANG. 5~ We shall state it and add a few comments.

1. In any motion with constant stretch history, if A1 has distinct eigenvalues then A1 and A2 determine the tensor L1 as a unique solution to the system of linear equations (cf. (9.18), (9.20) and (9.26)):

A t = L1 + L T ( 9 . 2 9 )

A2 = AlL1 +LTA1.

5~ C.-C., Arch. Rational Mech. Anal., 20,329-340 (1965). A simple criterion to determine when A1 has three or two distinct eigenvalues in incompressible materials is available. See HUILGOL, R.R., Quart. Appl. Math., 29, 1-15 (1971).


Viscometric flows always fall into this category because the eigenvalues of A1 are ~ , - ~ / a n d 0. So does the velocity field in the eccentric disk rheometer (2.29), because the corresponding eigenvalues are ~ r 0.

2. If A1 has two equal eigenvalues which are distinct from the third eigenvalue and A12 - A2, then

1 i x : ~ A I T W , (9.30)

where W is an arbitrary, skew-symmetric tensor which commutes with A1. We shall see in Problem 9.D below that this is always an extensional flow.

3. If A1 has two equal eigenvalues which are distinct from the third eigenvalue and A 2 ~ A2, then the triad of A1, A2, A3 determines a unique L1. Clearly, this flow is non-viscometric.

4. If A1 has all equal eigenvalues, then L1 obeys (9.30) and A 2 - A2. Con- sequently, one has an extensional flow again - see Problem 9.D below for a proof.

In summary, in any motion with constant stretch history the first three Rivlin- Ericksen tensors A1, A2 and A3 determine the strain history Ct(t- s) uniquely.

9.4 Sufficient Conditions Suppose we are given an arbitrary strain history C t ( t - s ) and wish to know whether the corresponding motion has constant stretch history or not. That is, does it satisfy (9.1)? The equation (9.23) provides the answer. For, if we can solve it for a skew- symmetric tensor Z(t), then we can find an orthogonal tensor Q(t) such that (cf. Problem 2.D):

d Q Q T _ Z(t) Q(0) - 1. (9.31) dt

Then, it is trivial to prove that the strain history obeys (9.1). Suppose we are given just the velocity field v(x, t). In this situation, it is not a

trivial matter to find the strain history as we have seen in w and w Thus appealing to (9.23) is not likely to be productive.

Is it then conceivable that the first two or three Rivlin-Ericksen tensors, as the case may be, will determine whether the flow has constant stretch history? To show that they will not do so, consider the following velocity field: 51

V -- [)~lX -- ~ ( t ) y ] i -b [~2Y "~" r -b )~3zk, (9.32)

where ,ki, i = 1, 2, 3, are unequal constants. The above velocity field gives rise to

A1 0 0 ) [A1] = 2 0 A2 0 .

0 0 A3 (9.33)

51HUILGOL, R.R., Rheol. Acta, 15, 120-129 (1976).


Clearly, this first Rivlin-Ericksen tensor has distinct eigenvalues. We note tha t dA1 /d t -- 0 as well. Next, we find that

2 0)

�9 (9.34) (A1 2)~~ 2)r . 202

These two combine to determine a unique tensor, say L1 as a solution to the system (9.29). This solution matrix is given by

A1 --w 0 ) [I-~1] : (M )~2 0 . (9.35)

0 0 A3

However, this is also the matrix of the velocity gradient L of the flow (9.32). Here we are faced with the following situation: one may calculate the tensor A3 according to the general formula

d (9.36) A3 - ~ A 2 + A2L q- L TA2.

If the flow (9.32) gives rise to a motion with constant stretch history, then this value of A3 must agree with that obtained from the formula (9.20), viz.,

A3 -- A2L1 4- LITA2. (9.37)

It is clear tha t the two values differ because dA2/d t ~ O. Hence, the velocity field (9.32) does not give rise to constant stretch history. What the above example suggests is tha t a new sufficient condition must be found. We are now in a position to state it:

Given a velocity field v, let there be a skew-symmetric solution Z to the following equation:

-- L - ZL + LZ. (9.38)

Then the flow has constant stretch history. 52 The condition (9.38) is also necessary in that , in all motions with constant stretch history, it holds true.

The proof is trivial. For, if Z obeys (9.381, then we define L1 from (9.7), i.e., we have

L 1 : L - Z. (9.39)

Next, it follows from (9.38)-(9.39) that

]:~1 - - i L l - LIZ. (9.40)

Hence the first Rivlin-Ericksen tensor obeys

A1 -- L1 -[-LT' (9.41)

A1 -- ZA1 - A1Z.



Introduce the notation A0 - 1, and suppose that for all integers m - 1 , . . . , n~

A m = A m - i L l + LTAm-1,

= ZAm - AmZ. (9.42)

Then, on using the formula (cf.(4.24)),

A,~+~ - A,~ + A,~L + LTAn,

we discover tha t

Also, we obtain

An+l - AnL1 + LTA~.

(9.43)

(9.44)

/kn+l -- ZAn+I - An+l Z. (9.45)

The above proof, based on induction, shows that since (9.42) is t rue for m = 1, it is t rue for all m = 1,2, .....

The consequence is tha t

An(t) - q( t )An(O)qT( t ) , (9.46)

where Q(t) is an orthogonal tensor which obeys

Q Q T = z, Q ( O ) = 1. (9.47)

Hence, we have the following Taylor series expansion for the tensor C t ( t - s):

c (t - = (- 1

nt A~(t), (9.48) r im0

where the Rivlin-Ericksen tensors A.(t) obey (9.46). It is easy to see tha t the strain tensor obeys (9.1), or the velocity field gives rise to a motion with constant stretch history.

Clearly, if a motion has constant stretch history, the tensor Z must satisfy (9.38).

9.5 A p p l i c a t i o n s

One may well ask" what has been accomplished by introducing a category called motions with constant stretch history? There are at least two advantages. Firstly, the definition of a viscometric flow can be generalised from (7.3) to tha t in (7.5); the latter has led to the surprising result tha t there exists at least one viscometric flow in which the slip surfaces bend. Also, it has been found tha t a viscometric flow need not be steady in a given frame of reference.

P r o b l e m 9.C Consider the two dimensional flow 53

[x sin ~t + cos ~/ v -- ~ y -~ct] i + ~[x cosec -ys in~ct] j, (9.C1)

,,,

53CHAURl~, A., C. R. Acad. Sc. Paris, Set. A 274, 1839-1842 (1969).


where ~ is a constant. Solve for a skew-symmetric tensor Z from (9.32) and prove that this is a viscometric flow. 54

P r o b l e m 9 . D

Use the principle of mathematical induction to prove that a flow is a simple extensional flow if and only if 55

A12(t) = A2(t) (9.D1)

for all t. Verify that the following velocity field

v -- [Ax- w(t)y]i + [Ay + w ( t ) x ~ + #zk, (9.D2)

where the two constants A and # are unequal, is a simple extensional flow.

Indeed, the flow decribed in (9.C1) is the simple shear flow viewed from a rotating frame of reference. 56 Turning to the motion (7.23), we see that it too is unsteady; it is much more than this. We shall prove later in w 11 that it is unsteady everywhere.

Secondly, the concept of a motion with constant stretch history permits a different characterisation of the simple extensional flow from the manner in which it has been done in w This is the content of Problem 9.D. In addition, the flow in (9.D2) should be contrasted with that in (9.32) which does not possess constant stretch history.

Apart from the above two generalisations, the study of the kinematics of the flows of viscoelastic fluids has benefited from the discovery of motions with constant stretch history, at least, in two more ways: the discovery that doubly viscometric flows exist and the proof that a flow with a constant velocity gradient has constant stretch history. The latter result has led to the demonstration that the flow in the eccentric disk rheometer and its generalisation both have constant stretch history.

We now turn to another insight gained from the study of flows with constant stretch history.

9.6 Strong and Weak Flows

To simplify matters, consider a steady homogeneous flow. This flow has a constant velocity gradient and, consequently, the deformation gradient tensor computed with respect to t ime t - 0 has the form

F ( t ) - e tL. (9.49)

The velocity gradient L has three eigenvalues A1, A2, A3 which are all real, or one is real and the other two are complex conjugates. Confining attention to isochoric motions in which t r L - - A1 + A2 + A3 -- 0, one finds that the set of eigenvalues may be divided into two different categories: that set in which all eigenvalues have non-zero real parts and the second in which all eigenvalues have zero real parts.

54HUILGOL, R.R., Rheol. Acta, 15, 577-578 (1976). 55HUILGOL, R.R., Rheol. Acta, 14, 48-50 (1975). 56HUILGOL, R.R., Arch. Rational Mech. Anal., 76, 183-191 (1981); see also the Problem 7.A

in w


Clearly, if all eigenvalues have non-zero real parts, at least one must have a real part which is greater than zero, if the motion is non-trivial. We shall call a flow strong if some eigenvalue has a positive real part and weak if all eigenvalues have zero real parts. 57 The reason behind this classification is the following. Let the Jordan canonical form of the matrix of L be given by the following:

[L] : [J][A][j-1], (9.50)

where [A] is diagonal, consisting of the eigenvalues of L. Then (cf. (2.25))

[F(t)] = [J l [et^l[ j -11. (9.51)

Now, the infinitesimal vector dX adjoining neighbouring particles changes according to the rule dx(t) = F ( t )dX and thus the interparticle distance will depend on the exponential function of the matrix of the eigenvalues of A. It is known tha t the tensor L and consequently, the matrix of A, is either nilpotent or it is not.

If it is nilpotent, then the sum of the eigenvalues, the sum of their squares as well as their product are all zero- see Problem 2.A. This means tha t all the three eigenvalues are zero. Hence, nilpotent velocity gradients are associated with weak flows and, in these flows, the distance grows linearly or quadratically in t - see (2.17) and (2.19). Viscometric flows and doubly superposed viscometric flows fall into this categroy.

Weak flows also arise when one of the eigenvalue is zero and the other two are imaginary conjugates; for instance, this occurs in the eccentric disk rheometer flow (2.29). The resulting motion is then essentially oscillatory and the average interparticle distance remains steady.

Finally, in strong flows, the interparticle distance grows exponentially large with t ime because at least one of the eigenvalues has a positive real part. Extensional flows are obviously considered as strong.

10 Effects on Local Kinematics from Translation and Rotation

Let O be an observer with a clock and an orthonormal triad of rigid base vectors, and let O measure the coordinates of a particle X at t ime to and call them Xa. Let O* be another observer similarily equipped, in a motion relative to O. At t ime t - to, the two observers set their clocks to be in agreement and align their axes so tha t the reference coordinates of X are the same for O*.

Now, let Mi(X, t), X E BR, t E Z be the motion of B, so tha t

x,(t) = M i ( X a , t ) (10.1)

are the components of the position vector occupied by X at t ime t, as measured by O. If O* moves relative to O then at t ime t, the measured coordinates of X will

57TANNER, R.I. and HUILGOL, R.R., Rheol. Acta, 14, 959-962 (1975). For an equivalent classification, see TANNER, R. I., Engineering Rheology, Oxford Univ. Press, 1985, pp. 189-190.

10 Effects on Local Kinematics from Translation and Rotation 61

X

0 , 0 " . . l~f~-- , / x

*

FIGURE 10.1. Frames of reference in relative motion.

be (see Figure 10.1) x~(t) = c,(t) + 2,(t), (10.2)

where a~i(t) are the components of Mi(X, t) with respect to the base vectors of (9*. However, since the two sets of base vectors are both orthonormal at any time t, there exists a unique orthogonal tensor function Q(t) of t such tha t

~ ( t ) - Q~(t )z~( t ) , (10.3)

where xj (t) are measured by the observer O. So (10.2) reads

x~ (t) = c~(t) + Q~j(t)xy(t). (10.4)

We shall write (10.4) in direct notation as

x* (t) = c ( t )+ q(t)x(t) , (10.5)

in order to facilitate the subsequent calculations. Since

M * ( X a , t ) = c ( t ) + Q ( t ) M ( X a , t) (10.6)

for all X E BR, t C Z, we may calculate the following:

~; ( x , t) = &~ T ~ = O ~ ( t ) 5 ~ ( x , t ) (10.7)

o r

F" = QF. (10.8)

Similarly, suppressing the dependence of q on t, one finds tha t

B* = F * F *T - Q B Q T, C* - F*TF * ---- C. (10.9)


However, the relative deformation gradient follows the rule:

F ; ( t - s) -- F* ( t - s)F* (t) -1 - Q ( t - s )Ft ( t - s)Q(t) T, (10.10)

and hence C ~ ( t - s) = Q ( t ) C t ( t - s)Q(t) T. (10.11)

So a pattern begins to emerge. Some strain measures, e.g. B, C t ( t - s) transform so that their components change with the observer, while others such as C do not. In other words, certain measures transform as "tensors", while others do not. We call a scalar ~, vector u or a tensor of second order A objective if

~o*--~o, u * - Q u , A * - Q A Q T. (10.12)

As may be anticipated from (10.11), the Rivlin-Ericksen tensors are objective, i . e . ,

A~(t) = Q(t)A~(t)Q(t) T, n = 1,2, 3, . . . . (10.13)

We shall prove this after showing that the velocity gradient L is not. To see this, note tha t the two velocities, in the reference description, are related through (cf.(10.6))

,~* (X, t) = c(t) + Q(t)~(X, t) + Q(t )x(X, t). (10.14)

Now express the velocity in a spatial field form and note tha t

L* = Vx. v* = V x V * - V x - x - (QL + Q)QT = QLQT + QQT, (10.15)

because (10.5) can be inverted to yield

x = QT(x* - c). (10.16)

Hence L* is not objective, but is altered by the spin arising from the skew-symmetric part QQT (cf. (2.45)) associated with the rotation of O* relative to O. Because of (10.15), we have tha t when L - D + W , the symmetric part D* of L* is given by

, D* - ~ A 1 - Q D Q T, (10.17)

while the skew-symmetric part meets

W* - Q W Q T + QQT. (10.18)

What (10.17) says is tha t the rate of deformation tensor D is objective and that if it is not zero in one frame, it can never be zero elsewhere. On the other hand, by choosing Q appropriately, we can let W* take on any value we wish. Thus, the spin tensor is not objective.

The foregoing results due to NOLL 58 show, in particular, that A1 is objective (cf. (10.17)):

A~ = QA1Q T. (10.19)

58NOLL, W., J. Rational Mech. Anal., 4, 3- 81 (1955).

10 Effects on Local Kinematics from Translation and Rotation 63

We now show by induction tha t all An, n = 1, 2, 3 , . . . are objective. Let all the Rivlin-Ericksen tensors An up to a positive integer n >_ 1 be objective.

Then

A~+I -- d A * L* L*TA * dt ~ + A ~ + - --n d QT = ~ ( Q A = Q T) + QA= (QLQT + QQT)

+ ( Q L T Q r 4- Q Q T ) Q A n Q T

= Q(k + + LTA~)Q T + + Q A ~ Q T + Q A ~ Q T

+ Q A ~ Q T Q Q T + Q Q T Q A ~ Q T .

(10.20)

Because QQT _ 1, we have

Q _ _ Q Q T Q , or QT _ _ q r q q r . (10.21)

Thus the last four terms in (10.22) cancel, yielding the result:

A~+ 1 - Q A n + I Q T, (10.22)

and hence, if all the Ak, k -- 1 , . . . , n are objective, An+l is also objective. Since the result is t rue for n - 1, it must be true for n - 2, 3 , . . . etc.

An interesting explanation concerning the way the An appear as the non-vanishing kinematical quantities can be given as follows. 59 Equation (10.18) says that , given L, one can choose an observer O* such tha t W * -- 0 for a fixed particle X i.e., O* can choose a rate of rotation such tha t L* -- D* for a specific particle X in the frame of reference of O*. Or O* chooses Q such tha t

Q - - Q w (10.23)

for a chosen particle X, and thus L* - L *T - 0 for X. So O* sees tha t A~ as the non-zero kinematical measure of the velocity gradient, since A~ = 2D*.

Now suppose tha t the observer O measures the acceleration gradient L2. Can , .

O* choose Q(t) such tha t L~ - L~ T - 0 for the particle X which already has L* - L *T -- 07 We now show tha t this can be done. From (10.14), we can derive that

a" (x , t) = e(t) + q ( t ) a ( x , t) + 2q(t) (x, t) + t). (10.24)

Hence the acceleration gradient L~ is given by

L~ -- QL2Q r + 2QLQ r + QQT. (10.25)

If we now demand tha t L~ be symmetric, it follows tha t

2Q - Q(L T - L2) + 2 Q L T W -t- 2 Q W L + 2 Q W 2, (10.26)

59RIVLIN, R.S. and ERICKSEN, J.L., J. Rational Mech. Anal., 4, 323-425 (1955).


where we have used QQT + 2QQT + QQT = 0, (10.27)

derived from (10.21), as well as (10.23). Now, employ (10.23) and (10.26) in (10.25). It follows that

1 1 2 T 1 , 1A.2 L~ = Q(~A2 - ~A1)Q - ~A2 - "~ 1 , (10.28)

which shows that both A{ and A~ appear as the non-vanishing kinematical quantities when Q and (~ are properly chosen. Indeed, the paper by RIVLIN and ER- ICKSEN established that one can choose Q, (~ , . . . , Q(~) such that L~ are objective quantities and that L~ is a function of A ~ , . . . A~. We have, of course, demonstrated this by a detailed examination of L* and L~ only.

As far as the translation vector c is concerned, we can always choose it so that a specific particle X remains at rest, i.e., x* = 0 for that par t ic le- see (10.5). Now that we have chosen Q, (~ , . . . , Q(n), we can choose c, ~ , . . . , c (n) so that this specifically chosen particle has zero velocity, zero accelera t ion , . . . , zero ( n - 1)th acceleration.

We thus see how the local kinematics is affected by focussing attention on a single particle and choosing a second observer with the required rotation and translation. In the next section we examine how the global kinematics, i.e., the kinematics of the whole body, is affected by the motion of the second observer.

11 Zorawski Velocity Fields - Global Effects of Trans la t ion and R o t a t i o n

Suppose we consider a velocity field defined over the whole body using the spatial description. If it does not depend on t, the velocity field is steady. It is equally possible that this velocity field is not steady in the first reference frame. Under what conditions will it be steady in another reference frame, i.e., is it possible for the second observer to undergo a rotation and translation such that the velocity field is steady in the second observer's frame of reference? This is a global question in that we seek an answer for the whole body rather than for a particular particle, s~

The question was posed originally by ZORAWSK161 and its relevance to viscoelastic fluids lies in that we can understand why the unsteady shear flow due to CHAURI~ 62 is really a steady, simple shear flow in a second frame of reference; equally, one can fathom why the YIN-PIPKIN motion 63 is intrinsically unsteady, i.e., why it is not steady in any frame of reference.

Before we proceed to discuss a solution to the Zorawski problem, it is desirable to dispel the notion that a homogeneous velocity field 6(t) added to an existing velocity field v(x, t) is equivalent to superposing a rigid motion on the body. For

J l i

6~ section is based on HUILGOL, R.R., Arch. Rational Mech. Anal., T6, 183-191 (1981). 61See w in TRUESDELL, C. and TOUPIN, R.A., The Classical Field Theories, Encycl.

Phys., Vol. III/1, Springer-Verlag, Berlin, 1960. 62See Problem 9.C. 63See Problem 7.A.

11 Zorawski Velocity Fields- Global Effects of Translation and Rotation 65

instance, the flow in the eccentric disk rheometer is not equivalent to a rigid rotation superposed on simple shearing, as we have already seen in w

Returning to the problem at hand, let vi = v i (x j , t ) be the velocity field measured in one frame of reference and let v~ -- v~(x m, t) be tha t in a second frame which is undergoing a rigid rotation and translation with respect to the first one. The two sets of coordinates are assumed to be related through (cf. (10.4)): ~4

x,(t) - b,(t) + R~j(t)x;(t) , x~(t) - c,(t) + Q,j( t )x j ( t ) . (11.1)

Given vi -- v~(xj, t), define a new function

I : (~ i , t) = ~ , ( ~ ( ~ i , t),t). (11.2)

Set fl,j -- R, mRjm, a , - b , - fl,jb.i. (11.3)

If (11.2) is differentiated with respect to t, while x~ is held fixed, we obtain

O.f~ Ovi Ovi (bj + Rjkx~). (11.4) - ~ = ot + ~

The above expression contains x~ and using (11.1)2 along with (11.3), one finds that (11.4) becomes

Of~ Ovi Ovi "- Ot + "~xj (~'i~'xk + aj). (11.5)

We now differentiate both sides of (11.1)1 with respect to t and find tha t

x , - b, + i~,~; + R,~xi. (11.6)

This can be given the following format:

v~(~(xk, t), t) - b~ + ~k~k + R~,,~ (~k, t). (11.7)

Clearly, the right side is a function of the type f~* (x~, t). Hence, keeping x~ fixed in it, we obtain

oo, oo, o (b, + R~k~ + (~k t)) cot + -~xj (f~jkxk + aj) = -~ l~jv j ' . (11.8)

- - - - "" X * " "bi T Rik k + Rijvj + Rij OvJ & .

Now, we wish tha t the velocity field v~ be steady in the second frame of reference, i.e., 0v~/0t -- 0. Put t ing this expression equal to zero, expressing v~ through (11.6) in terms of xi and t, and employing (11.3) as well, we find tha t a given velocity field vi(xj , t) is steady in a second frame of reference if and only if the Zorawski condition

Ov~ Ov~ ( f ~ k x k + a~) -- a~ -- f ~ x ~ -- f Z ~ v j - - 0 (11.9) ~ + ~-~

64The two orthogonal tensors EL and Q in (11.1) are inverses or transposes of one another.


is satisfied. The advantage of the above condition is tha t it is expressed in terms of the original velocity field v i ( x j , t). What the above condition means is tha t , if given vi -- v i ( x j , t) , we can find a t ime dependent vector a and a t ime dependent skew-symmetric ternsor n such tha t (11.9) is satisfied, then the given velocity field is s teady in a second frame of reference. If this set {12,a} does not exist, then the given velocity field is unsteady everywhere or intrinsically unsteady.

Now, n is skew-symmetric and hence it has three independent components and the vector a has three; thus we have to find six quantities from the three equations in (11.9). It is clear therefore tha t only in exceptional circumstances will a given unsteady velocity field be s teady in a second frame of reference on a global scale.

In order to find such fields, let us assume tha t we have smooth velocity fields of the type

v , ( x m , t) = a , ( t ) + L , j ( t ) x j + g i (xm, t), (11.10)

where the last t e rm is such tha t it is quadrat ic or of higher degree in xm and t, i.e.,

g,(O,t) -- O, g, , j(O,t) -- 0, (0,t) -- 0. (11.11)

Subst i tu t ing the expansion (11.10) into (11.9) and grouping together terms which are independent of xj , those tha t are linear in xj and those tha t are higher, we find three separate equations:

hi - L i j aj = &i - ~ j ai ,

('lij - Lik~"lkj + l"t~kLkj -- ~,~j, (11.12)

ag~ + gi , j ( f l jkXk + aj) -- f l i jg j -- O.

Ot

Let us examine these three equations. First, if the velocity field has the simple form vi = hi( t ) , then we simply take a i -- ai and • = 0. This class of velocity fields is not interesting and so one should s tudy homogeneous, t ime dependent velocity fields in detail. Secondly, if a given homogeneous velocity field is such tha t one can solve for N from it through (11.12)2, then we may assume tha t (11.12)1 is solvable for a, because of the assumed smoothness of a and L. Clearly, the last equation is trivially satisfied within the context of homogeneous, t ime dependent velocity fields, because gi -- 0. Indeed, even in the more general context, it is the second equation which must possess a solution for 12. Assuming tha t it does, then we solve for a from the first and see if the last one is satisfied - it acts like a compatibi l i ty condition.

11 .1 U n s t e a d y F l o w s w i t h A ~ ( t ) = O, n >_ 3

We shall i l lustrate the use of the above procedure by examining two flows in which all Rivlin-Ericksen tensors of order three and higher vanish. The first one is derived from Problem 9.C, with the velocity field:

'Y [x cos ~ - y sin '~ft] j [x sin ~t 4- y cos ~;t] i -4- -~ v(x , t ) - L ( t )x = -~ (11.13)

11 Zorawski Velocity Fields - Global Effects of Translation and Rotation 67

where ~ is a constant. This is a homogeneous velocity field and so we take a - 0 and g -- 0 in (11.12)1 and (11.12)3. It is easy to show that the skew-symmetric tensor 12 which satisfies (11.12)2 has the form

,(0 1 !) - - - - 1 0 , ( 1 1 . 1 4 )

In] : 0 0

and that this is unique. Because a = 0, a simple calculation shows that the field v~ is steady in the second frame of reference and that

X ~ v , (x~n) -- R k i ( L k , j - ~ k j ) R j m m (11.15)

where 12 and R are related through (11.3)1. Solving this equation, we get

2 [R(t)] -- - s i n ( ~ )

0

Inserting this in (11.15), we find that

sin(~) O) cos( ) o �9 0 i

(11.16)

x * - 0 , y * - - q x * , ~ * - 0 , (11.17)

which proves that the velocity field (11.13) is a steady simple shearing flow in a rotating flame of reference.

Let us now turn to the homogeneous, unsteady velocity field (6.6), viz.,

fc - X/(y _ t z ) , ~l - z , 2 - - O. (11.18)

Once again, the vectors a and g are both zero and it is found that there does not exist a solution f~ to (11.12)2. Hence, this unsteady flow is intrinsically unsteady or unsteady in all frames of reference.

That this must be so is also seen by the following argument. Assume, for a moment, that the velocity field (11.18) is steady in a second flame. Then, it will still remain homogeneous; its velocity gradient will be constant and so, in the second frame, the motion will be one of constant stretch history. Now, because the relative strain tensor C t ( t - s) is objective, this tensor will transform into one associated with a motion with constant stretch history in the first flame of reference. However, in the first frame, the flow (11.18) is not associated with such a motion. Hence, it must be intrinsically unsteady.

P r o b l e m 11.A In one-dimension, let the velocity field be v - - v ( x , t ) . Taking f~ -- 0, find the

Zorawski condition applicable to such a field. Show that the velocity field

1 (11.A1) v ( x , t ) - - -~ + f ( e X / t ) , t >_ X, O <_ x < oo,

is steady in a second frame of reference which is undergoing a translation with respect to the first. Hence, prove that in this new frame,

v * ( x * ) - - f(eZ*), x* - x - I n t . (11.A2)


11.2 Compat ib i l i t y Cond i t ions

There are occasions when the given velocity field is such that (11.12) cannot be solved easily for {Ft, a}. However, (11.9) may be used to prove that the velocity field is unsteady everywhere by demonstrating that the given velocity field does not satisfy a second set of compatibility conditions, which we shall derive next.

As usual, these conditions are necessary for a given velocity field to be steady in a second frame of reference and if the velocity field does not meet them, it cannot be steady anywhere. To derive these conditions, we turn to (11.9) and find its derivative with respect to the coordinate Xm and obtain

Ot .... + vi,jm (~jkXk + aj) + Vi,jf~jm -- ~im -- f~ijVj,m = 0 . (11.19)

The above equation is really the condition satisfied by the velocity gradient Vi,m of a velocity field which is assumed to have satisfied the Zorawski condition. Hence, the symmetric part of the velocity gradient Dim and the skew-symmetric part Wire will satisfy separate equations, which are

oqDim + D ~ , ~ ( ~ k x k + aj) + D ~ g ~ - ~ D ~ = O,

Ot OW~,,, .-.--&-- + W~,,,,~(~jkxk + aj) + W~,~j~ - f ~ W ~ -- a~,,,.

(11.20)

In two dimensions, the last equation reduces to a single equation, because we may take

w 3 ( 0 - 1 ) w ( 0 0 1 ) (11.21) [ W ] - - - ~ 1 0 , [12]---~ 1 '

where w3 = a)3(X, y , t ) is the vorticity and w = w(t) is the angular velocity of the second frame of reference with respect to the first one. Then, (11.20)2 reduces to the single equation

Ow3 Owl ~ Owl 0 t -]- " ~ X ~'1 -1- W P2 -- &' (11.22)

where

/~1 - - a l - 2 y, /~2 -- a2 + 2 x . (11.23)

We shall use the above to show that the viscometric flow arising from the motion (7.23) is intrinsically unsteady.

From (A1.42) in the Appendix, we know that in cylindrical coordinates, the physical component of the vorticity is given by

ov v oO ~o~(~, O) - ~ + - - 20 + r r ~ r '

(11.24)

where we have used the fact that the velocity component v - - r0 . Let us recall from (7.A4) that

_ 2 t~2(0+~) + ~(1- - t2~ 2) ln [ rV/ l+ t2~ 2] ..... -- ;/. 1 + t2~/2

(11.25)

11 Zorawski Velocity Fields- Global Effects of Translation and Rotation 69

It follows therefore that

r 0) -- 2 2tzy2(O + / ~ ) + ~/(1 -- t2~/2) l n [ r v / 1 + t2zy 2] - ~/

1 + t2~/2 "~(1 - t2~ 2)

+ i+t2, 2 "

(11.26)

Given this expression for w3, it is clear that the derivatives Ow3/Ox and 0W3/0y cannot contain any terms involving In r, whereas Ow3/Ot must do so. A glance at (11.23) shows that fll and f12 are independent of In r as well, since al, a2 and w depend on t only. Hence (11.22) cannot be satisfied, and consequently the flow arising from (7.23) is intrinsically unsteady.

11.3 Material Description of Zorawski Velocity Fields

Although Zorawski velocity fields are identifiable through their spatial forms, it is possible to give a material description 65 of such velocity fields. The latter follows from the fact that (cf. (1.3")) if the velocity field is steady in the moving frame, then in that frame the following should hold true:

,~* (X, t) -- F* (X, t)-~* (X, 0). (11.27)

This is really an assertion that the material description of the velocity field and the deformation gradient in the moving frame must be related as above, if the spatial description of the velocity field is steady in the moving frame. If we write (11.1) in the form

x* (X, t) = Q( t )M(X, t) + c(t), (11.28)

where M is the motion (cf. (1.1)) and make the assumption that Q(0) -- 1, then it follows quite easily that (11.27) takes the following form:

0 M ( X , t)

0 t - f ~ ( t ) M ( X , t) - a ( t ) -

cot t=o where the vector a is defined in (11.3) and the assumption (cf.(1.1)) that M ( X , 0) - X has been used.

It should be noted that (11.29) contains the velocity field in the material description and the deformation gradient, and is the necessary and sufficient condition for the material description of a motion to be a Zorawski velocity field in the spatial sense. It follows therefore that if, given a motion M(X, t) there exist a skew-symmetric tensor f~(t) and a vector a(t) which satisfy (11.29), then the motion is associated with a velocity field which is steady, in the spatial sense, in some moving frame of reference.

P r o b l e m l l . B i i

65HUILGOL, R.R., Proc. IXth Int. Cong. Rheol., 1, 285-296 (1984).


Using (11.28), prove that the motion (cf. (7.23))

x (X , t ) - r0(1 + ~2t2)- l [a(a) - t-~b(a)] 4- z0k ( l l .B1)

is the material description of the motion of an intrinsically unsteady velocity field.

12 Local Change of Reference Configuration

During a given motion, the configuration of the body as well as that surrounding a generic particle changes. Hence, by choosing two separate configurations at two specific times and treating them as two reference states, one may calculate the deformation gradient with respect to each configuration. From these gradients follow the strain tensors and others, as needed. We shall return to these issues after a discussion of the change in the deformation gradient at a particle when its neighbourhood is changed, in the reference configuration, so as to preserve the density of the material locally. Such a change may be perceived, if need be, to be in the form of an imaginary experiment. It will be seen that this exercise leads to a thorough understanding of the symmetry inherent in a given continuous medium, and has much relevance to the formulations of constitutive relations. 66

Thus, let X and X ' be two adjacent particles and let X and X + dX be their positions in a fixed reference configuration. Let the motion of the neighbourhood of the generic particle X be given by

x(t) = M ( X , t ) , (12.1)

or X goes to x and X ' goes to x + dx at time t, so that F, B, C, etc. are all well defined. Now let the neighbourhood (See Figure 12.1) be changed in the reference configuration so that X remains at X, while X ' goes to X + dX'. We assume that the gradient of this mapping H which appears in

d X ' = H dX (12.2)

is a p r o p e r u n i m o d u l a r m a t r i x . Or, det H -- 1. Now, if the motion is such that X again goes to x, and X ' goes to x + dx, then

we have a new motion M ' compared with the old M:

M : ( X o r ) X ~ x ; M : ( X ' o r ) X + d X ~ x + d x . (12.3)

M ' : (Xor) X ~-~ x ; M ' : (X'or) X + dX' ~-~ x + dx. (12.4)

Obviously, V x M and V x M ' will be different. They will be different, because two shapes about X are being mapped onto the same shape about X in the position x in space. We calculate F ' = ~7xM' from the following. Since,

dx -- F dX = F ' dX I = F ' H dX, (12.5)

66Based on NOLL, W., J. Rational Mech. Anal., 4, 3-81 (1955); Arch. Rational Mech. Anal., 2, ~ 97-226 (1958).

12 Local Change of Reference Configuration 71

X

1[

FIGURE 12.1. A change in the local reference configuration around a particle leading to an identical shape after deformation.

we have

F ' H - F or F ' - - F H - 1 .

Therefore, we obtain other kinematical tensors of interest, viz.,

B'- F ' F 'T -- FH-I(H-1)TF T,

C ' - - F ' T F ' - ( H - 1 ) T C H -1,

(12.6)

(12.7)

(12.8)

Hence

F ' t ( t - s) - F ' ( t - s ) F ' ( t ) -1

-- F ( t - s )F( t ) -1 - F t ( t - s). (12.9)

c ' ~ ( t - ~) - c ~ ( t - ~), (12.10)

or the strain measure C t ( t - 8) is not sensitive to unimodular changes of the neighbourhood of the particle.

Now, let H be a proper orthogonal tensor. Then H -1 - H T, and under this condition,

B'--B (12.11)

or B is not sensitive to rotations of the neighbourhood of the particle. Suppose tha t we are interested in measuring a certain constitutive property P de-

pending on the kinematical variable F(T). Moreover, let it so happen tha t some unimodular changes of local shape about X leave this quantity 7 ~ unchanged, though clearly F(T) is altered. For example, in an elastic fluid under isothermal conditions, the pressure is a function of the density. Hence, any density preserving, or unimodular, changes of shape about a particle will leave the stress at the particle unaltered. This explains why we have insisted that det H - 1 in this section.

Returning to the general case, this set G~ - { 1 , H I , H 2 , . . . } of unimodular transformations (finite or infinite in number) then preserves P, or P is unaffected by such alterations of the neighbourhood. Expressing the functional relationship between P and F(~) as

p(~) - P (F(~ ) ) , ~ < t, ~ e Z, (12.12)


we have tha t for all F(T)

P ( F ( T ) ) - :P(F(T)H1) - P (F(T)H2) , (12.13)

etc. Let us replace F(T) in (12.13) by F(T)HI , i.e., there is now a new deformation gradient. The first and last terms in (12.13) will change and the constancy of P means that

~ ( F ( T ) H I ) -- ~ ( F ( T ) H I H 2 ) . (12.14)

However, the left side in (12.14)is equal to 7~(F(T)) from (12.13). It follows therefore that

H I H 2 E G~, if H1, H2 e G~. (12.15)

Next, replace F(T) by F(T)H11 in (12.13). Then the first two terms in (12.13) show that

~ ( F ( T ) H ~ 1) = P ( F ( T ) H l l H 1 ) - ~(F(~')) (12.16)

or H1-1 e Gp, if H1 E G~. (12.17)

Hence the set G~ forms a group 67 and its elements are non-singular, unimodular matrices. We shall call this set G~, a P-symmetry group, i.e., the group of changes of the neighbourhood of X which leave :P invariant.

Different materials have different symmetry groups: there are isotropic, transversely isotropic, orthotropic solids; there are crystal classes; there exist subfluids and there are fluids. These materials may be classified according to their symmetry groups. In this treatise, we shall be dealing mainly with fluids and these are characterised by having the proper unimodular group as their symmetry group. The reasons for this will be made clear in Chapter 3.

Turning to isotropic solids, the symmetry group is the proper orthogonal group if the solid is in an uhdeformed state. However, if an isotropic solid undergoes a homogeneous deformation such tha t it is extended in one direction and suffers a uniform compression in the other two directions orthogonal to the first one, then it is clear tha t this deformation has altered the solid from an isotropic one to a transversely isotropic one. Hence the deformation or the reference configuration may affect the symmetry group. It is this aspect we shall examine next.

That is, we wish to find the group Gp with respect to another reference configuration of X. To achieve this, let us pick a second configuration of the body as tha t occupied by it during a motion and ask how the new group is related to the old one. Let X and X1 be the two reference positions of X, and let X ~ be a second particle in the neighbourhood of the particle X as before. With respect to X, X ~ will be assumed to be at X + dX, and with respect to X1, X ~ will be at X1 + dX1. Again, let X and X ~ go to x and x + dx respectively. See Figure 12.2.

Now, dx -- F dX -- F1 dXl. (12.18)

If the motion from X to Xl is such tha t

dXl---- N dX, (12.19)

67See, for example, HERSTEIN, I.N., Topics in Algebra, BlaisdeU Pub. Co., Waltham, Mass., 1964.

12 Local Change of Reference Configuration 73

dx

X / ~ ~ / x

Xl

FIGURE 12.2. Two separate reference configurations for a body during its motion.

or N is the gradient of the mapping from X to Xl at a fixed time TO, then by chain rule

F1 = V x l x = V x x V x t X - F N -1. (12.20)

Let the property P be given with respect to the two reference configurations by P and Pl respectively. That is:

-- ~(F(T)) -- ~Ol(FI(~-)) -- ~1 (F(T)N-1) , (12.21)

where we emphasise that the measurement of F (resp. FI) depends on X (resp. X1). Now, arguing as previously,

P(F(~-)) - - ~(F(T)H1)

-- ~I (F(T)N -1) - ~I(F(T)H1N-1) . (12.22)

Replace F(T) by F(T)N in (12.22). Then

7~(F(~-)N) -- ~ ( F ( T ) N H 1 )

- ~ I ( F ( T ) ) - ~ I ( F ( T ) N H I N - 1 ) . (12.23)

Hence, it follows that, if Hi E Gp, then 6s

NH1 N - 1 E GT)I, (12.24)

where Gpl is the new :P-symmetry group. It follows that

N G p N -1 - Gp,, (12.25)

6SThis result appears in OLDROYD, J.G., Proc. Roy. Soc. Lond., A200, 523-541 (1950). See eq. (49).


or the change in the reference configuration leads to a change in the :P-symmetry group to within a conjugation. 69

Now, if Gp --L/+, the proper unimodular group, then Gp 1 - - H + also for all N in (12.25). This is because if det H1 = 1, then det N H I N -1 = 1 for all non-singular N. In words, a proper unimodular group is mapped onto itself by conjugation.

This s ta tement leads to the following important fact. Any and all proper unimodular changes of reference configuration leave Ct ( t - s) invar iant , though objective motions do not do so. Hence, it fo l lowsthat C t ( t - s) appears in the constitutive equation of a simple material, whatever the symmetry group of a simple material may be.

Finally, it must be noted tha t in many instances, the symmetry group of a material may be extended to include those matrices which have a determinant equal to :t=l. The means to achieve this extension and the consequences, if any, will be explored in Chapter 3 where we discuss constitutive equations in detail.

Appendix to Chapter 1" Basic Results from Tensor and Dyadic Analysis

In this Appendix, we list the basic tools from tensor and dyadic analysis tha t are used in fluid flows. Some authors prefer the tensor notation, usually the Cartesian tensor notation, while others prefer dyadic notation. We shall provide a summary of both.

T e n s o r A n a l y s i s

Let {x '} be a curvilinear orthogonal coordinate system and let { X g } be a Carte- sian coordinate system. The measures of the distance in these two systems are given by the covariant metric tensors g~j and 6KL, respectively. These are defined, by using the summat ion convention, through the square of the length of an infinitesimal element, viz.,

dl2 = ~KL d X g d X L - g ~ j dx~ dxJ, (AI.1)

and considering X g = X g (xi), one has tha t

OX K OX L gij -- ~KL Ox i OxJ ' (A1.2)

where ~KL is the Kronecker delta. We reiterate tha t the summation convention will be employed throughout this treatise. Using

X l = x l c o s x 2, X 2 = x a s inx 2, X 3 = x 3, (A1.3)

69This result is due to NOLL, W., Arch. Rational Mech. Anal., 2, 197-226 (1958). For an explanation of the concept of conjugation of a group, see HERSTEIN, I.N., Topics in Algebra, Blaisdell Pub. Co., Waltham, Mass., 1964. Also, see SMITH, G.F. and RIVLIN, R.S., Trans. Amer. Math. Soc., 88, 175-193 (1958), and COLEMAN, B.D. and NOLL, W., Arch. Rational Mech. Anal., 15, 87-111 (1964) for a discussion of the symmetry groups of the 32 crystal classes and anisotropic solids.

Appendix to Chapter 1: Tensor and Dyadic Analysis 75

where ( x l , x 2 , x 3) stand for (r, 0, z) of the cylindrical coordinate system, one can show that

gll -- 1, g22 -- r2, g33 -- 1, gq -- O, i ~ j. (AI.4)

Similarly, for the spherical coordinate system with (r,O, r ( x i , x 2 , x a) and

X 1 - x l s i n x 2cosx a, X 2 - x l s i n x 2sinx 3, X 3 - x l cosx 2, (A1.5)

we have that

gll -- 1, g22 -- r2, gaa -- r2 sin2 0, gij -- O, i ~ j. (A1.6)

The property that gij -- 0, i ~ j , is valid in orthogonal coordinate systems only. One defines the contravariant metric tensor gq in orthogonal coordinate systems

through

~ 1/gq, i -- j , (A1.7) g*~ -- ( 0 , i ~ j ,

so that g'~ is also diagonal. Now let v(x) be a vector field. Then, with respect to the basis {gi} at the point

x in space, which is related to the Cartesian basis {ig} at x through

O X K ' " g i - Ox i ig (Al.8)

the contravariant components v' (x) of v(x) sartisfy

v(x) - v i (x)g, (x). (A1.9)

The reciprocal basis gi(x) is defined by the usual dot product:

g (x) (x) - �9 j ~ (A1.10)

were 6~j is again the Kronecker delta. Then (A1.9) tells us that

(x) - v ( x ) . gJ (x). (A1.11)

The covariant components vi of v(x) are defined through

v ( x ) = v,(x)g'(x) (Al.12)

and - v ( x ) (x). (A1.13)

In terms of these bases, the covariant and contravariant metric tensors are, respectively, given by

gij = g i ' g j , gq = g i . gJ. (A1.14)

The set {g,} is called the natural basis at x, and {g ' ) the reciprocal basis. Since they are not necessarily of unit length, we shall introduce an orthonormal basis ei at x via

e, = g , / v ~ - g 'v~_~, i = 1,2,3, (AI.15)

where the underscoring suspends the summation. This convention will be adopted in this treatise. The set {ei} is called the natural orthonormal basis.

Using this natural orthonormal basis, we define the physical component of a vector v in the direction ei to be

 = - (A1.16)

For a second order tensor T, one defines the contravariant, covariant and mixed components through

T - T/j gi | gj - ~ j g i | g5 _ T/jgi | gJ _ Ti j gi | gj, (A1.17)

where gi | gj is called the outer product of the vectors gi and gj; the other outer products are similarly defined. The set of nine such products forms a basis for the space of linear transformations. Analogous to (Al.16), we define the physical components T of the second order tensor T to be

T - T~j v / g ~ g ~ - ~ j / y / g ~ g ~ (Al.18)

where the summation convention is again suspended. Note tha t in a Cartesian coordinate system, the physical and tensorial components of a vector (or a tensor) are identical.

Notation for Manipulating Components

It is preferable to adopt a suggestive notation for indicating the physical components of a vector v or a second order tensor T when one is learning to manipulate the indices. The following are helpful.

v < l > ) [ v l - ,

v < 3 >

T < l l > [T]= T < 2 1 >

T < 3 1 >

T < 1 2 > T < 1 3 > ) T < 2 2 > T < 2 3 > . T < 3 2 > T < 3 3 >

(Al.19)

For an example of the use of such a notation, see (A1.34) - (A1.37) below. However, after one has come to terms physical components, the above cumbersome notation may be dropped. See w where such a strategy is indeed employed.

Covariant Derivative

The covariant derivative v i �9 of a contravariant vector field vi(x) is defined by ;3

v i _ i vk ;J "O~xJ + j k '

(A1.20)

Appendix to Chapter 1: Tensor and Dyadic Analysis 77

while the derivative vi;j of a covariant vector field vi(x) is

v~;j = OxJ i j vk .

Here j k -- k j are called the Christoffel symbols of the second kind and the

summation convention with respect to repeated indices again applies. We list the symbols in three coordinates:

(i) In Cartesian coordinates,

so that covariant differentiation reduces to partial differentiation. (ii) In cylindrical coordinates,

(A1.22)

with the others being zero. Note that, we have used X 1 - - r, x 2 = 0, x 3 -- z here. (iii) In spherical coordinates, the non-vanishing symbols are:

112 - - - r , { 13 }3------r sin2 0 , { ! 3 }

/ 22 i ! 1 = ( 1 3 / = ( : l ) = r

3 [ 3 2 --cot0.

= - sin 0 cos O,

(A1.23)

Here, we have used X 1 ---- r, x 2 = 0, x 3 -- r One can define the covariant derivative of a second order tensor T ij through

(A1.24) T i J ; k - - Ox k + k m k m "

Similarly, the covariant derivative of ~ j is given by

T ~ j ; k = Ox k -- i k j k T im , (A1.25)

with the derivatives of mixed tensors obtained by combining the rules in (A1.24) and (A1.25).

P rob l em AI.1

1. Let z -- x -4- i y , and its conjugate ~ constitute a set of coordinates in the Cartesian plane. Determine the new basis and reciprocal vectors and the corresponding metric tensors.

2. Find the relationships between the physical components of a vector and its covariant or contravariant components.

3. Determine the Christoffel symbols and show that covariant differentiation reduces to partial differentiation.


Applications We now comment on certain applications.

1. Raising and lowering indicex. By this, one means the operation of converting a covariant index into a contravariant one and vice versa. For instance,

v ~ -- giJvj, vi -- gijv j , ~ J -- gikgJlTkl. (A1.26)

2. Divergence operator. This is a differential operator which reduces the rank of the tensor by one. So a vector becomes a scalar on the application of this operator. Hence

div v -- v ~ (A1.27) ; i ,

div T = (T~J;j)g, , (A1.28)

where the fact t h a t div T is a vector has been used.

3. Curl operator:. We shall mention the curl of a vector field here only. We let curl v -- ca where ca is a vector such tha t

w ~ -- r (A1.29)

r ~jk -- e~Jk/V~, g - det g~j, (A1.30)

1 if i, j , k are an even permutat ion of 1,2,3, e ~jk -- 0 0 if i , j , k is not a permutat ion, (A1.31)

- 1 if i, j , k is an odd permutat ion.

One may obtain the covariant component of ca from ~ - gijw J. Note tha t (A1.31) means tha t

e 1 2 3 - - e 2 3 1 - - e 3 1 2 : +1, e 1 3 2 - - e 3 2 1 - - e 2 1 3 - - - 1 , (A1.32)

and tha t others are zero. Since e ~jk is a Cartesian tensor, e ~jk -- eijk.

4. Physical Components of Derivatives. The covariant differentiations listed above are all expressible in terms of the Christoffel symbols, the contravariant or covariant or mixed components, and their derivatives. In practice one usually needs the physical components of the derivatives, in te rms of the physical components of the vectors or tensors. In such cases, it is preferable to perform the differentiations and then obtain the physical components of the derivatives.

. .

As an example, consider the component T~; j in the cylindrical coordinate

~j _ A ~ system. Let T ;j so tha t we may choose i - 1, and obtain

A1 0T lj 1 -- Ox---~-~-(jk}TkJ-l-(}kIT ik. ( A 1 . 3 3 )

Using the non-zero components of j k from (.41.22), one has

T ' " A ~ - OT~" OT'~ OT'~ r T ~176 + . (A1.34) - 0 r + 00 + 0 - - ~ - r

Appendix to Chapter 1" Tensor and Dyadic Analysis 79

Since

A 1 - A n - A < r >, T ~ - T < rr > , T ~ - T < rz >,

T ~ ~ T < O O > = T e ~ 2, (A1.35)

one obtains the physical component of A ~"

OT < rr > 1 0 T < rO > A < r > = + -

Or r O0 1

+ - ( T < rr > - T < O0 >), r

OT < r z >

Oz (A1.36)

a result, and others similar to it, will be useful later in w

5. Divergence and curl of vectors. We list below the divergence and curl of vectors in the three coordinate systems.

div v:

Cartesian" Physical components - u, v, w.

Ou Ov Ow + oy~ + 'Oz" (A1.37) 0"~'

Cylindrical: Physical components- u, v, w.

Ou u 10v Ow " ~ + - r + 7 " ~ + Oz " (A1.38)

Spherical: Physical components - u, v, w.

Ou 2u 10v v 1 Ow + - - + - + - c o t 0 +

r r O'0 r r s in 0 0 r (A1.39)

-- curl v:

Cartesian"

[~] =

Ow Ov

Oy Oz Ou Ow

Oz Ox Ov Ou

oz Oy

(A1.40)

C y l i n d r i c a l '

[~] -

1 cow Ov

r O0 Oz Ou Ow

Oz Or Ov v l O u

"~r ~ r t O 0

(A1.41)

Spherical:

10w w 1 Ov r - ~ + - - c o t ~ -

r r sin 8 0 r [ w ] _ 1 0u w (3w

r sin 0 0r r Or Ov v 10u Or { r rO0

(A1.42)

6. Components of the first Rivlin-Ericksen Tensor. The physical components of the first Rivlin-Ericksen tensor A1 are listed in Cartesian, cylindrical and spherical coordinates below, in terms of the physical components of the velocity field v.

Ou Ou Ov 2 ~ -~ + ~

a~ [A1] - �9 2 ~

.

0u 0w 0-7 + Ov 0w ------4-

Oz

(A1.43)

Cartesian:

Cylindrical:

[Al l =

Ou 1Ou Ov v Ou 0w 2 - ~ ~oo ~or ~ ~ + - ~ -

2 ( 0 v ) Ov 10w �9 7 N + ~ ~ + 7 ~ - ~ "

�9 2 0 w Oz

(A1.44)

Spherical:

[AI] =

Ou (lOu Ov

�9 -; -~+

v) r

(A1.45)

1 Ou Ow w

r s i n 0 0 r ~ Or r 1 { 1 Ov Ow

+ wcotO) r sin 0 0r ~ -

) r sin O + u sin 8 + v cos 0

Finally, we note the the velocity field v gives rise to the vorticity ~o through curl v -- a0. Since we know the physical components of w in various coordinates, the physical components of the spin tensor W - �89 (L - LT),where L is the velocity gradient, can be found from

1 [ 0 - w < 3 > w < 2 > ] [ W ] = ~ w < 3 > 0 - w < l > .

- w < 2 > w < l > 0 (A1.46)

Appendix to Chapter 1" Tensor and Dyadic Analysis 81

Dyadic Analysis

In tensor calculus, the covariant derivative plays a crucial role as is obvious from the foregoing. In dyadic calculus, this position is occupied by the gradient operator V.

In Cartesian notation, the operator X7 is defined through

V - 1 ---0 0 0 (A1.47) Ox + J"~ + k-~z"

This operator acts on a scalar, a vector or a tensor of any order.

Gradients

For example, if r is a scalar and v is a vector, their gradients become a vector and a matrix, respectively. That is, we get

0r 0r Vr -- i + j ~ y + koz , (A1.48)

Vv : i 0v + _-~0v + k0V (A1.49) Ox Joy Oz"

Let us assume that the vector field v has the components (u, v, w) along the (x, y, z) directions, respectively. Because the triad t , j , k is fixed both in magnitude and direction, we find that

Ov Ou Ov j Ow (A1.50) Ox - "~xi + Ox + "~X"X k,

with similar results in the other two directions. Hence

Vv Ou. Oujt _~zk t Ov -- ~x l i + "~" + + ~xiJ

O kk + ~ j j + kj + ax + jk + Oz "

(A1.51)

The important point to note is that the operator V operates from the left, so that the Cartesian component Vvij is 0vj/Oxi.

In cylindrical coordinates (r, 0, z), the basis vectors are er, e0, ez. These are related to the Cartesian triad through

e,. = tcos0 + js in0,

eo - - - ts in0 + jcos0, (A1.52)

e z - - k o

The operator V has the form

0 1 0 0 V - - ~ - ' ~ - "1"- e 0 - --I-- ez

o r (A1.53)


From (A1.53) we see tha t

~----(er ,eo %)

0 N ( ~ , ~ o , ~ )

0 ~z (er, eo, ez)

= (0 ,0 , 0),

= ( e 0 , - e~, 0),

= (o, o, o).

(A1.54)

Now, let a vector field v have the physical components (u, v, w) in the cylindrical coordinates. Then

Vv Ou Ov Ow

-- "~r er er 4- -'~r er eo 4" "--~r er ez

1 0 u l Ov 1 0 w + - -- v]e0er + - + u]e0e0 + - [ ~ ~ [ N ~-b'~ e ~

Ou Ov Ow + ~ z e z e r + ~zeze0 + ~ z e z e z .

(A1.55)

The important point to remember is tha t the above are the physical components of Vv.

Lastly, let us consider spherical coordinates (r, 0, r The basis vectors are (er, e0, er These are related to ( i , j ,k ) through

er = i sin 0 cos r + j sin 0 sin r + k cos 0,

e0 = l c o s 0 c o s r 1 6 2

er = - i sin r + j cos r

The operator X7 has the form

0 1 0 1 0 v - e ~ + e 0 ; ~ + ~, - - .

r sin 0 0r

Now, it follows from (A1.57) tha t

0

0 0--~ (e~, e0, er a

o-7 (e~, eo, ~,)

= (o.o, o).

= ( . o , - -~. o),

-- (er sin O, er cos 0, - er sin 0 - ee cos 0).

(A1.56)

(A1.57)

(A1.58)

Appendix to Chapter 1: Basic Results from Tensor and Dyadic Analysis 83

We shall now determine the components of Vv when the vector field has the physical components (u, v, w) in the directions (r, 0, r The gradient is given by

V v Ou Ov Ow

- -b--~e~e~ + ~-e~eo + -~--r e~er

1 0 u Ov Ow -I-- - v ) e 0 e r -I- (u 4- -I- er

1 0u 0v 0)ecee + r sin 0 [(~-~ - w sin 0)ecer + (~-~__ - w cos

0w + (u sin 0 + v cos0 + -~7)ecer

u 9

(A1.59)

Matr ix Multiplication

In Cartesian coordinates, if we are given a vector equation v = Au, we write it simply as

v~ = a~juj. (A1.60)

That is, in two dimensions, we obtain

Vll

vzl

If we introduce the notation

: [allUl -{-al2u2]i,

: [a21u 1 -{-a22u2]j.

a-(bc) -- (a-b)c,

(A1.61)

(A1.62)

for any three vectors a, b and c, we find that eq. (A1.62) has the dyadic form

v = u - A , (A1.63)

because

i - i : j - j : l , i - j : j - i : 0 ,

u - u l i -{- u2j, (A1.64)

A = a l l i i -{- al2j | + a21~ + a22jj.

The above technique is used frequently in fluid mechanics in the following way. For example, if we wish to calculate the acceleration field, we can write it as

0v a = 0t + (v-XT)v. (A1.65)

Divergence and Curl

The divergence of a vector v is defined through V-v. The curl is defined through tlie vector product V • Since the results of these are already given in (A1.38)-(A1.43) above, they will not be repeated here.

84 1. Kinemat ics of Fluid Flow


2 Balance Equations for Smooth and Non-Smooth Regions

Part A: S m o o t h V e c t o r Fie lds on S m o o t h R e g i o n s

In w we shall derive Reynolds' transport theorem and use it in w167 to obtain the equations for the conservation of mass, the balance of linear and angular momenta and the conservation of energy. In proving these results, no at tempt is made to display hypotheses which require minimum smoothness assumptions on the integrands or the regions on which they are defined. That is, all functions are of class C 2, the regions are bounded and possess a piecewise C 2 boundary, so that an external unit normal may be defined at almost all points of the boundary. The major result of w is the existence of the stress tensor which plays a crucial role in continuum mechanics.

In w equations of motion using the stress tensor defined in the material description are derived for these have relevance to shock and acceleration wave problems.

13 Reynolds' Transport Theorem

Let ~(x, t) be an nth order tensor field defined over the volume ]2 occupied by the body B at time t. Then, in Cartesian coordinates, Reynolds' transport theorem asserts that

-~ �9 d x d y d z - (~,+r div v) dxdydz, (13.1) V ~

where the material derivative (~ is given by

0 r q) = - ~ + {Vr (13.2)

86 2. Balance Equations for Smooth and Non-Smooth Regions

In indicial notation, this means that

O(~ix...i,, + r (13.3) r = 0 t

The difficulty in proving Reynolds' theorem arises from the fact that the volume ]) occupied by the body is, in general, changing with time t. This is overcome by converting the integral to one defined over the volume in the reference configuration.

Thus, let l)n be the material volume occupied by the body in its reference configuration BR. Assuming that the motion x(X, t) is twice continuously differentiable, we may write

r t) - r t), t) = r t). (13.4)

Then, using the transformation rule for a multiple integral, we have

d f d f LJ(x(x, t)), -.~ (~ dx dy dz - -~

v v n

(la.~)

where the absolute value of the Jacobian, ]J], is used. As is well known, the Jacobian is the following determinant"

J (x(X, t)) -

0x 0x 0x

Oz Oz

(13.6)

The matrix in (13.6) is thus nothing other than the deformation gradient F. We shall assume, as in w that the motion x(X, t ) is such that this determinant is positive.

Next, because ])n is independent of t, we may differentiate the integrand on the right side in (13.6) with respect to t to obtain

_'J- -~ (~ a d X d Y dZ = (~ J) d X d Y dZ. (13.7)

Now, the rule for the material derivative leads to

~tt ~(X, t ) d O(x(X, t), t) dt 00

= W + ( r e } v , (13.s)

13 Reynolds' Transport Theorem 87

where the notation of (13.2) has been employed. Also, the rule for differentiating a determinant leads to three determinants:

d J

dt

d [ Ox cgx Ox (~) -~)

Oz Oz Oz

+

Ox Ox Ox

dt X O Y ! . ~ c9~ (o~1 O..~z --_0 z O..$z OX O Y OZ

(13.9)

O_.q_x O z O x OX O Y OZ

+

~ ~ J ~(~) Let us consider the first element in the first determinant. Here, we may change

the order of differentiation to obtain (cf.(1.14))

0 (+) "~ ~ - OX " (13.10)

Now, d x / d t - u ( x , y , z , t ) is the velocity component in the x-direction which means that

el Ox _ Ou _ Ou Ox Ou o~ + (13.11) -~ Y-~ o x - o~ o x + - ~ o--2 o~ o x

Similar procedures have to be applied to the other two elements in the first row of the first determinant. We thus derive

d Ox _ Ou _ OuOx OuOy + O u O z

"~ ( - ~ ) OY -- 'Ox OY + - ~ OY ~ OY ' d Ox _ Ou OuOx OuOy f O u O z

-~(-gh) - o z = o ~ o z + Oy OZ o ~ o z

(13.12)

If (13.11) and (13.12) are substituted into the first determinant, a fairly straightforward calculation using the properties of a determinant shows that

(~) ~(~) 0u 0_~ 0_~ 0_~ ox oY oz Ox J' (13.13) O._$z ----0 z O_~z OX O Y OZ

with similar results for the other two. The final outcome is

d J - ~ = g div v. (13.14)

It follows that (13.7)-(13.8) and (13.14) provide a proof of Reynolds' theorem announced above in (13.1). Another way of looking at Reynolds' result is that

+ ~ . - ~(r ~.)- (r + r ~i~ v) ~ . (13.15)


That is, we use the product rule on �9 dr, differentiate the tensor field ~ and take the material derivative of the infinitesimal volume to be

~ ( ~ ) = [div v l ( ~ ). (13.16)

Now, using the spatial representation, we find that

0 r 0 r (~ + ~ div v - ~ - + {Vr + ~ div v = --~ + div (r (13.17)

Applying the divergence theorem, one ha~

/ / -~ ~ dv -- (~ + r div v) dv ---- --~ dv + ( r n dS, (13.18)

l) )2 l) s

where S is the surface bounding the volume ]) and n is the unit external normal to this surface. The above is another form of the transport theorem of Reynolds.

14 C o n s e r v a t i o n of M a s s

In continuum mechanics, the conservation of mass is assumed to be a postulate. In other words, for all material volumes ]), i.e., those containing the same particles for all times,

p e,, = 0, (14.1)

where p = p(x, y, z, t) is the density at the point (x, y, z) at t ime t. By the transport theorem just proved, the above equation reduces to

/ [O--~ + div(pv)] dv = O. (14.2)

V

Since the size and shape of the material volume is arbitrary, a necessary and sufficient condition for the conservation of mass is the continuity equation:

Op - ~ + div(pv) -- ]) + p div v = O, (14.3)

where, as usual, the superposed dot denotes the material derivative. In incompressible materials, only isochoric, i.e., volume preserving motions are

possible and, since p is a constant everywhere, conservation of mass implies and is guaranteed by

However~

div v -- vJ,j -- 0. (14.4)

1 div v -- t rL -- -~trA1, (14.5)

where L is the velocity gradient and A1 is the first Rivlin-Ericksen tensor. Thus an assertion equivalent to the conservation of mass in isochoric motions is tha t t rL --

15 Balance of Linear Momentum 89

trA1 = 0. This condition demonstrates why numerous examples of velocity fields in Chapter 1 were either assumed or shown to obey these kinematical conditions.

Also, by the transformation rule for multiple integrals employed in the previous section, we know that the infinitesimal volume dv in the spatial configuration is related to the corresponding infinitesimal volume dV in the material configuration through

dv = J dY. (14.6)

Hence, if PR is the density in the reference configuration, conservation of mass is equivalent to the condition

p dv -- PR dV, (14.7)

which in turn yields = pj. (14.8)

Clearly, if the density is finite in the reference configuration, it will be finite in the current one provided the Jacobian J is positive and finite.

In view of (14.8), another necessary and sufficient condition for isochoric motions is

J = det F = 1, det (3 = 1. (14.9)

The latter condition is another reason why we shall demand later that in all flows of incompressible viscoelastic liquids, det Ct(t - s) = 1.

15 B a l a n c e of Linear M o m e n t u m

The forces that act on the body or its part are divided into two categories: those that act by contact with the surface, called surface tractions, and those that act at a distance, called body forces. Let V denote the volume occupied by the body B at time t, t the contact force per unit area on its surface S exerted by the outside world, and b the body force per unit mass. Then Newton's second law of motion in an inertial flame of reference states that the rate of change of the linear momentum is equal to the external forces on the body, i.e.,

d f pv dv = f t ds + f pb (15.1) ,5 P'

Using Reynolds' theorem, one finds that the left side is

f [ ( ~ t + P d i v v ) v+pa] dv. (15.2) I)

If one assumes that mass is conserved, one can see immediately that the equation of motion for a continuous medium now becomes

V 3


We wish to convert the surface integral in (15.3) to a volume integral through the divergence theorem in order to obtain a differential equation for the balance of linear momentum. To achieve this, consider a particle X which occupies a point x in the body at t ime t, and through this point let a plane be drawn such that n(x) is the unit normal to the plane. (See Figure 15.1.)

B . X

B+

FIGURE 15.1. Illustration of the normal unit vector at x, protruding from one part of the body into another.

Suppose the body is now considered to be divided into parts: B + into which the unit normal n is directed and B- on the other side. Then, B + exerts a force on its counterpart B- and the basic assumption which has already been made is tha t this force per unit area is the stress vector t. Now it is matter of experience that this stress vector at a point depends on the direction of the unit normal. For example, consider a bar subject to a tensile force. (See Figure 15.2.)

FIGURE 15.2. A tension bar to demonstrate that the stress at P depends on the direction of the unit normal.

It is clear from the above situation that the force per unit area at the point P in the bar changes with the direction of the normal vector n or the stress vector is a ~ n c t i o n of n.

Returning to the general situation, let us make this dependence of the stress vector on n explicit:

t -- t ( x , t, n) . (15.4)


The real problem is how does this vector t depend on n? Is the dependence linear in n or is it of a nonlinear 1 kind? Leaving aside the proof for the moment, let us accept Cauchy's result that the stress vector and the unit normal are related linearly through the stress tensor T. This is embodied in the following statement:

t (x , t, n) = T(x , t)n. (15.5)

Tha t is, over the body, a stress tensor field T is defined, and if one wishes to find out the force per unit area t exerted by B + on B- , then a plane with a chosen unit normal n is to be drawnthrough the point in question. The formula (15.5) then yields the answer.

Of course, if the point lies on the boundary of the body, then the unit normal n will be the external normal and the stress vector t becomes the traction vector mentioned in (15.1) already. The relation (15.5) can be used to find how the stress vector at the boundary is related to the stress tensor at tha t boundary point.

Substi tuting the formula (15.5) into (15.3), we obtain:

fpadv=fTndS+Jpbdv. (15.6)

Appealing to the divergence theorem, the surface integral may be turned into a volume integral:

/ T ~ j n j d S : / T ~ j , j dv, (15.7)

s V

where Tij , j is the divergence of the stress tensor field. Hence we have the first law of motion due to Cauchy:

/ p a d v = f d i v T d v + f p b d v , (15.8)

from which one obtains the differential equations of motion:

div T + pb = pa,

Tij,j + pbi = pai. (15.9)

We shall now present a proof 2 of Cauchy's formula (15.5). First of all, we make the following continuity asumption:

For each fixed unit vector m in the body, the stress vector t (x , t , m ) is a continuous function of x.

The reasons for this assumption will be made clear in w below and for the moment we shall accept it and proceed. Next, choose a point x in the body at

i i i i

l An example of nonlinear dependence is t ----- f ( x . n)n, where f is an arbitrary, scalar valued function.

2The proof here is due to GURTIN, M.E., The Linear Theory of Elastzcity, Encycl. Phys., V i a / 2 , Springer-Verlag, Berlin, 1972. See w

t ime t, select a plane through it and draw the unit normal n to this plane. Let the stress vector at this point x, exerted by B + on B- , be given the suggestive notation t(n). Let the stress vector exerted by B- on its counterpart be denoted by t(_n). The first step in proving (15.5) is to show that these two stress vectors are related through

t(n) (x, t) -- - t ( _ n ) ( x , t), (15.10)

which is Cauchy's reciprocal theorem; it is the continuum mechanics version of Newton's third law.

Now, let x be the centre of the parallelepiped Pe. See Figure 15.3. Let its top and bot tom surfaces 3e and 3~- be squares of length e with unit external normals n and - n respectively. Let the height of Pe be e 2. Let ,~ be the union of the four lateral surfaces of Pc.

II

~176176176176176176176176176176176 ~176176176176 .......... i ..........

E ~r

FIGURE 15.3. The parallelepiped used to prove Cauchy's reciprocal theorem.

Then the surface S(Pe) of P~ is the union:

s(p ) = u s : u (15.11)

and the volume and areas are given by

V(P~) = r A(S+) _ A(S[) = e 2, A(&) - 463. (15.12)

Let us now assume tha t the density, acceleration and body force fields are bounded so tha t we have the following:

k ( t ) - - s u p IP(Y, t){a(y, t) - b(y, t)}l < oo. y E P e

(15.13)

Then, from the equations of motion (15.3) applied to this parallelepiped, we find that

f t(.) dS! 2(P.)

<_k(t)V(P~). (15.14)


Hence, from the assumed continuity of the stress vector field in the body,

lim 1 / ~ o ~'~ t(n) d S - - O.

2(P~)

(15.15)

Again, from the assumed continuity of t(m) for each fixed unit vector m in the body, we have that

lim 1 / ~ 0 ~ t(• d S - t(• (x, t). (15.16)

Also,

lim 1 / ~-+0 ~'~ t(n) d S - O. (15.17)

3~

What has been proved is that , as ~ -~ 0, the body force on the parallelepiped and the inertia of the mass in the parallelepiped vanish along with the forces on the lateral surfaces, leaving only the stress vectors t(n)(x, t) and t(_ n)(x, t). Hence the equation of motion (15.3) has been reduced to the equation

t(n ) (x, t )-b t(_n)(X, t) -- O, (15.18)

which is the proof of (15.10). The second step in the proof of (15.5) is based on constructing a tetrahedron

with its vertex at x. This is drawn so that three of its sides are mutually orthogonal and, in a Cartesian coordinate system, they form the planes xi -- 0, i = 1, 2, 3. See Figure 15.4.

T x

. . , ~ 1 7 6 1 7 6 ~ 1 7 6 ~ ~ 1 7 6 "* . . . . . . . . . . . . . . . .

1

F

x 2

FIGURE 15.4. Proof of the existence of the stress tensor using a tetrahedron.

The inclined fourth surface is assumed to have the outward unit normal n such that n # +ei , i -- 1, 2, 3, where the latter are the unit vectors along the coordinate a x e s .

Let the altitude of the tetrahedron be h, the area of the inclined surface ~:h be A. If the projections of n onto the three axes are nl , n2, n3, then the area A~ of the face S~ normal to the x~ direction is n i A - A i , i - 1,2, 3. Assume that the fields

p, a and b are bounded as before and tha t t(n) is the stress vector field on the face with the unit normal n and ti , i = 1, 2, 3, are the stress vector fields on the faces, with the unit normals in the - x i directions. Then, in a manner similar to the derivation of (15.18), we can consider what happens to the equations of motion (15.3) applied to the te t rahedron as h -+ 0. We obtain

t(n) dS + ti dS <_ k ( t )V(h) , (15.19) "-- Si

where Y(h ) is the volume of this tetrahedron. It is a fact tha t Y(h) -- hA l3 . If we now divide by A and then let h -~ 0, we obtain as before tha t

3

t(n) (x, t) + Z t, (x, t)n, = 0. (15.20) i--1

The crucial point is tha t the three stress vectors ti do not depend on n because they are defined to be the stress vectors which act on the three coordinate planes passing through x at t ime t. Now, we may write (15.20) as

3

t(n)(X, t) + Z t i (x , t)(ei �9 n) -- 0. (15.21) i--1

This equation shows tha t t(n ) is linear in n, provided n ~ -t-ei, i -- 1, 2, 3. On the other hand, because of (15.10), the above holds even when n -- :t:ei. To see this, let n----el. Then (15.21) becomes

t ( e l ) -~- t l = 0 . (15.22)

This is nothing but (15.10) because we have defined the vector t l to act in the - X l direction and so

t l - t ( -e , ) . (15.23)

Thus, we conclude tha t the stress vector t(n)(x, t) is linear in all unit vectors n, or there is a stress tensor T(x , t) such tha t

t(n)(x, t) = T ( x , t ) n . (15.24)

Indeed by the property of a dyadic product, it follows tha t

3

T -- ~ ti | ei. (15.25) i--1

To put (15.24) in a more familiar context, let us adopt the convention tha t T/j is the i-th component of the stress vector acting on the outward (positive) side of the plane xj--const . Consider the plane x --- 0 through the origin. Choose n -- 1, the unit vector in the x-direction. Then t(_i) -- t l in the notation of (15.23) and by (15.22) we obtain

- t l x -- Txx, - t l ~ -- Tyx, - - t l z - - T z x . (15.26)

16 Balance of Angular Momentum 95

Similarly, we can obtain the components of t2 and t3. Returning to (15.21), we derive that t(n)k - Tkmn,~, where the quantities Tkm are independent of n and depend on (x, t) only.' By the quotient law of tensors, Tkm are the components of a Cartesian tensor of second order, and in general tensor notation,

t ~ n ) - Tk '~n,~, t ( ~ ) - I n . (15.27)

Now that the existence of the stress tensor has been demonstrated, the equations of motion in the form given in (15.9) are valid.

There are situations where some of the assumptions made here may be relaxed. In particular, one may ask whether the smoothness of the boundary of a domain is necessary for the divergence theorem to hold. Equally, one may question the need for the stress vector to be a continuous function of the position vector for every fixed unit vector in the body. We shall take up these matters in Part B of this Chapter.

16 Balance of Angular Momentum

Assuming that there do not exist any internal angular momentum, body couples, and couple stresses in the body, the balance of angular momentum equation with respect to an inertial frame for a material volume ]2 says that the rate of change of the angular momentum is equal to the external torque on the body. There are two torques: those arising from the surface tractions and those from the body forces. Hence, this balance equation may be written as

-~ x x (pv)dv -- x x t dS + x x (pb)dv, (16.1)

V s P

where x is the position vector of a particle from the fixed origin of the coordinate frame, and S is the surface of the volume ]?. Using the transport theorem, we find that the left side of (16.1) takes the form

' i J i - - x x (pv) dv -- (]9 + p div v ) ( x x v) dv + x x (pa) dv, (16.2) dt

when the identity x x v = v x v = 0 is used, along with the fact that the material derivative of the velocity vector v is the acceleration vector a. The continuity equation (14.3) shows that the first integral on the right is zero and hence the rate of change of the angular momentum is equal to the momentum of the inertia, i.e.,

~ xx(pv) dv= xx(pa) gv. (16.3) V ~

Examining the first integral on the right side of (16.1), the existence of the stress tensor T means that we may write this integral as

I x x t d S - / ( x x Tn) dS. (16.4) J q ]

3 S


Introducing Cartesian coordinates for convenience, the right side of (16.4) is

ei.ik xj Tkm nm dS, (16.5)

s

where Eijk is the alternating tensor - see (A1.31) - (A1.32) in the Appendix to Chapter 1. By the divergence theorem, we have

/ ~ i j k x j T k m n m d S = / (~ijkxjTkm),m dv ~2 s

= f + av (16.6) v

-- / (eqk Tkj + eijk xj Tkm,m) dr.

Introducing the axial vector t A corresponding to T through

the balance of angular momentum equation now reads

/ x • (pa) d v - / (tA + x • divT) dv + / x • (pb) dv. (16.8)

Assuming that the balance of linear momentum holds in 1), i.e.,

equation (16.8) reduces to

dirT + pb = pa, (16.9)

I t A dv = 0. (16.10)

v

Since l) is arbitrary, we conclude that t A -- 0 or that

6ijk Tkj = 0. (16.11)

Because of the property of the alternating tensor, it follows from the above that the stress tensor is symmetric, i.e.,

T - T T. (16.12)

Hence, in the absence of internal angular momentum, body couples and couple stresses, the necessary and sufficient condition for the balance of angular momemtum, when the conservation of mass and balance of linear momentum equations are satisfied, is that (16.12) shall apply. This is Cauchy's second law of motion.

In this treatise it will always be assumed that the stress tensor is symmetric and thus the second law of motion is automatically satisified. Even so, there are occasions when one is called upon to calculate the angular momentum of a fluid mass and equate it to the external torque to gain additional insight into the flow field.

17 Balance of Energy 97

17 Balance of Energy Let K: be the kinetic energy of the body and ~ its internal energy so that for any material volume 1),

S1 12 f K: - : p l y dr, ,f, -- pe dr, (17.1)

V

where e is the specific energy per unit mass. The mechanical power exerted by the outside world on the body is given by the rate at which the surface tractions and the body force do work on the body. These two are

i t . v dS + S P b . v dv. (17.2) s V

Let the etttux of energy out of S be q per unit area and the energy supply be r per unit mass. Then one postulates the balance of energy equation as

d{S(1 )}aS -~ -~pv. v T pe dv -- V

( t - v - q-n) dS + f (pr + pb . v) dv. (17.3) V

Using the existence of the stress tensor, one obtains:

) } / ~-~ p v - v + p e dv = ( v - T n - q - n ) dS a (17.4)

+ f p ( b - v + r ) d v .

Assuming that mass is conserved and using the transport theorem, the left side of (17.4) can be written as

f p(v. +~) dr, (17.5) a

where a is the acceleration and the superposed dot is the material derivative. The divergence theorem may be applied to the surface integral on the right side of (17.4) and, using Cartesian tensor notation to make the derivation easier, we find that

f (vi 7~j nj + qi n~) d S - f (T~j,j vi + T~j vi,j -qi,~) dv. s V

(17.6)

Hence, the equation (17.4) now becomes

i P ( v - a + ~ ) d v - - f ( v - d i v T + t r T L T - d i v q ) dv V v (17.7)

+f p ( v - b + r ) c l v ,


where L T is the transpose of the velocity gradient. Using the balance of linear momentum equation and the symmetry of the stress tensor implied by the balance of angular momentum, one finds that (17.7) may be simplified to

f f(tr T D - div q + pr) dv, (17.8)

~2 V

where D is the symmetric part of the velocity gradient. From this follows the differential equation of energy balance:

pk = tr T D - div q + pr. (17.9)

The reader will note that the conservation of mass equation was basic. This is needed in obtaining Cauchy's first law of motion, and these two together are needed for Cauchy's second law of motion. All three are necessary to obtain the energy equation in the form given.

18 Other Forms of Equations of Motion

The stress tensor T is a spatial field and, in this treatise, this tensor will figure prominently. However, when dealing with wave propagation problems, one finds that the Piola-Kirchoff stress tensor TR is very useful. This tensor is defined through

T - J - 1 T R F T ; T k i n - J -1TRkaxm (18.1)

The usefulness of this tensor lies in the following: if dS and n are the infinitesimal surface area element and the outward unit normal in the spatial configuration, respectively, and dSR. and N the corresponding ones in the reference configuration, then the contact force fc is given by

fc - [ T n dS = [ T a N dSR, (18.2) O r J

2 .SR

where s is the surface of the body in the reference configuration. The stress vector t(N)R in the reference configuration is, by definition

t ( N ) R - TRN. (18.3)

The equations of motion for T R become

Div TR + pRb = pRJt, (18.4)

where Div is the divergence operator in the reference configuration. Equation (18.4) may be written in indicial notation as

TR k~ (18.5) ,a + PR bk -- PR ~k,

where PR is the density in the reference configuration. In terms of TR, the balance of angular momentum reads

T R F T " F T T ; Takaxm,a -- xk,a TR ka. (18.6)

19 The Length of a Curve 99

One may construct other forms of stress tensors such as:

,~ _ F - 1 T R ; ~ o , ~ _ X , ~ TRm~ ~ m �9 (18.7)

J T - C T C . (18.8)

The last two are not very useful in viscoelastic fluid flow problems.

P a r t B: D i v e r g e n c e a n d S t o k e s ' T h e o r e m s for R e g i o n s B o u n d e d b y F r a c t a l C u r v e s

Some of the fluid-fluid interfaces, such as in viscous fingers, 3 or fluid-air interfaces as in an extrusion process, are jagged and may be considered to be 'fractals'. If this is correct, then it is necessary to understand what is meant by a fractal and to seek how one may interpret the laws of continuum mechanics in regions bounded by fractal curves or surfaces. As is obvious from the material in the previous sections, the divergence theorem plays a crucial role in obtaining the partial differential equations for the conservation of mass, the balance of linear and angular momenta and the balance of energy. Thus our at tent ion will be focussed here on this theorem as well as Stokes' theorem.

To make the presentation easier, we shall confine our at tention to two dimensional regions bounded by fractal curves. We summarise the procedures to calculate the length of a fractal curve, its Hausdorff and box (or Minkowski) dimensions, the construction of a unit tangent and normal to such a curve. Thus, first of all, the concept of the length of a (fractal) curve is presented in w tha t of the Hausdorff dimension of a set in w and, in w a method to calculate the box dimension of a set is given. Because irregular curves need not possess tangents, and regions bounded by them need not have external unit normals, in w we discuss how one may define the existence of such vectors to an arbi t rary curve through an application of geometric measure theory. Next, in w we delineate those regions in which the divergence theorem holds and in w we consider Stokes' theorem. It is seen tha t these two theorems apply even if the boundary of a domain is a fractal curve, and we show how to interpret these theorems numerically. A number of examples is provided throughout to illustrate the ideas presented here. Finally, the existence of a stress tensor in such regions is not yet well understood and we make some conjectures about this mat ter in w

19 The Length of a Curve

To begin with, let us assume tha t we are dealing with a two-dimensional region bounded by a piecewise C 2 curve, which is a Jordan curve. We need the curve to have an inside and an outside and be a simple closed curve and these are embodied in the notion of a Jordan curve. In this case, any s tandard textbook on calculus tells us how to find the length of the curve, the tangent and unit normal to the curve at all regular points. Let us now suppose tha t the boundary 0fl is a continuous

3FEDER, J., Fractals, Plenum, New York, 1988.


curve. For example, suppose it looks like the viscous fingers taken from the book by FEDER. (See Figure 19.1.)

FIGURE 19.1. Viscous fingering in a circular Hele-Shaw cell: air displacing glycerol (left) and water displacing a mixture of scleroglutan in water (fight), after FEDER.

In such a case, the problem of defining its length is not straight forward and here we turn to the t reatment by ALEXANDROV and RESHE~NYAK. 4

Essentially, the argument in the above book depends on the proof tha t a contin-

uous curve can be approximated to any degree of accuracy desired by a sequence of polygonal lines (see Figure 19.2).

~

FIGURE 19.2. The length of a curve approximated by secants.

Basically, this means that the curve can be thought of as a connected thread of straight lines. Given this, the idea of the length of a curve is quite easy to formulate. We simply project a polygonal segment, called a secant approximation,

of the curve onto the x and y axes, measure the lengths of these projections and find the length of the polygonal segment from them. Adding all the lengths of the segments together, we arrive at the length of the curve in the limit.

4ALEXANDROV, A.V. and RESHETNYAK, Yu.G., General Theory of Irregular Curves, Kluwer Academic, Boston, 1989.

20 Fractal Dimension 101

As a simple example, consider the Koch curve (see Figure 19.3). Here, the construction of the curve itself suggests how to find its length, for one begins with a straighline segment of length 3 units. The middle third, of length 1, is removed and replaced by a steeple consisting of two inclined lines of length 1 each; and at each successive step, the middle third of every line is similarly altered. We thus observe that the curve begins with 3 intervals of length 1 each (corresponding to j -- - 1 ) , turns into 4 intervals of length 1 each (j -- 0), and so on. That is, at each stage, the length Lj of the curve is

i j -- 4J+13 - j , j - - - i , 0 , 1 , . . . (19.1)

Of course, as j -~ oo, the length Lj --. oo. Not only is the Koch curve is non- rectifiable overall, every part of it is also of infinite length.

Speaking of curves that one meets in connection with viscous fingures (cf. Figure 19.1), it is easy to see that these curves resemble very closely a chain of straight lines and hence the length of such a curve is,theoretically at least, easy to define. Practically speaking, it is easy to measure as well.

..... E0

E l

E 2

E3

E

FIGURE 19.3. Non-rectifiable Koch curve constructed by replacing the middle third length at each step.

20 Fractal Dimension

Traditionally, in continuum mechanics, whenever one speaks of the dimension of a body, we know intuitively what we mean. In those cases when the body occupies the entire line, or the whole plane or the whole space, the meaning of the dimension is clear. The meaning is equally clear when we map, say, the real line or a part of

it onto a piecewise smooth curve; in this case, the dimension of the curve is clearly one.

Let us now consider the real line between [0,1] which is of dimension one and remove a finite number of points from it. Clearly, the dimension is unaffected for we get a chain of broken lines. Now, consider the situation when we remove a countably infinite number of points (for example, the set of all rational numbers in the above interval), or an uncountably infinite set of points from the above set. Since the dimension of a point is zero and tha t of the line is one, we may well ask what the dimension of the points tha t are left over may be. Intuitively, one feels tha t the number lies between zero and one. Can it be a fraction? Using the terminology of MANDELBROT 5 when the dimension is not a whole number, such a set is called a fractal set.

Hence we are led to look at the definition of the word dimension. Use of concepts such as linear independence, vector basis and so on from linear algebra are not of much use in the above situation. Wha t is needed is an understanding of the dimension based on geometric measure theory, i.e., on length, area, volume, etc. Here we shall present a heuristic introduction to this idea based on the mathematicaly rigorous t rea tment in the two books by FALCONER. 6

Take the real line segment between [0, 1] and cover it with N + 1 equal intervals, each of length 1IN. The resulting length is given by ( N + 1)IN. If we take the limit as the number of intervals N --. oo, we get the number 1. This is, simply put, the dimension of the line interval.

Let us now consider how to generalise this notion. Denote an interval of the real line or a curve in the plane by E and take enough subsets which overlap the set E. There is no reason why each of these subsets should have the same length. All tha t is needed is tha t the latter intervals, collectively, cover E (see Figure 20.1).

'~L ~ J ~ j ~ j ~ J ] J

r~ E

FIGURE 20.1. A covering of E by other subsets of the real line. Each subset U~ is of length 6 at most.

Thus, let Ui be a non-empty subset of the real line 7~ such tha t E C (J Ui, the union of all these subsets, and let the length of each Ui obey

o < IU, I _< (20.1)

We call U ui a 5-cover of E. There will be many different covers, since each set Ui has to obey (20.1) only, and one can form many collections from such sets. Now

5MANDELBROT, B.B., The Fractal Geometry of Nature, Freeman, San Francisco, 1982. 6FALCONER, K.J., The Geometry of Fractal Sets, Cambridge University Press, Cambridge,

1985; Fractal Geometry, Wiley, New York, 1990.

define the Hausdorff measure of E through

oo

7/~(E) -- inf E IUil ~. (20.2) i----1

Tha t is, we find the least value of the sum over all possible ~-covers of E and this gives us T/~(E). Then, as 6 --, 0, we get

7 / 8 ( E ) - limT~}(E). (20.3) ~- - ,0

We call this number T/s, the s-dimensional Hausdorff measure of the set E, and observe t ha t the Hausdorff measure changes with the exponent s.

There are some immedia te consequences of the way the measure has been defined. Firstly, the value of 5 will ul t imately become less than 1 for 5 must tend to 0. In this case, because ]U~I 8 _< 68 _< 1 ( - 5~ it is clear tha t T/s is a non-increasing function of s for all 0 _< s < oo. In fact, it is easily seen tha t 7/~ (E) is the number of points in the set E and thus, for the line segment we are dealing with, T/~ - oo. This is reflected in Figure 20.2.

Secondly, because ~ < 1, we observe tha t for an arbi t rary set E, whenever t > r, the following is true:

iv, i < lull (20.4) i i

Hence, by taking the infima of both sides, we find tha t 7 ~ ( E ) < ~ t - r T ~ ( E ) . If we now let 6 --* 0, we find tha t ?-/t(E) -- 0 for all t > r provided ~ r ( E ) < oo. On the other hand, if 0 < 7~* < oo and t > v, then it follows tha t 7~ v = oo, because 6,-v --~ 0 as 6 --, 0. Thus, there is a special value of s at which T/s (E) suffers a jump. See Figure 20.2. We call this number the Hausdorff dimension, DH(E), of the set E.

O0

I-Is

11 I II [

0 DH(E ) n

FIGURE 20.2. Hausdorff dimension DH(E) of the set E. The Hausdorff measure suffers a jump at the value of the dimension.

We may summarise the above through

0 if DH < s; (20.5) U'(E) oo if 8 < DH.

There are sets where DH may be 0, e.g., a single point, or oo. These do not concern us here.

We now record two fundamental properties of the Hausdorff measure T/s for they are useful in dealing with self-similar fractals. These are:

(i) If E and F are two disjoint sets, then

7"ls (E U F) = 7-l~ (E) + 7-lS (F). (20.6)

(ii) If A > 0, then 7~ 8 (AE) = AsT~ (E), (20.7)

where AE = {Ax : x E E}. It is the set obtained by scaling E by a factor of

We shall apply the above properties to two examples: the middle thi rd Cantor set and the Koch curve. The middle third Cantor set may be constructed as follows 7 - see Figure 20.3.

0 1/3 2/3 1

I / I I I /

E L E R

E 0

E l

E2

E3

FIGURE 20.3. Midle third Cantor set obtained by discarding the middle third of each line segment successively.

Let the set E0 - [0, 1] be given. From this, remove the middle th i rd and obtain the set E1 = [0, 1/3] U [2/3,1]. Remove the middle third from each of the two intervals above and arrive at E2 = [0, 1/9] U [2/9, 1/3] U [2/3, 7/9] U [8/9, 1]. Continue in this manner and observe tha t each set Ej consists of 2J intervals, each of length 3-J for j -- 0,1, 2, .... As j --* oo, we are left with the Cantor set E which may be defined as

(X)

N E = j = 0 Ej , (20.8)

7See, for example, FALCONER, K.J., The Geometry of Fractal Sets, Cambridge University Press, Cambridge, 1985, p. 14.

or E contains those points which are common to all E j , j = 0, 1 , - - - . Because the length of the set Ej is given by 2 j 3 - j , the length goes to zero as j --+ oo. Now, fix a value for s and consider the following (cf. (20.2))"

7~s3-j (E) < 2 j3 - is . (20.9)

Tha t is, we have chosen a total number of 2J sets, and each set Uj obeys IUjl _< 3 - j . Now, if

s -- log 2 / log 3, (20.10)

then 7~ s (E) _< 1 as j --, oo, for the right side is exactly 1 with this fractional value for s. Tha t is, we have obtained an upper bound for ;t/s (E). The trick is to prove tha t 7~S(E) __ 1 as well, so tha t 7~S(E) -- 1. The lower bound is quite difficult to obtain in many instances, al though by using the properties in (20.6) and (20.7), we can establish tha t the Hausdorff dimension is indeed log 2/log 3 in the present context.

This result follows because the Cantor set E splits into a left part EL -- E N [0,1/3] and a right part ER = E N [2/3, 1]. As Figure 20.3 shows, E = EL tO ER and EL and ER are similar to E, except tha t they are scaled by a factor of 1/3. Thus, by (20 .6) - (20 .7) ,

7-lS (E) - 7-I~ (EL ) + 7-lS (ER) - 2 7-l~ (E). (20.11)

We know tha t 7~ s _ log 2/log 3 already. If it can be proved 8 that 0 < 7~ e, then one may divide (20.10) by 7~ s and find tha t DH - log e/log 3.

One may delete, instead of the middle third from each set, other fractions and obtain a different Cantor set and this will have a different Hausdorff dimension. 9 The point worth noting is tha t the middle third Cantor set has zero measure (i.e., length), but has a positive Hausdorff dimension.

Let us tu rn to the Koch curve and recall the length Lj of the curve from w This is

Lj - 4J+x3 - j , j - - 1 , 0 , 1 , . . . (20.12)

Because of the self-similarity of this curve, it is easy to show tha t its Hausdorff dimension is log 4/log 3. As is obvious from the foregoing, a continuous curve possessing a Hausdorf dimension DH such tha t 1 < DH < 2, is not only non- rectifiable, i.e., each curve is of infinite length, but every segment of such a curve is itself of infinite length. Nevertheless, in a manner similar to the construction of the Koch curve, one may generate curves with a length Lj - 6 j+l 5 - j and a Hausdorff dimension log 6/log 5, and so on. These fractal curves may be used to generate snowflakes or particles with jagged edges as follows. One begins with an equilateral triangle and on each side of the triangle, a fractal curve of dimension log 4/log 3, or log 6/log 5, or log 5/log 4 etc., is then easily constructed. The resulting snowflakes or particles have the regular and irregular shapes illustrated below in Figure 20.4.

8See FALCONER, K.J., Fractal Geometry, Wiley, New York, 1990, pp. 31-32, where it is shown that ~/8 >_ 1/2.

9FALCONER, K.J., The Geometry of Fractal Sets, Cambridge University Press, Cambridge, 1985, p. 15.


FIGURE 20.4. Snowflake curves of dimensions log 4/log 3, log6/log 5, and log 5/log 4, from top to bottom respectively.

21 The Box or Minkowski Dimension 107

An immediate application of generating these particles with jagged edges lies in s tudying the drag experienced by them, which has relevance to sedimentat ion problems. 1~

So far we have considered curves. W h a t if the set E is an area or a volume? Here the covering sets Ui are defined so tha t their union still overlaps E, whereas their sizes are determined by their diameters. Recall t ha t the diameter of any set U is denoted by IUI and

IUI -- s u p { I x - Y I : x , y e U}, (20.13)

where I x - Yl denotes the distance between x and y, and thus the diameter of a set is the largest distance between any two points in the set. The Hausdorff measure is now defined as before (20.2) and one can calculate the dimension of the set as earlier.

An interesting point is t ha t when the dimension of a set is an integer n > 2, the Lebesgue and the Hausdorff measures of a set are not the same. In fact,

?-/"(E) - c ~ s (20.14)

where the constant cn depends on the integer dimension of the set and is given by

1, n : 1; c= -- ~ /4 , n - 2; (20.15)

4g/3 , n - - 3 .

Hence the two measures are proportional to one another if the dimension of the set is an integer. However, there is no Lebesgue measure of a non-integral dimensional set and hence one uses the Hausdorff measure here.

21 The Box or Minkowski Dimension

In the previous section, it has been made clear tha t the calculation of the Hausdorff dimension of a given set is not a trivial task. If we wish to find the dimension of an experimental ly generated curve such as a viscous finger or the ex t rudate profile of a polymer, an approximate method is required. In this section we investigate such an approach, which is based on the idea of a length scale. T h a t is, we say tha t the length of a curve is given by

L~ ~ cs - s , (21.1)

where c and s are constants. Taking the logarithm, we obtain

s - lim log L~. (21.2) ~--,0 --log

This number s can then be taken as the dimension of the set under consideration. Let us make this a bit more precise.

I~ R.R., PHAN-THIEN, N. and ZHENG, R., J. Non-Newt. Fluid Mech., 57, 49-60 (1995).


Suppose we are given a set E and let Af(E, 6) denote the smallest number of closed balls (resp. intervals) of diameter (resp. length) 26 needed to cover E. If

D B - lim log .Af (21.3) ~-,0 log(1 / c)

exists, then this DB is the practical definition of the fractal dimension of the set E. The dimension D s in (21.3) is called the box dimension or the Minkowski dimension. It must be noted tha t the Hausdorff dimension is always less than or equal to the box dimension. This is because of the way the two numbers are defined: the box dimension uses balls of the same radius, whereas the Hausdorff dimension does not require the size of the balls to the same as well as being the infimum of the relevant sum. Nevertheless, for self-similar sets, the Hausdorff and box dimensions are equal. 11

Returning to the definition (21.3), let us choose balls of radius 6 = 61 and cover the given set E. Count the number of balls. Reduce the size of 6 and repeat. Continue this process till the size of 6 becomes very small. Plot the values of log j~c as the ordinate and log (1/6) as the abscissa. The slope of the line as e --* 0 is the box dimension DB of the set E. It must be noted tha t it is possible to replace the balls by squares or cubes (or boxes) if so desired.

As an illustration in using this ball counting method, let us consider an example from the book by BARNSLEY. t2 Here, the aim is to estimate the fractal dimension of a set of dots in a woodcut. In Figure 21.1, a graph of log Af against log (1/6) has been drawn. The points lie, approximately, on a straight line with a slope of around 1.2. Of course, if DB ~ 1, then we see tha t the points would lie on an ordinary curve.

3.45

z t~O

0.69

D Q

1.1 log(l/e) 4.2

FIGURE 21.1. Fractal dimension of a woodcut obtained by the use of its box dimension.

11FALCONER, K.J., Fractal Geometry, Wiley, New York, 1990, pp. 117. 12BARNSLEY, M.P., Fractals Everywhere, Academic Press, San Diego, 1988.

22 Unit Tangent and Unit External Normal 109

Because the slope is 1.2, it means tha t these points lie on a fractal, non-smooth curve. Leaving aside for the moment whether the dots in the woodcut lie on a fractal curve or not, the above example shows the problems one faces in practice. Any a t t empt to find the box dimension means tha t we obtain an upper bound to it always and, if lucky, the t rue one. Even if the calculations are crude, we shall see later tha t the upper bound to the box dimension does mat ter as far as the divergence and Stokes' theorems are concerned, and thus the box dimension of a set is very impor tant in applications. Consequently, in any problems involving fractal curves, it is essential to determine an upper bound as precisely as possible.

22 U n i t T a n g e n t a n d U n i t E x t e r n a l N o r m a l

In this section, we shall examine how one may define a tangent vector and a normal vector to a curve, wi thout using calculus, and using geometric measures such as length and area. Before we introduce these notions, let us review some results from calculus. As is well known, if the given curve is of class C 2, then one defines the unit tangent and normal as follows. Let the curve be parametrised by the arc length s, so tha t the curve is described by the function x - x(s). Then the unit tangent vector t is given by

dx - - - - - t . ( 2 2 . 1 ) ds

If the curvature n is not zero, the unit normal n is given by

1 dt n = - - - . (22.2)

ds Note tha t the above definition does not always lead to the determination of the external unit normal; however, demanding tha t the bounding curve be a Jordan curve in two dimensions means tha t the region in question has an inside and an outside and hence we can find the unit external normal explicitly. Nevertheless, the definitions do not address the question of how to define a tangent and a normal if the curve is not of class C 2. Because boundaries in continuum mechanics have vertices and edges, it is necessary to define the concept of a tangent and normal without using the idea of differentiability, and we turn to this next.

Consider a point x on the boundary of the set ~. The first question tha t arises is how one can determine whether the given point x lies on the boundary. This is because the curve in question may have gaps (e.g., the Cantor set) or be full of kinks (e.g., the Koch curve) so tha t one does not know where the point is. Thus, whether x lies on 0 ~ or not is easy to answer. Here one uses the concept of the density D(O~, x) of the set 0 ~ at the point in question. 13 The density is obtained from drawing a circle B(x , p) of radius p with the point x as its centre. Now, define

Length of O~ in B(x , p) if s - 1, limp_~0 2p ' (22.3)

n(o~2, x) = 7-/~ [0~ N S ( x , p)] if s ~ 1 limp-..,o (2p) ~ ,

13See FALCONER, K.J., Fractal Geometery, Wiley, New York, 1990, pp. 69-71.


where s is the dimensional exponent. See Figure 22.1. The figure shows tha t in ordinary circumstances, when (22.3)1 aplies, the limit is clearly 1. Thus, in general, if this limit is not zero, then obviously the point x lies on 0 ~ and not otherwise.

FIGURE 22.1. A boundary point determined by showing that the ratio of the length of the curve to the diameter of the circle is 1.

So, we now assume tha t the point x lies on the curve in question. Let a unit vector r be drawn through this point. With this point as the centre, draw a circle B(x , p) of radius p. Also, construct a sector S(x , r , r of angle 2r through x, with r bisecting the sector. See Figure 22.2.

B ( x , p )

Part of

FIGURE 22.2. Construction of tangent to a curve. Note that the curve does not lie in the set B - S .

Essentially, for every sector angle r > 0, a negligible par t of the curve (i.e., par t of the set 0~/) must lie in B - S, in order tha t I" is the tangent vector. Tha t is

lim p-1 {Length [O~'l n (B - S)]} - 0. p-*0

(22.4)

In Figure 22.2, no par t of the curve 0f / l i es in B - S, and thus the set 0 ~ has a unit tangent at the point x.

We shall show tha t the above definition of a tangent makes sense by demonstrating tha t a square does not have a tangent at the vertex because a negligible port ion of the curve does not lie in B - S . See Figure 22.3 and note tha t the condition (22.4) is not obeyed; indeed, the limit is 1 because the length of [0f~ n ( B - S)] is the same as tha t of the radius p of the circle.

22 Unit Tangent and Unit External Normal 111

[Oa n (B -- S)]

FIGURE 22.3. Tangent at a corner does not exist because a significant amount of the curve lies outside the sector defined by the vector 7".

If the Hausdorff dimension s is different from 1, then we have to replace (22.4) by its analogue 14 from (22.3)2, i.e.,

lim p-S ~s [OFt n (B - S ) ] - O. p---,O

(22.5)

We shall now discuss the construction of the unit external normal. As before, let us assume tha t the point x lies on the boundary of the set ~. Erect a unit vector v at this point. Let us consider all those points y such tha t the dot product ( y - x ) - v > 0. See Figure 22.4.

FIGURE 22.4. Construction of the unit external normal. The region hatched by lines sloping upwards to the right is B + n ~; that hatched with lines sloping the other way is B - n i l .

Clearly, the collection of these points y lies in the upper half plane. Denote this set by P+ . Now, draw a circle with x as the centre and radius p and denote this

t4see FALCONER, K.J., Fractal Geometry, Wiley, New York, 1990, pp. 77-79.

by B(x , p). Let us now create the set B + which is common to both B and P+"

B + (x, p) - B(x , p) n P+. (22.6)

That is, B + is the semi-circle in the upper half-plane. Let us denote by P - the set of all those points y such tha t (y - x) - v _< 0. Then, as before, we define we define the semi-circle B - in the lower half-plane through

B - ( x , p ) - B(x ,p ) n P - . (22.7)

If (cf. Figure 22.4),

Area o f [B + n fl] lim ...... = 0 (22.8) p-,o Area o f B "'+

and

lim Area o f [B- n ~] = I, (22.9) p-~o Area o$ B -

we say tha t v is the exterior unit normal to ~ /a t x. If this exists, it is unique. For the specific si tuation depicted in Figure 22.4, it is easily seen tha t these two limits are true. Hence, the vector v, in Figure 22.4, is the unit external normal at x.

Now, just as the tangent does not exist at the vertex of a square, an external unit normal does not exist at the vertex, because the limit corresponding to (22.8) is 0, while tha t relative to (22.8) is 0.5 and not 1. (See Figure 22.5.)

B-n

FIGURE 22.5. A unit external normal vector does not exist at a corner.

We emphasise tha t the construction of the external unit normal, as described above, is valid if the boundary 0 f / i s one-dimensional. For higher integral dimensions, see ZIEMER 15 or F E D E R E R . t6 If the dimension s is not an integer, then there is no definition of an external normal similar to the one for the tangent.

From a practical point of view, the above constructions do not lead to a measure- theoretic unit tangent or a unit normal to fractal curves whose Hausdorff dimension lies between 1 and 2. This is because, almost everywhere, such curves have no

15ZIEMER, W.P., Arch. Rational Mech. Anal., 84, 189-201 (1983). 16FEDERER, H., Geometric Measure Theory, Springer-Verlag, New York, 1969.

23 Flux across a Fractal Curve: The Divergence Theorem 113

measure-theoretic tangents 17 and no measure-theoretic unit normals either, provided one assumes tha t the normal is orthogonal to the tangent. Nevertheless, the divergence theorem holds for regions bounded by such curves. This will be discussed next.

23 Flux across a Fractal Curve" The Divergence Theorem

To begin, let us assume tha t we are dealing with a bounded set ~t in a plane and a vector field defined on it. To be precise, let v be a continuously differentiable vector field defined on Ft. Let n be the unit external normal to the piecewise smooth boundary OFt of ~t. Then the net flux f of v across the boundary is defined through

f -- I v . n ds, (23.1)

0~

where ds is the length measure along the boundary. The following problem suggests itself:

P r o b l e m : How is the flux in (23.1) defined if the boundary 0~t is not smooth; in particular, if the boundary is a fractal curve, and what happens if the vector field is not smooth?

So how does one deal with the following question: Consider the steady flow of an incompressible fluid in a channel of width say, 3 units. Draw a Koch curve across it as described in w On physical grounds, the flux across such a curve must be equal to the incoming volume. Can we prove this result? One way of proving this is to show tha t the divergence theorem holds for regions bounded by fractal curves.

The precise conditions under which the divergence theorem holds has recently been answered in a series of papers, is This work is based on two separate issues: the first is the dimension of the curve bounding a given region of finite area; the second is the smoothness of the vector field in the region. To explain these two separate entitites, we assume that"

(i) The box dimension DB of the curve is bounded from above, i.e., DB < d, where d is a number such tha t 1 < d _~ 2. It is possible to relax the condition on DB to read DB _~ 2, provided the function Af(E, ~) of w obeys a certain growth condition as c --. 0.

(ii) The boundary curve must, in itself, have zero area.

(iii) The vector function v on the region is assumed to have a H61der exponent a such tha t a < d - 1. Recall tha t a scalar valued function f is said to be of

17FALCONER, K.J., Fractal Geometry, Wiley, New York, 1990, pp. 80-81. lSHARRISON, J.C. and NORTON, A., Indiana Univ. Math. Journal, 40, 567-594 (1991);

HARRISON, J.C. and NORTON, A., Duke MaSh. Journal, 67,575-588 (1992); HARRISON, J. C., Bull. Amer. Math. Soc., 29, 235-242 (1993).

H61der class a, or f E C a, if there is a constant M > 0 such tha t

I f ( x ) - f (y)] < M I x - yl a (23.2)

for all x, y in the domain of f. This idea has an obvious extension to a vector function v, for we can demand tha t each component of the vector field obeys (23.2).

(iv) The vector field belongs to the space W1'1(~), i.e., the vector field is integrable over fl along with its first order, weak (or distributional), partial derivatives.

Next, it is necessary to find a sequence of smooth approximators r to the fractal curve 0ft. These are straight line segments and are such tha t the area between the fractal curve and the approximators goes to zero as k --, oo. We shall now indicate how to construct such an approximator.

Consider the k-squares Q of the form

where j , l are integers from 4-oo. The domain fl, bounded by the curve Of}, is now going to be covered internally by a grid of these k-squares, such tha t we omit all those which touch the boundary Of} or tha t touch a k-square which touches the boundary. Take the union over k, discarding any squares which are contained in the larger ones. See Figure 23.1.

Call the boundary of this region r Then, as k --, oo, the divergence theorem holds. That is

/divvda = k-~o~lim /v-nds= /v-nds. (23.4) f} Onk a~

F I G U R E 23.1. Approx ima to r of a region bounded by a fractal curve.

It must be noted tha t if the vector field v is smooth, the existence of the integral of its divergence over the region fl poses no difficulty. On the other hand, when the

24 Stokes' Theorem 115

vector field is not smooth and the bounding curve is a fractal, the result embodied in (23.4) holds because the boundary integral is sensitive to cancellation, and approximators to a fractal curve have normal vectors that change directions a lot and so create a lot of cancellation in the integral, t9

In the case of three dimensional regions, the number d in (i)-(v) above lies between 2 and 3; as well the boundary surface, in itself, must have zero volume. Also, the k-squares are replaced by k-cubes, which are easy to define as an extension of (23.3).

As an aside, it is worth noting that the Minkowski or box dimension of the boundary plays a crucial role in the existence of solutions to the Dirichlet and Neumann problems for elliptic operators, 2~ just as it does in the divergence theorem.

24 Stokes' Theorem

To simplify matters, let us consider a two dimensional region bounded by a fractal curve. Because such a curve has no tangents almost everywhere, there is a need to establish the validity of Stokes' theorem in the present context. In the classical form, if a vector field f(x, y) = P(x, y)t + Q(x, y)j, then this theorem says that

P dx + Q dy = Ox Oy dx dy, (24.1)

where sufficient smoothness of the boundary and the vector field are assumed, the line integral is taken so that domain ~ lies to the left as the contour C is traversed.

In the present context, we shall assume once again that the vector field and the boundary obey the conditions (i)-(v) of w above. Then, the result in (24.1) holds. 21 The difficulty here is that the left side of (24.1) is a line integral of the

/ ,

type: ] f" t ds and the right side is an area integral. While these two can be d c

evaluated by using the approximator regions of w it is possible to calculate the line integral in a different, albeit more suggestive, manner. The procedure is as follows. 22

Suppose x0 and Xg are the end points of the arc C. Fix an integer kl ~_ 1 and construct as many ki-squares (cf.(23.3)) as necessary to cover the arc. Let T1 be the union of kl-squares that contain the initial point. Let the arc exit from T1 and let xi be the last exit point. Next, let T2 be the union of kl-squares that are not in T1 that contain xi . Let x2 be the last exit point from T2. See Figure 24.1 below for a pictorial representation of this idea applied to the Koch curve in Figure 19.3, when k - - 1.

19NORTON, A., Personal communication dated September 30, 1991. 2~ M.L., Trans. Amer. Math. Soc., 325,465-529 (1991). 21HARRISON, J.C. and NORTON, A., Indiana Unzv. Math. Journal, 40, 567-594 (1991);

HARRISON, J.C. and NORTON, A., Duke Math. Journal, 67, 575-588 (1992); HARRISON, J.C., Bull. Amer. Math. Soc., 29, 235 - 242 (1993).

22HARRISON, J.C., Proc. Amer. Math. Soc., 121, 715-723 (1994).


FIGURE 24.1. Entry and Exit Points of a Curve.

Inductively, continue this process till the end point XN lies in the union TN of kl-squares which are not in TN-1 but contain the previous point XN-1. Join x~, i -- 1 , . . . , N by a series of straight lines. This is called the secant approximator C~ 1 to C. Increase k and repeat, so tha t the secant approximations get more refined as k increases. Compare Figure 24.2 with Figure 24.1 to see the improvement in the approximation from k - 1 to k - 3.

Provided the vector field f is defined not only on ~ but in its neighbourhood as well, so tha t the line integrals along the secant approximators may be evaluated, it follows tha t

/ f - t ds -- ~-~oolim / f . t ds. (24.2)

Thus it is seen tha t the second basic theorem of vector analysis holds for regions bounded by fractal curves. The extension to a volume bounded by a surface with a fractal area has also been proved although a numerical integration scheme is yet to be devised.

I m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m u m m m m m m m m m u i l l l i m i m B B R B i i i l m B i i B i n n n i m B l i l m i i i B i B B n n m i i B B i u m i m ~ n m m i i n u i m m m n n i i m i m m i m m n B i m m m n l i i u i u l i m i B m i l i M m a i u i i m l l u n R i i m i m n n n m m i u m l m n m i m i m m m i m i i m l i i n l a p - q l i m m i i m m u m l l i m m m m i i n m i m i l i n n muummmmmmmnnmmmmmumnnmmmmo- ln~qmmmnuunmnmmmunnummmmnmnumn u u l m m u m m u u u u m m m m m u u u u m n m m l a . U k m l m m m u u u u m u u m m u u u u m m m m n m u u n u i m i i m m u m m u i m i m n m i m o m m m ~ m u ~ i m i ~ i m ~ i n i B m m i n i m n m m i i m m m i m m n | mmmmmmmmmmmmmmmmmmmnmr~mrJmmmk~mr~-mmmmmmmmmmmmmmmmmmmmmm mmmmmmmmmmmmmmmmmmmr.,. ,'.'mmmmm.' -mk--mmmmmmmmmmmmmmmmmmmm rmmmmmmmmmmmmmmmmmmm~mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm mmmmmmmmmmmmmmmmmmm~]mmmmmmmmmmmmmmmm,mmmmmmmmmmmmmmmmmmmm mmmmmmmmmmmmmmmmmmm,mmmmmmmmmmmmmmmmm, mmmmmmmmmmmmmmmmmmmm mmmmmmmnmmmmmmmmmmm.- ~mmmmmmmmmmmmmr ,.mmmmmmmmmmmmmmmmmmmm mmmmmmmmmmmmmmmmmmmmm~Immmmmmmmmmmm.mmmmmmmmmmmmmmmmmmmmmm .,,,,m,,mm-.',,,,mm,m,,~,,,,,m,m,,m~-m,mmm,,mml~.'~,,,m,,mm, mmmmmnm~-nm,-mmmmmmm-,Jmmmmmmmmmmmm ~-mmmnmmr m-'~mmmmmmm mmmmmmmlmmmmmmmmmmm~mmmmmmmmmmmmmmmmm&mmmmmmmmmmmmmmmmmmmm mmmmmmm.,mmm'~mm~mm.~mmmmmmmmmmmmmmmr.mmP~mm, mmmm, dmm-mmmm mmm.~mm-.mmmm~Im-.~am,r mm,.mm.mmmm~mm.~.mm i ' . ' 4 * ' . ' ~ m m m m m - . ' ~ m - , ~ m m i m m m m m m m m m m m m m m k ~ _ a ~ _ _ o m m m m m _ , ~ m , , J m

FIGURE 24.2. Entry and Exit Points of a Curve.

P a r t C: A d d i t i o n a l D e v e l o p m e n t s C o n c e r n i n g the Stress T e n s o r a n d Appropr ia te Shape for a B o d y and its B o u n d a r y

In {}15, the stress tensor has been proved to exist in a continuous medium. The proof used a coordinate based approach, viz., a tetrahedron was employed with an orthonormal tr iad of vectors lying along the three coordinate directions. In w a coordinate free proof will be presented.

25 A Coordinate Free Proof of the Existence of the Stress Tensor 117

Also, it has been assumed that the stress vector in the body is a continuous function of the position x for each fixed unit vector n. On the other hand, it is well known that in the solution of many problems in continuum mechanics, the stress tensor is not a continuous function of position: shock wave problems, concentrated force problems, flows around corners, etc., are all examples where the stress tensor is not continuous. Hence the question of the existence of this tensor needs to be studied and in w we collect together material from the scientific literature to throw some light on this difficult topic.

Lastly, there are questions regarding the shape a body and its boundary may possess, so that the laws of continuum mechanics are applicable to the body. Es- sential to this has been the need for the divergence theorem to hold. Because of the recent developments concerning the divergence and Stokes' theorems in regions bounded by fractal curves, this has opened a new area of research on the shape of a body and its boundary. We make some plausible conjectures on this topic at the end of w

25 A C o o r d i n a t e Free P r o o f o f t h e E x i s t e n c e of t h e

S t r e s s T e n s o r

In this section, we shall present a coordinate free proof of the existence of the stress tensor. 23 Suppose that the stress vector t(n)(x, t) has been defined at all points of the body, including its boundary, to be a continuous function of (x, t) for all unit vectors n. To simplify the notation, put

s(x, n) -- t ( , ) (x , t ) . (25.1)

Then, we have to prove that s(x, n) is linear in n for all x. This will be done by extending the vector function s(-, n) to all of T@ and proving its linearity in n.

For convenience, suppress the dependence on x and extend s(-, n) to all of 7~ 3 as follows. That is, we define

~(.,0) =0,

~(.,v) -Ivl~ - , ~ , v # o. (25.z)

Let c be scalar. If c > 0, omitting x for convenience, we have that

(e,) ~(~v) = I~1~ = ~1~1~ = ~(v). (25.3)

If c < 0, (25.2) yields ~(~) = ~(l~l(-v)) = I~l~(-~). (25.4)

Let u = v / I v I be the unit vector derived from v. Then, from (25.2)2,

~(-v) = I~1~(-~)=-Ivl~(~), (25.5)

23This proof is essentially due to NOLL, W., and appears in GURTIN, M.E., The Linear Theory o~ Elasticity, Eneyel. Phys., Via/2, Springer-Verlag, Berlin, 1972. See w

where the lat ter follows from Cauchy's reciprocal theorem (15.10), because we may identify u with the unit normal which appears there.

We can now see from (25.4)-(25.5) tha t

= -I li l (u) = =

because c < 0. Hence, it follows from (25.2)-(25.6) tha t

= ( 2 5 . 7 )

for all c, whether c is zero or non-zero. Thus, s is a homogeneous function of v. This is the first step.

The second step in the argument is to show tha t s is an additive function, i.e.,

S(V 1 ~- V2) = S(V1) -1- S(V2) ( 2 5 . s )

for any two vectors y 1 and v2. This par t of the proof is sub-divided into two parts. First of all, let us consider the situation when Vl and v2 are dependent, i.e., v2 -- /3Vl, for a non-zero constant /~. Then (25.8) follows from a repeated application of (25.7).

The final step is to prove tha t (25.8) holds even if the vectors Vl and v2 are linearly independent. In this case, define a new vector v3 such tha t

"43 -- --(Yl -~- V2)- (25.9)

Choose an arb i t rary point x0 in the body and construct a plane through it, con- ta ining the two vectors Vl and v2. Clearly, the vector v3, defined by (25.9), lies in this plane.

Next, draw through x0 two planes with Vl and v2 as normals respectively. Fix r > 0 and draw a th i rd plane normal to v3 through the point x0 + r See Figure 25.1 below for an explanation of the geometrical construction so far.

v2

FIGURE 25.1. A wedge used in the proof of the existence of the stress tensor.

Complete the wedge by drawing two planes, parallel to the first plane through x0 containing both v l and v2, such tha t each plane is at a distance x/~ from x0. Assume tha t r is small enough so tha t the wedge lies inside the body.

Let us denote the areas of the faces with the three vectors v~, i -- 1, 2, 3, normal to them as ai, i -- 1, 2, 3, respectively. The areas of the parallel faces are a4 and as.

25 A Coordinate Free Proof of the Existence of the Stress Tensor 119

It is obvious from the construction of the wedge that the areas and the volume v are such tha t

ai -- 0 ( 6 3 / 2 ) , i - - 1,2,3;

[v~[ = [v31, i - - 1 , 2 ; ai a3

a4 -- a5 -- 0(62),

v - O ( W 2 ) .

Then, with an obvious notation for the plane faces,

(25.10)

'V3' ~ / 'V3' ~ / O(E1/2 , t(n ) d a - t(n) da + a3 63 i--1 Ai i=l Ai

(25.11)

as 6 ~ O. Moreover,

/ t(n) d a - Z Iv~l a3 i=1 i=1 a- '7 t(=) da

Ai Ai

(25.12)

because of (25.10)2. Now consider the plane with the area al, for example. On it, the external unit normal is VI/IVll. Hence, using (25.2)2 , we find that, on this plane,

( Vi ) -- S(X, Vl) (25.13) [VIIS X , ~

Using similar arguments, it follows that

IV31 ~ / t ( n ) da-- ~31a--~./s(x, vi)da-~-O(61/2) a3 i-1 Ai i-1 Ai

(25.14)

as 6 -~ 0. On the other hand, we know from (15.14) that the force on the wedge is bounded by its volume, i.e.,

5 /~l/t(n) da- - 0 (r 5/2)

"'- Ai

(25.15)

as e --. O. Using this in (25.14), employing (25.10)1 and the continuity of the stress vector in x so that each surface integral divided by its area has a limit, we find that

3 y ~ s(x0, vi) = 0 (25.16) i=1

as 6 --. 0. Now (25.7) shows that s ( -v3 ) -- - s (v3 ) and thus, using (25.9), we obtain

$(X0, V 1 -~- V2) --- $(X0, Vl) -~- S(K0, V2) (25.17)

for all vectors v I and v2, whether they are linearly dependent or independent.

In conclusion, the function s(x, v) is homogeneous and additive in v. It is thus linear in v, by definition, and Cauchy's theorem on the existence of the stress tensor follows. For other proofs, see STIPPES 24 and ~ILHAVY. 25

26 Stress Singularities and Shape of a Body

26.1 Continuity of the Stress Vector

In the preceding sections, we have made a number of assumptions about the stress vector in order to derive the stress tensor from it. The assumptions are:

(i) The stress vector on the smooth surface of the body or its part depends on the unit external normal.

(ii) The stress vector is a continuous function of the position vector.

(iii) The integral form of the linear momentum balance equation holds for every part, including the whole, of the body.

(iv) The body force and inertia are bounded.

From these assumptions, two consequences follow. These are:

(a) The contact forces are area bounded, i.e.,

t(n dS a~

< CA(OP), (26.1)

where C > 0 is a constant and A(OP) is the area of the part (or whole) of the body in question.

(b) The contact forces are volume bounded, i.e.,

t(n) dS <_ KV(7)) , (26.2)

where K > 0 is a constant and V(P) is the volume of the part (or whole) of the body in question.

Using (i)-(iv), we have already seen that there is a stress tensor field defined over the body and that the divergence theorem holds for this tensor because, implicitly, we have assumed that conditions guaranteeing the latter have been met.

Omitting the proofs, we shall summarise the major advances in arriving at the existence of stress tensor fields in bodies when the stress vector is not continuous or

i , i i

24STIPPES, M., J. Elastzczty, 1, 175-177 (1971). 25~ILHAV~r, M., Rend. Mat. Acc. Lmcei, Serie IX, 1,259-263 (1990).

26 Stress Singularities and Shape of a Body 121

is unbounded. The examples below deal with a scalar version of Cauchy's (stress) theorem, for if a scalar version holds, then the vector version follows quite easily, since each component of a vector is a scalar. Thus, we shall initially examine balance laws of the type

/ 1(x, - /b(x) (26.3) i ] J

s v

and seek conditions such tha t the scalar field f is related to a vector field f through

f (x , n) = f(x) �9 n, (26.4)

or f is linear in n. The first work to deal with this question 26 established tha t (26.4) holds for almost every x in the domain of the body, provided the balance law (26.3) is true for every tetrahedron in the domain of the body. Also, in deriving this result, it is found tha t the vector field f(x) need not be continuous, only integrable. Hence, the existence of the stress tensor in bodies containing shock waves, for example, follows.

Let us next introduce the notation F for the flux and assume that F is area and volume bounded as in (26.1) and (26.2) above for all surfaces in the body. Also, let the flux over the union of $1 U $2 of two disjoint surfaces $1 and $2 obey

F(S1 t.J $2) = F(S1) T F(S2), (26.5)

when the two surfaces, each of finite area, lie in the same plane and have the same oriented unit normal. See Figure 26.1 for a pictorial explanation.

2

S1

FIGURE 26.1. Flux leaving the disjoint surfaces 5'1 and 5'2.

Such a flux F is called weakly balanced. 27 Now, let us suppose tha t associated with the flux F is its density, f , i.e., the following holds:

f (x , n) = lim F ( D ( x , n)) ~ 0 A(D(x, n)) '

(26.6)

26GURTIN, M.E., MIZEL, V.J. and WILLIAMS, W.O., J. Math. Anal. Appl., 22, 398-401 (1968).

27GURTIN, M.E. and MARTINS, L.C., Arch. Ratwnal Mech. Anal., 60, 305-324 (1976).


for all oriented disks D of radius r, with centre x and the oriented normal n and area A(D). Here, it must be noted tha t x lies on the surface S of the volume V. If the density is defined through (26.6), then the formula for the flux

F - / f ( x , n ) dS (26.7)

s

holds for all plane surfaces in the body. Then, it has been shown tha t the density f is linear in n almost everywhere. Indeed, a density, which is not linear along the entire X l axis but linear everywhere else in three dimensions, has been constructed to emphasise the almost everywhere statement. 2s On the other hand, if f is continuous in x, then f is linear in n everywhere. The above results show why in w we assumed that the stress vector is a continuous function of the position vector x.

26 .2 Se t s o f F i n i t e P e r i m e t e r

In the preceding paragraphs, we have seen tha t it is possible to begin with a weakly balanced flux F and define its density which depends on the unit normal. Is it possible tha t we just define a quantity, called the flux F, and expect it to have a density? Wha t is the most general kind of a body or its part for which (26.7) holds? In this sub-section, we summarise some of the known results.

Suppose the region ~/is bounded and has a topological boundary. 29 For example, consider the flag with a pole. Tha t is, we may consider a pipe flow problem when the domain consisits of a square cross-section with an extra "string"attached to it. See Figure 26.2 below.

FIGURE 26.2. A flag with a pole. The topological boundary of such a domain includes the bottom point,

The topological boundary consists of the edge of the flag as well as the bot tom end point of the pole. Clearly, in dealing with a concept such as flux, the bot tom point is irrelevant. Hence one introduces the measure-theoretic boundary 3~ OMfl through the idea tha t a point x E OM~ if every disk (resp. ball) with the point as centre has some area (resp. volume) in common with both ~/and its complement. A

28MARTINS, L.C., Arch. Rational Mech. Anal., 60,326-328 (1976). 29For a nice introduction, see ARNOLD, B.H., Intuitive Concepts in Elementary Topology,

Prentice-Hall, New Jersey, 1962. a~ ZIEMER, W., Arch. Rational Mech. Anal., 84, 189-201 (1983).

26 Stress Singularities and Shape of a Body 123

similar notion has already appeared before - see (22.3). Here, for example, consider a set ~ C 7~ 2. Then

OM~ -- n 2 n {x- d ( n , x ) # 0 n d(n 2 - fl, x) # 0},

where the density d is defined, for example, through

Area of ~ M D(x, r) d ( ~ , x ) - lim (26.9)

r-~0 Area of D(x, r)

and D is the disk with centre x and radius r. The above definition means tha t the measure-theoretic boundary of the flag shaped region with the pole is just the edge of the flag, because a disk of radius r, with the bo t tom point as its centre, has no area in common with f~ as r --. 0. In general, if the measure-theoretic boundary of a set has a finite length (or surface area) computed with respect to the Hausdorff measure, 31 then the set ~ is said to have a finite perimeter.

These sets have some nice properties. The first one is tha t they have a measure- theoretic external normal - see w above for a description - almost everywhere. For the flag shaped region, the normal exists everywhere except at the corners. The second consequence is tha t for every weakly balanced flux F, there is a density f which satisfies (26.7).

We have come a full circle here. Wha t has been stated above can be summarised as follows: if the body and its boundary correspond to a set of finite perimeter and the flux is weakly balanced, then the formula (26.7) holds. In addition, this density is linear in n and has a vector field f(x) associated with it according to (26.4). A drawback to a body belonging to the class of sets of finite perimeter has been found to lie in the possibility tha t such a set may have a boundary which in two dimensions may have a non-zero area or, in three dimensions, have a non-zero volume. Thus, the concept of a fit region 32 has been introduced which is a set of finite perimeter but which has a negligible boundary. This last assumption appears in w in connection with the divergence theorem.

Even the fit region has some drawbacks. First of all, a fit region is too large 33 if one is interested in delineating the properties of the stress vector to allow for the existence of singular stresses in the body. That is to say, certain system of forces may exist in bodies with boundaries less complicated than those with finite perimeter and these forces cannot be extended to this larger class of sets. However, this position is in need of revision because the divergence theorem holds for regions with fractal boundaries.

26.3 Shape of a Body and its B o u n d a r y

The research collected together in the above summary is not sufficient to deal with the existence of the stress tensor in bodies with fractal boundaries. Nevertheless, it seems to us tha t the following approach will work:

31The Hausdorff measure of the length of a curve means that the exponent s in w above is 1 and in the case of area, s = 2.

32NOLL, W. and VIRGA, E.G., Arch. Rational Mech. Anal., 102, 1-21 (1988). 33See ~ILHAV~, M., Arch. Rational Mech. Anal., 116, 223-265 (1991).


(i) Replace the given body with the fractal boundary by its approximation, using the k-squares or k-cubes as depicted in Figure 23.1.

(ii) Because this is an internal approximator to the given body and because the flux across the boundary of the approximator approaches that across the boundary of the given body as k --, oo and the divergence theorem holds provided the vector field obeys a global H61der condition, one solves the continuum mechanics problem in the approximator region for large values of k.

(iii) Note that Stokes' theorem is valid in the internal approximator region as well.

3 Formulation of Constitutive Equations-The Simple Fluid

In this Chapter, we shall derive the constitutive relation for the incompressible simple fluid, because much of the research into the fluid mechanics of viscoelastic liquids has been driven by this model of fluid behaviour. Even though, historically speaking, the simple fluid was not the first, subsequent development has shown that earlier models of viscoelastic fluids are equivalent to it or are special cases of it. Additionally, it is now possible to present the derivation of the constitutive relation of the simple fluid in such a way tha t one can apply the arguments behind the derivation of the constitutive equation for elastic fluids to tha t of the simple fluid in easy steps, while at the same t ime maintaining the necessary amount of rigour in the presentation.

In w we begin by demonstrat ing why constitutive relations are necessary and then list the two basic principles used again and again in their formuation. These are the objectivity of the stress tensor and the frame indifference of the constitutive operator. In a presentation which may be considered to be the opposite of the tradit ional approach, in w the constitutive equation of an elastic material is postulated and the restrictions on it due to the symmetry of a material are derived. These are then specialised to the case of an isotropic solid and an elastic fluid and deliver the s tandard results.

In w the restrictions on the constitutive equation of an elastic material due to the objectivity of the stress tensor and the frame indifference of the operator are found. These are applied to the case of the isotropic solid, as an example. In w the integrity basis for the constitutive relation for an elastic fluid is used to derive the classical result tha t the stress is a pressure which depends on the density of the fluid alone.

In w the concept of the simple material is introduced and this is used to derive the restrictions tha t must be placed on it, if it were to describe a fluid. The method used is very similar to tha t in w and shows tha t the stress in a

126 3. Formulation of Constitutive Equations-The Simple Fluid

compressible simple fluid must depend on the history of the relative deformation gradient and the current density of the fluid.

In w the objectivity restrictions on the constitutive relation of a simple material are obtained and from this a reduction to the case of the simple fluid is made. Finally, in w the constitutive relation of the incompressible simple fluid is determined from that of the compressible simple fluid.

In Appendix A, the concept of an integrity basis for scalar valued, isotropic functions of vectors and tensors is introduced. This is then shown to yield vector or tensor valued functions of vectors and tensors, by a clever use of linearity inherent in the definition of the scalar products created specifically to use the relevant integrity bases. The extension of these polynomial forms to functions of the arguments is shown to be possible in many cases.

In Appendix B, we discuss briefly the procedure to obtain approximations to the constitutive equation of the incompressible simple fluid for small deviations from a state of rest. These approximations lead to the models arising from linear viscoelasticity, or to the order fluids, or to n-integral, m-order fluids.

27 Constitutive Relations and General Principles of Formulation

27.1 Constitutive Relations- Are They Necessary?

In Chapter 2, we assembled the basic laws of continuum mechanics: the continuity equation, the equations of motion and the balance of energy equation. If we now wish to solve initial-boundary value problems, which in fluid flows mean the determination of the density, velocity and temperature fields in the body due to the the prescription of intitial and boundary conditions, we note that there are five equations for these five unknowns: the continuity equation, the three equations of motion and the energy equation. However, when one examines the equations of motion, it becomes apparent that unless the stress tensor is related in some way to these five unknowns, we cannot solve for them; the situation concerning the energy equation is even more complex because one must know how the internal energy, the stress tensor and the heat flux vector are related to these variables.

To make the difficulties tranparent, consider the case of the two dimensional form of the equations of equilibrium. Here, we find that we have to examine the set

OTz z OTz y

Ox+Oy OT~ OT~ �9 + Ox Oy

-- O,

= 0 . (27.1)

For instance, these equations have great importance in the two-dimensional elas- tostatic problem arising in linear elasticity. It is known that if we define a function (I), called the Airy stress function, such that

02~ 02r 0 2 r Txx--Oy 2, Ty v - Ox 2, Txv- OzOy' (27.2)

27 Constitutive Relations and General Principles of Formulation 127

then the equations of equilibrium 1 are trivially satisfied. However, this function (I) does not provide any information on the state of displacements within the elastic body. An extra relation is needed, a relation which ties the stresses to the displacements. Such relations are called cons t i t u t i v e re lat ions .

Once the need for such relations is granted, there is the next order of difficulty, because bodies in the physical world behave differently under identical forces. Whereas a solid will undergo a shear deformation, a liquid will flow under the same stress. Liquids appear to be incompressible under a wide vartiety of conditions, whereas gases are not so. Indeed the possible variety of behaviours is as large as the number of materials which exist in nature or can be manufactured.

Even within the fluid mechanics of viscoelasticity, the topic of the present treatise, the following experimental observations are well known:

(i) The liquid cannot withstand shear stress; it flows, though it may do so very slowly;

(ii) In viscometric flows, the shear stress is, usually, a nonlinear function of the shear rate;

(iii) The normal stresses in these flows are not equal to one another;

(iv) A viscoelastic liquid behaves like an elastic solid in fast deformation processes, e.g., a lump of it thrown against a wall may bounce back like rubber;

(v) If a strain is applied and if the material is held in this configuration, the stresses relax gradually to zero;

(vi) The fluid exhibits creep under a steady load;

(vii) On exit from a die, an appreciable swelling is noted;

(viii) If a rod is ro ta ted in a container of a viscoelastic liquid, the liquid climbs up the rod.

One could list many other examples of such striking behaviour. Essentially what one notes is this:

1. The previous history of the motion determines the present stress.

2. Elastic recoil, creep and stress relaxation occur.

3. The relation between the stress tensor and the velocity field is highly nonlinear, even in situations where the history of strain is significantly repetitive.

While it is t rue tha t a single equation cannot possibly describe all liquids, it is desirable tha t a general set of guidelines exist so tha t the methods for formulating specific equations to deal with individual materials can be formulated, unders tood and manipulated. It is to these we turn next.

1A stress-density function which satisfies the continuity equation as well as the equations of motion in a compressible medium, when the body forces are absent, has been discovered by FINZI, B. For details, see TRUESDELL, C., Arch. Rational Mech. Anal., 4, 1-29 (1959), or w167 in TRUESDELL, C. and TOUPIN, R.A., The Classical Field Theories, Encycl. Phys., Vol. III/1, Springer-Verlag, Berlin, 1960.


2 7 . 2 G e n e r a l P r i n c i p l e s

Although by 1950, the basic ideas behind the formulation of constitutive relations for linear elasticity, visous liquids and finite rubber elasticity were well understoood, the extension of these to the level of general principles applicable to all materials undergoing large deformations had not been made. In a remarkable piece of work, OLDROYD 2 discovered these principles and proposed that the rheological equations of state, or constitutive relations, must be based on:

(i) the motion of the neighbourhood of a particle relative to the motion of the particle as a whole in space;

(ii) the history of the metric tensor, i.e., the strain tensor, associated with each particle;

(iii) convected coordinates imbedded in the material and deforming with it;

(iv) physical constant tensors defining the symmetry of the material.

Then the constitutive relations for a homogeneous continuum would be a set of integro-differential equations connecting the histories of the stress tensor, the strain tensor and the temperature, along with the physical constants. In addition, he remarked that an incompressible fluid has no special reference state, i.e., a reference time such that the configuration at that time has a permanent significance

N

in any subsequent motion. As an example, he chose the stress flux T related to the stress tensor T through

= T + T L + L TT, (27.3)

where L is the velocity gradient tensor. The constitutive relations exhibited were a set of linear equations connecting the stress tensor and its flux with the rate of deformation tensor and its flux.

A number of questions arise from the above ideas: Why convected coordinates? Why not use the stress tensor by itself? Why should the flux depend on the rate of deformation which is objective (cf.(10.17)), whereas the flux itself may not be so - that is, where is the proof that the flux is an objective tensor? Lastly, how can one include those tensors which define the symmetry of the material explicitly into the constitutive relations?

Without going into the history of the subject, we note that NOLL 3 and GREEN and RIVLIN 4 along with SPENCER 5 addressed the questions raised and found the solutions. Briefly, the answers are as follows. Convected coordinates are just one set of coordinates needed to describe the motion; the kinematically useful tensor is the deformation gradient; the constitutive relation may be defined in terms of the

i

2OLDROYD, J.G., Proc. Roy. Soc. Lond., A200, 523-541 (1950). aNOLL, W., J. Rational Mech. Anal., 4, 3-81 (1955); Arch. Rational Mech. Anal., 2, 197-226

(~958). 4GREEN, A.E. and RIVLIN, R.S., Arch. Rational Mech. Anal., 1, 1-21 (1957); 4, 387-404

(1960). 5GREEN, A.E., RIVLIN, R.S. and SPENCER, A.J.M., Arch. Rational Mech. Anal., 3, 82-90

(1959).

27 Constitutive Relations and General Principles of Formulation 129

stress tensor explicitly; the objectivity of the stress flux follows from the following assumption: O b j e c t i v i t y o f t h e S t r e s s Tensor : In all relative motions (cf. w the stress tensor is objective. Tha t is,

T* - Q T Q T. (27.4)

P r o b l e m 27 .A If the stress tensor T is objective and L is the velocity gradient tensor, which

N

transforms according to the rule (10.15), show tha t the flux T in (27.3) is objective.

P r o b l e m 27 .B Let a constitutive equation satisfied by T and the velocity field v be the following

implicit equation: T~j,kvk -- v~,kTkj -- ~ k V k , j -- O, (27.B1)

where v is a steady velocity field in two dimensions. Prove tha t (27.B1) has the explicit form 6

T - h ( r T, (27.B2)

where h(r is an arbi trary function of the stream function r

We now make the additional assumption: 7 F r a m e I n d i f f e r e n c e of the Const i tut ive Operator: The constitutive operator is the same for all observers in relative motion (cf. w Given the above two hypotheses, one may start with the constitutive relation for the stress tensor depending on the history of the deformation gradient and from it deduce the dependence of the stress tensor on the strain history. Lastly, explicit procedures for the incorporation of the tensors defining the symmetry of the material exist.

2 7 . 3 R o t a t i o n s o r R o t a t i o n s a n d R e f l e c t i o n s

There is a great deal of controversy concerning whether in the relative motions discussed in w one may include reflections as well. That is, can the orthogonal tensor Q, which accounts for the orientation of the two sets of axes, be improper? The mat ter would not be so important, were it not for the fact that , in an objective motion, the deformation gradient transforms as F* - QF, whereas under changes to the local reference configuration (cf. w the same tensor changes from F to F H . If we permit Q to take on the value - 1 , then because ( - 1 ) F -- F ( - 1 ) , we have to accept tha t the symmetry group of a material includes the tensor - 1 . The physical meaning, as is apparent from w is that the neighbourhood of the particle can suffer such a deformation and this is impossible. Indeed, FOSDICK and SERRIN s remark: '%he body cannot be expected to ever take on configurations

6RENARDY, M., J. Non-Newt. Fluid Mech., 50, 127-134 (1993). 7For a discussion of the role of objectivity in the formulation of constitutive relations, see pp.

171-179 in HUILGOL, R.R., and PHAN-THIEN, N., Int. J. Engng. Sci., 24, 161-261 (1986). aFOSDICK, R.L. and SERRIN, J., J. Elasticity, 9, 83-89 (1979). See the footnote on page 85.


which would require the vanishing of the local volume elements or mirror reflections of the reference configuration. This point cannot be too strongly emphasized."

Thus, in this treatise, the assumption is made that only rotations are permitted in discussing the effects due to the changes of the reference frame, or in w Q(t) is a proper orthogonal tensor function of t. Secondly, only proper tensors H are used to define the symmetry group of a material - see w in connection with this. Nevertheless, as will be apparent from (29.14), (30.3), (32.9) and (33.4) below, after the various restrictions on the constitutive equations are obtained, it is possible to extend the domain of the restrictions to improper orthogonal and/or unimodular tensors because these extensions make no new demands on the constitutive relations.

27.~ Simple Materials and Simple Fluids

In this treatise, the basic structure used for the development of the constitutive relation for a viscoelastic fluid, from the perspective of continuum mechanics, is that of the simple material. 9 Secondly, the reduction of the constitutive equations appropriate for an isotropic, simple solid or a simple fluid follows that developed for elastic materials. 1~ In connection with these matters, one may gain additional insight by examining the recent resolution of the controversy regarding the development of the constitutive relation of a simple fluid from that of a simple material. 11

N o t e : In this treatise, we are concerned with homogeneous materials only. Thus, the continuum approach of this chapter, or the averaging processes used in the next chapter based on microstructures, deliver constitutive equations which are relevant for homogeneous solids or fluids.

28 Symmetry Restrictions on Constitutive Equations of Elastic Materials

In linear elasticity, the stress tensor is a function of the strain tensor which is the symmetric part of the displacement gradient tensor J, which in turn is related to the deformation gradient F through (5.2), i.e., F = 1 + J. In general, the determinant of the displacement gradient tensor is not always positive. Hence, in defining the constitutive equation for the stress tensor in an elastic material, it is preferable to use F because the latter will always have a positive determinant, which follows from the conservation of mass equation (14.8). In addition, the polar decomposition of F = R U = V R , as stated in (1.36), leads to positive definite matrices for U and V, because F is non-singular. All of these will have important repercussions as we shall see.

Hence, we begin by postulating that the constitutive equation for the stress tensor T in an elastic material can be expressed as a function of the deformation

, , ,,,,,,

9NOLL, W., Arch. Rational Mech. Anal., 2, 197-226 (1958). I~ W., J. Rational Mech. Anal., 4, 3-81 (1955). 11See HUILGOL, R.R., Rheol. "Acta, 27, 351-356 (1988); 28, 253-254 (1989).

28 Symmetry Restrictions on Constitutive Equations of Elastic Materials 131

gradient F, which is determined with respect to a fixed reference configuration; tha t is, we write

W = f (F) , (28.1)

where f (F) is a symmetric tensor valued function of F. In w we introduced the concept of a symmetry group and discussed how the change of the reference configuration affects the deformation gradient. Let us now suppose tha t a non- singular, proper tensor H belongs to the symmetry group G of the material in question. Then, as described earlier in w - see (12.13), the stress tensor must remain invariant if F is replaced by F H , i.e.,

T = f ( F ) = f (FH) . (28.2)

Now F in non-singular and hence is an element of s the linear group of all non- singular, proper tensors. Clearly, the symmetry group G is a subgroup of s because its elements are non-singular, proper tensors and ~ is a group in its own right. The problem tha t one has to solve is to find the way the stress tensor depends on the deformation gradient, when a symmetry group is given. Obviously, the answer must come from group theory and we turn to this again.

This theory 12 tells us tha t if one keeps F fixed and forms all the products F H , where H runs over all the elements of G, we obtain a left coset of ~ i n / : + . If we look at (28.2), it is clear tha t the restriction on the function f(-) is equivalent to saying tha t the value of f(-) is constant on the left coset to which F belongs. 13

Group theory also tells us tha t other elements of / :+ , namely the other deformation gradients, either belong to one coset or other; tha t is, any two cosets of G in s are either identical or have no elements in common. So, the symmetry group has the ability to divide s into different cosets, on each of which f(-) is constant. Thus the problem of finding the restriction on f(-), inherent in (28.2) above, is solved if we can find how f(-) depends on F in each coset.

In a few cases, one can find this dependence quite easily because of the special nature of the group G. The following examples demonstrate the procedure one can adopt when the elastic material is an isotropic solid or when it is an elastic fluid.

28.1 I s o t r o p i c So l id s

It is well known that , in an isotropic elastic solid, the symmetry group is the proper orthogonal group, at least with respect to one fixed reference configuration. Using this, let us consider an arbitrary, deformation gradient tensor F. This has the unique polar decomposition F -- V R , where V is positive definite and symmetric and R is orthogonal. Now the condition (28.2) says tha t

f (F) = f ( F H ) (28.3)

for all orthogonal H. Thus all F, which belong to the same coset as F does, are derivable from F in one and only one way. That is, the two are related through

~" - F H - V R H , (28.4)

12See, for example, HERSTEIN, I.N., Topzcs in Algebra, BlaisdeU Pub. Co., Waltham, Mass., 1964.

13WANG, C.-C., Arch. Rational Mech. Anal., 32,331-342 (1969).


where H and R H are both orthogonal. This means tha t F and F belong to the same coset provided they have the same s t re t ch t e n s o r V, because it is this latter tensor tha t defines the coset or is the representative element of the coset. To put it another way, one may choose a positive definite and symmetric tensor V and form the coset {V, VQ1 , V Q 2 , . . . } , where Q1, Q2, etc., are orthogonal. Then, all tensors belonging to /~+ lie in one of these cosets, and any two tensors F and ~" belonging to the same coset have the same generating element V. So,

f (r) = f (v ) (2s.5)

in an isotropic elastic solid, as is well known. What (28.5) also means is tha t in an isotropic solid, the stress function which depends on the deformation gradient is unaltered if F is replaced by V or by FI-I, where H is a proper orthogonal tensor.

To explain this further, consider the constitutive equation for the neo-Hookean solid: 14

f (F) -- - p I q- # F F T. (28.6)

Because V 2 - F F T - see (1.37), we see tha t this relation obeys (28.5). Also, if we replace F by any other tensor from the same coset, e.g., by VQ1 , then (28.5) is again satisfied.

On the other hand, had (28.6) been expressed as a function of V, i.e., as

f (V) -- - p 1 + # V 2, (28.7)

and we replace V by V Q I in (28.7), we do not find tha t the equation remains invariant; similarly, replacing V by F in (28.7), we do not recover (28.6). These inconsistencies arise because V is the square root of F F T and not F 2. Using this fact shows tha t f (V) = f(VQ1 ) : f (V). To put it another way, given a constitutive relation in terms of F, it is easy to substi tute from left to right in (28.5). On the other hand, given a function of V, one has to recognise the relation between the two tensors F and V in order to prove tha t the relation satisfies the symmetry restriction (28.3).

28.2 Elastic Fluids

We may recall from the ideal gas theory tha t the pressure in the gas is proportional to the density. Equivalently, because of the conservation of mass equation (14.8), we may say tha t the pressure is proportional to the reciprocal of the determinant of the deformation gradient. This means that , for the ideal gas, the symmetry group must be such tha t the left coset, to which both F and F H belong, must be generated by [det F] I . We shall now derive this result in a different way.

Turn to the general case of an elastic fluid and define its symmetry group to be the proper unimodular group. 15 Then (28.2) must hold for all proper unimodular H. As in the case of the isotropic solid, we now find the representative element of a coset to which F belongs. This is quite simple to do, for what ties F and

14RIVLIN, R.S., Phil. Trans. Roy. Soc. Lond., A240, 459-490 (1948); KUBO, R., J. Phys. Soc. Japan, 3, 312-317 (1948).

15NOLL, W., Arch. Rational Mech. Anal., 2, 197-226 (1958).

28 Symmetry Restrictions on Constitutive Equations of Elastic Materials 133

together when they belong to the same coset is that their determinants have the same values. This is because ~" - F H , where H is unimodular. So, given a real number d E (0, co), we choose

Fa - dl/31 (28.8)

to be the representative element in three dimensions. Then every F which belongs to the same coset as Fa has the unique representation

F - F a l l (28.9)

for some unimodular H. Thus

f (F) - f ( F a ) - f ( d 1 / 3 1 ) - f([det F]I) . (28.10)

This shows quite clearly that the stress in an elastic fluid depends on F through det F and nothing else. The manner in which it depends on the determinant is found by an appeal to the principle of objectivity. We shall discuss this procedure in the next section.

P r o b l e m 2 8 . A Let

0 det F 0 ) f (F) - a(det F) det F 0 0 , (28.A1)

0 0 0

where a is a function of det F. Show that the above equation satisfies f (F) - f ( F H ) for all unimodular H. Does this result contradict (28.10)?

One other important aspect of the symmetry group of the fluid being the proper unimodular group is the following. In w it has been shown that this group is unchanged during a motion, for the conjugation leaves it invariant. This means that all configurations of an elastic fluid are undistorted. Hence, there is no point in considering separate reference configurations for a fluid. Equivalently, a fluid particle has no preferred state and one is justified in saying that the constitutive operator for the elastic fluid is the same with respect to all reference positions, including the current one at time t.

Clearly, this is not so for elastic solids, for the deformation may affect the symmetry of the solid - see w above for an example. This means that for a solid, the constitutive operator may change from one reference configuration to another. Because this treatise is concerned with fluids only, this point will not be discussed any further here. 16

28.3 Invariance Restrict ions for Singular F

An intriguing example ~7 of the solution to the equation g(F) - g ( F H ) for all unimodular H, when F is itself singular, will now be given. The important role

16For a discussion of these matters, see COLEMAN, B.D. and NOLL, W., Arch. Rational Mech. Anal., 15, 87-111 (1964).

17We are grateful to the late Professor PIPKIN, A. C. for this example.


played by the representat ive matr ix element of a coset in (28.9) above is occupied by a vector here.

Suppose tha t F -- u v T is a matrix, where u and v are vectors, with the vector v ~ 0. Then, for any w ~ 0, there is a unimodular H such tha t v T H -- w T. To unders tand this, consider the t ransposed equation H T v -- w. Let the polar decomposition of H T -- V R . Then, we select R and rotate v to lie in the same direction as w. Suppose tha t in component form, v -- (1 0 0) and w -- (c~ 0 0). Then, one may map v onto w by choosing the unimodular matr ix V -- diag[a, 1/c~, 1], which causes H T to be unimodular.

Now, the invariance condition g ( F ) = g ( F H ) gives

g ( u v T) -- g ( u w T) (28.11)

for all non-zero v and w. This means tha t g ( u v T) depends only on u. Thus,

g(uv = h(u).

Next, if k is any scalar, (ku)v T = u ( k v T) and

h(u) = h(ku)

for all k. Hence we derive tha t

(28. 2)

(28.13)

(TU~) " (28.14)

For any such function, g ( u v T) -- g ( u v T H ) when H is unimodular , indeed non- singular.

As a specific example of the foregoing, let B - F F T and define

B g(F) = trB' (28.15)

whenever det F = 0 and F ~ 0. Then, F = UV T means tha t B - - u ( v T v ) u T and the independence of g from v means tha t g (F) - u u T / ( u �9 u), which does not depend on v at all. Tha t is, the function g(F) depends on the singular par t u of F, which is appropr ia te since this vector is the generating element.

29 Objec t iv i ty Restr ic t ions on Const i tu t ive Equat ions of Elast ic Mater ia ls

As already s ta ted in w the two basic principles in formulating a consti tut ive equation for the stress tensor are:

(i) The stress tensor is objective, i.e.,

T* = Q T Q T (29.1)

for all proper orthogonal tensors Q. In what follows, we shall suppress the dependence of the orthogonal tensor on t ime t for convenience.

29 Objectivity Restrictions on Constitutive Equations of Elastic Materials 135

(ii) The constitutive operator is frame indifferent, i.e., it is the same for all observers in relative motion. For elastic materials, this means tha t if T -- f (F) in one frame of reference, then in any other frame of reference, T* = f(F*), where T* and F* are the stress tensor and deformation gradient, respectively, as measured in the second frame of reference, at the same particle in the body.

Hence, we find that , if T - f (F) as in (28.1), then

T* = f ( q F ) (29.2)

for all rotations Q, for the rule for transformation applied to F* is given by (10.8) and it says tha t F* = QF. Combining (28.1), (29.1) and (29.2), we obtain a restriction on the constitutive relation for all elastic materials"

Q f ( F ) Q T -- f ( q F ) , (29.3)

and this is valid for all rotations Q. , ,

P r o b l e m 29 .A Consider the constitutive relation f (F) in Problem 28.A. By choosing Q =

d i a g [ - 1 , 1 , - 1 ] show tha t this constitutive relation does not satisfy the objectivity restriction (29.3), unless the function a is trivially zero.

Now, let us recall the polar decomposition F -- RU, where R is a rotation and U is positive definite and symmetric. Because Q is arbitrary, we can choose it so that it is always equal to the inverse of R; tha t is, we may choose Q - R T. Then, a simple manipulation of (29.3) gives rise to the following equation:

f ( F ) - R f ( U ) R T. (29.4)

The above relationship is misleading at first glance, because the left side depends on F, whereas the right side does not appear to do so. This is easily put right by observing tha t both U and R depend on F, because

U 2 - F T F , R - F U -1. (29.5)

Turning to (29.4), one now finds tha t the constitutive relation (29.1) must satisfy

f ( F ) - F U - I f ( u ) U - I F T, (29.6)

where it must be noted tha t (29.5)1 is understood to apply. Let us now define a new function

U - I f ( u ) u -1 - g(C) , (20.7)

where we have used the result tha t the tensors C and U are related through (1.38), i.e.,

C - U 2 - FTF . (29.8)

Combining (29.6)-(29.8), we find tha t the stress strain law in elasticity satisfies the following restriction:

f(F) -- Fg(C)F T. (29.9)


The relationship above is a consequence of the objectivity restriction (29.1) on the stress tensor and the frame indifference (29.2) of the constitutive operator. It applies to all elastic materials regardless of the symmetry of the material.

We shall now examine an interesting point about (29.9). Using it, it is easy to see that

f (QF) = Q F g ( C ) F T Q T, (29.10)

because C is unchanged due to the change in the observer (cf. (10.9)). On the other hand, it follows that

Q f ( F ) Q r -- Q F g ( C ) F T Q r . (29.11)

Comparing (29.9) with (29.11), it is clear that the restriction (29.3) imposes no condition on the function g of the tensor C. Hence, any restrictions that arise on the function g must come from the symmetry of the material.

P r o b l e m 29.B Using the assumption that the symmetry group of an isotropic solid is the proper

orthogonal group, show that (29.9) can be reduced to the statement tha t the stress tensor in an isotropic solid is a function of B = V 2. See (28.5) for the derivation based on symmetry alone.

29.1 Isotropic Solids

Let the stress tensor in an isotropic solid have the constitutive equation (28.5). As we have just seen, we may rewrite it as

T - h(B). (29.12)

Then the objectivity of the stress tensor and the frame indifference of the constitutive operator mean that

T" = Q T Q T, T" = h (B ' ) . (29.13)

However, we know that the tensor B" = Q B Q T - see (10.9). Thus, (29.12)-(29.13) lead to the following restriction on the function h(B):

Q h ( B ) Q T = h(QBQ T) (29.14) where Q is any proper orthogonal tensor. The above restriction holds for all orthogonal tensors, proper or improper, because replacing Q by - Q leaves (29.14) invariant. We call the function h(B), an isotropic function of the tensor B when the function satisfies (29.14) for all orthogonal tensors.

Using the function basis - see (A3.18) in the Appendix A, we find that the full expansion of an isotropic function such as h(B) is given by

h(B) -- a01 -I- a I B q- a2B 2, (29.15)

where the scalar-valued coefficients are functions of the three invariants of B. Quite often, (29.15) is used in a different form, because B is invertible~ and hence B 2 can be written as a polynomial in 1, B and B-1 through an application of the Cayley- Hamilton theorem. In conclusion~ one may use

h(B) -- fl01 +/~1B -{-/~2 B-1 (29.16)

instead.

30 Integrity Basis for an Elastic Fluid 137

30 Integrity Basis for an Elastic Fluid

It has been demonstrated in (29.9) tha t the constitutive equation for the stress tensor in an elastic material, with the use of the objectivity restriction, has to satisfy"

T - f ( F ) - F g ( C ) F T, (30.1)

where g(C) is a symmetric tensor-valued function of the positive definite and symmetric tensor C. Let us now determine the form of g(C) for an elastic fluid. This means tha t the invariance of f (F) under the proper unimodular group has to be employed and thus we find tha t

F g ( C ) F T - FHg(HTCH)HTF T, (30.2)

since C = FTF . Since F and F T are non-singular, we find tha t g(C) must satisfy

g(C) = H g ( H T C H ) H T (30.3)

for all proper unimodular tensors H. Here we note tha t equation (30.3) is unaltered if H is improper, i.e., its determinant is - 1 . Thus, we may extend the domain of applicability of (30.3) to all unimodular tensors.

Let us now seek how g depends on its variable C. We need an integrity basis - to find the exact form of this dependence for functions obeying (30.3) for all unimodular tensors. The basis tha t is available is says tha t g(C) is given by

g ( c ) = - p c • c , (30.4)

where i6 is a scalar valued polynomial function of det C, and the product C • C has the indicial form

(C x C)~j - 6,pq6jrsCp~Cqs. (30.5)

In the above equation, 6~jk is the usual alternating tensor, and using the property of the product of two of these tensors, we find tha t

C x C - - 2 C 2 - 2(trC)C + [(trC) 2 - (trC2)]l. (30.6)

From the Cayley-Hamilton theorem, we see immediately tha t the right side is equal to 2(det C ) C -1, because the tensor C is non-singular.

Thus, g (C) has the representation 19

g ( C ) - - - - p C -1 (30.7)

where p - 2(det C)i~, or p is a polynomial function of det C. Equivalently, p is a polynomial in det F. Substi tuting the form (30.7) into (30.1), and noting tha t (:3-1 _ F - I ( F T ) - 1, one finds tha t

f ( F ) - - p 1, p = p(det F), (30.8)

lSFAHY, E. and SMITH, G.F., J. Non-Newt. Fluid Mech., 7, 33-43 (1980). We have reduced the results in eqs. (25)-(28) to the case of a single matrix.


which says that the stress in an elastic fluid is proportional to the unit tensor. That is, it is the same in all directions and the magnitude of the stress depends on the determinant of the deformation gradient or, equivalently, on the density of the fluid. Hence, defining the symmetry group of a fluid to be the proper unimodular group has delivered the classical result for elastic fluids and thus one is justified in postulating that all simple fluids have the same symmetry group.

31 Restrictions due to Symmetry: Simple Fluids

Following NOLL, 2~ a simple material is defined to be a material in which the stress tensor at time t is a functional of the history of the deformation gradient F(T),--oo < T _< t. That is

T(t) -- G(F(T)), - o o < T < t. (31.1)

The simple material is a compressible simple fluid if and only if the constitutive equation (31.1) is invariant under the proper unimodular group in the following sense:

a(F(T)) -- a(F(T)H), (31.2)

where H has determinant equal to I. The above definition follows naturally from that for elastic fluids. For example, the single integral model,

G(F(T)) -- - p (de t F ( t ) ) l (31.3) t

+ / ~[t-- T, det(F(t)] (F( t ) - I ) TF(T) TF(T) F( t ) - l dT J

- - 0 0

satisfies (31.2) for all proper, unimodular H. We shall put (31.2) in a different form for further elucidation of the restriction inherent in it.

Recall tha t the relative deformation gradient Ft(~') is defined through (cf. (1.55))

F t ( T ) - F(T)F(t) -1.

Then we can define a new functional

~(F(T))- 7"~(Ft(T), F(t)).

Hence, equivalent to (31.3) is the following:

(31.4)

(31.5)

TI(Ft(T), F(t)) - -p(det F(t))l (31.6)

+ ~[r, det (F(t))] Ft(r)TF~(r) dr.

Returning to the invariance requirement (31.2), it is easy to prove that it will be satisfied by (31.5) if and only if the new functional satisfies

~(Ft(r) , F(t)) = ~(Ft(r) , F(t)H) (31.7)

2~ W., Arch. Rational Mech. Anal., 2, 197-226 (1958).

31 Restrictions due to Symmetry: Simple Fluids 139

for all proper unimodular H. This is because replacing F(~-) by F(T)H leaves Ft(T) invariant (cf.(31.4)), while altering F(t) to F( t )H; and conversely the only way to leave Ft(T) unchanged, while F(t) becomes F( t )H, is to replace F(~-) by F(~-)H for all T E (--OO, t]. To emphasise this point further, we observe that the constitutive relation in (31.6) satisfies (31.7), because det F = det FH.

If we examine the restriction in (31.7) yet again, we see that a simple material is a fluid if and only if, for each fixed relative deformation gradient history Ft(T), a funct ion of F(t) is equal to the same function of F ( t )H for all proper unimodular H. To interpret this statement, in (31.6) above, we fix the relative deformation gradient and think of the resulting equation as a function of the current value of the deformation gradient. Because this function is unaltered when F(t) is replaced by F( t )H, since these two tensors have the same determinants, we may accept (31.7) as defining a simple fluid.

Returning to the general case, fix a history Ft(T), and define a function

?~(Ft(T), F(t)) = f(F(t)) . (31.8)

Then (31.7) says that for a fluid, the new function must obey

f(F(t)) = f ( F ( t ) n ) (31.9)

for all proper unimodular H. We now have to answer the question as to how this new function depends on F.

That is, what is the representative element of the coset to which F belongs? The answer has already been provided in w above. Thus, in a simple fluid,

f (F(t)) = f([det F(t)] 1). (31.10)

Hence a simple material is a simple fluid if and only if

?'~(Ft(T) ,F(t)) -- ?~(Ft(T), [det F( t ) ] l ) . (31.11)

The above result has been obtained through the use of the symmetry group 21 and not by appealing to the principle of objectivity. It says that in a simple fluid, the dependence on the current value of the deformation gradient is through the determinant of this gradient; equivalently, we call a simple material a simple fluid if

T(t) - G(F(T)) -- ?:/(Ft(T), p(t)), (31.12)

where p(t) is the density of the fluid element at time t. Again, as in the case of the elastic fluid, the simple fluid has no preferred states

because its symmetry group is the proper unimodular group. Hence, the same constitutive operator applies to all histories of the relative deformation gradient and the current value of the density.

One point worth noting is that in deriving the major results in this section, we have assumed that the unimodular tensor H in (31.2) may depend on the current t ime t. Indeed, a glance at (31.3) shows that the tensor H can be a function of t,


without affecting the validity of the constitutive relation. It must be emphasised, however, that H cannot depend on T. If it were to do so, the constitutive relation (31.3) would not satisfy the restriction (31.2). 22

32 Objectivity Restrictions on Constitutive Equations of Simple Materials

In w we have examined the restrictions that the principle of objectivity imposes upon the constitutive equation of an elastic material. In this section, we shall derive the corresponding ones for the simple material defined by (31.1). Because of the history dependence of the constitutive relation, the dependence of the rotation Q on the time units T and t have to be spelled out clearly and this will occur in the sequel.

Following the methodology of w we now examine the consequences of the following restriction:

q ( t ) O ( F ( T ) ) q ( t ) T - G(Q(T)F(T)) (32.1)

for all rotational histories Q(T),--oo < T _< t. We shall use the polar decomposition of F(T) -- R(T)U(T) and choose Q(T) --

R(T) T. Then, we find that (32.1) leads to the following form of the restriction:

G(F(T)) -- R ( t ) ~ ( U ( T ) ) R ( t ) T

= F ( t ) U ( t ) - I G ( u ( T ) ) U ( t ) - I F ( t ) T (32.2)

where the fact, that U is a symmetric tensor, has been used. Using the definition C - U 2, we shall define a new functional T ~ through

7 ~ ( C ( T ) ) - U ( t ) - I G ( u ( T ) ) U ( t ) -1. (32.3)

Combining (32.2) and (32.3), we obtain that in a simple material

G(F(T)) -- F(t )T~(C(T) )F( t ) T , - o o < T < t. (32.4)

The extraordinary similarity between the restriction (29.9) for an elastic material and the one just derived must be noted.

P r o b l e m 3 2 . A Show that the condition (32.1) imposes no restriction on the constitutive operator

7~(C(~)). Hint" See (29.10)-(29.11).

P r o b l e m 32.B Prove that

C ( ~ - ) - F ( t ) T c t ( T ) F ( t ) , (32.B1)

22This point of view is expressed by OLDROYD, J.G., as well. See eq.(6) in OLDROYD, J.G., Proc. Roy. Soc. Lond., A200, 523-541 (1950).

32 Objectivity Restrictions on Constitutive Equations of Simple Materials 141

where the right relative Cauchy-Green tensor Ct(T) = Ft(T)TFt(T) as defined earlier in (1.56).

, , ,

Hence, using (32.B1), we find that in all simple materials,

f \ G(F(T)) = F(t)'P~F(t)Tct(T)F(t))F(t) T. (32.5)

This says that in all simple materials, the stress depends on the history of the right relative Cauchy-Green strain tensor and the current value of the deformation gradient. That is, we may define a new functional

~ ( F ( T ) ) - ~T:~ (Ct(T), F ( t ) ) . (32.6)

This new functional must obey the restriction (32.1) and so we find that

Q(t)~(Ct(T), F(t)) Q(t)T - ~(Q(t)Ct(T)Q(t)T, Q(t)F(t)) (32.7)

for all orthogonal tensor functions Q of t. The above result follows because of the way the tensor Ct(T) transforms under a change of frame- see (10.11).

There is no contradiction between the functional T~(C(T)) having no restrictions imposed on it, while the new functional "J:~(Ct(T),F(t)) has to meet (32.7). The reason is the way the two functionals are defined, or the manner in which they depend on the variables in question.

32.1 Simple Fluids Turning to simple fluids, it has already been established that the stress tensor is determined by the history of the relative deformation gradient and the current value of the density of the fluid - see (31.12) above for emphasis. Because the history of the relative deformation gradient yields the history of the right relative Cauchy-Green strain tensor, a comparison of (31.12) with (32.6) shows that the constitutive equation for a simple fluid 23 must have the form

(328)

This relation must be objective and so we find that it obeys

for all proper orthogonal tensor functions q( t ) . The constitutive relation (32.8) and the restriction imposed upon it through (32.9) form the basis of a great deal of research in viscoelastic fluids.

23NOLL, W., Arch. Rational Mech. Anal., 2, 197-226 (1958).


33 The Incompressible Simple Fluid

Incompressibility of a fluid is a restriction on the class of kinematically possible velocity fields tha t the fluid body can experience. In the real, physical world, no fluid is incompressible. However, many of the fluids behave as if they are incompressible under quite a large variety of flow conditions. Hence there is a need to examine the form of the constitutive relation for incompressible simple fluids and we shall see how the relation (32.6) has to be modified to account for incompressibility. The first modification to (32.6) is trivial because the density of the fluid cannot change in such a fluid and so p drops out of (32.8) as a variable. That is, because

p ( t ) - PR, det F ( t ) - - 1, det C ( t ) - 1, det C t ( T ) - 1, (33.1)

none of these may enter the constitutive equation of an incompressible fluid as variables.

There is a second modification that must occur and this arises from the following principle, which is fundamental in Lagrangian Mechanics. In this approach to the classical mechanics of point masses and rigid bodies, a force on the body, called the force of constraint, is associated with a kinematical constraint. The assumption made in analytical mechanics is that this force of constraint does no work in any virtual displacement. 24 In analogy with this, we assume that the stresses in an incompressible material have a stress tensor of constraint such tha t the power expended by this extra stress tensor is zero in any isochoric motion. The stress power due to a symmetric stress tensor T is given by t r TD, where D is the symmetric part of the velocity gradient. In incompressible fluids, the isochoricity condition equivalent to (33.1) is that the velocity field has zero divergence or tha t t r D -- O. Hence we are led to seek a resolution of the following:

P r o b l e m : Find the general form of a symmetric stress tensor T so that t r T D -- O,

when t r D -- 0 as well. The answer to this can be found in a number of ways 25 and it states simply that

T is a multiple of the unit tensor 1. In sum, we postulate tha t in an incompressible fluid, the total stress consists of two parts: a part that produces no stress power in an isochoric motion and another one that does, with the latter being determined by the kinematics of the motion.

Thus, the total stress tensor T has the form

T = - p l + S, (33.2)

where p is called the pressure and S is called the extra stress. In a state of rest, a fluid is under pressure and this is positive. However, the pressure leads to a compressive stress tensor and, by convention, compressive stresses are assigned the negative sign. Thus, we assign a negative sign to the pressure term in (33.2).

We shall now see how p is determined in practice. The case of static equilibrium shows that the pressure has to be found from the equations of equilibrium and not

24See, for example, ROSENBERG, R.M., Analytical Dynamics of Discrete Systems, Plenum, New York, 1977, Chap. 8-9. For a thorough treatment of constraints in continuum mechanics, see ANTMAN, S.S. and MARLOW, R.S., Arch. Ratwnal Mech. Anal., 116, 257-299 (1991).

25For a simple proof, see LANGLOIS, W., Amer. J. Phys., 39, 641-642 (1971).

33 The Incompressible Simple Fluid 143

the constitutive relation. The obvious extension to a fluid undergoing any motion is made by postulating that the pressure has to be determined from the equations of motion and not by any constitutive relation.

Of course, the extra stress S is determined by the constitutive relation. Turning now to (32.6)-(32.9), we see immediately that in an incompressible simple fluid the constitutive equation must have the form 26

S ( t ) - ~ ' < C t ( T ) ) , --OO < T < t, (33.3)

and that this must obey the restriction

for all proper orthogonal tensor functions Q(t). We see once again that the domain of restriction in (33.4) may be extended to

all orthogonal tensors because replacing Q by - Q poses no extrra burden on the constitutive relation in (33.4). In this treatise, we shall see that (33.3) and (33.4), valid for all orthogonal Q(t), form the foundations for all that is to come.

We shall call the constitutive functional ~:" an isotropic functional of its argument Ct(T) because the functional obeys (33.4). This nomenclature should not be confused with the question of the symmetry of the fluid which has a different symmetry group. This point must be understood because the restriction (33.4) does not arise from the symmetry of the fluid; rather it arises from the objectivity of the stress tensor and the frame indifference of the constitutive operator.

Finally, it is easy to show that in an incompressible simple fluid, the extra stress tensor must be a functional of Ft(T) only as follows. One begins with the relation (cf.(31.1))

S(t) = r - o o < T _< t, (33.5)

where det F(T) -- 1 for all T. Since the above functional must satisfy the restriction (cf.(31.2))

~(F(T)) -- r (33.6)

for all proper unimodular H, we may choose H -- F(t) -1 for the latter is also unimodular. Since the relative deformation gradient Ft(T)---- F(T)F(t) -1, we see that F (T)H -- Ft(~-). Thus, an incompressible simple material is an incompressible simple fluid if and only if

r -- r (33.7)

As an example, consider the constitutive relation of finite linear viscoelasticity:

s ( t ) - u ( t -

= u ( t -

( 3 3 . 8 )

26For an application of the idea behind the concept of a simple fluid to fractional derivative models, see PALADE, L.I., HUILGOL, R.R., ATTAN]~, P. and MENA, B. J. Rheol., (submitted).


The latter may be written as a functional of F(T) as follows (cf.(31.3)):

t

g(F(T)) = f #(t -- T)(F(t)-I)TF(T)TF(T)F(t)-:dT. w O O

(33.9)

Here, if we replace F(T) by Ft(T), we must change F(t) to the identity tensor 1 because Ft(t) = 1. Hence, (33.6) is satisfied, and (33.8) or (33.9) describes an incompressible simple fluid.

Of course, we may choose the tensor H = F ( t ) - I H 1 , where Hi is any other unimodular tensor, for use in (33.6). It will be seen that (33.9) remains invariant, yet again, because we have to replace F(T) by Ft(T)H1, and we must change F(t) to equal the tensor Hi .

However, it is not t rue tha t in a simple fluid we may begin with a functional of Ft(T), and demand tha t

n ( F t ( T ) ) = n ( F t ( T ) H 1 ) , (33.10)

where H1 is any unimodular tensor. The explanation comes from the way (33.7) has been derived. That is, as in connection with (28.7) earlier, care needs to be exercised in interpreting (33.6). One may, in (33.7), go from "left to right" readily, whereas going the other way requires a careful examination of the argument leading to (33.7).

Appendix A to Chapter 3: Exploiting Integrity Bases

Consider a scalar valued function of a vector, say f (v) , and suppose that for all orthogonal tensors Q this function obeys the restriction:

f (v ) = f (Qv) . (A3.1)

Such a function f is said to be an isotropic function of v. Let us now pose the following

P r o b l e m : In which specific manner does f depend on v? The answer to this is simple because the only property of v that remains invariant

under all possible rotations and reflections is its length, or the dot product v - v . Let us now consider the case when f is a function of two vectors u, v and let it

obey

f (u , v ) = f ( Q u , Qv), (A3.2)

or f is an isotropic function of its two arguments. The invariants for this problem are the three dot products

u - u , v - v , u - v ; (A3.3)

that is the two lengths and the angle between the two vectors remain invariant when both vectors are rotated and/or reflected simultaneously.

Appendix A to Chapter 3: Exploiting Integrity Bases 145

The uses of these invariants are as follows. Assume that f is a scalar valued polynomial in the components of the vector v and subject to the constraint (A3.1). As we have seen, the only invariant is v - v. Thus we can claim that

f (v ) -- F ( v - v ) , (A3.4)

where F is a polynomial in its argument. Used thus, the invariant v - v is called an integrity basis. 2~ In a similar manner, if f is a polynomial in the components of the two vectors u and v and satisfies (A3.2), then

f (u , v) -- F ( u - u , v - v , u - v ) , (A3.5)

where F is a polynomial in its arguments. 2s Suppose tha t f is a scalar valued polynomial in the components of a symmetric,

second order tensor S and let this function satisfy the following:

f (S) - f ( Q S Q T) (A3.6)

for all orthogonal tensors Q; tha t is, f is an isotropic function of S. In this case, it is easy to see tha t the only invariants are the eigenvalues of the tensor S. Equivalently, the invariants are

tr S, tr S 2, tr 8 3. (A3.7)

The above set is derivable from the invariants Is , I I s and H i s listed in (1.52) earlier; conversely, the earlier set of invariants can be obtained from the above three. Again, the invariants in (A3.7) form the integrity basis if f is assumed to be a polynomial in the components of S and obeys (A3.6). We list below two more sets of integrity bases.

(i) f is a scalar valued, isotropic function of a symmetric second order tensor S and two vectors u and v. The basis is

tr S, tr S 2, tr S 3, u- u, u . v , v - v , u - S u , u - S2u,

v - S v , v - S 2 v , u - S v , u - S 2 v . (A3.8)

(ii) f is a scalar valued isotropic function of two symmetric tensors A and B. The basis is

tr A , tr A 2, tr A3 , t r B, tr B 2, tr B 3,

tr AB, tr A2B, tr A B 2, tr A2B 2. (A3.9)

Vector Valued F u n c t i o n s

Suppose tha t w - g(v) is an isotropic, vector-valued function of the vector v, i.e., let it obey

Qg(v) = g(Qv) (A3.10)

i i i

27See WEYL, H., Classical Groups, Princeton Univ. Press, New Jersey, 1946, pp. 30. 28See WEYL, H., op. cit., pp. 31-36 for a detailed proof.


for all orthogonal tensors Q. Now, define a scalar valued function f of two vectors u and v through

f (u , v) = u - g ( v ) , (A3.11)

where u is arbitrary. Hence,

f(qu, qv) = qu-g(Qv). (A3.12)

However, (A3.10) says that

f (Qu , Qv) - Q u - Q g ( v ) - u - g ( v ) , (A3.13)

the latter following from the fact that u - w ----- Q u - Q w for any orthogonal tensor Q. It follows that f (u , v) - f (Qu , Qv) for all orthogonal tensors Q, or f is an isotropic function of the vectors.

If we assume that g(v) is a polynomial in v and that f (u , v) is also a polynomial in its arguments, then the integrity basis for the latter is given by (A3.3). However, by its definition in(A3.11), the function f is linear in u. Hence, f cannot depend on u - u . Consequently,

f (u , v ) = u - h ( v - v ) v , (A3.14)

where h is a polynomial in its argument. It follows therefore that

g(v) = h ( v - v ) v . (A3.15)

S y m m e t r i c T e n s o r Valued F u n c t i o n s

As the next example, consider the task of finding the integrity basis for a symmetric tensor valued function of a symmetric tensor, say g(B), such that it is isotropic, i.e., let it obey

Qg(B)Q r = g(QBQT). (A3.16)

Now, form a scalar valued isotropic function of two symmetric tensors through

f ( A , B ) = tr [Ag(B)]. (A3.17)

Using the linearity of f in A and the integrity basis in (A3.9), it is seen immediately that f is a polynomial in t rA, t r A B and t rAB 2. Hence, g(B) must have the form

g(B) - g01 + gl B + g2B 2, (A3.18)

where the coefficients go, gl, g2 are polynomials in the three invariants (cf. (A3.7)) of B.

The principle 29 on which the above argument has been based is the following: in order to find the vector valued, isotropic function of a vector or a symmetric tensor valued, isotropic function of a symmetric tensor, form an artificial scalar product with a second vector or another symmetric tensor. Find the relevant integrity basis for this newly formed scalar valued, isotropic function. Noting the linearity of this

29PIPKIN, A.C. and RIVLIN, R.S., Arch. Ratwnal Mech. Anal., 4, 129-144 (1959).

Appendix A to Chapter 3: Exploiting Integrity Bases 147

artificial scalar product in the second vector or tensor, discard the nonlinear terms in the integrity basis involving the second vector or tensor.

Then, using yet again the linearity of the scalar valued function in the second vector or tensor, one is led immediately to the polynomial expansion of the original function with the scalar valued coefficients depending on the appropriate invariants.

. , , . .

P r o b l e m A3.1 Let f be a vector valued, isotropic polynomial of a symmetric tensor S and a

vector v. Use the integrity basis in (A3.8) to prove tha t

f(S, v) = [f01 + f l S + f2S2]v, (A3.19)

where the scalar valued coefficients are polynomials in the six invariants involving only S and v in the list (A3.9).

P r o b l e m A3 .2 Suppose tha t v is an isotropic, vector valued function of a symmetric second

order tensor A, i.e., if v - g(A) and Qg(A) - g(QAQ T) for all orthogonal tensors Q. Show tha t such a vector valued function does not exist unless it is trivially zero. Hint" Let )~1,A2,A3 be the eigenvalues of A and let el , e2, e3 be the corresponding orthonormal eigenvectors. Write g(A) as a function f of its eigenvalues and eigenvectors. Show that the isotropy restriction on g means tha t f must satisfy

Qf(A1, )~2, ~3; ea, e2, e3) : f()~l,/~2, ~3; q e l , qe2, qe3) . (A3.20)

Hence, deduce tha t f is identically zero. 3~

The procedure spelled out above can be used to deal with the case of multiple vectors and tensors. All one needs is the relevant integrity basis and these have been published in detail. For functions which are isotropic, or transversely isotropic or have crystal classes as their symmetric groups, see the review arti- cle by SPENCER. 31 For functions which are invariant under the full unimodular group, see FAHY and SMITH. 32 The latter work has been used earlier in w

Integrity Basis and Function Basis

So far, the emphasis has been on polynomial forms of the various equations. Thus, all equations have polynomial expansions with scalar valued coefficients, which are also polynomials in their arguments. If the functions in question are invariant under the full orthogonal group or its subgroups, i.e., transversely isotropic and crystal classes, then it is known 33 tha t the integrity basis is also a function basis. Hence, we may assume tha t the various expansions of isotropic functions which occur in

3~ are grateful to Professor SPENCER, A.J.M., for the above simple argument. 31SPENCER, A.J.M., in Continuum Physzcs, Vol.1, Ed. ERINGEN, A.C., Academic Press,

New York, 1971. See Chapter 3. 32FAHY, E. and SMITH, G.F., J. Non-Newt. Fluzd Mech., 7, 33-43 (1980). 33WINEMAN, A.S. and PIPKIN, A.C., Arch. Rational Mech. Anal., 17, 184-214 (1964).

this treatise are functions of their arguments, rather than polynomials. However, such a proof does not exist for unimodular functions and hence this assumption still remains as far as the results in w are concerned. In practical terms, however, this should not be seen as a hindrance in accepting the results of w regarding the constitutive equation for an elastic fluid.

Appendix B to Chapter 3: Constitutive Approximations

The constitutive relation for the incompressible simple fluid is given by (33.3), which is repeated here for convenience:

T ( t ) = ~ ' ( C t (T)). (33.3)

In this appendix, 34 we shall consider ways of developing approximations to the constitutive functional when a given motion may be considered as a perturbation about the state of rest. Here C t ( r ) is "close"to 1, the strain history in the rest state, where the word close is defined in terms of the smallness of the norm of the history. Thus, we need to choose a function space of strain histories with a norm, or magnitude.

To this end, given the tensor C t ( t - s), we define the difference G(s) from a state of rest through

G(s) -- C t ( t - s ) - 1, 0 < s < oo. (B3.1)

Now, let us define a new stress functional

Hence, approximating the constitutive functional about the state of rest reduces to considering the development of the functional ~ about G(s) - 0. Now consider the collection of all difference histories, i.e., all symmetric tensor valued functions G(s), 0 _~ s < oo, and define an L2 norm 35 on the set through

llGil = - /o

= - (B3.3)

Note that the latter is the Schur norm of G(s), defined earlier in (5.3). In (B3.3), h(s) is a monotonically decreasing function of s such that h(0) --

1, h(s) ---. 0 as s ---, co sufficiently fast so that I1" II < oo for almost all difference histories. Physically speaking, the norm in (B3.3) embodies the notion that the memory of the fluid fades in time and that distant events should thus have less

34The development here follows that in HUILGOL, R.R. and PHAN-THIEN, N., Int. J. Engng. Sci., 24, 161-261 (1986).

35See COLEMAN, B.D. and NOLL, W., Arch. Rational Mech. Anal., 6, 355-370 (1960); Rev. Mod. Phys., 33, 239-249 (1961); Erratum, Rev. Mod. Phys., 36, 1103 (1964).

Appendix B to Chapter 3: Constitutive Approximations 149

bearing on the present state of stress than those in the recent past. Now, the set of all difference histories with a finite norm is not a vector space because 1 + G ( s ) must always be positive definite and det(1 + G(s)) = 1, since the fluid is incompressible. To overcome this difficulty, let us imbed this set of G(s) into a larger set S = { J ( s ) ' l l J ( s ) l I < c~, J(s) = J(s) T, 0 ~ s < cx:).} This set S is a normed vector space which is complete, i.e., it is a Hilbert space.

Next, although S is complete, we have to extend the domain of the constitutive functional 7g of the incompressible fluid, to include elements of this space. To this end, recalling tha t the fluid can sustain isochoric motions only, we take any J (s) E S and create a unimodular tensor from it through

J(s) = [det(1 + J(8))-1]1/3(1 -~- J(8)). (B3.4)

Thus, the domain of T / m a y now be conceived as consisting of all elements J(s) of S.

This procedure creates its own problems, because it is quite easy to find finitely normed elements J(s) E S such that (1 + J(s)) is singular. For example, let 0)

1 + J(s) - - /~s 1 0 , (B3.5) 0 0 1

which is singular whenever ( ~ 2 _ c~2)s 2 _ 1. Hence, J(s) may not exist, even if J(s) does. Thus, a solution to all of the above problems lies in developing theories of fading memory for compressible fluids on cones, 36 so that (1 + J(s)) is always positive definite. Then, one insists that the domain of the constitutive functional 7g of the incompressible fluid consists of those elements J(s) of S, obtained from the positive definite 1 + J(s) according to (B3.4).

The second option is, yet again, to assume that the fluid is compressible. Then, the domain of the constitutive functional may be extended to S, without any hindrance. From the results so obtained, one retains only those which have relevance for incompressible materials. For instance, from the results for a compressible fluid, one discards terms involving the unit tensor, for these quantities may be absorbed into the pressure term; for an application of this idea, one may compare (B3.10) and (B3.13) with (B3.14)1,2 below. In addition, incompressibility forces certain lower order terms to be moved into higher order approximations. As an example of this, see the remarks preceding (B3.13) below. To summarise the above, it will be assumed tha t the theories of fading memory we describe are properly constituted along one of the above lines.

Returning to the matter under discussion, postulating that 7s is once Frdchet differentiable at the origin, we obtain

-- n ( 0 ) + n(01G(s) + o(llGIl(8)).

Without loss of generality, we may put

(B3.6)

n ( 0 ) = 0, (B3.7)

36COLEMAN, B.D., and MIZEL, V.J., Arch. Rational Mech. Anal., 29, 18-31 (1968).


because, in a state of rest, the stress in a fluid is a hydrostatic pressure and, in an incompressible fluid, this can be arbi trary to within a constant. In (B3.6), 5T/(01G(s)) is the Frdchet differential evaluated at the zero history in the direction G(s) and is a linear functional of the latter. Using the Riesz representation theorem for linear functionals, the definition of the inner product

OO (J1, J2) - t r{J1 ( s )J2(s )}h2(s) ds, (B3.8)

and the objectivity restriction (33.4), we obtain

~0 ~176 din(OlG(s)- g(s)G(s) ds. (B3.9)

Now, we define an incompressible fluid through the constitutive equation

OO S ( t ) - # ( s )G(s ) ds. (B3.10)

The above relation is tha t of f inite linear viscoelasticity, with the kernel function # delineating one fluid material from another. This model approximates the stress in the corresponding simple fluid, in any motion, to within o(l[G[[(s)) of the difference history.

Now, if the norm of the deformation gradient is infinitesimal in the sense of w then the difference history has the form G(s) : 2 E ( t - s) - 2E(t), where E is the infinitesimal strain tensor. Comparing the constitutive relation (B3.10) with tha t of linear viscoelasticity, 37

S(t) - G(s)[2E(t - s ) - 2E(t)] ds, (B3.11)

we see tha t the material function ~ is related to the relaxation modulus G as follows:

#(s) : G(s) , G(s) - - #(a) da. (B3.12)

We now make the following additional assumptions and/or observations:

(i) a bounded second order Frdchet derivative of 7t: exists at the rest state, and it is noted tha t this differential is bilinear in its arguments;

(ii) incompressibility of the fluid means tha t t r G is always expressible in terms of the second and third order invariants 3s of G(s). Hence, any term proportional to t r G ( s ) will appear in a model at a higher order.

The above conditions permit one to obtain a constitutive relation of second order viscoelasticity. Assuming further tha t the bilinear operator has an integral

ii

37For a thorough disussion of the linear theory, see TSCHOEGL, N.W., The Phenomenological Theory of Linear Viscoelastic Behavior, Springer-Verlag, Heidelberg, 1989.

3SSee PIPKIN, A.C., Rev. Mod. Phys., 36, 1034-1041 (1964).


representation, 39 one finally derives

s ( t ) -

/0 /0 -}- O~(81,82)G(81)G(s2)ds1 a82,

0t(81,82) -- 0~(82,81). (B3.13)

It is possible to obtain higher order approximations and these are available in the literature just cited.

The stage is now set to obtain expansions of the functional depending on C t ( t - s ) itself. For, terms proportional to 1 can be introduced into the extra stress S at pleasure and all linear terms in C t ( t - s) may be absorbed into the first integral. Hence, we may adopt the following

s ( t ) - f o

S(t) -- f o # ( s ) C t ( t - s)ds (B3.14)

-}-f0 ~176 f0 ~176 O~(81, 82)Ct( t - 81)Ct( t - 82)dsld82,

instead of (S3.10) and (B3.13) respectively.

O r d e r Fluids

Instead of assuming that a viscoelastic fluid has a memory which fades in time, let us propose that its memory is of instantaneous duration. That is, let the constitutive relation be an isotropic function of the first N Rivlin-Ericksen tensors, i.e.,

S = f ( A 1 , - - - , A N ) . (B3.15)

The physical dimensions of the A~, i = 1,---, N are T -~, where T denotes time. Hence, a sequence of approximations to (B3.15) may be developed, correct to order T - 1, T - 2, etc. These are given by: first order:

S = v/oAI; (B3.16)

second order: S - ~oA1 + flA12 + "IA2;

third order:

S -- (second order) + a0(trA2)A1 + O~ 1 (AIA2 + A2A1) + 0~3A3,

fourth order:

(B3.17)

(B3.18)

-- (third order)+ [a3tr(A1A2) + a4trA13]A1 + [asA12 + aeA2]trA 2

+aT(A2A2 + A2A 2) -I- asA,~ -I- ceo (A1 A3 + A3A1) + choA4. (B3.19)

39See SAUT, J.C and JOSEPH, D.D., Arch. Rational Mech. Anal., 81, 53-95 (1983).

It is possible to show a connection between the above order fluids and the integral models within the L2-norm theory. To achieve this, note tha t

N G(s) = Z ( -1 )=s=A= n! + o(slV). (B3.20)

n-1

Hence, under certain conditions, a functional of G(s) should be approximated by a function of A1 , - - - , AN. This argument becomes clear when it is recalled tha t the L2-norm is intended to ensure tha t little weight is given to events in the distant past, i.e., a large t ime lapse s. Thus, if the Taylor expansion for G(s) approximates the history quite well for a "long period of time," then the integral models will all reduce to functions of the Rivlin-Ericksen tensors.

One way to elongate real t ime is to let a fluid particle experience the same history but a slower rate, i.e., to retard the process. This may be acheived by replacing 4~ the t ime lapse s by as , where 0 < (~ < 1. Then we have a new difference history

N S An (~ ( s ) - G(as) -- ~ (-1)n n^

n! + ~ (B3.21) n-1

where

At, =o~'~An, n - 1 , . . . , N . (B3.22)

Using this new difference history, the integral models will deliver the following expansions:

Sl -- ~0h l + o(a), (B3.23)

S2 -- VOhl -I- ~A12 -I- " / i2 "~- 0(0~2),

and others as required. Hence, the order fluids listed in (B3.16) - (B3.19), respectively, represent the stresses in a simple fluid correct to within error terms of order o(a) , . , . , o(a 4) when the strain history is retarded by the factor a.

The above procedure, while no doubt valid, is not totally satisfactory, because it delivers the order fluids as appproximations through a special modification of the strain history. Faced with this, one seeks the solution to the following task: to find a solution of the problem of encompassing within the concept of a simple fluid with fading memory, these order fluids and other models without recourse to any artificiality. Here, we have in mind the Newtonian fluid or the Oldroyd-B fluid which is a special case of the following 1-integral, 1-order model:

/ OO

S - r/oA 1 q- /t(s)G(s) d8. (B3.24)

It has been shown by SAUT and JOSEPH 41 tha t a fading memory theory based on LF spaces and their duals delivers the desired models. Tha t is, in this modified

4~ COLEMAN, B.D. and NOLL, W., Arch. Rational Mech. Anal., 6, 355-370 (1960). 41See SAUT, J.C. and JOSEPH, D.D., Arch. Rational Mech. Anal., 81, 53-95 (1983), and

RENARDY, M., Arch. Rational Mech. Anal., 85, 21-26 (1984) as well.


theory of fading memory, the material function tt(s) in (B3.10) is shown to have the representation

k

#(s) = y ~ #,6 (') (s) +/2(s), (B3.25) i - -1

6 (~) is the i - t h derivative of the Dirac delta function. Using this in (B3.10), where one obtains:

k oo

S - y ~ #,A, + ~0 ft(s)G(s) ds. (B3.26) i - -1

Other models may be derived by appropriate choices of the memory kernels.

4 Constitutive Equations Derived From Microstructures

It is clear from Chapter 3 that the continuum approach to the formulation of constitutive equations for viscoelastic materials has been most helpful in setting up the theoretical framework for all simple fluids. Although NOLL i considered simple materials and fluids of the rate type, i.e., those that involve stress fluxes or convected derivatives, traditionally simple fluids are considered to be defined through an explicit constitutive relation for the stress tensor. Hence, the emphasis on these aspects in Chapter 3.

Both integral and differential constitutive models have been used with varying degrees of success in numerical schemes dealing with viscoelastic fluids. Single integral models, although relatively easy to use when one wishes to match them with experimental data, are more difficult to deal with numerically, since an integration backwards in t ime (in a Lagrangian sense) is required to march ahead. There have been significant progress in stream line integration schemes, but an efficient three-dimensional scheme dealing with integral constitutive equations has yet to be devised.

Thus, those equations which define a stress rate as a function, perhaps nonlinear, of the first Rivlin-Ericksen tensor and/or its flux are very useful in numerical work. Such a constitutive relation can then be considered to be a part of a system which includes the balance laws and this system can be used, as a whole, to solve the resulting boundary/ini t ial value problem. Such constitutive relations arise quite naturally in microstructural theories and it is to them that we turn next.

In the microstructural approach, a relevant model for the microstructure of the fluid is postulated and the consequence of this is explored at the macrostructural level, with appropriate averages (ensemble or volume) being taken to smear out microstructural details. The advantage of this approach is that the particular con-

iNOLL, W., Arch. Rational Mech. Anal., 2, 197-226 (1958).

156 4. Constitutive Equations Derived From Microstructures

stitutive equations and the material functions arrived at are expected to be explicitly relevant to the fluid concerned; and if a particular phenomenon cannot be predicted by the model due to a missing physical cause, this can be corrected at the microstructural level by inserting the relevant physical detail. This iterative model-building process, from microstructure to constitutive equations, to flow predictions, to comparison with experimental data, and then back to a revised microstructure, has seen a flurry of activity over the past five decades, beginning with the early works of KDTIN, 2 KUHN and KUHN, 3 GREEN and TOBOLSKY, 4 and JAMES. 5 Excellent reviews of the different aspects of this research area can

be found in LODGE, s FLORY, 7 YAMAKAWA, s DE GENNES, 9 BIRD et al., i~ DOI and EDWARDS, ll LARSON, 12 0 T T I N G E R , 13 and TANNER. i4 The main emphasis in this Chapter will be to describe some of the popular microstructural models for polymer solutions and melts.

In w simple macromolecular models for dilute polymer solutions are introduced, starting with a brief review of the gross physical properties of a polymer chain in a non-flowing system. In these dilute polymer solutions, the interaction between different polymers, or the inter-polymer interaction, can be neglected, and therefore the only forces that come into play are the hydrodynamic forces and the Brownian forces exerted on the polymer by the surrounding solvent molecules. The connection between the mobility of the polymer and the strength of the Brownian forces is made through a fluctuation-dissipation theorem, and the full constitutive equation is derived for a number of microstructures. The theoretical basis for macromolecular models for dilute polymer solutions has been well established and is rigorous, simply because there is no ambiguity in the physical laws which are in operation at the microstructural level (i.e., Newton's second law, equipartition of energy), while the differences between the models of dilute polymer solutions are really those of fine tuning; that is, the degree of complexity incorporated into the microstructure. The clear understanding of the physics behind models of dilute polymer solutions is the main reason for the attention given to the constitutive models of these fluids in this section.

, ,

2KUHN, W., Kolloid-Zeit., 68, 2-15 (1934). 3KUHN, W. and KUHN~ H., Heir. Chim. Acts, 28, 1533-1579 (1945). 4GREEN, M.S. and TOBOLSKY, A.V., J. Chem. Phys., 14, 80-92 (1946). 5JAMES, H.M., J. Chem. Phys., 15,651-668 (1947). 6LOGDE, A.S., Elastic Liquids, Academic Press, London, 1964. 7FLORY, P.J., Statistical Mechanics of Chain Molecules, Wiley-Interscience, New York, 1969. 8YAMAKAWA, H., Modern Theory of Polymer Solutions, Harper & Row, New York, 1971. gDE GENNES, P.G., Scaling Concept in Polymer Physics, Cornell University Press, New York,

1979. i~ R.B., CURTISS, C.F., ARMSTRONG, R.C., and HASSAGER, O., Dynamics of Poly-

meric Liquids: Vol. II. Kinetic Theory, John Wiley & Sons, New York, 2nd Ed., 1987. liDOI, M. and EDWARDS, S.F., The Theory of Polymer Dynamics, Oxford University Press,

Oxford, 1988. i2LARSON, R.G., Constitutive Equations for Polymer Melts and Solutions, Butterworth Pub-

lishers, Boston, 1988. 13(~TTINGER, H.C., Stochastic Processes in Polymeric Fluids, Springer, Berlin, 1996. 14TANNER, R.I., Engineering Rheology, 2nd Ed., Oxford, New York, 1988.

34 Dilute Polymer Solutions 157

In w167 two basic models for concentrated polymer solutions and melts are reviewed. Network models, which are considered first, have their origin in the molecular theory of amorphous polymeric networks (see TRELOAR 15 for a review), although the network junctions are supposed to be the entanglement points of the polymers rather than the more permanent chemically cross-linked points in a rubberlike solid. An alternative and more appealing approach to the modelling of dense polymeric systems is to look at the motion of a polymer chain in a tubelike environment - see w This leads to reptation models, which are linked with the names of Edwards, de Gennes and Doi, who have made major contributions.

Finally, in w recent progress in constitutive modelling of suspensions will be reviewed. These models for suspensions consist of an evolution equation for the microstructure, which is idealised either by a unit vector or a second-order tensor field, and a stress rule, allowing the calculation of the stress tensor from the unit vector field. While the theoretical framework for the description of dilute suspensions is complete, the understanding of the mechanics of moderately to highly concentrated suspensions is rather sketchy, and therefore recent research on suspension rheology has been directed towards complex flows and non-dilute regimes.

34 Dilute Polymer Solutions

3~. 1 General Physical Characteristics

In this section, we consider primarily linear polymer solutions in the infinitely dilute limit, where the polymer-polymer, or inter-polymer interaction can be neglected, so that only one polymer chain need be considered in isolation. Now, a typical polymer chain consists of a large number of sequentially bonded monomer units of molecular weight of the order 102, with a total molecular weight (M) of the order 107, giving an idea of the number of monomer units in a polymer chain (of the order 105). The number of the monomer units is also called the polymerisation index. If a polymer is made up of the same repeating units of monomers, then it is called homopolymer; otherwise, if it is made up of different types of monomers arranged in some sequence, then it is called copolymer.

Apart from its molecular weight, there are other gross physical properties of a polymer: the radius of gyration RG, which is a measurable quantity, typically of the order a few hundred AngstrSms, providing the average size of the polymer; the hydrodynamic radius RH, which is a measure of the size of the polymer as it drifts through a solvent; and the second osmotic virial coefficient A2, which is proportional to the excluded volume effect, a measure of polymer-polymer interaction. All of these gross characteristics of the polymer vary as a power of its molecular weight, with the front factor and the exponent being weak functions of the molecular weight.

From experimental data on a number of solvent-polymer systems, it has been found that some of the gross features of the polymer, such as the second osmotic virial coefficient A2, are proportional to 1 - 9IT, where T is the absolute tempera-

15TRELOAR, L.R.G., The Physics of Rubber Elasticzty, 3rd Ed., Clarendon Press, Oxford, 1975.

ture, and ~ is called the theta temperature. 16 This temperature is a function of the particular polymer-solvent system. At temperatures near the theta temperature, the intra-polymer interaction does not have a large effect, and simple idealised models derived for dilute polymer solutions are expected to perform well.

Idealised models for linear polymers assume that the polymer consists of n segments (which may or may not correspond to a monomer unit), and has an end-to- end vector R. In the limit of large n, if the polymer has sufficient degrees of freedom and there is no flow, then the probability density function P ( R ) must be Gaussian, according to the central limit theorem, where P ( R ) d R is the probability of finding a polymer chain with an end-to-end vector lying between R and R ~- dR. In that case, P ( R ) is determined solely by the mean (R) and the variance ( ( R - <R))2) of the stochastic process R. 17 Furthermore, since there are neither flow nor external forces acting on the polymer chain, there is no way to distinguish between the ends of the chain, and therefore (R) -- 0. Thus the only relevant quantity is (R2), where R 2 - R - R. If this quantity is proportional to n, then the linear polymer is said to be flexible. Otherwise it is said to be stir, in which case the probability density function may not even be Gaussian. The important point here is that <R2) 1/2 is the relevant length scale carrying information about the molecular weight for a flexible linear polymer.

Thus, at large n, and in the absence of all external forces, the asymptotic probability density function of the end-to-end vector of a flexible linear chain must be Gaussian:

P ( R ; n ) -- ( 3 e x p ( - 3R2 27r<R2>n) 3/2 2<R2>n) , (34.1)

where the subscript n indicates the number of segments in the chain. Suppose that we have two flexible chains, of nl and n2 segments, connected to form a single chain with n : nl -}- n2. Then, from the definition of conditional probability we have

P (R ; n) = f P ( R - R'; n I )P (R ' ; n2)dR'. J

This implies that = + (34 .2 )

i.e., the mean square of the end-to-end vector must be an additive function of the number of segments, or the chain length, for a flexible chain. Alternatively, we can write

<R2>n - n b 2, (34.3)

where b is a constant length scale. The relation (34.3) is also taken as the definition of a flexible chain.

16FLORY, P.J., Prmczples o.f Polymer Chemzstry, Cornell University Press, Ithaca, New York, 1953.

17Here and elsewhere, the angular brackets denote an ensemble average with respect to the probability density function of the process concerned; i.e.,

<R) = f RP(R)dR. J


The simplicity of the probability density function and the relation for the mean square end-to-end vector is the reason for replacing the real polymer chain by the so-called Kuhn effective chain, is with N (Kuhn) segments, each of length b, such that the full extended contour length of the chain is L - N b , and the mean square of the chain is given by (R2>N -- N b 2. Both parameters, b and N, are therefore coarse-grained parameters introduced for mathematical convenience. The effective chain model would be a reasonable approximation to the real chain provided the number of segments is large, and provided we are interested only in macroscopic phenomena that do not depend on the detailed structure of the chain. Let us suppose that the fully extended contour length of the chain is L -- N b , then by eliminating N we have

(R2>N -- Lb. (34.4)

Since L is expected to be independent of the temperature, this relation serves to define the temperature dependence of the parameter b through the temperature dependence of (R2>N. The probability density function may be rewritten as

-- exp ( - ~ - ~ j . (34.5)

3~.2 Random- Walk Model

Gaussian Bond Probability

R2 R

R

r4 R

N

N+I

FIGURE 34.1. Random-walk model of a polymer; P~j are the bond vectors.

The simplest model for a linear polymer chain (without flow) is the r a n d o m - w a l k

model, a schematic diagram of which is shown in Figure 34.1. Here the polymer chain is replaced by an effective chain, with N Kuhn segments connecting N + 1 sequentially bonded units, each located at a position vector r j , j - 1 , . . . , N + 1.

18KUHN, W., Kolloid-Zezt., 68, 2-15 (1934); KUHN, W. and KUHN, H., Helv. Chim. Acta, 28, 1533-1579 (1945).


Each bonded unit may represent one or several monomer units. The end-to-end vector of a segment is defined to be

R j -- rj_l_l -- r j , j -- 1 , . . . , N ,

and the end-to-end vector of the polymer chain is denoted by R. For simplicity we may assume tha t these segments are identical, each having a Kuhn length of b.

The potential of the mean force U - U ( r l , . . . , r N + l ) is defined so tha t the probability density function for {rk} is given by

{ lU(rl )} (34.6) P ( r , , . . . , rN+, ) -- Q - l e x p - ~ - ~ , . . . , rN+, ,

where kT is the Boltzmann temperature in energy units, and Q is a constant so that

P ( r l , . . . , r N + l ) d r l . . . drN+l 1.

The potential of the mean force is supposed to represent all the monomer- monomer interaction on the chain through the solvent. Note tha t inter-polymer interaction is not considered for an idealised chain, or a chain in infinite dilution. This monomer interaction can either be short-range, i.e., between neighbouring monomers, or long-range, i.e., between different monomers far removed along the chain, but perchance happen to 5e nearby spatially. This idea is represented by the statement

N

V ( r l , . . . , rNq_ 1) :~ - -~ uj ( r j + l , r j ) -~- W ( r l , . . . , r N + l ) . (34.7) j--1 ~- ,~ �9 long range

short range

The short range bond probability is defined as

{ 1 } pj(Rj) -- q-~l exp ---s (no sum on j) , (a4.8)

where qj is a normalised constant, giving the following expression for the chain probability density function

N { 1 W ( r l ) } . (34.9) P ( r l , . . . , rN+l ) --" Q-11-I qjpj(Rj)exp - - . ~ , . . . , rN+l j = l

If the segments are identical, then the bond probabilities are the same, i.e., pj (Rj) -- p(Rj) . Furthermore, if far-field effects can be neglected, i.e., the temperature is at or near the the ta temperature for the polymer-solvent system, then we have

N

P ( r i , . . . , rN+l) -- H p(Rj)- j = l

Then, the end-to-end vector N

R -- ~ Rj (34.10) j--1


will have the probability density function

-

P(R;N) -- f__ R dR1 . . , dRN. (34.11) J j - -1 j----I

The chain with no far-field interaction is called a random-walk chain, since the evaluation of P(R; N) is equivalent to a random walk process of N steps, each step having a probability density function p(Rj). To solve for P(R), we introduce the formal expression for the generalised Dirac delta function 19

, (j.~ Rj-R)- (21)3/exp [i(j.~ I R,-R)-k] dk (34.12) into (34.11), and obtain

(27r)3 exp(--iR �9 k) p ( a j ) exp (iaj. k) dRj ,

so that the Fourier transform of P ( R ; N ) , or the characteristic function for the random process R, is given as

/5(k; N ) - [ / p ( R j ) e x p ( i R j . k) dRj] N . (34.13)

P r o b l e m 34.A The simplest model for the bond probability is that the bond length is uniformly

distributed over a sphere of radius b:

1

Show that the characteristic function for the end-to-end vector is

/5(k; N) _ L[sin(kbkb) ] N. (34.A1)

The inverse Fourier transform of this expression is rather unwieldy; however, if N is large, then

[ sin(kb) ] -- [ 1 -~ . I 1 (kb)2+ . . . . )N~exp( -Nk2b2)

Show that, for a spherically-symmetric function, the Fourier transform pair is reduced to

f~ f ~ P(k) -- J dO I dRexp(ikRc~ 21rR2 sin

. ~ a 0 . . . . (34.A2) t 2sin/eft . .

J ]o 4rR kR P(R)dR,

19JONES, D.S., Generalised Functions, McGraw-Hill, New York, 1966.


and f~ foo

P ( R ) = 1 / d O ] dkexp(- ikRcosO). f : ' (k)27rk2sinO (27r) 3 J ~ J 0 (34.A3)

__ 27r2R1 f o s i n k R [ : , ( k ) k d k "

Show that the distribution P(R; N) is Gaussian in the limit of large N.

Clearly, different models for the bond probability functions will yield different probability functions for the end-to-end vector. But in the asymptotic limit of large N, a Gaussian distribution for the end-to-end vector must be recovered, with a mean square value given by (34.3). A direct derivation is given in the following exercise.

P r o b l e m 34 .B From the definition of the end-to-end vector, (34.10), show that if Rj , j --

1,---, n, are completely independent processes, then

N - (34 B1)

j = l

If all the segments have the same length, then show that this leads to (34.3).

We can go one step further and identify the effective segment as being made up of a large number of smaller segments (say, monomer units), each of length of As and extended length b. Then it is justified to assume that the bond probability is Gaussian:

p ( R j ) - - 2~bAs exp 2bAs '

where the two parameters b and As, in analogy with the total extended length and the Kuhn length, may be regarded as temperature-independent. The total number of links of size As is now denoted by n, where h A s -- L, the fully extended contour length. The probability density function for the entire effective chain is Gaussian:

( 3 ) 3n/2 ( ~-~ 3 ( r j + l - r j , 2 } (34.15) P ( r l , . . . , rn+l) -- 2~bAs exp - 2bAs '

j--1

where rj is the position vector of the 'monomer' unit j . It is clear that the sequence {rj, j - 1 , . . . , n + 1} represents a discretisation of

a continuous random process r(t) such that rj - r(t j) , where tj = ( j - 1)As is the arc length along the chain, which plays the role of a time-like variable in this diffusion process; the end points of the chain are at t -- 0, and t - L. The limit to the continuous process can be taken formally by considering

n ---+ oo, A s ---+ O, h A s = L.

Thus, in this limit, we obtain the formal result

E ( r j + l -- r j ) 2 n (91" L 12 j--i A8 -- E -~ A 8 - Ir(t) dr,

j=l it--tj

and the probability of having a continuous chain having configuration between r(t) and r(t) + dr(t) is given by

P(r(t))dr(t) -- exp - I/~(t) 12 dt Dr(t), (34.16)

where the normalised constant has been absorbed into the 'differential' Dr(t). This is recognised as the Wiener measure, 2~ devised to treat Brownian motion. In this notation, some of the integrals can be related to path integral formulation in quantum mechanics, allowing useful analogies to be made between the two areas of research. We will not make use of the path integral representation in this book, however.

P r o b l e m 34 .C The probability of a chain having an end-to-end vector of R is given from (34.15)

by

P(r l -- 0, rn+l -- R; n) --

.... 2bAs j--1

The integral in rl can be carried out immediately. The integral in r2 is written as

/ d r 2 (.27rb3~s)3 exp { 3r2 3(r3 - r2)2 2bAs 2bAs j "

Re-write the argument of the exponential function and show that this yields

( 27rb32As ) 3/2 exp { 3r32 2b2As } '

which can be combined with the integral in r3 to obtain

/ ( 3 ) 3 / 2 ( 3 )3/2 { 3r2 3(r4 _ r3)2 } dr3 ~ 27rb2A'"~ exp 2bAs- 2bAs "

Show that this yields

( 3 ~ ~ ) ~ o x ~ { ~r~ 2b3As }"

Finally, show that

"<~1-0'~+1-~;n'-( ~ ) ~ ~ n ~ ox~{ ~}-~o~ . ,~4.~1~ Since nAs = L, this leads to the Gaussian distribution for the end-to-end vector as noted in (34.5).

2~ K. and McKEAN, H.P., Diffusion Processes and their Sample Paths, Springer-Verlag, Berlin, 1965.


Diffusion Equation

As a Gaussian process, the probability density function of the end-to-end vector satisfies a diffusion equation. An elementary but direct derivation of this is given below.

Suppose we have a polymer chain, of end-to-end vector R - b and n segments. The probability of n + 1 segments having an end-to-end vector R is the probability of the last segment n 4- 1 having a bond vector of b, conditional on the first n segments having an end-to-end vector of R - b:

P (R; n + 1) - / P ( R - b; n)Pb(b)db.

This is generally true for Markovian processes - namely what happens at any given instant depends only on the instantaneous state of the system, but not on its previous history; Pb(b) is usually referred to as the transition probability, which depends on both the current and the next states of the process. Here we assume that b is completely independent of the current state. The equation above can be expanded in a Taylor series:

OP "R, o) + o

/ Pb(b) (P(R; n) - b " VP(R; n) + 2 bb " XTVP(R; n) + O(b3)) db.

The average with respect to Pb(b) is taken, assuming that the process b is independent of R, and noting that for the added random segment,

f f b2 Pb(b)bdb = O, P b ( b ) b b d b - --~-1,

to obtain

OP (R;n) - V P(R;n).

On

The solution to this, subject to the "initial condition"

(34.17)

lim P(R; n) -- ~(R) n---,0

is the Gaussian distribution (34.5). The important result for this section is that a flexible chain will have a mean

square extension proportional to the number of effective segments in the chain. If the chain has some hindered rotation, for example, the bond angle is fixed, then the chain will be stiff, but in the limit of a large number of segments, the chain will be flexible, with a different effective bond length. 21

21FLORY, P.J., Statistical Mechanics of Chain Molecules, Wiley-Interscience, New York, 1969.

Radius of Gyration

The radius of gyration is defined as the root mean square distance from the centre of gravity of the chain to all the effective units in the chain:

n+l s2 - 1 2

-- n + 1 E (rj - re) , (34.18) j--1

where the center of gravity is defined as

rc =

1 n+l

n + l E r J �9 j--1

(34.19)

P r o b l e m 3 4 . D Show that the definition (34.18) leads to

n + l

s2 -- 1 E ~2-r~'2 n + 1 r

j----1

Furthermore, from the definition of the centre of gravity, and that

2 2 2 r~ + rj - 2r~ �9 rj -- r~j,

where rij -- rj - ri, show that

82 _ 1 ~n+~ 2 1 "2(n + 112 ' 7 1 r,j - (n + 1) 2 E

�9 "= l < i < j < _ n + l

This result is due to Lagrange. 22

r, 2. (34.D1)

From Lagrange's result, (34.D1), the average radius of gyration is

1 (s2) : ( n + 1)2 E (r'2)"

l < i < j < n + l

However, the distribution of rij is also Gaussian, from (34.15), with a variance of

Thus, 1

(s2) -- (n + 1) 2 E (J --i)As2" l~_ i<j~_n+l

Denoting k - - j - i - 1 , . . . , j - 1, for j - 2 , . . . , n + 1, we have

n As 2 ~ j-1 As2 ~ - ~ j ( 4 + l ) - - n + 2 (s2) -- (n + 1) 2 E E k -- 2(n + 1)2 z_~ ,- - 6(n + 1)

j = l k = l j = l

nAs 2.

22See LAMB, H., Sta t ics , 3rd Ed., Cambridge University Press, Cambridge, 1928, p. 166.


At large n, Debye's relation 1 - (34.20)

is recovered from the general formula. It should be noted tha t although the Gaussian distribution is asymptotically cor-

rect for a long chain, it does not exclude a non-zero probability for the chain length to be greater than the fully extended length. A more exact t rea tment produces a Langevin distribution, 23 which vanishes for R >_ n A s as required.

3~.3 Forces on a Chain

r 2 r~

1

Ri

1

-N+I

(a) Rouse Bead and Spring Model

rl

R

- r 2

(b) Elastic Dumbbell Model

FIGURE 34.2. (a) Rouse model and (b) the elastic dumbbell model for a polymer chain.

In simple models for dilute polymer solutions, such as the Rouse bead-spring model shown in Figure 34.2a, 2a a polymer chain is discretised into a number of effective segments, each of which has a point mass (bead) undergoing some motion in a solvent (which is t reated as a continuum). Each bead achieves an acceleration in response to the forces acting on it due to the solvent, the flow process, and the surrounding beads, and consequently the chain will adopt a configuration in a statistical sense. The task here is to relate the microstructure information to a constitutive description of the fluid. When there are only two beads present, the model is referred to as the elastic dumbbell model shown in Figure 34.2b, 25 which has been most popular in elucidating the main features of the rheology of dilute polymer solutions.

The forces acting on the beads include:

H y d r o d y n a m i c forces . These arise from the average hydrodynamic resistance of the motion of the polymer through a viscous solvent. Since the relevant Reynolds number based on the size of the polymer chain is negligibly small,

23FLORY, P.J., Statistical Mechanics of Chain Molecules, Wiley-Interscience, New York, 1969. 24ROUSE, P.E., Jr., J. Chem. Phys., 21, 1272-1280 (1953). 25KUHN, W., KoUoid-Zeit., 68, 2-15 (1934).


the average motion of the chain is governed by Stokes equations, and Stokes resistance can be used to model this effect. The chain is usually t reated as a number of discrete points of resistance, each having a frictional coefficient. In the simplest model, a frictional force of

F~ d) - (~ ( v , - i ' ,) , (34.21)

is assumed to be acting on the i - t h bead, which has a velocity i ' i - vi relative to the solvent, and (~ is a constant frictional coefficient, which is usually taken as 67rr/sa , where r/8 is the viscosity of the solvent, and a represents the size of the bead. To obtain a more realistic model of the nature of the dependence of the frictional forces on the configuration and the deformation of the polymer chain, ~ can be allowed to depend of the length of the segment i,26 or indeed it may be considered to be a second-order tensor, 27 which reflects the physical idea tha t the resistance to the motion perpendicular to the chain is much higher than tha t along the chain.

This simple frictional model neglects hydrodynamic interaction with other beads, usually called the free-draining assumption; the hydrodynamic interaction arises because of the solvent velocity containing disturbance terms due to the presence of other beads. An approximate theory taking into account hydrodynamic interaction has been derived by KIRKWOOD and RISEMAN, 28 and in a more complete form by ZIMM. 29

T e n s i o n in t h e cha in . A chain in equilibrium will tend to curl up into a spherical configuration, with the most probable zero end-to-end vector. However, if the chain ends are forcibly extended, then there is a tension or a spring force, arising in the chain, solely due to the fewer configurations available to the chain. To find the expression for the chain tension, we recall tha t the probability density function is proportional to the number of configurations available to the chain (i.e., the entropy), and thus the Helmholtz free energy of the chain is30

F r ( r l , . . . , r=+l) = A ( T ) - k T l n P ( r l , . . . , r , + l ) , (34.22)

where A ( T ) is a function of the temperature alone. The entropic spring force acting on bead i is

F~S) _ OFt k T 0 In P Ori (rl,... , rn+l)- Ori (rl,... ,rn+l). (34.23)

For a Gaussian chain of N effective Kuhn segments, each segment, of extended length b, is made up of N1 sub-segments of length As, where N1 = b/As;

26HINCH, E.J., Coll. Int. CNRS No. 233, Polym. Lubrification (1975). 27PHAN-THIEN, N., MANERO, O., and LEAL, L.G., Rheol. Acta, 23, 151-162 (1984). 28KIRKWOOD, J.G., and RISEMAN, J., J. Chem. Phys., 16, 565-573 (1948); 22, 1626-1627

(1954). 29ZIMM, B.H., J. Chem. Phys., 24, 269-278 (1956). a~ P.J., Statistical Mechanics of Chain Molecules, Wiley-Interscience, New York, 1969.


this is indeed a good approximation if the deformation of the chain is small when compared to its extended length. Now, the probability density function is given by (34.15) and the force acting on bead i is (i ~ 1, n + 1)

F~S)_ 3 k T 3 k T ( R ~ - R~ ) (34.24) - bAs (ri+l - 2ri + ri-1) -- ~ -1 .

This implies that the beads are connected by linear springs of stiffness

3 k T 3 k T (34.25) g = b A s - N1--"~"

A distribution which takes into account the finite segment length in a better manner is the Langevin distribution, and this results in the so-called inverse Langevin spring law for the chain tension"

L ~-~ = coth - k'~ = Nib ' (34.26)

where F is the magnitude of the force, r the magnitude of the extension, and the Langevin function is defined as L(x) = coth x - x . A useful approximation of the Langevin spring law is the Warner spring 31

3 k T 1 (no sum) (34.27) Hi = Ylb~ l J, ' (p~/L,)2,

where Li = N1 As is the maximum extended length of segment i. This stiffness approaches infinity as R~ --, Li.

B r o w n i a n Forces. Brownian forces are the cumulative effect of the bombardment of the chain by the solvent molecules. These forces have a small correlation time scale, typically the vibration period of a solvent molecule, of the order 10-13s for water molecules. If we are interested in time scales considerably greater than this correlation time scale, then the Brownian forces acting on bead i, F~ b), can be considered as white noise having a zero mean and a delta autocorrelation function:

= o ,

no sum). (34.28)

Equation (34.28) states that the strength of the Brownian forces is the measure of the integral correlation function over a time scale which is considerably greater than the correlation time scale of the Brownian forces:

F OO

(no sum). (34.29)

The strength of the Brownian forces is not an arbitrary quantity determined by a constitutive modelling; it is in fact related to the mobility of the Brown- ian particle. This point is taken up next.

3tWARNER, H.R., Jr., Ind. Eng. Chem. Fund., 11,379-387 (1972).


3~.~ Fluctuation-Dissipation Theorem

Langevin Equation

There are several fluctuation-dissipation theorems, relating the strength of the fluctuat ing quanti ty to the macroscopic 'mobility' of the phenomenon concerned. 32 The following development is pat terned after HINCH. 33

All micro-mechanical models for a polymer chain in a dilute solution can be written in the form

m ~ + ~ + K x - F (b) (t), (34.30)

called the Langevin equation, where the state of the system is represented by the finite-dimensional vector x, such tha t its kinetic energy is � 89 • and its generalised linear momentum is m.~, with re(x) being a generalised inertia tensor. 34 The system is acted on by a frictional force, which is linear in its s tate velocities, a restoring force (possibly nonlinear in x), and a erownian force F(b)(t). We assume tha t the frictional tensor coefficient ~(x) is symmetric. 35 This system can be conveniently s tar ted from rest at t ime t - 0.

Since the Brownian force has only well-defined statistical properties, the Langevin equation (34.30) must be understood as a stochastic differential equation. 36 It can only be ' so lved 'by specifying the probability distribution W(u , x, t) of the process ( u = ~, x) , defined so tha t W(u , x, t )dudx is the probability of finding the process at the s tate between ( u , x ) and ( u + du, x + dx) at t ime t. Prescribing the initial conditions (u(0) ---- u0,x(0) = x0) for (34.30) is equivalent to specifying a delta probability at t ime t - 0:

W ( u , x , 0) = - uo) (x - x 0 ) .

The description of the stochastic process (u , x} through its distribution W(u , x, t) is said to be a phase space description. In some cases, the dependence of W on x, or u can be eliminated, by integrating out the unwanted independent variable. In these situations we have either a velocity space, or a configuration space description, respectively.

Furthermore, the existence of the temperature T of the surrounding fluid demands tha t the distribution in the velocity space must tend to a Maxwellian dis- tr ibution as t --~ oo, in the sense tha t the surrounding solvent molecules follow this distribution at equilibrium, according to the kinetic theory of gases. This is equivalent to the requirement

lim (~(t)~(t))= kTm -1, (34.31) t - -~ o o

32LANDAU, L.D., and LIFSHITZ, E.M., Fluid Mechanics, translated by J.B. Sykes and W.H. Reid, Pergamon Press, New York, 1959.

33HINCH, E.J., J. Fluid Mech., 72, 499-511 (1975). 34The inertia tensor m is defined through the kinetic energy, and therefore there is no loss of

generality in considering only symmetric m. a5 All models for dilute polymer solutions satisfy this requirement. 36LIN, Y.K., Probabilistic Theory of Structural Dynamics, McGraw-Hill, New York, 1967.

i.e., each mode of vibration is associated with a kinetic energy of �89 This is sometimes known as the principle of equipartition of energy. We now explore the consequence of this on the Brownian force.

There are three t ime scales in this system:

1. Tr, the relaxation t ime scale of the chain in its lowest mode; this t ime scale is of O (]r K-11);

2. T,, the inertial relaxation t ime scale o f the chain; this t ime scale is of O ( I r a - ~ - 1 [ ) ;

3. To, the correlation t ime scale of the Brownian force - this is of the same order as the relaxation t ime scale of a solvent molecule.

Fluctuat ion-Dissipat ion Theorem

In general we have Tc << Ti << Tr, but the est imate of Ti varies a great deal depending on the value considered to be the relevant mass of a segment. To derive a fluctuation-dissipation theorem for the Langevin equation (34.30), we need to consider only events at t ime scale ~i. In this t ime scale, re(x) and 4(x) can be replaced by their local values, i.e., regarded as being constant, and the s ta te vector can be re-defined so tha t (34.30) becomes

m ~ + r177 -- F (b) (t), (34.32)

subjected to the initial rest s ta te

x ( 0 ) = 0 = , ( 0 ) .

The solution is given by

~0 t :~(t) -- exp {m -1- ~(t'-- t)}-m-lF(b)(t ') dt', (34.33)

leading to the expectat ion

/ot/o t ( • 1 7 7 exp {m -1- 4 ( t ' - t ) } - I l l -1" (F(b)(t')F(b)(t"))

�9 m - l - e x p { ~ . m - l ( t ' ' - t ) } dt H dr'.

If ~-c << Ti, the white noise assumption for the Brownian force can be used, so tha t

(F (b) ( t ' )F (b) ( t " ) ) - 2~(t' - t " ) f ,

giving

~0 t 2 d t ' exp{m- l . f , ( t ' - t ) } .m- l . f .m- l . exp{~ .m- l ( t ' - t ) } .


This can be integrated by parts to yield

(•177 - - 2 e x p { - - m - 1 - ~ t } - ~ - 1 . f . m - 1 . e x p { ~ . m - l ( g _ t)}

+2~ -1 - f - m - ' - ~ - 1 - m - ( • 1 7 7 ~ - m -1. (34.34)

We now invoke the assumption of equipartition of energy, which assumes tha t each mode of vibration is associated with a kinetic energy of l kT . Thus, in the limit t ~ OO~ 37

f = kT~. (34.35)

This is the fluctuation-dissipation theorem, relating the strength of the Brownian force to the mobility of the Brownian system; any dependence on the configuration of f is inherited from tha t of ~.

Diffusivity and the Stokes - Einstein Relation

The diffusivity of the process is also inherently connected to its mobility. To understand this, we start with the calculation of the velocity correlation function. Since a white noise is a stat ionary process, 3s ~(t) is also a stationary process in the limit of t -+ oo, with the auto-correlation function:

Ri(T) -- lim (• + T)~,(t)). t----~ oo

We consider the case where T :> 0; the remaining case is simply the reversal of the role of t + T and t. Thus,

(*(t + T)*(t)) --

2 { m -1 �9 t - m - i . f . m-i. x p { r m - i ( t , _ t ) } ,

which can be rewritten as

(~(t + T)~.(t)} -- exp {- -m -1- (T}-{~( t )~ ( t )} .

Taking the limit t ~ c~, we obtain the velocity correlation function:

R i (T) -- exp {- -m -1 - t~T}-kTm -1, T >_ O. (34.36)

The diffusivity of the process is defined through

D - - lim 1 d t - ~ 2d'~ (x( t )x( t ) ) ,

by making the analogy with the Gaussian distribution. Equivalently,

D - lim 1 f t t+oo 2 Jo ( •177 - T) + • -- T)• dT,

37Here, limt-,oo denotes the limiting process for t >> r i , but t ~ r r so that the state equation remains linear.

38The correlation function of such a process at two different times depends only on the time difference. See, for example, LIN, Y.K., Probab~l~stic Theory of Structural Dynamzcs, McGraw- Hill, New York, 1967.

yielding

From (34.36),

/oo 1 [R~(T) + RT(T)] dr. D - ~ (34.37)

D = k T ~ - 1, (34.38) which is the classical Stokes-Einstein relation, relating the mobility of the process to its diffusivity.

3~.5 Fokker-Planck Equation As mentioned before, solving the Langevin equation means specifying the probability distribution function of the process. The equation governing the time evolution of this distribution function is called the Fokker-Planck equation, and to this we turn our attention next.

We consider the Langevin equation (34.30), with u -- ~, written in the equivalent increment form:

Au(t) - - ( / ~ u + H) At + B(At) , (34.39)

where/9 = m-1~, H = m - 1 . Kx, At is a time scale considerably larger than the correlation time of the Brownian force, but much smaller than the inertial time scale, and B(At) is the random process

f t + A t

B(At) -- m -1- F (b) (t)dt. (34.40) J t

From the statistics of the Brownian force, 39 we have

(B(At)> = 0,

(B(At)B(At) ) -- 2m - 1 . f . m - l A t -- 2kTm -1- ~ - m - l A t . (34.41)

P r o b l e m 3 4 . E Since u and x vary on a time scale considerably greater than that of the Brownian

force, they are completely uncorrelated to the Brownian force, i.e., <xF(b) ) -- 0 =

<uF(b) ) . Show that

(Au> = - ( t 3 u + H) At, (34.E1)

and (AuAu> -- 2kTm - 1 . ~. r e - ' A t + O(At2). (34.E2)

Next, we make use of the Markovian nature of the Brownian process to write a~

W(u, x, t + At) =

^ ~ g B ( h t ) is a s ta t ionary process, and its dependence on t has been suppressed. 4~ section is pa t t e rned after C H A N D R A S E K H A R , S., Rev. Mod. Phys . , 15, 1-89 (1943).

W ( u Au, Ax, t ) r A u , - Ax; Au, Ax)d(Au)d(Ax), (34.42) x x

where W(u ,x , t) is the phase space probability distribution function, and r Au, x - Ax; Au, Ax) is the transition probability that u and x suffer increments Au and Ax during the time At from the current state { u - Au, x - Ax}. From the definition of u,

r x; Au, Ax) = r x; Au)~(Ax -- nat),

and thus the integration over Ax can be performed, after changing x - uAt to x, to obtain the following

uAt , t + At) = f W ( u - Au, x , t ) r Au, x; Au)d(Au). (34.43) W ( u , x +

We now expand both sides of this equation, keeping only terms of up to O(At, Au2) �9

ow ow (u,x, t) - W(u, x, t) + A t - ~ - ( u , x, t ) + Atu �9 - -~-

f [ ow 2 ~ a(au) W(u,x, t ) - A u . - ~ u (u,x, t )+ ~uAu. ~ a u

[ 1 02r (u, x; An)] or (u,x; Au) + ~AuAu �9 ~u0u x r Au) - A u . ~

The right hand side is expanded, and noting the following results

r Au)d(Au) -- 1, x;

Aur x; Au)a(Au) -- <Au>,

AuAur x; Au)d(Au) -- <AuAu>,

f o .<An> a u - ~ ~r (u,x; Au)a(Au) -- ~

zXuAu. ~162 (u x; Aula(~u)- OuOu. <AuzXu> OuOu ' '

we obtain

+ u - - - ~ - = 0u " 2At " 0u At 0u .... 2At

where the arguments of W and terms of O(At) and higher have been omitted. In the limit of At --, 0, and using the results (34.E1 - E2), we derive 41

OW OW O { k T m _ l . ~ . m_ l . 0 W 1 K x ) W } (34.44) Ot + u " 0x = Out" oqu + m - (~ 'u+ .

41Note t h a t none of the d y n a m i c a l tensors ( inert ia , fr ict ional and stiffness tensors) are allowed to depend on u.

174 4. Const i tu t ive Equat ions Derived From Microstructures

This diffusion equation is the Fokker-Planck equation governing the evolution of the probability distribution function in phase space; it is a generalisation of the classical Liouville equation for a dynamic system with Brownian excitation.

3~. 6 Smoluchowski Equation

In all models of dilute polymer solutions, the inertia tensor is considered to be negligible, i.e., the time scale of interest is considerably greater than the inertial time scale. In this limit, the velocity space can be integrated out, leaving the distribution function as a function of the configuration space only. Whence, the Fokker-Planck equation reduces to a form called the Smoluchowski equation. The passage to this equation can be derived by a method similar to that discussed in the previous section. 42 However, the limit m --. 0 is a singular limit, as evidenced from (34.44), and it would be safer to derive the Smoluchowski equation by considering a singular stochastic perturbation scheme. 43 The need for a rigorous treatment of the singular nature of the problem is even stronger when the dynamic tensors (inertia, frictional and stiffness tensors) are allowed to depend on x. 44

Now, if we define

m = e2M, A = M - 1 . ~ , B = M - 1 . Kx,

F (b) -- e - l c z ( t ) , C - M - I ( k T r 1/2,

then the Langevin equation (34.44) can be written as

~: = e- ly,

:9 = - e - U A y -- e - l B + • - 2 C z ( t ) , (34.45)

where y is the 'normalised' velocity and z is the 'normalised' Brownian force. From the statistics of the Brownian force,

(z(t)) = o, (z(t + T)z(t)) = 2 d , 0 - ) 1 . (34.46)

Identifying y with u in (34.44) yields the following phase space Fokker-Planck equation for the process {x, y}:

OW Ot

OW 0 = -e- 'yi-~x ~ + -~r [(e-2Aijyj + e-lBi) W]

02W + e - 2 C i k G k 0yi0y j . (34.47)

42CHANDRASEKHAR, S., Rev. Mod. Phys., 15, 1-89 (1943). 43 A mathematical discussion of singularly perturbed stochastic differential equations is provided

in BLANKENSHIP, G. and PAPANICOLAOU, G.C., SIAM J. Appl. Math., 34,437-451 (1978). 44The following is patterned after unpublished lecture notes of ATKINSON, J.D. and PHAN-

THIEN, N., University of Sydney, 1978.


We wish to look for a perturbation solution

W(x, y, t) -- Wo(x, t) + eWl(X,y,t) + e2W2(x,'y, t) + O(c3), (34.48)

and in particular, we wish to find the equation governing the evolution of r - W0, in the limit of c --~ 0 (or m --. 0).

We find it easier to deal with the adjoint of (34.47), which is sometimes called the backward Fokker-Planck equation:

OW 02W OW -1 cOW (s Jr s ~ -F- s . (34.49) Ot - - e Y~ ~ - Oy~ Oyj

Substituting the perturbation solution (34.48) into (34.49) and equating like terms, we find

02Wo OWo s - C~kCjk OyiOyj - Aijyj Oyi = O,

OWo OWo s W1 = -Yi Oxi + B~ Oyi = s176

0W1 0W1 0W0 _ / ~ 2 W 1 ~_ 1 ~ 1 W 2 - - --Yi OXi + Bi Oyi +

OWo 0t '

(34.50)

with the operators El, ~2 defined as indicated. The general solution to (34.50)1 is clearly W0(x, t), and when this is substi tuted

into (34.50)2, we find that

0w0 W~ (x, y, t) = 0(x, t) + A~lyj

Ox~ (34.51)

where 0(x, t) is arbitrary. Next, the condition for solvability of (34.50)3 is tha t

/22W1 + OWocot should be orthogonal to W*, where W* is any solution of the adjoint

equat ion/2[W* - -0 , which is

02W * 0 s - C~kCjk Oy~Oyj + - ~ (AiiyjW*) = 0. (34.52)

P r o b l e m 3 4 . F Look for a solution to (34.52) of the form

(1 ) W* = 0* (x, t) exp -~Di jy~yj

and show tha t D -- ( C - c T ) -1 . A. From this solution, show that

f y~W* - 0 dxdy

and

/ yiyj W* dxdy -- / D~ I W* dxdy.

(34.F1)

(34.F2)

(34.F3)


This result can also be derived without knowing the form of the solution by multiplying (34.52) by YiYj and integrating over y and x, using integration by parts, and applying the divergence theorem whenever needed.

We note that

OWo OWl OWl OWo L2w1 + - ~ - -y~ 0x( + B, 0W + &

O0 O ( OWo) OWo = -Yi'~x ~ - YiY~~ Af~ Ox i + A~ 1Bj Oxi +

From the condition for solvability of (34.50)3, viz.,

OWo 0t

cOWo f o and the results from Problem 34.F, ( 3 4 . F 2 - F3), we find that

f { OWo OWo ot " + A~ i s j o ~

o (A;:OWo "----- - D: x i cOxj ) } W'dxdy - 0

or OWo OWo

+ A~IBj Oxj o (A;:OWo

"--'-- -- D~I"~x i Ox}' ) -- O.

The adjoint of this is the governing equation that we seek:

cOWo 0 [Af~ 0 ] 0 (A~IBjWo). (34.53)

Using the definition of A, B, C and D and taking the limit where m --, 0, we find that W0 obeys the Smoluchowski equation:

Ot = Oxi kTr ~ + r , (34.54)

where we have re-labelled W0 by r the probability distribution function in configuration space.

P r o b l e m 34.G Starting from the Langevin equation in the configuration space, in the limit of

m --~ 0, :~ __ __~-1 . i x -[- ~ - 1 . F(b)(t), (34.G1)

show that

and

( A x ) 1 (34.G2) At = - ~ - "Kx

(AxAx) 2At

= kT(,- 1 + O(At). (34.G3)

The Smulochowski diffusion equation in the configuration space can therefore be written in the form

c3r lim 0 [ ( A x A x ) 0 r <Ax> ] 0t - ~t~0 ~ " 2At cox At r ' (34.G4)

which is physically more appealing, since the relations between the diffusivity and the drift velocity to the averaged displacement and its correlation are made explicit:

(AxAx> <Ax> D - lim v = lim . (34.G5)

At-,o 2At ' ~t--,o At

These results should be compared with (34.38) and the Fokker-Planck equation.

3~. 7 S m o o t h e d - O u t B r o w n i a n Force

In the limit of m -~ 0, the Langevin equation is reduced to (34.G1). If the right hand side of this equation is regarded as a deterministic quantity, and the Smolu- chowski equation (34.54) is compared with the classical Liouville theorem of conservation of probability,

0r 0 -~- § ~ - ( • 1 6 2 - 0, (34.55)

then we obtain

:~ __ __~- 1 . Kx -- kT(,- 1 01nr

0x (34.56)

Thus, in the limit of no inertia, we can take the Brownian force as being given by

F (b) (t) - - k T 01n r Ox ' (34

called the smoothed-out Brownian force, ana apply the Liouville theorem to (34.G1) to generate the correct diffusion equation for the probability function in configuration space. This practice has been adopted in most text books dealing with the kinetic theories of polymer solutions. 4s

3 ~ . 8 T h e S t r e s s T e n s o r

There are several ways to derive the expression for the stress tensor contributed by the polymer chains in a dilute solution. One is the probabilistic approach by BIRD et al. 46 In essence, in this approach, the number of polymer chains straddling a surface and the net force acting on that surface by the chains are calculated.

45BIRD, R.B., CURTISS, C.F., ARMSTRONG, R.C., and HASSAGER, O., Dynamics of Poly- meric Liquids: Vol. H. Kinetic Theory, John Wiley & Sons, New York, 2nd Ed., 1987; TANNER, R.I, Engineering Rheology, 2rid Ed., Oxford, New York, 1988; LARSON, R.G., Constitutive Equa- tions for Polymer Melts and Solutions, Butterworth Publishers, Boston, 1988; YAMAKAWA, H., 1971, Modern Theory of Polymer Solutions, Harper & Row, New York, 1971.

46BIRD, R.B., CURTISS, C.F., ARMSTRONG, R.C., and HASSAGER, O., Dynamics of Poly- meric Liquids: Vol. II. Kinetic Theory, John Wiley & Sons, New York, 2rid Ed., 1987, w


F 1 Fi ~ r i ~ F 3

RI

h F1

F 3

,/~rN+ 1

R i 7

r i+ l

~ F i

FIGURE 34.3. Connector forces in a Rouse chain.

The force per unit area can be related to the stress tensor. Another approach is to calculate the free energy of the chain from its entropy, and the rate of work done can be related to the dissipation due to the presence of the chains from which the expression for the stress tensor can be derived. 47 In this section, the expression for the stress tensor is derived by using a mechanistic approach treat ing the chain as a continuum.

Consider a model for a polymer chain, for example, the bead-spring chain shown in Figure 34.3. The tension in the i - t h Kuhn segment is denoted by f~. If the chain is Gaussian, then

f{ -- H~R~, (no sum), (34.58)

where R~ is the end-to-end vector, and H~ is the stiffness of the segment i, given in (34.25) - the bead-spring model in this case is said to be the Rouse model. Using the approach of LANDAU and LIFSHITZ, ts and BATCHELOR, t9 we regard the fluid as an effective continuum made up of a homogeneous suspension of polymer chains, also regarded as a continuum. Then the effective stress in the fluid is simply the volume-averaged stress:

(T) - ~ TdV---- ~ TdV-t-~ TdV,

where T is the total stress, V is a representative volume containing several chains, and is made up of a solvent volume V, and a polymer volume ~ Vp. In the solvent volume, the stress is simply the solvent stress, and we have

1 1 1 ~ /v. T(8)dV -- ~ /v T(S)dV - ~ ~ /vp T(8)dV

47LARSON, R.G., Constitutive Equations for Polymer Melts and Solutions, Butterworth Pub- lishers, Boston, 1988, w

4SLANDAU, L.D., and LIFSHITZ, E.M., Theory of Elasticity, 2rid Ed, Pergamon Press, New York, 1970.

49BATCHELOR, G.K., J. Fluid Mech., 44, 545-570 (1970).


With a Newtonian solvent, we obtain

1V/v T ( S ) d V - -P I 1 + 2~8D, (34.59)

where Pl is the hydrostatic pressure, v18 the solvent viscosity, and D the strain rate tensor. The remaining terms are given by

1 ~ ~ T(8)dV = 1 / { -p l + , . ( v u + vu )} dv P yp

'~ [ nu) dS. (34.60) = - p 2 1 + ~ ~ (un+ p

Since the chain is modelled as a series of discrete beads, where the interaction with the solvent and other segments takes place, the surface of the chain p consists of the surfaces of the beads, s~ On the surface of a bead, the velocity is regarded as uniform, and can be taken out of the integral. Thus

~p un dS - u ~sp n dS - O

by an application of the divergence theorem. The contribution from (34.60) is therefore only an isotropic stress, which can be lumped into the hydrostatic pressure.

Next, if we consider the chain as a continuum as well, then from the force equilibrium we must have V - T : 0 in the chain, and thus ~ k = O(T~jxk)/Oxj. The volume integral can be converted into a surface integral, and the contribution to the effective stress from the polymer chains is given by

T = 7 Z x T - n dS - - v x T - n dS, (34.61) p P

where Sp is the surface of a representative chain in V, T- n is the traction arising in the chain due to the interaction with the flow, and v is the number density of the chain (number of chains per unit volume); the passage to the second equality is permissible because of the homogeneity assumption which allows us to just consider one generic chain. We can now replace the integral in (34.61) by a sum of integrals over the beads:

~ x T - n d S - - ~ ~ x T - n d S . i ead i

On bead i, x can be replaced by r~ and taken outside the integral, and the remaining integral of the traction on the surface of bead i is therefore the drag force, which the bead i exerts on the solvent, and which is proportional to the relative velocity of the bead to that of the solvent:

= - r u,).

5 ~ t h e c o n n e c t o r s a re e n t i r e l y f i c t i t ious , t h e y a re a l l o w e d to c ross one a n o t h e r ; t h e c h a i n

m o d e l is s o m e t i m e s ca l l ed t h e phantom chain for t h i s r e a son .

In the absence of inertia, this force is equal to the connector forces plus the Brown- ian forces acting on the beads:

N+I

f x T . d S - -rlfl+r2(fl +rN§ �9 __ . . . - - L - i t i

S~, i--1 N N+I

: Z ,f, Z - - a-iJ: i �9

i=1 i=1

Next, the ensemble average with respect to the distribution function of {R~} is taken. The contribution from the Brownian forces is only an isotropic stress, as can be shown either by using the expression for the smoothed-out Brownian forces, or by integrating the Langevin equations directly. We demonstrate this by using the smoothed-out Brownian forces here, leaving the alternative proof as an exercise. Now, from the expression for the smoothed-out Brownian force,

N+I

Z /riF~b) / i--1

1 / 01n r .. "drN+l = - k T ri v,~"i i - -1

= - k T r i - - d r l . . , drN+l -- k T ( N + 1)1 i= l Ori '

by an integration by parts. Thus, apart from an unimportant isotropic stress that has been absorbed into

the hydrostatic pressure, the stress contributed by the polymer chains is

N N

r (p) = v Z <R,f~) - v ~ <H~R,R,>, (34.62) i =1 i = l

which is sometimes known as the Kramers form for the polymer-contributed stress. 51 Note that the polymer-contributed stress is always symmetric, since the connector

force is always parallel to the end-to-end vector of the segment. The total stress tensor in a dilute polymer solution is therefore

N

<T> = - p l + 2waD + u Z <H{R,R,>. (34.63) /=1

3~.9 E las t i c D u m b b e l l Mode l

Langevin Equation

The simplest model designed to capture the slowest, and in many ways, the most important relaxation mode of a polymer chain, is the elastic dumbbell model, first proposed by KtVrIN, 52 as depicted in Figure 34.2b. Here we are only interested in

, , , ,

51BIRD, R.B., CURTISS, C.F., ARMSTRONG, R.C., and HASSAGER, O., D y n a m i c s o f P o l y -

m e r i c L i q u i d s : Vol. i I . K i n e t i c T h e o r y , John Wiley & Sons, New York, 2rid Ed., 1987, w 52KUHN, W., K o l l o i d - Z e i t . , 68, 2-15 (1934).


the end-to-end vector of the polymer chain, and all the interaction with the solvent and the chain itself is localised into two beads, located at the ends of the chain at r l and r2. Each bead is associated with a frictional factor r and a negligible mass m. We will assume a Gaussian chain, with a constant spring stiffness given by

3kT H - - Nb 2'

where N is the number of effective Kuhn segments in the dumbbell, each of which has an extended length b. Furthermore, the frictional coefficient ( = 67rrlsa is assumed to be a constant, where r/s is the solvent viscosity and a represents the radius of the beads. The model is also called the linear elastic dumbbell model to emphasize the linear force law being used.

From Newton's second law, we have:

mi~l - - ( ( U l - 1"1) ~- H(r2 - r l ) q- F~b)(t) (34.64)

and mi~2 -- r (u2 -- f'2) + H ( r l - r2) + F (b) (t), (34.65)

where ui = u(ri) is the fluid velocity evaluated at the location of the bead i, and

F~ b) (t) is the Brownian force acting on bead i. The fluctuation-dissipation theorem

(34.35) can be used to relate the strength of the Brownian forces F~ b) to the mobility of the beads:

(F~b)(t)) -- 0, (F~b)(t +T)F~.b'(t)) -- 2kT(6(T)6 , j l . (34.66)

Let us now define the centre of gravity and the end-to-end vector of the dumbbell through

1 (34.67) R (c) - - ~ ( r2 + r l ) , R - - r2 - r l ,

and expand the fluid velocity about the centre of gravity of the dumbbell; for example, we have

1 U 1 = U (c) I R - V u (~) + R R " V V u (~) + o ( I a ] 3)

- 2 8 '

where the superscript (c) denotes an evaluation at the centre of mass, and higher order terms have been neglected. 5a Then, we find

and m R : r ( L R - I~) - 2 H R + F (b) (t),

where L - (Vu (c)) T is the velocity gradient and

1 (F~b) + F(b)) , F(b)= F~b)_ F~b) F(b,c) - -

(34.69)

(34.70)

53Neglecting these te rms is a reasonable assumption, provided tha t the velocity field does not vary rapidly over a length scale of O(R) .


are the Brownian forces acting on the centre of mass, and the end-to-end vector, respectively. From the fluctuation-dissipation theorem (34.66), we obtain

( F ( b ' c ) ( t ) / - O , ( F ( b ' c ) ( t + T ) F ( b ' c ) ( t ) l - - k T ~ 5 ( T ) l , (34.71)

( F ( b ) ( t ) ) - 0, ( F ( b ) ( t + T ) F ( b ' ( t ) ) = 4 k T ~ i ( T ) i . (34.72)

When m is negligible, the Langevin equations for the system are

1 1~ (c) -- u (c) + ~ R R " V V u (c) + (:-lF(b,c)(t), (34.73)

1~ - - L R - 2H~- 1R + e- iF(b)( t ) . (34.74)

Average Stretching

We now assume tha t the imposed flow is homogeneous, i.e., L is at most a function of time. In this case, on the average the centre of the dumbbell is convected along as if it were a fluid particle: s4

(34.75)

The average end-to-end vector evolves in time according to

d ~-~ (R) = L ( R ) - 2Uff-i (R) , (34.76)

flow stretching restoring force

which consists of a flow-induced stretching plus a restoring mechanism due to the connector spring force. Indeed this equation has been used as a basis for delineating between strong and weak flows. 55 Strong flows are those in which the flow-induced

d deformation overcomes the restoring force allowing ~-7(1~) to grow exponentially in

time; otherwise the flows are weak. The flow is therefore strong if

1 EKg(L) > 2"~' (34.77)

where Eig(L) is the maximum eigenvalue of L, and )~ is the Rouse relaxation time:

~Nb 2 A - 4 H = 12kT" (34.78)

A sightly different flow classification is discussed in TANNER. 56 The connection with the flow classification discussed in w should be noted. There, the fluid is considered as a continuum without any microstructural feature; the flow is regarded

54The ensemble average opera t ion commutes wi th the t ime derivative. 55TANNER, R.I., Engineering Rheology, 2nd Ed., Oxford, New York, 1988; Trans. Soc. Rheol.,

19, 557-582 (1975). HINCH, E.J., Coll. Int. CNRS No 233 Polym. Lubrication (1975). 56TANNER, R.I., Engineering Rheology, 2nd Ed., Oxford, New York, 1988.


as being strong if any of the eigenvalues of L has a positive real part. Two adjacent fluid particles will then be pulled apart at an exponential rate. Here, the microstructure as represented by the dumbbell has a restoring mechanism that allows it to resist the deformation. For the flow to be strong, at least the real part of one of the eigenvalues of L must exceed 1/2)~, if this restoring mechanism is to be overcome by the imposed deformation.

P r o b l e m 34.H Show that the solution to (34.76) is given by

fo t 1 t,)]~_lF(b)(t, ) dr' (34.H1) R(t) -- e- t /2~Fo(t)Ro + exp [ (L- 1)(t

where Ro is the initial value of R, ,k is the Rouse relaxation time, and Fo(t) is the displacement gradient satisfying (cf.(1.17))

F 0 ( t ) - LF0(t). (34.H2)

Thus, show that (R(t)) -- e-t/2~Fo(t)Ro, (34.H3)

and ] ( R ( t ) ) l 2 - e - t / ~ R o �9 C o ( t ) - R 0 , (34.H4)

where Co(t) - FT(t)Fo(t) is the Cauchy-Green tensor. Show that if Ro is chosen randomly with a uniform distribution, then a further average with respect to this distribution will yield

( ~ - ~ ) 2 - l e - t / ~ t r C~ 3 , (34.H5)

where h(t) is the average separation at time t, and h0 is the average separation initially. TANNER 57 called this parameter the dimensionless stretching parameter Sg, since this can be a measure of the microstructural distortion.

Constitutive Equation

Let us now denote the distribution function of the process {R (c), R} by (I)(R (c), R, t). Then the diffusion equation for the process, from the Smoluchowski equation (34.54), is

0r _ 0 [ k T 0....~r u(C)~] Ot -- OR( c ) 2(: OR( c )

The relevant boundary condition on (I) is that

( I ) - 0, when either ]R (c) ] or IRI--* oo. I I

(34.79)

(34.80)

57TANNER, R.I., Trans. Soc. Rheol., 19, 557-582 (1975).

Since R (c) is independent of R, we can write 5s

(I)(R (c), R, t) -- r (c), t ) r (R, t). (34.81)

We first note that

because of the normalisation conditions

/ r l, f C(R(~),t)dR(~) - 1.

Furthermore, from the divergence theorem, we have

fv R 0 . [ kT O@ _ u(C)@] dR(C) _ f B [ kT O@ (o) 0R(c) 2~ 0R(c) R(o) 2~ 0R(c)

@(R(C), R, t )dR (c) ,

-- u (c) (I) 1 dS,

where VR(c ) is a volume of radius [R (c) ] --, oo, and SR(c) is its bounding surface. From the no-flux boundary condition at JR(C) I ~ e~,we conclude that

/ 0 . [ k T 0@ _u(C)(I)]dR(C)_0" 0R(c) 2r 0R(c)

In a similar manner,

0 < o . (L.- o] d R - 0 .

Thus, by integrating (34.79) with respect to either R or R (c), and using the preceding results, the diffusion equations for r (c), t) and r can be obtained a s

and

0r O [ k T 0r u(C)r (34.82) = 0R( )

'0t = OR r 0 R - L R - R r . (34.83)

The diifusivities in both processes, R (c) and R, are extremely small, of the order kT/( - Nb2/12A, which is about 10-1~ if N1/2b "~ 10-6m and )~ N 10-3s. However, the relevant length scale in the process R (c) associated with (34.82) is that over which u (c) varies considerably (a macroscopic length scale), and therefore the process R (c) diffuses extremely slowly. For all practical purposes, R (c) can therefore be regarded as being convected along with the fluid. , ,

P r o b l e m 34.I

58The re are theor ie s t h a t do no t a s s u m e this. However , these are not cons i s t en t , in t h a t the i n t e r - d e p e n d e n c e b e t w e e n R (c) a n d R impl ies t h a t the B r o w n i a n forces on the b e a d s m u s t be cor re la ted .

By applying the divergence theorem as needed, and using the far-field boundary condition (34.80), show that

and

f.... 02r ~ t ~ t O R - OR d R = 2 x 1

/ R R s . (ARO) dR - - / (A . R R T R R . AT) r

Thus, show that

OR

4 k T ( L R - 2 - ~ R ) 0] dR -- -'~--1

+L-<RR> + <RR). L r 4H <RR) - --~-- .

Since the total stress in the fluid is given by

(T> - - P 1 + 2~/sD + ~-(P),

where the polymer-contributed stress

3 v k T ~'(P)- uH<RR)- N b 2 <RR>

(34.I1)

(34.I2)

(34.I3)

(34.84)

L - < R R > - <RR>-L T } - 1Nb21 .

Thus the polymer-contributed stress evolves in time according to

.(P)+A{d.(p)-L..(v)-.(V).L T} = G o l ,

where the modulus Go is given by

Go = ukT.

(34.85)

(34.86)

A-- ~ - - ~Nb2 4 H 12kT '

we find that

d <RR>+A ~-~<RR>-

is proportional to <RR), we wish to look at the evolution of this quantity. One way to accomplish this is to pre-multiply the diffusion equation (34.83) by R R and integrate it over the R-space , which was done in Problem I. Using the result (34.13) and the fact that time derivative commutes with ensemble average, we find that

d <RR> = 4 k T --dr - ~ I + L-<RR> + (RR)-L T _ -~4H <RR>.

Since the Rouse time constant has been defined as


It is customary to re-define the polymer contributed stress as

r (p) = (701 + S (p) ,

and absorb the isotropic stress into the pressure term. Then

S(P)+)~{ds(P)-L.S(P)-S(P).L T} =2ripD,

where rip is the polymer-contributed viscosity

(34.87)

(34.88)

rip = GoA- lvcNb2. (34.89) Iz

The terms inside the curly brackets in the constitutive equation (34.88) is said to be the upper convected derivative, one of the many convected derivatives introduced by OLDROYD. 59 We denote this derivative by A/At, i.e.,

----~S = d LT At ~ S - L - S - S - .

The constitutive equation (34.88) is called the Upper Convective Maxwell model (UCM), despite the fact that MAXWELL 6~ only introduced the linear version of this in his influential work on the kinetic theory of gases.

When both the polymer- and the solvent-contributed stresses are combined into one, i.e.,

S -- 2risD + S (p), (34.90)

and noting that

we find

A - - -1 = - 2 D , At

AS ( A D ) (34.91) S-~-)~l- ~ - - - 2ri S + , ~ 2 - ~ ,

where ri = Us + rip is the total viscosity, )~1 = A is the Rouse relaxation time, and )~2 -- Aris/ri is the retardation time. The constitutive equation (34.90), or (34.91) is the Oldroyd-B model. 61

From the definition of the relative strain tensor (1.56) and the fact that F(t) - L(t)F(t) , we find that

a - C;I( )L T dt

Now, it is fairly easy to show that the integral of (34.85) is

f = Go = C01 + OO

(34.92)

59OLDROYD, J.G., Proc. Roy. Soc. Loud., A200, 523-541 (1950); A245, 278-297 (1958). 6~ J.C., Phil. Trans. Roy. Soc. Loud., A157, 49-88 (1867). 6tOLDROYD, J.G., 1950, op. cit.


This integral form of the UCM model is known as the Lodge equation; 62 it is a special case of the K-BKZ model.

From the equation of change of (RR), it is found that

The form

r ( P } - / F ( ~ } R \ - vr A ( R R ) + l / k T X (34.93) \ / 4 A t

r(p) = v~ A (RR> (34.94) 4 At

is called the Giesekus form for the stress tensor, 63 ignoring the unimportant isotropic term. It is clearly valid for elastic dumbbell microstructures; for other type of microstructures, such as the rigid dumbbell model, there may be other constraints that should be considered carefully.

Main Features of the Oldroyd-B Model

The Oldroyd-B model qualitatively describes many features of the so-called Boger fluids. 64 In a steady state simple shear flow, this constitutive equation predicts a constant viscosity, a first normal stress difference which is quadratic in the shear rate, and a zero second normal stress difference. In an unsteady state shear flow, the stresses increase monotonically in time, with no overshoot that is usually observed with some dilute polymer solutions. In an elongational flow, the elongational viscosity becomes infinite at a finite elongational rate of 1/2A - see the following Problems.

P r o b l e m 3 4 . J In an oscillatory shear flow where the shear rate is

--- ~0 e iwt

show that the UCM model (34.88) predicts the steady state response:

where

S ~ ) - Soe i~t, S~ p) - Noe 2i~t, other stresses Si (p) - 0, (34.J1)

~pTo 2~pATo 2 (34. J2) S o - l + i A w ' No-- ( l + 2 i A w ) ( l + i A w ) '

regardless of the magnitude of 7o" From this, deduce that the dynamic properties of the Oldroyd-B fluid are given by

Dynamic and storage viscosities:

~' - r h + ~?P ~7" = ~?P)~w (34.J3) 1 + ~2r ' 1 + )~2022 '

62LODGE, A.S., Trans. Faraday Soc., 52, 120-130 (1956); Elastic Liquids, Academic Press, New York, 1964.

63GIESEKUS, H., Rheol. Acta, 2, 50-62 (1962). 64Dilute solutions of polymers in highly viscous solvents; see BOGER, D.V. and BINNINGTON,

R., Trans. Soc. Rheol., 21,515-534 (1977).

Storage and loss moduli:

GoAw G' = G~ G"= ~Tsw + (34.J4) 1 + A2w 2' 1 + A2w 2"

, , ,

P r o b l e m 34 .K Consider the response of the UCM model in the start-up shear flow from a zero

stress state where L12 = ~/(constant), other components of L are zero. Show that the non-zero stresses are given by

S~)(t) = ~7p7 [1 - e-t/x], (34.K1)

S~Pl ) (t) - 27}vA~ 2 [1 - e -t/~] - 2%~/2te -t/x. (34.K2)

Deduce tha t the viscometric functions for the Oldroyd-B model are given by

= ~/v + ~s, (34.K3)

Nl=2VlpA~ 2, N 2 - - 0 . (34.K4)

P r o b l e m 34.L Consider the response of the UCM model in the start-up elongational flow from

a zero stress state where

Show tha t

1~ 1 ) L = d i a g e ' - 2 ' - 2 e "

[1- S[~ ) (t) - i - 2A~ (34.L1)

[1 _ (34.L2)

Conclude that if either A~ < - 1 or A~ > 1/2, then at least one component of the stress increases unboundedly in time. This ks a reflection of the linear spring adopted in the model: it allows the end-to-end vector of the dumbbell to increase without limit in a strong flow. For - 1 < A~ < 1/2, show that the elongational viscosity is given by

The Trouton limit of ~E --' 3~0 is recovered when ~ -~ 0.

Although the UCM model ks derived for homogeneous flows, it ks also used in non-homogeneous flows. The implicit assumption is that the length scale over which the velocity field varies rapidly ks considerably greater than any relevant microscale, so tha t the microstructure only sees a localksed homogeneous flow. Note tha t when V V u (c) is non-zero, the centre of gravity of the dumbbell no longer moves just like


r

10 3

10 2

101

10 0

10 -1

I ! �9

Solution B2 j / N 1

i o

Io ,4

I t

0

] m m - - - - w ~ n

Oldroyd-B _

! I 1 I 10 -2 10-1 10 o 101 10 2 10 3

,;,,, ~ s - 1 )

FIGURE 34.4. N1 versus shear rate and 2G' versus frequency for a Boger fluid. The dashed line is the prediction of the Oldroyd-B model; the solid line is the predidtion of the Maxwell model with two relaxation times.

a fluid particle: there is a net migration of the dumbbell particles from regions of low to regions of high shear rates.

From the previous Problems, it is clear that the predictions of the linear elastic dumbbell model are not adequate. In an oscillatory shear flow, the shear stress has been shown to be proportional to the amplitude of the shear rate (or the shear strain), irrespective of its magnitude. This is unrealistic: in practice this proportionality is only found when the shear strain is small. 65 The frequency response of the model is not adequate; but this is due to incorporating only one relaxation time in the model.

In a steady shear flow, the model predicts a constant viscosity, a quadratic first normal stress difference in the shear rate, and a zero second normal stress difference. The Boger fluids show little shear thinning over a large range of shear rates, but this is no doubt due to the high solvent viscosity that completely masks the contribution from the polymer viscosity; any amount 9f shear-thinning from the polymer contribution would not make any visible impact on the total viscosity of the fluid. In general, dilute polymer solutions usually show some degree of shear thinning, 66 when the dimensionless shear rate A~ exceeds unity; the amount of shear thinning is typically of the order of 25% over a decade in the shear rate, from

SSIn fact, in the limit of small strains, the response of all models must reduce to that of linear viscoelasticity.

66Some polymer solutions show both shear-thinning and shear-thickening behaviour; see, for example, BIANCHI, U. and PETERLIN, A., J. Polym. Sc~. A-~, 6, 1011 (1968).


,k~ ~ 1. The first normal stress difference in dilute polymeric liquids is observed to be quadratic at shear rates below a critical value, as it must be for any simple fluid. This critical dimensionless shear rate could be of O(1) for most dilute polymer solutions, but it could be as high as O(10) for Boger fluids, as shown in Figure 34.4 for the B2 solution. 67 The dashed lines in this figure represent the prediction of the Oldroyd-B model whereas the solid line is the prediction of the storage modulus for a two-relaxation-time UCM model.

A carefully prepared test Boger fluid, designated as fluid M1, and distributed by NGUYEN and SRIDHAR 68 to several rheological laboratories in a "round robin" a t tempt to compare independent determinations of the rheological properties of the fluid, has yielded important data. Of these, oscillatory shear data have been com- piled by Te NIJENHUIS; 69 a good agreement with the dynamic viscosity prediction has been found by using the Oldroyd-B model with a relaxation time ~1 -- 0.38 s and a retardation time ,k2 -- 0.24 s (see (34.91)). However, the agreement with the storage modulus is poor at high freqencies (w > 101 rad/s). The results indicate that M1 fluid is indeed more complicated than the Oldroyd-B model: in the linear viscoelasticity regime at least two relaxation times are required to describe dynamic data well. In fact, a model with three Maxwell elements in parallel has been proposed by BOGER and MACKAY 7~ as a suitable constitutive equation for the M1 fluid; however, the model does not predict the first normal stress data well.

The fluid is also slightly shear thinning above a shear rate of O(101) s-1. Its normal stress difference ratio N 2 / N 1 lies in the range from -0 .1 to -0 .2 , 71 although KEENTOK et al. 72 have reported a near zero second normal stress difference.

Thus, although there are problems with the Oldroyd-B model, especially in elongational flows, it provides a reasonable description for the Boger fluids, at least in ftow regimes that do not severely distort the microstructure. This, plus the fact that the Oldroyd-B model has a good physical basis in dilute polymer solutions, has made it a popular choice as a computational model in the last two decades. However, due to the prediction of unbounded stresses in strong flows, the model tends to be numerically unstable in flows in which there exist strong flow regions.

Nonlinear Dumbbell Models

There are several mechanisms, not present in the linear elastic dumbbell model, that are responsible for shear-thinning: finite extensibility, hydrodynamic interaction, configuration-dependent friction coefficient, excluded volume effects, internal viscosity. Some of these are discussed below.

67MACKAY, M.E. and BOGER, D.V., J. Non-Newt. Fluid Mech., 22, 235-243 (1987). 68NGUYEN, D.A. and SRIDHAR, T., J. Non-Newt. Fluid Mech., 35, 93-104 (1990). 69Te NIJENHUIS, K., J. Non-Newt. Fluid Mech., 35, 169-177 (1990) 7~ D.V. and MACKAY, M.E., J. Non-Newt. Fluid Mech., 41, 133-150 (1991). 71CHIRINOS, M.L., CRAIN, P., LODGE, A.S., SCHRAG, J.L. and YARITZ, J., J. Non-Newt.

Fluid Mech., 35, 105-119 (1990). 72KEENTOK, M., GEORGES(~U, A.G., SHERWOOD, A.A. and TANNER, R.I., J. Non-Newt.

Fluid Mech., 6, 303-324 (1980).

F i n i t e E x t e n s i b i l i t y

At high molecular extension, the number of configurations accessible to the polymer chain is greatly reduced, implying a high tension in the chain. A more exact theory shows tha t the correct force law is the inverse Langevin function (34.26). This nonlinear force law is unduly complex, in view of the approximate nature of the dumbbell model, and the preferred force-law is the Warner spring (34.27), 73 rewritten here for the dumbbell model:

H0 (34.95) F(S) - H ( R 2 ) R ' H(R2) = 1 - ( R / L ) 2'

where F (s) is the connector force, Ho = 3 k T / N b 2 is the Gaussian stiffness in the limit of small molecular extension, and L = N b is the maximum extension of the dumbbell. The additional parameter tha t enters into the fluid rheology is the ratio of the extended length L to the root-mean-square length at equilibrium (R2)~/2, or, conveniently L2/(R2>o = N .

The dumbbell model with the Warner force law is said to be the FENE model (Finitely Extendable Nonlinear Elastic). z4 The Langevin and the Smoluchowski equations retain their respective forms, i.e., (34.73)-(34.74), and (34.82)-(34.83), except that H is replaced by the Warner spring stiffness. The polymer-contributed stress tensor is now p(H(R2)RR>. Unfortunately, the non-linearity in H induces the non-closure problem usually encountered in many areas of statistical physics, and a closed-form constitutive equation is not possible unless an approximation is made. A well known one is due to PETERLIN, 75 who replaces (H(R2)> by H((R2>) to force a closure. This is also called the delta-function approximation adopted in many subsequent works. The error in the Peterlin approximation,

(H(R2)RR> ~ H(<R2>)(RR), (34.96)

is of the order

0H ((R2>)((R- (R>)RR> OR

which is expected to be small in a strong flow, where the distribution is sharply peaked; it is expected to be small in a weak flow also, where the microstructural deformation is negligible. Indeed some exact numerical solutions of the diffusion equation from WARNER, 76 FAN, 77 and LEE 7s support this. So, let us adopt this approximation and define the micro structural tensor by

cz--< RR N--~I. (34.97)

7aWARNER, H.R., Jr., Ind. Eng. Chem. Fund., 11,379-387 (1972). 74BIRD, R.B., CURTISS, C.F., ARMSTRONG, R.C., and HASSAGER, O., Dynamics of Poly-

meric Liquids: Vol. II. Kinetic Theory, John Wiley & Sons, New York, 2rid Ed., 1987, w 75PETERLIN, A., Makromol. Chem., 44, 338-346 (1961); KoUoid-Zeit., 182, 110-115 (1962). 76WARNER, H.R., Jr., Ind. Eng. Chem. Fund., 11, 379-387 (1972). 77FAN, X.-J., Rheology Research Center Report, No 91, University of Wisconsin, 1984. 78LEE, S.J., PhD Thesis, Department of Mechanical Engineering, University of Sydney, 1992.


Then, the polymer-contributed stress tensor is given by

, r(p ) _ 3 N G o or, (34.98) 1 - t r ct

where Go is the modulus already defined in (34.86). The time evolution of the microstructural tensor can be found by pre~multiplying the diffusion equation with R R and integrating over the configuration space. The Peterlin approximation is then applied to yield the following constitutive equation for the microstructural tensor:

AAc~ a 1 1. (34.99) At + 1 - - t r a = 3N

The constitutive equation therefore consists of a stress rule, i.e., (34.98), allowing the calculation of the stress tensor from a microstructural tensor, and an evolution equation for the latter quantity, i.e., (34.99). This constitutive equation predicts stress overshoot in the start-up of a simple shear flow, shear-thinning at steady state, and a finite elongational viscosity at high strain rates. The second normal stress difference in a simple shear flow is zero at all shear rates, however.

P r o b l e m 3 4 . M

Consider the response of the FENE model (34.98)-(34.99) in a simple shear flow. Show that the non-zero components of the microstructural tensor are given through

1 (34.M1) C~11 -- 2A~C~12 : 3N' l - t r c~

~ 2 2 1 c~33 (34.M2) 1 - t r c t 3 N 1 - t r

Show that o~12 is given by solving the cubic equation

1 )2 A~ a12 1 -{- ~ ~- 2A~C~12 -- 3N"

At low shear rates show that

O ~ l l - - O~22 - - O~33 - - 3(1 + N )

(34.M3)

+ O(A2~2), (34.M4)

+ O(AZ~a). (34.M5) ~12 -- 3(1 + N)

Thus show that N2 - 0 (at all shear rates), and the zero-shear-rate viscosity and first normal stress coefficient (r -- N1/~ 2) are given by

N N ~7o -- ~Ts + N + 1 GoA - ~78 + N + 17/~,, (34.M6)

2N2 A 2. (34.M7) r = (N + 1) 2 Go


Show that the asymptotic solution at high shear rates is given, to the order of leading terms, by

1 ( . ~ ) 1/3 O/ll - - 1 + ~-ff ( ) ~ ) - 2 / 3 , (34.M8)

1 ( . ~ ) 1/3 0/22 -- 0/33-- ~ ()~A/) -2/3 , (34.M9)

1 ) 1/3 O/12 ----.

12NA~,

leading to a viscosity and a first normal stress difference of

(34.M10)

(A~) -2/s , (34.Mll)

N1- 2Go (3-~~) 1/3 (,~,.~) 1/3. (34.M12)

P r o b l e m 3 4 . N Consider the steady state elongational response of the FENE model (34.98)-

(34.99). Show that

(1 - A) (1 - 2A~)(1 + A~) - N A ( 1 - A~), (34.N1)

where A - 1 - t r r ~ is the elongational rate, and the components of the microstructural tensor are given by

A A O ~ 1 1 : 3N(1 - 2A~A)' c~22 - c~33 - 3N(1 + A~A)" (34.N2)

At low elongational rates, show that

1 ( C ~ l l - - " 3(N + 1) 1 + 2N A~ + . . . ~ (34.N3)

N + I ]

O/22 = C~33 -- 1 (

3(N + 1 ) 1 N A ~ + . . . ' ~ (34.N4)

N + I ]

leading to a zero-strain rate elongational viscosity of

3N ~E,0 -- 3Vl8 + N + 1 zip -- 3r]~ (34.N5)

At high elongational rates, show that an asymptotic expansion in 1/A~ will lead to

1 C~11 - - " 1 6NA~ ~ " ' " (34.N6)

1(1 1 ) 9 N A 2 ~ 2 " ,

(34.N7)


leading to an elongational viscosity of

~TE -- 3~78 + 6Nrlp (1 \

1 ) 9NA~ t- . . . . (34.N8)

From the previous problems, it is clear that the FENE model is able to predict qualitatively many effects in dilute polymer solutions: shear-thinning, first normal stress difference, high elongational viscosity at high strain rates. Quantitative agreement is not good, however: the shear rate above which shear-thinning occurs is overpredicted, sometimes by an order of magnitude; the degree of shear-thinning in real dilute polymer solutions varies, unlike the universal shear-thinning of O('~ -2/3) as predicted by the FENE model; this is a clear indication that shear-thinning is due to a number of causes, not just finite extensibility. Despite these shortcomings, the FENE model should provide a reasonable starting constitutive model in numerical simulations of dilute polymeric fluids.

A variance of the FENE model is the Chilcott-Rallison model, 79 where the microstructural tensor A obeys

A A - ~ + u ( n ) ( A - 1) -- 0, (34.100)

where A is the relaxation time, H ( R ) -- 1/(1 - R 2 / N ) , R 2 -- tr A, and the stress rule is given by

_ c R r (p) ~rl, U( )A. (34.101)

Here, N is the ratio of the maximum extended length to the equilibrium length, and c is a measure of the volume fraction of the polymer. In a steady shearing flow, this constitutive relation predicts a constant viscosity and a first normal stress difference that is of O(A2~ 2) at low shear rates, and O(IA'~]) at high shear rates. The second normal stress difference is zero. In a steady elongational flow and at high elongational rates, the model predicts a finite elongational viscosity, UE "~ 2cN~ls.

H y d r o d y n a m i c I n t e r a c t i o n

Although the dumbbell model (and its multiple-relaxation counterpart, the Rouse model) is a nice, simple concept of a polymer chain, its qualitative predictions regarding the molecular weight dependence of the rheological properties are not correct. For example, since the Rouse relaxation time is proportional to (~ and N, both of which are proportional to the molecular weight M, we have the result A ~ M 2 for this free draining chain. Experimental data, however, can be fitted to a power-law A ~ M 3a, where the exponent ~ ~ 1/2 near the # temperature. A satisfactory explanation of this and other scaling laws has been proposed by KIRKWOOD and RISEMAN s~ based on hydrodynamic interaction. We now briefly consider the effects of hydrodynamic interaction in our dumbbell model.

We start with the same Langevin equations as in the linear elastic dumbbell case, with the exception that the fluid velocity is now given by the main flow plus a

79CHILCOTT, M.D. and RALLISON, J.M., J. Non-Newt. Fluid Mech., 29, 37-55 (1988). S~ J.G. and RISEMAN, J., J. Chem. Phys., 16, 565-573 (1948).


perturbation due to the presence of the other mass. For example, the fluid velocity at the point mass 1 is given by

Ul = Lr l + Vl, (34.102)

where L -- (Vu(C)) T is the velocity gradient tensor as before, and Vl is the perturbed velocity due to the presence of the second bead. This perturbed velocity can be approximated by

V1 -- ~ " F~ d), (34.103)

where n is the Oseen tensor (or Stokeslet)

1 ( R R ) (34.104) 12-- 8~78R I + ~ ,

and F (d) is the drag force acting on the solvent by the second bead:

F (d) -- - r (u2 + v2 - ~2) = -m~2 - H R + F (b).

Effectively, for the purpose of calculating the perturbed velocity at the first bead due to the second bead, the second bead is replaced by a point force. The Langevin equation becomes

mi~l -- r (ul + v l - ~1) + U(r2 - r l ) + F~ b) (34.105)

and mi~2 -- ~ (u2 + v2 - }2) + H ( r l - r2) + F (b) ,

which can be combined to yield

m (1 + r R(~) -- r (LR (~) - 1~ (~)) + (1 + ~[2) F (b'~) ,

(34.106)

(34.107)

where the dependence on the R(C)-mode has been integrated out, and

1 + (34.110) B - - I - ( 1 2 - - I - 8 ~ 8 R ~ .

The parameter h -- ~/87rv/8 is a length scale introduced by the hydrodynamic interaction. The equation of change for any dynamic quantity Q can be generated by pre-multiplying the Smoluchowski equation by Q and averaging it over the

0r 0 { 0 r 1 6 2 (34.109) Ot -- O R " 2 k T ~ - I B " " ~

m (1 - el l) R -- r ( L R - R ) - 2H (1 - ~fl) R + (1 - el l) F (b) . (34.108)

Here, R (c) is the centre of gravity, R is the end-to-end vector of the dumbbell, F (b'c) is the Brownian force acting on R(~)-mode, and F (b) is the Brownian force acting on R-mode. In the limit of vanishing bead inertia, the Smoluchowski equation in the configuration space can be written as:

configuration space, using integration by parts and applying the divergence theorem whenever needed. The result is

d OQ OQ

<o, ( 1111 +2kT OR~ IB~_~, .

Clearly, the non-closure problem has to be dealt with here in order to generate a closed-form constitutive equation. ZIMM st avoids this problem by replacing the Oseen tensor with its equilibrium-averaged value:

0 r 1 + - 1 - - - - - - 1 8rr/8 R " - ~ 0 67rr/s (R>0 3N1/2b "

FIXMAN s2 improves this approximation by using the equilibrium-averaged value of i t in the Smoluchowski equation to obtain a first-order approximation to the distribution function. This first-order distribution function is then used to average i t again. A second-order approximation to the distribution is solved for, and with it, the stress tensor to second-order accuracy. 0 T T I N G E R s3 also replaces i t with its pre~averaged value, but seeks to determine both <it> and the distribution function simultaneously, and therefore obtains a better approximate solution than FIXMAN.

If the average value of i t is used in the equation of change for Q - R R , we have

~H 4 k T (1 - r A <RR) + ---- (1 - r <RR) - - ~ At

or, in terms of the polymer-contributed stress,

A (1 - ~<it>)-i ~_~t.r(p ) _~_ T(p) _ G01, (34.112)

where A is the Rouse relaxation time and G0 -- v k T is the shear modulus. Thus

the 'effective relaxation time' is O ( I A ( 1 - ~ < i t ) ) - l l ) , which is greater than the ~ | | % .

\ 1 I 1

Rouse relaxation time. Furthermore, as the shear rate increases, R must necessarily increase leading to a relative reduction in the relaxation time. Thus the viscosity, which is proportional to the relaxation time, must decrease. Therefore hydrodynamic interaction is another source of shear-thinning.

If we do not use a pre-averaged value for it , then, by noting that

we find tha t

2 h ) 0 B R - 1 - - - ~ R, -B- -O,

OR

4 k T h

~R a

SlZIMM, B.H., J. Chem. Phys., 24, 269-278 (1956). S2FIXMAN, M., J. Chem. Phys., 45, 785-792 (1966); 45, 793-803 (1966). s3OTTINGER, H.C., J. Chem. Phys., 83, 6535-6536 (1985); 84, 4068-4073 (1986).


= r 1 - 1. (34.113)

The Peterlin approximation can be applied here, resulting in a constitutive equation for the polymer-contributed stress. In fact, there is no special difficulty in including a nonlinear force law if this approximation is adopted. PHAN-THIEN and TAN- NER s4 have found that the Peterlin approximation in the FENE dumbbell with hydrodynamic interaction results in a shear-thinning model, where the onset of shear-thinning occurs at a dimensionless shear rate A~ - O(1). Hydrodynamic interaction does not have a large effect in a strong flow, where the microstructure is considerably stretched.

O t h e r I m p o r t a n t Effects

There are other important effects that have not been discussed so far: excluded volume, variable friction coefficient, internal viscosity. A brief review follows.

Excluded volume is the term used to denote the fact the beads cannot occupy the same location in space. FIXMAN s5 has shown that the excluded volume effects can be modelled by allowing a potential to act between the beads, effectively increasing the linear dimensions of the dumbbell. Since this results in a nonlinear force law, excluded volume also leads to a shear-thinning model. In addition to the excluded volume effect, a physical chain should not be allowed to cross another (i.e., a non- phantom chain); however, the effects of non-phantom chains have not been fully elucidated.

Dumbbells with variable friction have also been studied. The main idea is that in a weak flow, the polymer chain coils up and the drag force on it must reflect the coiled dimension of the chain. As it stretches out in a strong flow, the drag on it must increase with the linear dimension of the chain. The coiled-to-stretched transition of the chain may be modelled by a frictional coefficient of the form s6

r = r (1 + ~R).

Indeed, the drag force on a stretched polymer chain must be necessarily anisotropic, since the stretched polymer chain would resemble a slender body. Thus, to improve the modelling of the coiled-to-stretched transition, the friction coefficient may need to be a second-order tensor; for instance, s7

if_ r177 + (r r177 R2 , where the frictional coefficient along the chain is r which is expected to be less than half of the frictional coefficient normal to the chain, r177 based on the slender- body theory for a rigid rod. The diffusivity is now anisotropic, being proportional to ~-1, and of a similar form to that of the hydrodynamic interaction case. The main feature of a model where r = r is the appearance of hysteresis loops in

, , | ,

s4PHAN-THIEN, N. and TANNER, R.I., Rheol. Acta, 17, 568-577 (1978). S5FIXMAN, M., J. Chem. Phys., 45, 785-792 (1966); 45, 793-803 (1966). S6HINCH, E.J., Coll. Int. CNRS No. 233, Polym. Lubrzficatwn (1975). TANNER, R.I., Trans.

Soc. Rheol., 19, 557-582 (1975). sTPHAN-THIEN, N., MANERO, O. and LEAL, L.G., Rheol. Acta, 23, 151-162 (1984).


elongational responses. This is due to the double-peaked distribution function in these flows; the ensemble average of a quantity will appear to have two stable states although there is only one unique distribution function, as Physically speaking, when the chain is fully stretched out and the strain rate is reduced to a level below that required to induce a coiled-to-stretched transition, it will take a considerable time, much greater than the chain relaxation time, to return to the coiled state. The reverse occurs when the strain rate increases beyond the level required for coiled- to-stretched transition, and this gives the appearance of a hysteresis loop. The use of the scalar configuration-dependent friction coefficient also leads to shear-thickening behaviour, which disappears if the tensorial form is used. s~

A polymer chain at low to moderate deformation will have a certain amount of solvent entrapped within its boundary. Thus, the chain tension induced by the deformation may not be entirely due to the loss of configurations available to it. A model of this so-called internal viscosity is due to KUHN and KUHN, 9~ who propose that the connector force law be given by

R R F(,) _ ~ + rhR �9 _--:-z-,

and r h is the internal viscosity of the chain. Mechanically, the beads are connected by a linear spring and a damper in parallel. The Langevin equation for the R-mode, neglecting the bead inertia, is

1 + 2r h R R ) 1F(b). ( R2 1~: -- L R - 2 H C - I R + ~-

Denoting e = 2rh/( and noting that

( R ~ ) - 1 s R R l + e = 1 l + e R 2 '

we find R R 2H

e eL �9 - ~ R - - t - ~ ( e) R l~ = L R - i + L ' I ~

e R R ) F(b) (34.114) -4-~- 1 1 1 + e R 2 "

The main feature of this is that the dumbbell sees an effective non-a~ne deformation given by

e R R - L" R 2 . L l + e

This also leads to an anisotropic diffusivity tensor of a form similar to that in the hydrodynamic interaction case, or configuration-dependent friction tensor case.

88The Smoluchowski equation is quasi-linear and has a unique solution. SgPHAN-THIEN, N., MANERO, O. and LEAL, L.G., Rheol. Acta, 23, 151-162 (1984). 9~ W. and KUHN, H., Helv. Ch~m. Acta, 28, 1533-1579 (1945). A different form for the

internal viscosity was proposed by CERF, R., Fortschr. Hochpolym. Forsch., 1, 382-450 (1959).


BOOIJ and VAN WIECHEN 91 have done a perturbation analysis in e and have shown tha t in a small strain oscillatory flow

lim 77' - r}~ + 0.4trip, O3--+OO

which agrees with some experimental data tha t ff(oo) is higher than ~}s. This also implies tha t the Newtonian viscosity at the inception of a s tar t-up of a shear flow is higher than rls. There is also a small amount of shear-thinning, of O(e). The numerical response of the model in a number of homogeneous flows has been considered; 92 in particular, it is shown that the steady state response in an elongational flow is

independent of the internal viscosity. It would be fair to conclude at this point tha t there is no single one-mode con-

stitutive equation tha t can qualitatively describe all observed phenomena in dilute polymeric liquids. A person interested in a numerical solution must thus make a judicious choice of a constitutive equation, keeping in mind the complexity of the model, the gross physical phenomena to be simulated, and the available computing resources. We would not, for example choose the Oldroyd-B model to simulate a nearly elongational flow. A constitutive equation of the FENE variety is usually a safe choice in this case, with an easy-to-understand microstructure and a simplicity to match.

3~ .10 R i g i d D u m b b e l l

There are cases where the polymer chain can be regarded as being rigid; for example, various DNA and protein molecules, tobacco mosaic virus, etc., have the appearance of slender rods tha t do not stretch appreciably in a flow. In these cases, it is expected tha t a rigid-rod microstructure would be more relevant in the constitutive description than the flexible microstructure. The simplest model for this is the rigid dumbbell model, where the polymer chain is idealised to two negligible point masses, located at r l and r2; each is associated with a constant frictional coefficient r as in the elastic dumbbell case, but now the distance between the two beads is kept constant at Irl - r 2 1 - L by a rigid, but phantom connector. 93

A great deal of work in this area has been done by BIRD et al., 94 but our approach is slightly different in one important aspect: instead of using the Giesekus form for the stress tensor, 95

r(p) = v( A (RR), (34.115) 4 At

91BOOIJ, H.C. and VAN WIECHEN, P.H., J. Chem. Phys., 52, 5056-5068 (1970). 92PHAN-THIEN, N., ATKINSON, J.D. and TANNER, R.I., J. Non-Newt. Fluid Mech., 3,

309-330 (1978). 93Note that the scalar L is the constant length of the dumbbell; it should not be confused with

the velocity gradient tensor L. 94BIRD, R.B., WARNER, H.R., Jr. and EVANS, D.C., Fortschr. Hochpolym. Forsch., 8, 1-

90 (1971); see also BIRD, R.B., CURTISS, C.F., ARMSTRONG, R.C. and HASSAGER, O., Dynamics of Polymeric Liquids, Vol. II. Kinetic Theory, ist Edition, John Wiley & Sons, New York, 1977. 95GIESEKUS, H., Rheol. Acta, 2, 50-62 (1962).


which is valid for elastic dumbbells, we use the Kramers expression for the stress tensor, v(RF(S)), which, in our view, is more fundamental. 96 The Giesekus form for the stress tensor does not involve the connector force, and it has been shown to be valid even for rigid dumbbells from a phase-space consideration. 9~' The Kramers expression involves the spring-force law, which is indeterminate for a rigid connector; however, the inextensibility of the dumbbell will provide the correct expression for the connector force, in much the same way the indeterminate hydrostatic pressure is determined from the incompressibility constraint in an incompressible fluid. It will be shown that the Kramers expression leads to the same expression as the Giesekus form, without having to go through the phase space formulation used by BIRD et al.

Langevin Equations

First, the Langevin equations for the beads, neglecting the beads' inertia, are

r - L r l ) - F (') = F~ b) ( t ) ,

r - L r 2 ) + F (') -- F(2 b) (t),

where F (s) is the connector force, not determined by a constitutive law, but by

a reaction to the inextensibility constraint of the dumbbell, and F~b)(t) are the Brownian forces, with zero means, and autocorrelation functions (via a fluctuation- dissipation theorem)

By taking the sum of the two Langevin equations, it is found that the centre of gravity of the dumbbell moves just like a fluid particle, on the average. This convected mode is unimportant, and can be integrated out of the diffusion equation. We concentrate our attention on the motion of the end-to-end vector of the dumbbell, R - r2 - r l -- Rp, where p is a unit vector along R. Now, the process R satisfies

( R - L. § : F? ' ( , ) -

where F (b) (t) is the Brownian force acting on R. Note that

(F(b)(t + TlF(b)(t)) -- 4kT~5(T)I. (34.117)

We are not imposing the rigid constraint R - L on the dumbbell at this stage. This has several advantages. Firstly, forcing the constraint at an early stage favours the use of a spherical coordinate system, which makes the integration over the configuration space unnecessarily complex. Secondly, the physical nature of the

96There is an isotropic component due to the Brownian forces, which has been absorbed into the hydrostatic pressure.

97BIRD, R.B., CURTISS, C.F., ARMSTRONG, R.C., and HASSAGER, O., Dynamics of Poly- meric Liquids: Vol. II. Kinetic Theory, John Wiley & Sons, New York, 2rid Ed., 1987, w

constraint is quite subtle: the dumbbell is rigid with respect to processes on the time scale for tumbling relaxation, but not so on the Brownian time scale. In fact, there are four time scales for the molecule: the Brownian correlation t ime scale Tc (the period of vibration of a solvent molecule, of order 10 -13 sec if the solvent is water), the inertial t ime scale ~-i -- m / ~ (of order 10 -1~ sec for the tobacco mosaic virus, with length 100 nm and molecular weight 107), the connector relaxation time scale ~-s = ff-1~7" F(S) (of order 10 -7 s if the Young's modulus of the molecule is 108-109 Pa, which is in the soft rubbery region), and the tumbling relaxation time scale Tt -- ~ L 2 / 1 2 k T (of order 10 -4 s). The condition ~'c << ~'i << (TIe,Tt) is necessary for the validity of the Langevin equations given above, allowing us to drop the inertia term and to assume delta-function-correlated Brownian forces. For flexible molecules, Ts "~ Tt , both being equal to the Rouse relaxation time, but in our case Ts << 7-t. If the inextensibility constraint were enforced from the start and Langevin equations were set up for the resulting five degree of freedom system (for example using spherical polar coordinates), this would be equivalent to assuming Ts << To. Since this does not appear to be so for typical rod molecules, we contend that this procedure is formally incorrect. 98 In their book, BIRD et al. 99 discuss both approaches: the incorporation of constraints at the start or the use of a very stiff spring, and favour the former. In fact, for the rigid rod (and all completely rigid molecules), both give the same result for the rheological properties; however, for flexible molecules with constraints (for example, Kramers bead-rod molecules), the results differ, although by only a few percent. 1~176 In the stiff-spring approach, the rigidity of the dumbbell is to be introduced at the stage where any fluctuation on the time scale Tc has been averaged out; this is the approach taken here.

By writing R = RIb 4-/~p and noting that p - p - 1, and lb- p - 0, it is found that

2 F(~) = ~ p F(b) (34.118) / ~ - RL "pp + ~ �9 (t) ,

and 1

15 -- L - p - L ' p p p + ~--~(1 - pp) -F(b) ( t ) , (34.119)

where we have assumed that the connector force is parallel to p, i.e., F (8) = F(~)p. 1~ Note that

F(S) 1 �9 1 1 = - ~ r + ~(~RL " p p p + ~ p p - F (b) (t). (34.120)

9Slf correct, the procedure would imply tha t the component along p of the Brownian force on the first bead must necessarily be felt instantaneously at the second bead, and therefore must be correlated with the component along p of the Brownian force on the second bead.

99BIRD, R.B., CURTISS, C.F., ARMSTRONG, R.C., and HASSAGER, O., Dynamics of Poly- meric Liquids: Vol. II. Kinetic Theory, John Wiley & Sons, New York, 2nd Ed., 1987, w 1~176 R.B., JOHNSON, M.W., Jr. and CURTISS, C.F., J. Chem. Phys., 51, 3050-3053

(1969); HASSAGER, 0., J. Chem. Phys., 60, 2111-2124 (1974). 1~ inextensibility constraint only determines the component of the connector force parallel

to p; it cannot determine the component perpendicular to p. From the physics, however, we can assume tha t the connector force has only a component parallel to p. There may be some artificial microstructures tha t allow a component force perpendicular to p; for these microstructures, the stress tensor will not generally be symmetric. We do not consider these cases.

From the Langevin equations (34.118)-(34.119), the 'drift velocity' in R,

lim < A R > At--,0 " ~ ~ - - R L ' p p - - F (s), lim <A-~t > --L-p-L'ppp,

A t - - . 0 c (34.121)

and the diffusivity in R

lim = At--.0 2At /

ApAp > 2kT (1 - pp) (34.122) 2kT lim ' a te0 2At ~ ~R 2

can be obtained. Here ( )c denotes an average over a time much less than T~ and Tt, but much greater than To. Note that the 'rotational' diffusivity (perpendicular to p) of the process p is

2kT Dr = ~R 2 . (34.123)

A Smoluchowski equation, or configuration space diffusion equation, for the probability distribution function r can now be written down in terms of R and p, but should be used with caution since these four variables are not in fact independent.

The equation of change for any quantity B(p) can be derived without having to go through the Smoluchowski equation by noting that

dt = Opk n k ~ p ~ -- nm=pmp=pk + " ~ (t) .

Thus, d OB

+ . �9

To evaluate the last two terms in the previous equation we should keep in mind the fast time scale of the Brownian force and use it to our advantage. First, we define

10B 10B Ck = D k - -- P,~Pk.

R Opk ' R Opm

Then, from (34.118)-(34.119), these vectors evolve in time as follows:

1 02B [ 1 (Szm_plpm) F(mb) ] Ck = -~ OpkOpl L, mPm -- LmnPmPnPl + - ~ (t)

and

1 OB [ 2F(S) 1 F(b)(t) 0 w - +

1 02B [ 1 F(b) Dk -- -~ Op~Op p,~pk L ~ p r - LrsprPsP~ + - ~ (5~r - P~Pr) (t)


1 0 B [ 1 (Sk _ pkp~) F(b)(t) ] Pm

l O B [ l (Smr _ pmpr) F(b)(t) ]

1 OB [ 2F(.)+1 F(b)(t)]. --R2Op pmpk RL,.sp,.ps-'~ (e',',.

Our plan is to write down formal solutions to the previous two equations at t ime t + At in terms of those at t ime t, where At is much larger than the fluctuation time scale Tc of the Brownian force, but small compared with the relaxation times % and Tt of the process, so tha t the two previous equations can be regarded as linear. Thus,

1 c32B ck(t + At) = Ck(t) + r 2 0pkOpl (Sire -pzp.~)B~)(At)

1 0 B pmB~ ) ( A t ) + EkAt, ~R 2 0pk

(34.126)

Dk(t + At) = Dk(t) + 1 02B B(b) (At) CR 20pmOpn PmPk (Snr -- PnPr)

+ 1 0 B

CR2 (:Opm 1 0 B B(b)(At)

P~ ( ~ - P~P~) B(2)(~t) + ;R--W Opm p~ ( ~ - P~P~)

1 0 B R 2 0 p m

--pmpkp~B(b)(At) + FkAt, (34.127)

where Ek and Fk are the 'drift ' terms tha t do not involve the Brownian force, and

f t+At

B(b)(At) -- F(b)(t ')dt '. J t

(34.128)

The average of the last two terms in (34.125) is

1 !

At)F(b) (t At)F@ (t \

This average contains terms linear in the Brownian force (Ck(t), Ok(t) and the drift terms Ek and Fk), which will not survive the averaging process, and terms tha t are quadratic in the Brownian force. These latter terms will contribute to the average, since 1~

i t+At

B(~) (t + ~t)r(~) (t + ~t ) - r(~) (t')r(~) (t + ~t)~t' 1 - - - x 4kT~l. J t 2

l~ factor of 1/2 arises because the delta correlation function overlaps the upper limit of the integral.


Thus

1 2 k T ( 1 [ 02B -- - ~ ~ ( S k , - PkPl)OpkOp,

- 2 OB]

In the stiff limit, R 2 can be taken outside the average, and the equation of change for B(p) is

-~ (B)=Lkm P m ~ - L m n pmpnpk'~k

2kT ( 02B + - ~ (Skl -- PkPt) OpkOpl 013 ) (34.129) - 2 p k ~ .

In particular, with B -- pipj,

013 02B Opk = 5~kpj + 5jkPi, OpkOpl -" 5ikSjl "+" 5jkSil,

and d 4kT

5-;, (PP) -- ( L - p p + p p - L T - 2L �9 pppp ) + ~ (1 - 3pp) q r t -

leading to

( ~ t ) 1 (pp) + 2L" (pppp) + (pp) -- g l , (34.130)

where A -- 1/6Dr, and A / A t is the upper convective derivative. This is exactly the expression given by BIRD et al. 1~ The relaxation time for the tumbling motion of p is A -- r in the same form as the Rouse relaxation time, except that the mean square distance Nb 2 is now replaced by L 2.

Kramers Stress Tensor

To derive the polymer-contributed stress tensor, we use the Kramers expression

r(p) -- v(RF(S)) ( l �9 l l ) (34.131)

-- u --~r + ~ R 2 L " p p p p + ~ R p p p - F (b) ,

using (34.120). If the connector is sufficiently stiff so that the fluctuations of R and p may be considered to take place on vastly different time scales, then the first term on the right hand side is

(R/~pp) -- (R/~)(pp) -- 0,

since (RR) - �89 2) -- O. In the same stiff limit, R 2 can be taken outside the average in the second term; this term represents the viscous dissipation. To

i

I~ R.B., CURTISS, C.F., ARMSTRONG, R.C., and HASSAGER, O., Dynamics of Poly- meric Liquids: Vol. II. Kinetic Theory, John Wiley & Sons, New York, 2rid Ed., 1987, w

evaluate the third term we use the same procedure as outlined previously. First, we define the third-order tensor A = Rppp . Then A evolves in time according to

h / t p p p + R (15pp + pI~p + ppl5)

R (L- p p p + p L - p p + p p L - p - 2L : p p p p p ) - 2r

~__~-1 (F(b)pp + pF(b)p + ppF(b) _ 2 p p p p - F(b)) .

The formal solution of this at t ime t + At in terms of that at t ime t, where At is much larger than the fluctuation t ime scale Tc of the Brownian force, but small compared with the relaxation times Ts and Tt of the process, is obtained and the average is taken as

which contains terms linear in the Brownian force, which will not survive the averaging process, and terms quadratic in the Brownian forces. Thus, we find, in a similar manner to the derivation of the equation of change for (B> ,

< A - F (b) > = 6 k T <pp>, (34.132)

and the polymer-contributed stress is given, from (34.131), by

r (p) = l v L 2 r �9 (pppp> + 3vkT (pp) .

viscous entropic

(34.133)

Note that the viscous term is instantaneous in the velocity gradient; the moment the flow is turned off, it disappears instantly. The entropic term has a relaxation time of the order A. This is identical to that obtained from the Giesekus expression (34.115), to within an isotropic stress.

To solve flow problems with the rigid dumbbell model, we must first solve for the probability distribution function, then the ensemble averages of p p and p p p p can be evaluated, followed by the stress tensor using (34.133). Alternatively, an approximation scheme similar to the Peterlin approximation can be introduced, expressing (pppp) in terms of (pp> to force a closure on (34.130). Some of these schemes are discussed by HINCH and LEAL. t~

P r o b l e m 3 4 . 0

I~ E.J. and LEAL, L.G., J. Fluid Mech., 71,481-495 (1975).

a) Show, by using (34.125), that the evolution of (pppp) is governed by

dt = Li,~(PmPjpk, pl) + Lj.~(p,npipk, pt)

+ L~m~pmpipjpt) + Lt.-~ (P~PiPjPk)

- 4Lmn(PmPnPiPjP~,Pt) + 2D,.{Sij(p~:p~)

+ ~.,~(p~p,~) + ~.<p~p~) + ~,~(p.p~)

+ 5jt(PiP~) + 5ta(PiPj)- lO(piPjpkpt)}.

(34.01)

b) Consider a slow flow, with a velocity gradient tensor L = ee, where e is a small perturbation parameter. The strain rate tensor is D -- ed. Look for a steady solution of the form

(PP) = (PP)o+e(pP) I+e 2(pp)2 + . . . ,

(PPPP) = (PPPP)o +e(PPPP)I + . . . ,

(PPPPPP) = (PPPPPP)o + . . . ,

where higher-order terms have been neglected. Show that

1 (Pr - -~5r

(P~PjPkPl)o = 1 (5~jSkl + 5~kSjl + 5~lSjk) ,

(PiPjPkPlPmP,~)O = 1 {5~j(SklS,n~ + 5kmS,a + 5knSlm) 105 + ~ , ~ ( ~ ~ + ~ ~ , + ~ j ~ )

+ ~ , , ( ~ ~ + ~ ~ + ~ ~ )

09iPj)l : ~)~(e i j + eji) - 6 )tdij,

1 (PiPjPkPl)a - "i'~{dijSkl + dik~jt "b dit~jk "+-djkc~it "t-djtcSik + dklSij},

2 A 09iPj)2 -- l"~A2(eikdkj -4- e~kdk~) - ~ (dmndmnSij -t.- 4dikdkj)

c) Thus show that the polymer-contributed stress is given by

1 ( 9 . ) r (p) = ~ 1 + ] -~#1+6A#2 D + 6 # 2 A 2 ( L D + D L T ) 1 5

( 1 2 A ) ( t r D 2 1 + 4 D 2 ) + + ] - 5 g ~ - 1--~ m .. . ,

(34.02)

(34.03)

(34.04)

(34.05)

(34.06)

(34.07)

(34.08)

__ 1 v(~L 2, and -- 3 v k T . where #1 ~ P2 d) In a steady simple shearing flow at low shear rate, show that

T~P2 ) = vkT4/ , N1 -- 6vkTA2~2, N2 - -0 , (34.09)

exactly as given by BIRD et al. 1~ equation.

from a perturbation solution of the diffusion

Extensive predictions of the rigid dumbbell model are given in the book by BIRD et al. 1~ Typically, in a steady shear flow the model produces shear thinning in the viscosity of O(~-2/3), and in the first normal stress coefficient of O('~-4/3). The second normal stress coefficient is exactly zero due to the mutual cancellation arising from the viscous and entropic terms. In a steady elongational flow at high strain rates, the unit vector p aligns with the flow, leading to a constant and finite elongational viscosity.

34.11 Rouse Model

Dumbbell models have only one mode of vibration, and consequently possess only one relaxation time. Real fluids, of course, have a spectrum of relaxation times. The simplest model having this feature is the Rouse chain shown in Figure 34.2a, where a polymer is modelled as a multiple-bead-spring system (a total of N + 1 beads and N springs); each spring has a constant stiffness of H -- 3 k T / N l b 2, where N1 is the number of Kuhn segments in the spring and each Kuhn segment has a length b, and each bead is associated with a constant frictional coefficient ~ and a negligible mass. The Langevin equations for the beads, neglecting the inertia, are

(~ (~1 -- Lrl )

(~ (f%- Lri)

(I 'N+I -- L r N + I )

-- H (r2 - r l ) + F~ b) (t),

= H ( r i + 1 - 2r, + r i-1) + F~b)(t),

~(b) = H (rN -- r N + l ) q- - N + i ( t ) ,

(34.134)

where F~ b) (t) is the Brownian force acting on bead i, 1 < i < N + 1, and L is the homogeneous velocity gradient tensor. The Brownian forces acting on the beads satisfy

In terms of the centre of gravity R (c) and the segment vectors Rj defined by

N+I (34.13 ) R(~) = N + 1 ~ r~, R i = r /+ l - ri,

a= l

I~ R.B., CURTISS, C.F., ARMSTRONG, R.C., and HASSAGER, O., Dynamics of Poly- meric Liquids: Vol. H. Kinetic Theory, John Wiley & Sons, New York, 2rid Ed., 1987, Example 14.4-1. 1~ R.B., et al., 1987, 0p. cit., w

the Langevin equations can be rewritten as

N + I 1 F ~ ) N + 1 ~ (t) -- F (b'c) (t),

r

(: (1~. - L R , ) + HA,jRj -/~,~F(b)(t)- F~ b) (t),

where A~j is the Rouse matrix, of size N x N defined by 1~

(34.136)

2, i f i - j ,

A ~ j - - 1 , if i - j •

0, otherwise.

(34.137)

Also, Bia is an N x (N + 1) matrix defined by t~

B~a -- ~ii+l,a - 5{~,

and F <b,c) (t) and F~ b) (t) are the Brownian forces acting on the R(C)-mode and the Ri-mode, respectively. Note that

R~ = ] ~ r ~ , and A~j - B~]~j~.

The Brownian forces have zero means and autocorrelation functions

2kT~ < F(b'c)(t + T)F(b'c)(t)> -- (N + 112 ,

From these results, the Smoluchowski equation for the distribution function (I)(RI,..., RN , can be written down:

0t = 0R(c---"-~ " (:(N + 1) OR(c) - LR(C)@

+ 0 0o ORi" Aij ORj

To form a constitutive equation, it is more useful to consider the normal modes of vibration of the system (34.136), ignoring the centre of gravity since it can be

I~ P.E., Jr., J. Chem. Phys., 21, 1272-1280 (1953). l~ this sub-section, Greek subscripts take values in ( 1 , . . . , N + 1), and Roman subscripts

take values in (1 , . . . , N). The usual summation convention is also assumed with respect to these subscripts, unless otherwise noted.


integrated out of the Smoluchowski equation in the end anyway. Let f~ik be the orthogonal matrix that diagonalises the Rouse matrix t so that

E 12ikAij~j, = aI~81r (no sum), (34.138) i,j

where ak are the eigenvalues of the Rouse matrix: 1~

( kTr ) (34.139) a k = 4 s i n 2 2(N4-1) "

The normal modes of vibration, Qj, are the vectors:

R~ -- ~ j Q j , Qj - ~ j R i . (34.140)

The Langevin equations for these normal modes, by substituting the above into (34.136), are

(~j - LQj + ~-ajQj = akj]3k~F~)(t) -- F~b'q)(t) (no sum). (34.141)

where F5 b'Q) (t) is the Brownian excitation on the mode Qj. Defining

i t+At

Bi(At) = F~b'Q)(t')dt ', Jt

we find that

(Bi(At)Bj(At)) t+At /t+At t t

(t')F5 (t") ) t'at"

it+At J t

k.T = 2:~-Ata~8~jl, (no sum).

Thus, with respect to the probability distribution of Bi(At),

( ) lim = i Q i a iQ i (no sum) ht-~O 'At" --'~- ' '

lira / AQiAQj ) / B i (At )Bj (At ) ) kT At--,0 2At = ...... 2At = Tai~ijl, (no sum),

and the Smoluchowski equation for the process {Qj} is given by (of. (34.a4))

0t - E . 0Qi" - ~ a , 0Q--7- - -~-aiQi r �9 (34.142)

i l l

I~ H.A., Physica, 11, 1-19 (1944).


By pre-multiplying this with Q j Q j (no sum), and averaging it over the Q-space, we find that

A'-7 (QJQJ) 4- a j Q j Q j - 2 a j l , no sum),

o r A 1

)~j~-~ (QjQj) 4- (QjQj ) - ~Nlb21,

where the j-relaxation time is defined as

(no sum), (34.143)

~Nlb 2 Aj -- 2Haj 24kTsin2LjTr/(N + 1)]"

(34.144)

For large N and small j , the large time constants are given approximately by

~Nlb2(N 4- 1) 2 ,kj ,.., 24kTTr2j 2

(34.145)

Now the polymer-contributed stress is given, from (34.63), by

r(v) = N N

i--1 i--1

N N

i = 1 i = 1

(34.146)

where 3vkT

I"~ p) -- v (HQiQi) - (QiQi) , (no sum). (34.147) Nab 2

Thus, the polymer-contributed stress consists of N different modes and, from (34.143), each mode obeys the UCM model with a relaxation time of ~j; the constitutive equation is written as

N

(T) -- - p l + 2r/,D + E 1"~v) i--1

)q---A r!P) + r~ p) -- G01 (no sum) At ' ' '

where Go - t,,kT is the modulus of rigidity. Alternatively, we can define

(34.148)

~.~v) _ S~v) + Go1,

and lump the isotropic stress with the pressure term. Then we obtain the familiar form of the UCM model:

)~i---~ S! p) + S~ p) = 2r/~P)D At ' (no sum), (34.149)


where

v r (34.150) ~!~ p) -- A, Go -- 24sin2[iTr/( N + 1)]

is the polymer viscosity contributed from mode i. The polymer-contributed stress can also be written in the integral form, recalling (34.92):

r (v) -- Go e -(t-r)/a~ C[ or i--1 i I ( T ) d T - - G ( t -- T ) C t I ( T ) d T ,

O 0

(34.151)

where

O(t) - Go Z e- t /a ' i--1

(34.152)

is the relaxation modulus. The set of {Ai, Gi - Go} constitutes the discrete relaxation spectrum of the Rouse model. Since the Rouse model is made up of several UCM modes, it inherits whatever shortcomings of the UCM model (constant viscosity, quadratic first normal stress difference, zero second normal stress difference, infinite elongational viscosity at a finite elongational rate). However, with multiple relaxation times, the dynamic response of the Rouse model is an improvement over the simplistic prediction of the elastic dumbbell model. Specifically, we have the following:

dynamic and storage viscosities:

GoAi ~1" G~ ~/'-- ~7~, -4- Z 1 + A2w 2' -- Z 1 -4- )k2Cd 2' (34.153)

i i

storage and loss moduli:

2 2 GoA~w GoAiw G" (34.154) G' = Y~ 1 + A2w2, -- r/~w + Z 1 + A2w2"

i i

At low frequencies, G ' ~ w 2, G"~.. w, while at high frequencies we find

G ! - ( G t t i ~ s ~ ) ,.,a w1/2. (34.155)

This is a distinctive feature of the Rouse model due to the spacing of the longest relaxation times, Aj ~ j -2 . In fact, the model also predicts the equality between G' and G ' - r/sw at high frequencies. In typical dilute polymer solutions, a scaling of

G ' ~ ( G " - rt, w ) ~ w 2/3

is often seen. For this to occur with the Rouse spectrum, a spacing of the longest relaxation times of Aj ,,~ j -3 /2 is required.

We shall now discuss the approach of ZIMM, 11~ who incorporates the hydrodynamic interaction between the beads by using an equilibrium pre-averaged Oseen

11oZIMM, B.H., J. Chem. Phys. , 24, 269-278 (1956).


J I ! I I j

-

-

,..q~ -2 I I I t

-1 0 1 2

log(t~)

g

FIGURE 34.5. Reduced dynamic moduli versus dimensionless frequecies of polystyrene in two 0 solvents, DECALIN and DOP. The dashed line has a slope of 1/2 as predicted by Rouse theory.

tensor. His t reatment involved a dimensionless hydrodynamic interaction length h*:

h . _ i 3a 2 a 7rNlb 2 -~ 0.98 N:/2b,

where a is the bead radius, and N1/2b is the equilibrium root mean square of a Kuhn segment. A value of h* of the order 0.3 is considered to be maximum; h* = 0 is the free-draining limit (no hydrodynamic interaction), and h* --~ oo represents a dominant hydrodynamic interaction, or the non-draining limit. Zimm's approximation should be adequate in a weak flow, such as small strain oscillatory flows. HIS t reatment is quite similar to Rouse's treatment; in fact the resulting equations are identical to those obtained with the Rouse model, except tha t the Rouse matrix is now replaced by a modified matrix Ai;, whose eigenvalues ~j determine the relaxation times, just like the Rouse model. Tables of the eigenvalues of Aij are provided by LODGE and WU, l i t who have determined them numerically. ZIMM et al. tt2 show that the spacing of the longest relaxation times changes from j - 2 behaviour in Rouse model to j-3/2 in the non-draining limit at large N. This implies tha t the

lllLODGE, A.S. and WU, Y., Rheology Research Center Report Nos. 16 and 19, University of Wisconsin, 1972.

112ZIMM, B.H., ROE, G.M. and EPSTEIN, L.F., J. Chem. Phys., 24, 4881-4885 (1956).


high-frequency behaviour of the dynamic and loss moduli changes from (34.155) to

V ~ G ! = ( G I ' - ~sO.l) ~ r 2/3, (34.156)

which agrees excellently with experimental data for dilute polymer solutions in poor solvents, as shown in Figure 34.5 for polystyrene in two 0 solvents. 113 In this figure, the reduced moduli

G' G~ = G " - r/,w = Go' G0

are plotted against the dimensionless frequency )~w, where )~ = (7 /0- ~Ts)/G is a measure of the longest relaxation time. In good solvents or when the polymer concentration is no longer dilute, the measured dynamic response of the fluid tends to favour the Rouse model; this is usually explained by a hydrodynamic shielding of a polymer chain due to the surrounding chains: the increase in the probability of self-intersection is compensated by a decrease in the probability of intersection with different chains. 1 t4

Note tha t both Rouse and Zimm theories predict that the modulus of mode i is independent of i. In practice, given the dynamic data, a set of relaxation times is chosen, usually two per decade in the inverse frequency range covered by the data, and the Gi in (34.154) are adjusted by a least square procedure to obtain a good fit to the data. The problem is actually ill-posed, in the sense tha t any small variation in the input dynamic data is greatly amplified in the output relaxation spectrum. Several methods to regularise this problem are discussed in WEESE. 11s

P r o b l e m 3 4 . P From the expression for the viscosity

N

j--1

show tha t the intrinsic viscosity defined as

[r/] = lim r / - T/s c-*0 C~T s

is given by k T N A r + 1) 2 ~-~ 1 (34.P1)

[~1 = Mr/s 27r2H z--, 7 ' 3

where NA is the Avagadro's number, c -- v M / N A is the mass concentration of the polymer per unit volume, and M is the molecular weight. It can be assumed tha t N is large so tha t the asymptotic form for the relaxation times is valid.

113jOHNSON, R.M., SCHRAG, J.L. and FERRY, J.D., Macromolecules, 5,144-147 (1972). 114FLORY, P.J., Principles of Polymer Chemistry, Cornell University Press, Ithaca, New York,

1953; FREED, K.F. and EDWARDS, S.F., J. Chem. Phys., 61, 3626-3633 (1974). 115WEESE, J., Comp. Phys. Comm., ?7,429-440 (1993).


Since N ~ M, deduce tha t the intrinsic viscosity is proportional to the molecular weight. This is known as Staudinger's rule. 116

Experimental data on intrinsic viscosities of dilute polymer solutions 117 suggest a power-law relationship of the form

[V] ~ M ",

where the power-law index varies between 0.5, for 0 solvents, to about 0.8 for good solvents.

The deviation from Staudinger's rule can be explained by Zimm's theory: the viscosity in Zimm's model is given by

N

- 78 -- u k T ~ Aj, j-1

where

= 2ga

are the Zimm relaxation times. THURSTON 118 has fitted the numerical values of LODGE and WU t19 to the power-law relation:

gj - K aj N + 1 '

where aj are the Rouse eigenvalues, and K and a are curve-fitting constants, related to the hydrodynamic interaction parameter h*, by

g - 1 - 1.66(h*) ~ a = - 1 . 4 0 ( h * ) ~

With this empirical fit, it may be shown tha t

[ ~ ] - g k T NAr + 1) 2+a Z j2+al ,-.., Ml+a. (34.157) 2~2~8MH

3

As h* varies from 0 (free-draining) to 0.25 (non-draining), a varies from 0 t o - 0.5, leading to the variation [~]--~ M at the free-draining limit to [V] ~ M1/2 at the non-draining limit, in excellent agreement with experimental data on dilute polymer solutions in ~ solvents.

3~. 12 T i m e - T e m p e r a t u r e Superpos i t i on Pr inc ip l e

Both Rouse and Zimm relaxation time spectra have the same dependence on the temperature. We can define the shift factor aT(T), a function of the temperature

116STAUDINGER, H. and HEUER, W., Bet. deutsch. Chem. Gesellschaft, 63, 222-234 (1930). 117NODA, I., MIZUTANI, K., KATO, T., FUJIMOTO, T. and NAGASAWA, M., Macromole-

cules, 3, 787-794 (1970). 11STHURSTON, G.B., Polymer, 15, 569-572 (1974). 119LODGE, A.S. and WU, Y., Rheology Research Center Report Nos. 16 and 19, University of

Wisconsin, 1972.


as

~

10 5;

1 0 4 -

1 0 3 _

10 2 -u

I I I I I I

G !

I I I I I I 10 -2 10 -1 10 0 101 10 2 10 3

Reduced Frequency a T co (s -1 )

FIGURE 34.6. Dynamic moduli for LDPE obtained by the time-temperature superposition principle to To -- 150~ The data of cover a wide range of temperatures, from 112~ to 190~

alone, as

Aj(T) _ Torl~(T) A j ( T ) - aT(T)Aj(To) (34.158) a T ( T ) - Aj(T0) -- TTls(To)'

assuming tha t a, N1 and b are independent of the temperature, and To is a reference temperature. From the dynamic properties taken at temperature T, for example r/' from (34.153), we obtain

r/'(w, T)aT(T)- rls(T) Tov(TO)Tv(T) - y ~ 1 +G~ (T0)~2 2(T0) = r/'(~, T 0 ) - rh(T0 ), (34.159) j Aj

where if) = aT(T)w is the reduced frequency. That is, if the quantity on the left hand side of the previous equation is plotted against 5), then the data at all temperatures should collapse on to a master curve. This master curve is also the plot of the reduced dynamic viscosity 77'(~, T o ) - ~/, (To) against ~ at the reference temperature To. This procedure, called the time-temperature superposition principle, has been known for a long time, 12~ and seems to work extremely well for both solutions and melts not near their glass transition temperatures; the Rouse model provides a molecular basis for the principle. Figure 34.6 is an example of the t ime-temperature shifting to a reference temperature of 150~ for a low density polyethylene melt as reported in LAUN; 121 each set of data taken at a temperature spans about 2-3 decades in the frequency. The combined data shifts to the reference temperature span nearly 7 decades in the frequency range. Lowering the temperature relative to To increases aT, and thus increasing the effective frequency D at the reference temperature. Raising the temperature has the opposite effect.

12~ J.D., J. Amer. Chem. Soc., 72, 3746-3752 (1950). 121Reproduced from LAUN, H.M., Rheol. Acta, 17, 1-15 (1978).


Note tha t the number density is given by v(T) -- n /V (T), where n is the number of chains in the volume V(T). Since neither the number of chains nor the mass of the chains depend on the temperature, we have

v(To) V(T) p(To) v(T) Y(To) p(T)

where p(T) is the density of the fluid at temperature T. The vertical shift

T0 (T0) Top(To) Tv(T) Tp(T) '

is not as important as the shift due to aT(T), since the effects of T and p(T) lead to a partial cancellation, because increasing the temperature reduces the density due to the expansion of the fluid. In essence then, the reduced dynamic viscosity is shifted vertically down by a distance aT, while the frequency is shifted to the right by the same distance, both in logarithmic scale, to produce the master curve at the reference temperature.

The shift factor has been fitted to the empirical relation, known as the WLF equation: t22

-Cl (T - To) (34.160) log aT = c,2 + ( T - To)

where Cl and c2 are curve-fitting constants and To is a reference temperature. This relation is found to work for many polymer solutions and melts.

There is more to the t ime-temperature superposition principle than just a way to produce a master curve in oscillatory shear flows. Consider a non-isothermal flow; the theory leading to the Smoluchowski equation for the Rouse model remains unchanged, and therefore the constitutive equation for the quantity (QjQj> also remains unchanged, i.e., an UCM equation. Thus, we have 123

,r~V) _ 3vkT vT _(v) (34.161) - - Nab 2 ( q ~ q J ) - v0T0-~0,

where the subscript 0 refers to the quantity measured at the reference temperature

To. The constitutive relation for ~.~v) is, from the constitutive equation for (q~q~) and the definition of ~.~v),

aT~jO {~ 'r~j ) -- -rj-(P) d (ln r, T) } +-(v) - Tj (34.162)

We now relate ~r~ v) to Ir~ ) in (34.162)to obtain

aT vT . A _ ( p ) v T ,.r~Po) _ vkT1, voToAJO-~ Tj~ + voTo

122WILLIAMS, M.L., LANDEL, R.F. and FERRY, J.D., J. Amer. Chem. Soc., 77, 3701-3707 (195s). 123PHAN-THIEN, N., J. Rheol., 23, 451-456 (1979).


or

a T A j O ' - ~ jO ~- " = vokTol.

If we were to define the reduced time and velocity gradient as

then we find

t - -- --- , L -- aTL,

a T

(34.163)

~ 0 ~ ~ 0 + - -0kT01, (34.164)

which is exactly the constitutive equation of r ~ ) at temperature To. In other words, the time-evolution of

T0v(T0) r~p)(t; L, T) - Top(To) r~p) (t; L T) Tv(T) Tp(T) '

in a time-varying temperature field T is exactly the time-evolution of

= t - - ; L -- arL, To), a T

but with a reduced time and strain rate, and at a constant temperature To. Since the dependence of the relaxation times on the temperature is uniform, the above result holds for the polymer-contributed stress. The integral equivalence of (34.164) shows that the strain tensor is calculated in a pseudo-time t /aT, as if the fluid particle carries with it an 'internal time clock' that samples the temperature field. This type of material is said to be thermorheologically simple. 124

As an example, consider a flow at temperature T, with strain rate ~ and let T (p) be any component of the polymer contributed stress tensor. Then,

T~176 T(P) (t; " - T(p) (~T; aTe, T o ) ,

so that if we plot the left hand side of this equation versus t /aT, then we would get the time response of the fluid at temperature To, but at a reduced time t - t /aT, and at a reduced shear rate ~ -- aT~/. Raising the temperature (with respect to To) has the effect of reducing aT, effectively probing the region of larger time than t, or smaller shear rate than ~ at temperature To. At steady state, we find the polymer- contributed viscosity, the first and second normal stress differences, the first and second normal stress coefficients, and the polymer-contributed elongational viscosity obeying

Top(To) n p(~,T) - ~p(aT~/, To), (34.165)

Tp(T) aT

(34.166) T~176 ] 2(7 T) = N1,2(aT~, To) Tp(T) ' '

124MORLAND, L.W. and LEE, E.H., Trans. Soc. Rheol., 4, 233-263 (1960).


lO 6

,~ 10 5 ~

8 104 ~

~ 103 o = 102

101

lo 0

I I I I I I I

I I I i I I I ~ 10 -3 10"2 10-1 10 0 lO 1 10 2 lO 3

Reduced Shear Rate (s -1)

r

10 4--J

10 6 ~

10 5 o r j

104 ~

103 rm

2; 101 ~

10 ~ o o

FIGURE 34.7. Reduced viscosity •//aT and first normal stress coefficient ~bl /a 2 as functions of reduced shear rate; the reference temperature is To -- 150~ The data cover at a wide range of temperatures, from 112~ to 190~ The solid lines are the predictions of the PTT model using eight relaxation times.

T~176 ~I/1'2('~' To) ---- 1I/1,2(aTty, To) (34.167) Tp(T) a 2

Top(To) T) - ~p,E(aT~, To). (34.168)

Tp(T) aT

The factor Top(To)/Tp(T) is not significant, and is often omitted. A typical set of results, spanning nearly eight decades in the reduced shear rate to a reference t empera tu re To -- 150~ is shown in Figure 34.7, for a low density polyethylene melt. 125

In summary, we have condensed the extensive l i terature on the consti tutive modelling of dilute polymer solutions in the past five decades in one single section, detailing the main development from a mechanics point of view, the principal models and the tools necessary to unders tand and to undertake new research in this exciting area. This condensed story is by no means complete; more references can be found in BIRD and 0 T T I N G E R , 126 BIRD and WIEST, 127 and the recent book by 0 T T I N G E R . 12s The strong impression tha t we wish to impar t is t ha t

, , , , , , , , , , ,

125Reproduced from LAUN, H.M., Rheol. Acta, 17, 1-15 (1978). 126BIRD, R.B. and 6TTINGER, H.C., Ann. Rev. Phys. Chem., 43, 371-406 (1992). 127BIRD, R.B. and WIEST, J.M., Ann. Rev. Fluzd Mech., 27, 169-193 (1995). t2St3TTINGER, H.G., Stochastic Processes m Polymeric Fluids: Tools and Examples for De-

veloping Simulation Algorithms, Springer-Verlag, Berlin, 1995.

35 Network Theories 219

the preferred approach to the modelling of dilute polymer solutions is through a Langevin description of the process in the configuration space, coupled with a relevant fluctuation-dissipation theorem that leads to a diffusion equation. The phase space description is equally valid, but more formal and more general than necessary for our purposes.

35 Network Theories

The Langevin description of the polymer chains discussed in the previous section can be extended, in principle, to polymer solutions and melts. The main problem is that our knowledge of the hydrodynamic interaction between different parts of the chain, and between different chains is severely lacking, due to the difficult topological constraints on a concentrated polymer system (crossing and knots). The alternative to a Langevin description is to overlook the fine details of the micromechanics, and only at tempt to model the gross features of the chain motion in a concentrated system. There are two successful constitutive theories: network and reptation theories. The network theories of polymeric liquids as described by GREEN and TOBOLSKY, 129 LODGE, 13~ YAMAMOTO, TM and WIEGEL and DE BATS 132 are really inspired by the success of the rubberlike elasticity theory, taken from JAMES and GUTH, and JAMES. 133 We describe the basis of the network theory in this section, and defer a discussion of reptation theories to the next section; our development is patterned after WIEGEL and DE BATS, and PHAN-THIEN. 134 A excellent review of the theory can also be found in BIRD et al., 13s and LARSON. 136

35.1 A f f i n e M o t i o n

A convenient starting point is the rubberlike elasticity theory, where the polymeric material is envisaged as a collection of tangled polymer chains, as shown in Figure 35.1. The entangled points, or junctions, are formed by either van der Waals forces, or chemical bonds, or simply interaction between different parts of the same chain, or different parts of different chains, the exact nature of which is not relevant here.

129GREEN, M.S. and TOBOLSKY, A.V., J. Chem. Phys., 14, 80-92 (1946). 13~ A.S., Trans. Faraday Soc., 52, 120-130 (1956); Elastic Liquids, Academic Press,

New York (1964). 131yAMAMOTO, M., J. Phys. Soc. Japan, 11,413-421 (1956); 12, 1148-1158 (1957); 13, 1200-

1211 (1958). 132WIEGEL, F.W., Physica, 42,156-164 (1969); WIEGEL, F.W. and DE BATS, F.Th., Physica,

43, 33-44 (1969). 133jAMES, H.M. and GUTH, E., J. Chem. Phys., 11,455-481 (1943); JAMES, H.M., J. Chem.

Phys., 15, 651-668 (1947). 134PHAN-THIEN, N., PhD Thesis, Department of Mechanical Engineering, University of Syd-

ney, 1978. 135BIRD, R.B., CURTISS, C.F., ARMSTRONG, R.C., and HASSAGER, O., Dynamics o] Poly-

meric Liquids: Vol. II. Kinetic Theory, John Wiley & Sons, New York, 2nd Ed., 1987, w 136LARSON, R.G., Constitutive Equations for Polymer Melts and Solutions, Butterworth Pub-

lishers, Boston, 1988, w


M boundary p

junctions ._ n e t w o r k s t r a n d

FIGURE 35.1. An idealised network of polymer chains.

The chain between two junctions, or a network strand, usually consists of about 103 skeletal bonds, for an average strand molecular weight of the order 104.137 If there exist enough junctions, the whole mass of polymer molecules will become linked together in a coherent network, that is, every junction is connected by at least two unbroken network paths to the boundary of the sample. To prevent this network from collapsing into a small volume under the action of Brownian motion, the boundary points must be constrained to lie on the bounding surfaces, and deform with the global displacement imposed on these surfaces. These boundary junctions are sometimes called fixed junctions, and the interior junctions are free junctions. If the deformation is small, the network strands will not be highly deformed, and their statistics will be Gaussian. 13s The entropic force arising in the Gaussian network strand due to the large amount of configurations available to it has been considered before in the previous section, and we have

3kT Fv, = Nv~,b2Pv~,, (no sum) (35.1)

where Nvu is the number of Kuhn segments in the strand, each of length b, and p~u = ru - rv is the end-to-end vector of a network strand. The subscripts v# represent the strand connecting junction v to junction #.

For a Gaussian network, the probability distribution function is

1 } P ( r l , r 2 , . . . ) -- Qexp -- Z ' ~ v Pv~ ''pv~` ,

/z>v

where Q is a normalised constant, and the summation is over all junctions - if there is no strand between junctions v and #, then Nv~ is set equal to oo for this pair of junctions. The most probable configuration of the network, denoted by {r~ ~ r (~ will maximise P, that is,

1 E (r;0,_ -- 0 p

laTFLORY, P.J., Proc. Roy. Soc. Lond., A351, 351-380 (1976). 13SjAMES, H.M., J. Chem. Phys., 15,651-668 (1947).


This linear equation involves both the fixed and the free junctions; it is clear tha t the position vectors of the free junctions are linear functions of the position vectors of the fixed junctions. Tha t is, the free junctions must move in the same manner as the fixed junctions on the average. This is an important conclusion due to JAMES and GUTH. 139 Thus, if the boundary junctions are given an a]fine mot ion ,

i'v = L- rv,

where L = (~Tu) T is a global velocity gradient tensor, then all the free junctions, and the end-to-end vectors between every pair of junctions, must also move in the same affine motion on the average, i.e.,

L- (35.2)

Non-affine motion of the network strands arises in the case of a non-Gaussian network, or when the boundary junctions move non-affinely. PHAN-THIEN and TANNER, 14~ and JOHNSON and SEGALMAN T M assume a non-affine motion of the form

hv~ : ( L - C D ) - p ~ - s (35.3)

where ~ is a constant, a n d / 2 = L - ( D is an effective velocity gradient tensor. The motivation leading to this special form for the effective velocity gradient is based on the notion tha t the network slips with respect to the effective medium, and this slip velocity is an isotropic function of the strain rate and the end-to-end vector of the network strand. Tha t is,

,bv~ ̀ - Wp= ,~ , - ? ' / ( D , P v ~ ) ,

where W is the vorticity tensor, and 7 / is an isotropic function of D and Pv~. This is the mathematical s ta tement tha t the slip velocity/~v, - L . Pv~ is frame- indifferent. Now the isotropic tensor can be expanded in terms of the invariants of D and Pv~, and a linear version of this is

h,~ = (L - CD)- p ~ - a p ~ .

The term a p v ~ is not important at the end; furthermore, as BIRD et al. 142 have pointed out, the inclusion of this te rm implies either the end-to-end vector of the network strand collapses to zero, or increases without limit when there is no flow imposed. This te rm is therefore excluded, leading to the effective velocity gradient s already mentioned. This is also the non-affine motion proposed by GORDON and SCHOWALTER, 143 in their constitutive modelling of dilute polymer solutions.

139JAMES, H.M. and GUTH, E., J. Chem. Phys., 11,455-481 (1943). 14~ N. and TANNER, R.I., J. Non-Newt. Fluid Mech., 2,353-365 (1977); PHAN-

THIEN, N., Trans. Soc. Rheol., 22, 259-283 (1978). 141JOHNSON, M.W., Jr. and SEGALMAN, J., J. Non-Newt. Fluid Mech., 2, 255-270 (1977). 142BIRD, R.B., CURTISS, C.F., ARMSTRONG, R.C., and HASSAGER, O., Dynamics of Poly-

meric Liquids: Vol. II. Kinetic Theory, John Wiley & Sons, New York, 2nd Ed., 1987, w 14aGORDON, R.J. and SCHOWALTER, W.R., Trans. Soc. Rheol., 16, 79-97 (1972).


35.2 Constitutive Equation

With all the frictional resistances located at the network junctions, the stress tensor contributed by the network strands can be calculated in the same manner as in the dilute polymer solution case; it is given by

3kTnvt, r - - E Nv~,b 2 (P~P~} ' (35.4)

where the angular brackets denote an ensemble average with respect to the distribution of Pv~, and nv~ is the number density of the network strands. We now dispense with the subscripts, and consider a sub-network made up of strands with the same number of Kuhn units. The total stress contributed by the network will be the sum of all the contributions from different strands.

Let ~(p , t) be the probability distribution of network strands such tha t r t)dp is the probability of strands (with N Kuhn units) having configurations ranging from p to p+dp. The conservation of probability, i.e., the Liouville equation, states that

0r 0 + - a - ( a 5 . 5 )

where g and h e are the rates of creation and destruction of network strands of the same type for the given configuration; both contain constitutive information and are considered to be functions of p. The specific form of the rate of destruction of network strands is due to YAMAMOTO, 144 who assumes that h is a constant as well; tha t is, the rate of destruction of network strands is proportional to the number of network strands present. Here, we leave the dependence of h on p and other deformation parameter unspecified.

There is nothing specific about the rate of creation of network strands, except that at equilibrium we must have

g = h0r

where the subscript 0 denotes the equilibrium value. On this basis, it has been suggested tha t g may be considered as a function of p.

Now, the stress tensor for this sub-network is given by

3kTn J I"= Nb 2 pp'Odp. (35.6)

After a change of integration variables into a reference configuration P0, with the help of the Jacobian

J -- det ( 0~~0),

we find (an isothermal flow is assumed here):

d m , , r - - .

dt 3kTn / Nb 2 { (i~p + pi~)r + pp~bJ + pp~2J} dpo.

144yAMAMOTO, M., J. Phys. Soc. Japan, 11,413-421 (1956).


From the assumed non-affine motion for the network strands and the Liouville equation, and

we find that

J - J t r ( 0 ~ ) - J t r Z : - 0 ,

d f (35.7) - - - I " - f_.,-r- ~.s _ 3 k T n dt -- Nb 2 ( g - he) ppdp.

To proceed further, specific forms for g and h must be nominated. Here, we assume (a different assumption is treated in Problem 35.A)

g = x 2~}/b 2 3P 2 '~

gl ((p2>, T) exp - 2 N b 2 ) ,

1 h - ~ H ((p2),T),

where T is the temperature and A is a time constant; note that any other isotropic form for g will not change the following results. The integration over the configuration space can now be performed and we find

{d } A ~-~-r- s - r s T + H ((p2), T) r = C 1 1 , (35.s)

where C 1 = n k T g l ((p2}, T). This is the generic form of the constitutive equation derived from network theory. The equivalent integral form to (35.8) is given by

f ~'(t) = 1 cr " ~ c l (tl) ( l f t l ) e x p - ~ H ( t " ) d t " B ( t , t ' ) d t ' , (35.9)

where B(t, t') is the 'Finger' strain tensor corresponding to the effective velocity gradient L:

d B(t, t') -- s t') + B(t, t ' )s dt

B(t ' , t ' ) = 1. (35.10)

Note that B can be written in terms of the effective deformation gradient E(t, t'), i.e., the deformation gradient corresponding to the effective velocity gradient 15, as (cf.(1.37) and (2.38)):

B (t, t') = E(t, t') E(t, t') T, (35.11)

where d E ( t , t') -- l :(t)E(t, t'), dt

E(t', t') - 1. (35.12)

P r o b l e m 35 .A Suppose the rates of creation and destruction of network strands are given, re-

spectively, by

1 ( 3p2'~ g - xQ ~xp - 21qb~ ] '

1 ( p2) h - X 1 + ~ - ~ ,

where c is a small parameter, Q is a Gaussian normalised constant, and A is a t ime constant. Normalise p by x/Nb 2, t ime and velocity gradient by A, and look for a series solution in the form

r ~E~r r - - 0

Show tha t

( O + s r Qexp ( _ 3 p 2 ) - r

+ l~ijpj ~)r -- --~)r -- P2r r -- 1 , 2 , . . .

Denote the second-order moment by B,

(35.A1)

(35.A2)

B - - <pp> -- E erB(r) ' r -0

where

Show tha t

B ( r ) - <PP>r = / PPCrdP.

d B ( 0 ) _ s ) _ B(0)s + B(0) _ 11, 3 (35.A3)

d n ( r ) - - ~ B ( r ) - B ( r ) ~ T 4- n ( r ) - - --<p2pp>r_ 1 r > 1 (35.A4) t ' - - '

d 5 "~<p2pp>o -- E.<p2pp>o -- (p2pp>os + <p2pp>o -- -~1

+ 2 s �9 <pppp>, (as .As)

d -~<PiPjPkPz>o -- Lira (PmPj PkPl>o -- Ljm <PiPmPkPz>o -- Lkm <PiPj PmPl>o

--l:.lm<pipjpkpm>O + (PiPjPkPl>o -- Aijkl, (35.A6)

where the components of A are equal to zero except when all indices are equal or pairwise equal, and

1 Al l l l ---- A2222 -- A3333 -- ~,

1 Al122 -- Al133 - A2233 - ~.

Returning to the general form of the constitutive equation (35.8), if we assume tha t G1 is proport ional to H, i.e.,

G G1 -- H,

and eliminate the isotropic pressure term through

G r = -----I +S,

l-f


then the constitutive equation for S is

A { d s - s + H( t r S T)S - 2GAD dt ' '

(35.13)

where the dependence on tr S arises because of the proportionality between p2 and tr S. Note tha t tr S is unrelated to the hydrostatic pressure, which is determined by the balance of momentum. To take into account the distribution in N, we allow for different sub-networks, where each has a different relaxation time and modulus. Thus the constitutive equation is described by the following set of equations

S - E S(0' (35.14) i

d S(i) _ s _ S ( i ) s } -~- H(t r S (0 T)S (0 - 2GiAiD. ~ ~ (35.15)

35.3 SomeSpecialCases

The parameters in the network model consist of a discrete spectrum of relaxation times and moduli {Ai, Gi}, the non-affine parameter ~, and the functional form for H. The moduli can be regarded as being proportional to the temperature, but there is no information on either the spacing of the spectrum or the dependence of the relaxation times on other molecular parameters. The t ime-temperature superposition principle will hold if we assume that

H( t r S ( 0 , T ) - r S(0), (35.16)

where at the reference temperature To at which the relaxation spectrum is measured we require tha t r -- 1.

When H -- 1 and r - -0 , we recover the rubberlike liquid model of LODGE. 145 PHAN-THIEN and TANNER 146 have suggested two empirical forms for Y:

C 1 + ~ t r S(0,

Y(tr S (0) --

s exp ( ~ t r S(0) �9

The difference in these two can be seen through the elongational viscosity. With the linear form, r/E is monotonic in the elongational rate, which approaches a constant at high values of the elongational rate. For the exponential form, r/E goes through a maximum and then decreases at high elongational rates, since the rate of destruction overwhelms the rate of creation of network strands. The linear form is algebraically simpler, and is often referred to as the PTT model. When e -- 0, the

145LODGE, A.S., Trans. Faraday Soc., 52, 120-130 (1956). 146PHAN-THIEN, N. and TANNER, R.I., J. Non-Newt. Fluid Mech., 2,353-365 (1977); PHAN-

THIEN, N., Trans. Soc. Rheol., 22, 259-283 (1978).

model reduces to the Johnson and Segalman's model, 147 which has been derived on purely continuum considerations. In addition, if ( -- 1, then the co-rotational model results; and, if ( = 0, the upper-convected Maxwell model, or the co-deformational model, or the rubberlike liquid model of LODGE, which is usually written in the equivalent integral form

f S - m ( t - t ' )B ( t , t ' ) d t ' , (35.17) O 0

where B(t, t') is the Finger strain tensor, and

Gi e t/h, (35.18) = -

i

is the memory function, is obtained.

P r o b l e m 35.B Show that in a small strain oscillatory flow where the velocity gradient tensor is

L = 6eJWtm, 6 << 1, j2 = -1 ,

the partial stress for the PTT model is given by

S(i) _ 6eio, t GiAi 1 + jwAi (m + mT) . (35.B1)

Thus, show that the complex viscosity is given by

CiAi (35.B2) ~* -- E 1 + jwAi

i

or, Gi)q

~7'=E. I + ( w A , ) 2 ' $

Ci ( ~ i ) 2 C'-- Z. 1 + (wAi) 2" (35.B3)

$

In a simple steady shear flow with shear rate -~, show that

GiAi rl = Z I + ((2 - ()Ai2"~ 2" + O(e), (35.B4)

i

2G~,x~ 2 N1 -- Z 1 + r - r 2 + O(e), (35.B5)

i

~ N , (35.B6) N 2 - - - 7 1

t r S ( i ) _ 2(1 r 2.2 - &7 + o ( 0 Gi -- 1 + r -- ~))~i~2" 2 "

(35.B7)

147JONHSON, M.W., Jr. and SEGALMAN, D., J. Non-Newt . Fluid Mech., 2,255-270 (1977).


Thus show that , up to O(E),

( x ) - v 4c(2- c) '

(35.B8)

1 ( x ) 35B9 ?}it(X) -- "~X~ 1 4 , ( 2 _ ~) ,

where ~1 -- N1/9/2 is the first normal stress coefficient. In an elongational flow, where the elongational rate is ~, show tha t the first

normal stress difference for the P T T model with the linear trace function is given by

3G{A(~(I + eX{) S l l - $ 2 2 - Z . [1 + eX~ - 2(1 - r [1 + eX~ + (1 - r = ~/E~/' (35.B10)

where 1

X ~ - - - - t r S (0 (35 .Bl l ) G~

is the normalised trace of the partial stress S(0, which satisfies the cubic equation

X [1 + eX~ - 2(1 - r [1 + eX~ + (1 - ~)A~] - 6(1 - r 2. (35.B12)

There is only one real solution to this, which is not written here for brevity. Show that in the limit of large elongational rates,

cX = 2 ( 1 - r 1 +

leading to the elongational viscosity

E 1 - r + " " (35.B13)

~E -- 2(1 -- r Z G..~A, _ 2(1 - r176 (35.B14) c s i

at high elongational rates, where r/0 is the zero-shear-rate viscosity.

The discrete spectrum of relaxation times and moduli should not be regarded as arbi trary adjustable parameters; given a linear relaxation spectrum of the fluid, {G~, Ai} are fully determined once and for all. The parameter ~ governs the shear response of the fluid, and should be determined from such a flow, for example, from the relation between 7' and 77 described through (35.B8). This parameter is typically of the order 0.2, from the experimental data on the second normal stress difference. The predictions of the P T T model in shear flow are given as solid lines in Figure 34.7, using r - 0.2, e - 0.015 and eight relaxation times determined from the oscillatory flow data, spanning 10-3s to 103s. 14s

14SAdapted from BIRD, R.B., CURTISS, C.F., ARMSTRONG, R.C., and HASSAGER, O., Dynamics o] Polymeric Liquids: Vol. H. Kinetic Theory, John Wiley & Sons, New York, 2nd Ed., 1987, Figure 20.5.1.


lO 6

10 5 o

~. 10 4

O .= 10 3

~ 2 o 10

m

I I I I I 10 -1 10 0 101 10 2 10 3

Time (s)

FIGURE 35.2. Comparison of the PTT model predictions with LPDE data in the starting up of an elongational flow, adapted from Bird et al., 1987, op. cit.

The parameter E does have a weak influence on the response in shear flow, since it only enters at O(E). The condition for neglecting O(E) terms is tha t t r S(0/G{ should be of O(1). From (35.B7), even at an infinite shear rate, trS({)/G{ ~ 4 when

: 0.2. However, from the results of the Problem 35.B it is seen tha t e governs the elongational response of the fluid; in the P T T model the elongational viscosity is O(e-1). From L D P E data as reported by WAGNER, 149 e : 0.015 is about the best value. Figure 35.2 is a comparison between the prediction and the data on L D P E in the s tar t-up of an elongational flow, using ( : 0.2, c : 0.015 and eight relaxation times determined from the oscillatory flow data, spanning 10-3s to 103s, 15~ the same set of parameters used to fit shear data in Figure 34.7.

The shortcomings of the P T T model are apparent in fast flows, for example, in a single step strain flow where the strain rate is @ : ~/6(t); ~/is the step strain. In this flow, it can be shown tha t the P T T model with c : 0 or the Johnson-Segalman model gives

S12 : (~(t) sin ('yv/(~(2 - (~))

v'r r

4 G ( t ) s i n 2 (_~ V/(~( 2 _ (~)) N1 : ~ ( 2 - ~)

(35.19)

149WAGNER, M.H., Rheol. Acta, 18, 39-50 (1979). 15~ from BIRD, R.B., et al., 1987, op. cit., Figure 20.5.3.


where G ( t ) is the relaxation spectrum of the model. The Lodge-Meissner rule TM

N1 = 1 (35.20) ~/S12

is no longer obeyed by this model, except at small strains. More seriously, the shear stress can become negative at large strain. Allowing c ~ 0 will fix this, but the likely problem is that the non-ai~ne motion adopted does not fully account for structural changes in a fast flow.

JONGSCHAAP 152 has shown that the model proposed by ACIERNO et al. 153 can be derived from a network theory, with a particular choice of the rates

of creation and destruction of network strands, which are deformed affinely. The partial stress of this model obeys

Ai~-~ ~, Gi ,] + " ~ / - 2AID, (35.21)

where ~tt is the upper convected derivative, and

G~ - G i o x ~ , A~ - A i o x 1"4, (35.22)

~_~ tr S(0 A~ = l - x~ - ax~ 2G~ " (35.23)

Here, x~ is a structural parameter, { G i o - N i k T , A~o} are the modulus and relaxation spectra at equilibrium; the empirical law (35.22) is motivated by the fact that the shear viscosity (GiA~) is proportional to the 3.4 power of the concentration; and a is a parameter of the model. A good fit to LDPE data has been obtained with a single value of a = 0.4.1 s4

There are other models which may fit into the network picture, but are not reviewed in this section because of space constraint. Most of these models conform to the pattern,

s - Z sr (35.24) i

---~S (0 + 1 S ( 0 = Gi l , (35.25)

where ~ is the GORDON and SCHOWALTER 155 non-affine convected derivative

~t d S( 0 _ (L - CD)S (0 - S (0 (L - CD)T s r (35.26)

151LODGE, A.S. and MEISSNER, J., Rheol. Acta, 11,351-352 (1972). For a derivation based on simple fluid theory, see (A5.15) below. t52jONGSCHAAP, R.J.J., J. Non-Newt. Fluid Mech., 8, 183-190 (1983). 153ACIERNO, D., LA MANTIA, F.P., MARRUCCI, G. and TITOMANLIO, G., J. Non-Newt.

Fluid Mech., 1, 125-146 (1976). 154ACIERNO, D., LA MANTIA, F.P., MARRUCCI, G., RIZZO, G. and TITOMANLIO, G.,

J. Non-Newt. Fluid Mech., 1, 147-157 (1976). t55GORDON, R.J. and SCHOWALTER, W.R., Trans. Soc. Rheol., 16, 79-97 (1972).


and {Gi,Ai} can be functions of the partial stress, e.g., tr S (i), the strain rate invariants, or both. This differential equation can be recast in the equivalent integral form

where B(t, t') is the 'Finger' strain tensor corresponding to the effective velocity gradient s = L - CD, as defined in (35.10).

Some notable successes are the White-Metzner, 156 Giesekus, 157 Leonov, 15s and Wagner models. 159 The Wagner model, with an in-built irreversibility mechanism due to the use of the minimum function in its memory kernel, may not fit in the class of simple fluids with fading memory.

36 Reptation Models

The network theory is quite adequate in producing a constitutive framework for polymeric liquids. However, it does not provide any information on the relaxation spectrum, nor the dependence of the gross properties of the liquid on its molecular weight; in particular it cannot explain the M 3"4 dependence of the viscosity. An alternative view of a polymer chain in a network of other chains, which is widely accepted now, is the reptation model of DE GENNES. 16~ In this model, a polymer chain is envisaged to be constrained in a tube-like environment created by the surrounding chains and deformed with the tube in a cooperative manner. However, the transversal motion of the chain is greatly impeded by the tube constraint, and a large scale rearrangement of the configuration of the chain can only be achieved by a snake-like motion, or reptation, along its own length. Since the configuration of the chain is determined not by the original tube it occupied, but by the new tube-like environment it just moves into, the orientations of the chains should be calculated simultaneously as a multi-bodied problem. In the Doi-Edwards theory, 161 each chain segment (of a constant primitive length) is assumed to deform independently, and this is the independent a l ignment assumption. Thus, the motion of a chain segment is brought about by a three-stage process: an instantaneous deformation, followed by a quick retraction to its original length, and finally by a slow reptation mechanism out of the tube it originally occupied. The net effect of the first two stages is to orient the chain without stretching it, and the last stage allows a relaxation mechanism. The stress generated by the deformation is calculated from the free energy expression, and is given by the Gaussian network expression. The Doi-Edwards theory is therefore very similar to the kinetic theories of rigid rods

156WHITE, J.L. and METZNER, A.B., J. Appl. Polym. Sci., 7, 1867-1889 (1963). 157GIESEKUS, H., J. Non-Newt. Fluid Mech., 11, 69-109 (1982). 158LEONOV, A.I., Rheol. Acta, 15, 85-98 (1976). 159WAGNER, M.H., Rheol. Acta, 18, 33-50 (1979). 16~ GENNES, P.G., Scaling Concepts in Polymer Physics, Cornell University Press, New

York, 1979. 161DOI, M. and EDWARDS, S.F., J. Chem. Soc. Faraday Trans., 74, 1789-1832 (1978); 75,

38-54 (1979).

36 Reptation Models 231

(rotation without stretching); it may be regarded as a blend between rigid-rod and network theories.

Since the full details of the Doi-Edwards theory can be found in their book, 162 we shall present only a brief summary of the model here.

36.1 D o i - E d w a r d s M o d e l

Following a deformation, the tube and the chain it contains are assumed to deform affinely, i.e., the end-to-end vector of the chain is given by

R = FRo,

where R0 is the end-to-end vector initially, and F is the deformation gradient. This is followed by a retraction process of the chain within the tube to bring its contour length back to its equilibrium value, without changing its direction. The time scale for this retraction process is small, compared with the t ime scale of the reptation process tha t soon follows. Thus, if we ignore the retraction process, then the end- to-end vector R of the chain deforms as if it were a rigid rod, and we obtain, after the retraction process,

R - - FRo (36.1) IFp01'

where P0 = R 0 / I R I is the unit vector along R0. The stress at the end of the retraction process is calculated by assuming a Gaussian network of n strands per unit volume:

( F p ~ 1 7 6 ) (36.2) 3 n k T <RR) -- 3 n k T 12 , S o = R2 I Fp0 0

where the angular brackets <->0 denote an ensemble average with respect to the equilibrium distribution of the chains. Implicit in the theory is that each chain deforms independently of others, which is the so called independent alignment approximation.

After the retraction process, the chain will reptate along its own length to sample different configurations. DE GENNES 163 assumes tha t this is a one-dimensional diffusion process and we outline the argument here. At t ime t - 0, suppose there is a chain of contour length L in a tube. Let x be the arc length variable along the contour, and r t; s) be the probability density function tha t the chain has moved a distance x along its length at t ime t, given tha t neither end has reached the point s on the contour. At time t - 0, the chain has not moved, and the initial boundary condition for r t; s) is

o; = (36.3) Since neither end has reached the point s on the contour, the boundary conditions a r e

r t; s) - O, r - L, t; s) = O. (36.4)

162DOI, M. and EDWARDS, S.F., The Theory of Polymer Dynamics, Oxford University Press, Oxford, 1988. 163DE GENNES, P.G., Scaling Concepts in Polymer Physics, Cornell University Press, New

York, 1979.

Implicit in this is tha t r -- 0 for x > s and x < s - L, tha t is the chain can only move forward along its length a distance s, or backward a distance L - s. This one-dimensional reptation process obeys

0 r = Dc 02r (36.5) Ot cgx 2 '

where Dc is the diffusivity of the process. From the Rouse model, we could identify Dc with the diffusivity of the centre of mass of the Rouse molecule, i.e.,

Dc = k T (36.6) Nr

where ~ is the frictional coefficient felt by the chain along its length, and N is the number of Kuhn units in the chain, each of length b. The root mean square the Rouse chain is N b 2 -- Lb; similarly, if the reptating chain consists of a number of units, each of step length a, then

La - N b 2. (36.7)

The step length a can be regarded as the diameter of the tube, and (36.7) serves as its definition.

Now, from a separation of variables, the solution to (36.5) can be found:

r t; s) -- ~ ~ sin s sin - - ( L - x ) e x p \ A , ] ' i

(36.8)

where )~1 will be seen to be the largest relaxation time of the reptation process,

L 2 / ~ 1 - 7r2D. (36.9)

The probability of the tube to remain somewhere between the points x = - L + s and x = s is simply the integral of the probability density distribution (the upper and lower limits on the integral can be replaced by •

f 8

P ( s , t) = r t; s ) d x s - L

4 sin s exp i odd

(36.10)

Since 0 < s < L, the fraction of the original tube tha t remains occupied by the chain is

--/L P(s, t)ds 1 P ( t ) - Y d o

i odd

(36.11)

where A~ = ) t l / i 2. Thus, the fraction of strands that enter the tube at time t ' and still remain in

the original tube at t ime t is P ( t - t~), by a shift in the time axis: note that the problem is linear. The fraction of strands that enter at time t ' + At ' and still remain in the original tube at time t is

8 ( t - t ' ) P(t - t' - At') -- P( t - t') + .~odd" 7r2i2)~i exp -- )~i At ' + O(At ') 2 (36.12)

The term underlined in (36.12) is the fraction that remains at time t of those strands that enter during a period At' at t ime t', to within an error of O(At ') 2. The stress contributed at the present time from all strands will be a linear superposition of the individual contributions:

S t -- odd24nkT<Fp~176 I I ( t-t')A, S(t) -- ~-~., ,Fp0,2 exp dr' - - (x) 0

t

-- m ( t - t ' ) q d t ' , - - ( : ~

(36.13)

where the memory function is given by

. , (t,) m ( t - t ' ) - - ~ ~Texp - A, ' (36.14)

i odd

G1 2 4 n k T G~ = "~5-, G1 = 57r2 , (36.15)

and

Q - 5 Fp~176 , IFpo o

(36.16)

called the geometric universal tensor. The factor 5 is inserted so that this tensor reduces to the strain tensor when the strain is small. Equation (36.13), together with the specific form of the memory function is the Doi-Edwards model in its simplest form. There are other forms, and the reader is referred to DOI and EDWARDS 164 for a fuller explanation. Apart from the two parameters G1 and A1 that govern

the linear relaxation response, the model contains no additional parameter. Its prediction of molecular diffusivity is excellent. Its rheological predictions, considering that there is no free parameter involved, is average: in a simple shear flow, the amount of shear thinning is excessive, and there is no normal stress overshoot in the start-up of the flow. The second normal stress difference is predicted to be - 2 / 7 of the first normal stress difference, which is not far from the experimental results on most polymer systems. A careful evaluation of Doi-Edwards theory in linear

164DOI, M. and EDWARDS, S.F., The Theory of Polymer Dynamics, Oxford University Press, Oxford, 1988.


viscoelastic regime, and in elongational flows have been made by LODGE, ROT- STEIN and P R A G E R , 168 and WAGNER. 166 WAGNER and S C H A E F F E R 167 introduce a slip-link modification to the Doi-Edwards theory to obtain a bet ter fit to elongational data. In general, the Doi-Edwards theory provides the correct quali tative t rends in the viscometric functions, but the viscosity tends to decrease far too rapidly. 168 HASSAGER t69 has also pointed out tha t the Doi-Edwards theory cannot describe the Weissenberg rod-climbing effect at small deformation. Despite its average performance in the predictions of various rheological functions, the Doi-Edwards theory provides an excellent framework under which the dynamics of entangled polymer chains can be effectively studied.

Various modifications and approximations of the Doi-Edwards theory are summarised in MARRUCCI , 17~ and LARSON; 17t we outline a few well known approximations below.

3 6 . 2 A p p r o x i m a t i o n s

The Doi-Edward model can be wri t ten in the familiar K-BKZ form 172

{ ou OU c } S ( t ) - - cr m ( t -- t ') 2 7 -1 C t l ( t ' ) -- t( t ' ) tit', (36.17)

where U = U ( I I , I 2 ) is the potential function, and/1 ,2 are the strain invariants:

I1 - tr C t 1, 12 = t r Ct. (36.18)

For the Doi-Edwards model, the potential function cannot be writ ten in a closed form for a general deformation; it must be obtained numerically for a given deformation. C U R R I E 173 has found tha t the Doi-Edwards potential function can be well approximated by

5 ( J - l ) U(I1,12) -- "~ In 7 '

leading to

f {' S(t) -- m ( t - t ') J - 1 C t l ( t ' ) - O 0

J = I1 + 2~/I2 + 13/4, (36.19)

't

5 Ct ( t ' ) ~ dr'. (36.20) ( J - 1 ) V / I 2 + 1 3 / 4 J

165LODGE, A.S., ROTSTEIN, N.A. and PRAGER, S., Adv. Chem. Phys., 79, 1-132 (1990). 166WAGNER, M.H., Rheol. Acta, 29, 594-603 (1990). 167WAGNER, M.H. and SCHAEFFER, J., J. Rheol., 36, 1-26 (1992); Rheol. Acta, 31, 22-31

(19~2). 168SAAD, H.H., BIRD, R.B. and CURTISS, C.F., J. Chem. Phys., 77, 4747-4766 (1982); BIRD,

R.B., SAAD, H.H., and CURTISS, C.F., J. Phys. Chem., 86, 1102-1106 (1982). 16~ 0., J. Rheol., 29, 361-364 (1985). 17~ G., in Transport Phenomena in Polymeric Systems, Ed. MASHELKAR, R.A.,

MAJUMDAR, A.S. and KAMAL, R., Wiley-Eastern, Delhi, 1-36 (1987). 171LARSON, R.G., Constitutive Equations l'0r Polymer Melts and Solutions, Butterworth Pub-

lishers, Boston, 1988. 172Independently proposed by KAYE, A., College of Aeronautics, Crankfield, Note No 134

(1962), and BERNSTEIN, B., KEARSLEY, E. and ZAPAS, L., Trans. Soc. Rheol., 7, 391-410 (1963). 173CURRIE, P.K., Proc. VIII Int. Cong. on Rheology, Naples, Vol. 1, 357-362 (1980).


Another simple approximation, due to LARSON. 174 is given by

s ( t ) - ~ m(t - t') s + (h - 3) c ; i ( t ' ) d t " ( 3 6 . 2 1 )

This should be compared with the Lodge model where the front factor of the Finger strain tensor is unity. Since/3 _> 3, the Doi-Edwards model predicts a much softer response than the network model; this is due to the retraction mechanism tha t is built into the model.

36 .3 D i f f e ren t i a l Mode l s

Differential versions of the Doi-Edwards model have also been proposed independently by MARRUCC1175 and LARSON. 176 The main idea is tha t the microstructure deforms just like a rigid rod. Indeed, the same idea can be applied to the s tandard network theory and we outline it here. Thus, we start with the s tandard network model, except tha t the network strands are allowed to deform in the manner described in the Doi-Edwards theory. If the retraction process has a small t ime scale, then the s trand deforms just like a rigid rod, tha t is, if the end-to-end vector of the strand is denoted by R -- Rp, where p is a unit vector directed along the chain, then

15 -- L p - L �9 ppp . (36.22)

The solution to this is

where

q (36.23) p - ~ ,

(~ - - LQ, Q - Q L . p p . (36.24)

The stress tensor is calculated from the formula

S = 3nkT(pp> - 3nkT / - ~ 2 r (36.25)

where the angular brackets denote the ensemble average with respect to the probability distribution ~, which is governed by

= g - hV, (36.26)

where, as before in the network theory, g and h e are the creation and destruction rates of the network strands. The stress integral can be converted into the reference configuration Q0 via the Jacobian g - det 10Q/0Q01, and we have, from the evolution of Q and ~,

- LS - SL T + 6 n k T D " (pppp> -- 3 n k T f p p (g - he ) dQ. (36.27) J

174LARSON, R.G., Constitutive Equations for Polymer Melts and Solutions, Butterworth Pub- lishers, Boston, 1988, w 175MARRUCCI, G., in Advances in Transport Process, Ed. MAJUMDAR, A.S. and

MASHELKAR, R.A., Wiley Eastern, New Delhi, 1-36 (1987). 176LARSON, R.G., J. Rheol., 28,545-571 (1984), w


If we assume that g is isotropic in Q, then the integral involving g is just an isotropic tensor,

3nkT p p g a q -

where G = nkT and ~ is a constant. Furthermore, if h -- l/A, then the constitutive equation for S is

) S + )~ ~ - + 6GD" (pppp) - G1. (36.28)

This is the differential equivalent of the Doi-Edwards model, with the independent alignment assumption.

P r o b l e m 3 6 . A Assume that the velocity gradient is small, show that

D" ( p p p p ) : 2 D + 2 ~ (LD + DL T + LTD + DL + D " D 1 ) + O(L3).

Thus, show that the steady-state solution to (36.28) is the second-order fluid given by

S - - G I + 6 G ) ~ D + 6 G , k 2 ( L D + D L T ) - 1 2 G ) ~ 2 ( L T D + D L + D ' D I ) 35

with an error of O(L3). Deduce that the viscometric functions at small shear rates are

12 71- 3GA, ~I/1- 34--25 GA 2, ~I/2--~-~GA 2. (36.A1)

The ratio of the second to first normal stress differences is N2/N1 -- - 2 / 7 , as shown in the Doi-Edwards model.

To obtain a usable constitutive equation a closure scheme must be adopted, expressing the fourth-order moment (pppp) in terms of the stress and its invariant. There are infinitely many ways to do this. If we choose the Peterlin approximation:

D : (pppp) -- D : (pp)(pp),

then we arrive at the differential constitutive equation proposed independently by MARRUCCI and LARSON:

AS 2___ D ) S4-A - ~ + 3 G "SS - 0 1 , (36.29)

which has been known to reproduce the Doi-Edwards response well. 177 Peterlin's approximation is valid in a strong flow, but does not have the correct behaviour in weak flows. For example, D" (pppp) ~ 2 D + O(D2), and this cannot be well approximated by D : (pp)(pp) _ O(D2). Furthermore, Peterlin's approximation gives a zero second normal stress difference at all shear rates.

177LARSON, R.G., Constitutive Equations for Polymer Melts and Solutions, Butterworth Pub- lishers, Boston, 1988, w


Several better closure schemes have been discussed by HINCH and LEAL; 178 simple one tha t has the correct strong and weak flow behaviour is

1 D " ( p p p p ) "-' ~. ( 6 ( p p } . D . ( p p ) - D �9 (pp)(pp} - 2(pp} 2" D1

4-2(pp) : D 1 ) .

With this closure scheme, a differential form of the Doi-Edwards model is

AS 2 ( 6 S . D . S - D . S S - 2 D . S 2 1 4 - 6 G D . S 1 ) } - G 1 . L 7 +

(36.30)

The response of this model in shear flow is remarkably close to tha t of the Doi- Edwards model. In particular the normal stress difference ratio is N2/N1 -- - 2 / 5 .

To improve on the N2 prediction, we can follow LARSON 179 and assume tha t an incomplete retraction process can be modelled by the non-affine deformation

I ~ - L I ~ - c~D �9 p p R , (36.31)

where a is a constant. Clearly the unit vector p evolves in t ime according to (36.22), and the chain retracts according to

/ ~ - (1 - a ) R D : pp. (36.32)

The solution to (36.22) is p = Q / Q , where q deforms affinely. In a similar manner to the derivation of (36.28), it can be shown tha t

S + A - ~ + 6 a G D . ~ p p p p -- G1. (36.33)

LARSON uses the Peterlin approximation and ignores the dependence on R 2 to arrive at the form

) s "NT" + g 0 D . s s - a l , (36.34)

which has been found to agree well with the 'exact ' Doi-Edwards model for a = 0.35. A better approximation, which captures both the weak and strong flow limits is

3GD" ~--b~pppp N l t r s ( 6 ( p p ) . D . ( p p ) - D" (pp) (pp)

- 2(pp) 2" D1 + 2(pp) �9 D 1 ) .

This leads to the constitutive equation

{ A S 2c~ (6S.D.S _ D . SS _ 2D . $21 + 2tr SD . S1) } : G1, S4-A - ~ + 5 t r S

(36.35)

178HINCH, E.J. and LEAL, L.G., J. Flu:d Mech., 76, 187-208 (1976). 179LARSON, R.G., J. Rheol., 28, 545-571 (1984).


which should be somewhat better than Larson's model in a steady shear flow. For instance, we note that this constitutive equation predicts the following:

S12= 1 - g c ~ G , V ~ + . . . ,

( ' / N~ - 2 1 - g ~ G,~2,~ 2 + . . . ,

' ( ' / N2 = - g ~ 1 - g ~ C,X2,~ 2 + . . . ,

leading to the normal stress ratio N 2 / N 1 = -2c~/5; the value c~ = 0.35 corresponds to N2/g~ = -0.14.

A comparison between the multimode Larson and P T T models has been made by LARSON. tS~ It is found that both models make relatively good predictions for all three set of HDPE data, in step shear, start-up elongational, and step biaxial extensional flows.

3 6 . ~ C u r t i s s - B i r d M o d e l

CURTISS and BIRD 181 consider a polymer chain as a sequence of freely-jointed beads and rods that diffuses predominantly along its length, from a phase space consideration. From the assumptions of anisotropic Stokes' law and anisotropic Brownian motion, together with the mild-curvature assumption, they arrive at the constitutive equation

I' } S -- n k T m ( t - t ' ) (pp)t , dt' + 2eD- G( t - t ' ) ( p p p p ) t , d t ' . (36.36) o o

Here F~(t')p'

p = iFt(t,)p, I,

where p ' is randomly oriented at time t', and the angular brackets (.)t, denote the average with respect to this distribution. The first term in this expression is recognised as the Doi-Edwards stress (except for a factor of three), which is not surprising, since the Curtiss-Bird theory is basically that of a rotating rod. The second term is new; it arises from the assumed frictional forces that the surrounding chains exert on the reptating chain, and e is called the link tension coefficient (in an elongational flow, the tension along the chain is proportional to e). The second term helps to improve the fit in steady flows, but often leads to an overprediction in start-up and oscillatory flows. 182 A detailed comparison of Curtiss-Bird and Doi-Edwards theories has been provided in SAAD, BIRD and CURTISS. 183

lS~ R.G., J. Rheol., 28, 545-571 (1984). lSlCURTISS, C.F. and BIRD, R.B., J. Chem. Phys., 74, 2016-2033 (1981). lS2BIRD, R.B., CURTISS, C.F., ARMSTRONG, R.C., and HASSAGER, O., Dynamics of Poly-

meric Liquids: Vol. II. Kinetic Theory, John Wiley & Sons, New York, 2rid Ed., 1987, w ls3SAAD, H.H., BIRD, R.B. and CURTISS, C.F., J. Chem. Phys., 77, 4747-4766 (1982).

37 Suspension Models 239

We note that Curtiss-Bird theory gives a much better fit to experimental data than the Doi-Edwards theory. In shear flow, and at lo~/shear rates, the ratio of the second to first normal stress differences is given by N 2 / N 1 - - 2 ( 1 - e)/7. Since the rod-climbing at small shear rates only occurs when N 2 / N 1 > - 1 / 4 , lsa an e-value greater than 1/8 is required, if rod-climbing effect is to be predicted by the theory, ls5 At high shear rates, the model predicts

and

T~--O(IAfI-1 ) , N1-----O(lnl~l) , e ~ O

i 2 - O (];~]-1/2) .

With the Doi-Edwards model (e = 0), a much higher degree of shear thinning

occurs at high shear rates (7 -- O (]~/]-3/2) N1 - O (1))

The model predictions in a steady elongational flow agree well with experimental data on HDPE with e ~ 0.08; is6 however, a strain-softening behaviour in elongational flow is predicted.

Extensions of Curtiss-Bird theory to polydisperse polymer melts, ls7 to non- isothermal flows, lss to reptating rope model is9 have been made.

37 Suspension Models

3 7 . 1 I n t r o d u c t i o n

The concept of a suspension is only meaningful when there are two widely different length scales in the problem: 1 is a typical dimension of a suspended particle and L is a typical size of the apparatus. When these two length scales differ by several orders of magnitude, 1 << L, we have a suspension rather than a collection of discrete individual particles suspended in a medium. We will be concerned with Newtonian suspensions, i.e., suspensions of rigid particles (mainly spheroids) in Newtonian solvents; for other types of suspensions, the reader is referred to the reviews by METZNER, 19~ and OHL and GLEISSLE. 191

Every field variable, e.g., velocity, pressure, stress components, etc., will consist of a fast-varying component and a slow-varying component; 'fast' refers to variations on the microscale l, and 'slow' refers to variations on the macroscale L. The process of averaging out the fluctuations at the microscale level is called homogenisat ion;

for example, see SANCHEZ-PALENCIA. 192 This analysis is carried out by quan-

lS4BEAVERS, G.S. and JOSEPH, D.D., J. Fluid Mech., 69, 475-511 (1975). lSSBIRD, R.B. and C)TTINGER, H.C., Ann. Rev. Phys. Chem., 43, 371-406 (1992). lS6BIRD, R.B., SAAD, H.H., and CURTISS, C.F., J. Phys. Chem., 86, 1102-1106 (1982). lSTSCHIEBER, J.D., J. Chem. Phys, 87, 4917-4927 (1987); 87, 4928-4936 (1987). lSsWIEST, J.M. and PHAN-THIEN, N., J. Non-Newt. Fluid Mech., 27, 333-347 (1988). lsgGUERTS, B.J., J. Non-Newt. Fluid Mech., 28, 319-332 (1988); 31, 27-42 (1989). 19~ A.B., J. Rheol., 29, 739-775 (1985). 191OHL, N. and GLEISSLE, W., J. Rheol., 37, 381-406 (1993). 192SANCHEZ-PALENCIA, E., Non-Homogeneous Media and Vibration Theory, Lecture Notes

in Physics, Vol. 127, Springer-Verlag, Berlin, 1980.

tifying all important variables and their relationships to microstructural variables (volume fraction, two-point correlation functions, etc.), followed by an averaging process (ensemble or volume averages) of these quantities to yield the observable, or homogenised, or effective values. An important example is the development of the kinetic theory of liquids. 193 Here, LEVY and SANCHEZ-PALENCIA have found that the homogenised stress in a Newtonian suspension is anisotropic, but linear in the strain rate. 194 Finding the anisotropic viscosity tensor requires solving the micromechanic problem for the geometry concerned. Essentially the same form of constitutive equation has been arrived at by ADLER et al., 19s in their study of general periodic suspensions. NUNAN and KELLER 196 have considered periodic suspensions of spheres in simple cubic lattices, and have shown that the effective stress contributed by the particles is anisotropic and linear in the strain rate, with a fourth-order viscosity tensor depending on two scalar functions of the volume fractions of the spheres; in fact, these are the intrinsic shear and elongational viscosities. Numerical values of these functions have been given for a number of simple cubic lattices, showing that these effective viscosities increase quickly with the volume fraction, and asymptote to e -1, where e is the dimensionless gap between two neighbouring spheres. It is clear that periodic suspensions are highly idealised; however, a good understanding of their mechanics provides much insight into the consitutive modelling of suspensions.

If the particles are small enough (#m sized), then they will undergo Brown- ian motion, and the micromechanics is described by a set of stochastic differential equations, together with some relevant fluctuation-dissipation theorems, and the full solution of the relevant equations can only be obtained by specifying the probability distribution of the system, through solving a Smoluchowski equation. The relative importance of Brownian motion can be characterised by a P6clet number, such as R e = O ( ~ s ' ~ a 3 / k T ) , where -~ is a typical strain rate, r/s is the solvent viscosity, a is the size of the particle, and k T is the Boltzmann temperature. At low P6clet numbers, Brownian motion is strong, and the particles' orientation tends to be randomised, leading to a larger dissipation than when the P6clet number is large, the Brownian motion is weak, and the particles tend to align with the flow most of the time. Thus, we expect shear-thinning with the inclusion of Brownian motion (increasing shear rate leads to an increase in the P6clet number).

Another consequence of I << L is that the microscale Reynolds number is small: the microscale Reynolds number of a mm sized particle moving in a 1 Pa.s fluid of the density of water at a typical velocity of 1 mm.s -1 is of O(10-3). The relevant governing equations in this case are the Stokes equations,

V.u : 0, - V p + r/V2u : 0.

Since the Stokes equations are linear and instantaneous in the driving boundary data, the microdynamics is also linear and instantaneous in the driving forces; only

193FRENKEL, J., Kinetic Theory of Liquids, Dover, New York, 1955. 194LEVY, T. and SANCHEZ-PALENCIA, E., J. Non-Newt. Fluid Mech., 13, 63-78 (1983). 195ADLER, P.M., ZUZOVSKY, M. and BRENNER, H., Int. J. Multiphase Flows, 11,387-417

(1985). 196NUNAN, K.C. and KELLER, J.B., J. Fluid Mech., 142, 269-287 (1984).


the present boundary data are important, not their past history. This does not imply that the overall response will have no memoryt nor does it imply that the Reynolds number determined from the macroscale is small. The implications of the lack of microscale inertia have been examined by HINCH and LEAL. 197 In particular, we note also that its inclusion may lead to a non-objective constitutive model, since the stress contributed by microscale inertia may not be objective. 19s In addition, although the particle Reynolds number is small and may be neglected in solving the instantaneous micromechanics, its effects may be cumulative and may lead to a reordering of the microstructure, t99 These effects have not been considered in detail.

Overall, solutions to the Stokes equations have been considered in great detail; we refer to Lamb's treatise, 2~176 where the general solution of the Stokes equations is given in terms of the spherical harmonics r Cn and p~. HAPPEL and BRENNER, and KIM and KARRILA 2~ 1 have shown how to obtain these harmonics for a given

boundary data. The linearity of the micromechanics implies that the particle-contributed stress

will be linear in the strain rate; in particular all the viscometric functions (viscosity, first and second normal stress differences), and elongational stresses will be linear in the strain rate. Several investigators have found Newtonian behaviour in shear for suspensions up to a large volume fraction. However, experiments with some concentrated suspensions usually show shear-thinning behaviour, e.g., see KRIEGER, 2~ but the particles in these experiments are in the #m range, where Brownian motion would be important. Shear-thickening behaviour, and indeed, yield stress and discontinuous behaviour in the viscosity-shear-rate relation have been observed, e.g., see METZNER and WHITLOCK, 2~ HOFFMAN. 2~ This behaviour cannot be accommodated within the framework of hydrodynamic interaction alone; for a structure to be formed, we need forces and torques of a non-hydrodynamic origin, for example, of the nature considered by BRADY and BOSSIS, 2~ and CHEN and DOI. 206

This instantaneous nature of the micromechanics has also important implications which can, in fact, explain the shear reversal experiments of GADALA-MARIA and ACRIVOS. 9~ They have found that if shearing is stopped after a steady state has been reached in a Couette device, the torque is reduced to zero instantaneously, since it is a linear functional in the strain rate; we may regard the fluid as having

197HINCH, E.J. and LEAL, L.G., J. Fluid Mech., 71,481-495 (1975). 19sRYSKIN, G. and RALLISON, J.M., appendix to RYSKIN, G., J. Fluid Mech., 99, 513-529

(1980). 199PETIT, L. and NOETINGER, B., Rheol. Acta, 27, 437-441 (1988). 2~176 H., Hydrodynamics, 6th Ed, Dover, New York, 1932. 2~ J. and BRENNER, H., Low Reynolds Number Hydrodynamics, Noordhoff Inter-

national Publishing, Leyden, 1973; see also KIM, S. and KARRILA, S.J., Microhydrodynamics Principles and Selected Applications, Butterworth-Heinemann, Boston, 1991. 2~ I.M., Adv. Coll. Int. Scs 3, 111-136 (1972). 2~ A.B. and WHITLOCK, M., Trans. Sac. Rheol., 2, 239-254 (1958). 2~ R.L., Trans. Soc. Rheol., 16, 152-173 (1972). 2~ J.F. and BOSSIS, G., Ann. Rev. Fluid Mech., 20, 111-157 (1988). 2~ D. and DOI, M., J. Chem. Phys., 91, 2656-2663 (1989). 2~ F. and ACRIVOS, A., J. Rheol., 24, 799-814 (1980).

no memory in this experiment. In the absence of Brownian motion, or when the P6clet number is large, the microstructure will be frozen in place, or slowly relaxes with a relaxation t ime considerably greater than the t ime scale of the experiment. Thus, if shearing is resumed in the same direction after a period of rest, but not necessarily the same value (GADALA-MARIA and ACRIVOS have reported only on the case where the resumed shear rate is the same as the previous shear rate), then the torque would at ta in its final value tha t corresponds to the resumed shear rate almost instantaneously, since the micromechanics will s tar t from the near equilibrium sta te where it has been left off; in this context, the fluid may be regarded as having an infinite memory. However, if shearing is resumed in the opposite direction, then the torque signal will not instantaneously assume its expected steady state value; instead, it at tains an intermediate value and gradually settles down to a steady state; in this experiment, the fluid is viscoelastic. This is because either the previous s tate of the microstructure is not truly a steady state, but only asymptotically so, and this s tate is unstable in the reversed flow. Any weak disturbance will cause the microstructure to "flip" over to a new equilibrium state tha t corresponds to the new shear rate. We could think of a dumbbell-like microstructure here; for example, in the case of a suspension of spheres, the doublet formed by two spheres can be considered as a "dumbbell". Thus, although the micromechanics is instantaneous, there will be a relaxation process associated with the s tar t -up of a flow due to the readjustment of the microstructure, but there is no relaxation when the flow is stopped. Thus, according to one's own pre~conceived idea, the fluid memory can be zero, fading or infinite!

The relaxation mechanism in the flow reversal can also be used to explain the nonlinear behaviour in small strain oscillatory flows. Here, given a sufficiently large amplitude in a small strain oscillatory circular Couette flow, the torque signal follows a sinusoidal curve in the first half cycle; when the flow reverses in the second half cycle, the relaxation mechanism sets in, causing the torque signal to depart significantly from the sinusoidal curve. Most importantly, GADALA-MARIA and ACRIVOS 2~ have found tha t the torque signals at different shear rates can be collapsed on to a master curve, if plotted against the strain ~t. This implies tha t the relaxation t ime associated with the start up of a flow is of O(~-1) , and this information is quite vital in setting up a relevant microstructural theory for concentrated suspensions.

37.2 Bulk Suspension Properties

Consider now a volume V which is large enough to contain many particles but small enough so tha t the macroscopic variables hardly change on the scale V 1/3, i.e., 1 << V 1/3 << L. The effective stress tensor seen from a macroscopic level is simply be the volume-averaged Stress: 2~

(a,j) = V a,j d V - - V cri.i dV + V ai.i dV,

2~ F. and ACRIVOS, A., J. Rheol., 24, 799-814 (1980). 2~ is patterned after LANDAU, L.D. and LIFSHITZ, E.M., Fluid Mechanics, Pergamon,

New York, 1982, and BATCHELOR, G.K., J. Fluid Mech., 44, 545-570 (1970).


where Vf is the volume occupied by the solvent, lip is the volume of the particles in V, and the angle brackets denote a volume-averaged quantity. If the solvent is Newtonian, we have

aij(x) - -p5 U 4- 2rl~Dij - a ~ ), x E V I.

Thus

1 a i jdY - - (p) 6ij 4- 2rl~ (Dij) - ~ vii dV. V

Furthermore, from the equations of motion in the absence of inertia and the body f o r c e ,

0 OXk ( X i a k j ) = a i j ,

and we find that

1 L 1L~ 1 L V o, av-v av-V ,t as,

where tj - - t:rjknk is the traction vector and Sp is the bounding surface of all the particles. In addition,

1 /v~ a!f)

i z P

to within an isotropic tensor which can be lumped into a generic hydrostatic pressure P, which is determined through the balance of momentum and the incompressibility constraint. The average stress is thus given by

1 Jfs {xitj (uinj 4-ujni)} dS, (aij) - -P'6ij4- 2rls (Dij) 4- ~ - rib ~ r p

s o l v e n t �9 ~" p a r t i c l e s

(37.1)

consisting of a solvent contribution, and a particle contribution; p' is just a scalar pressure (the prime will be dropped from hereon). The particle contribution can be decomposed into a symmetric part, and an antisymmetric part. The symmetric part is in fact the sum of the stresslets 3~ p) defined by

SU _ "21 fsp {xitj 4- xjti - 2~% (uinj 4- uinj ) } dS - E S}P) p

(37.2)

and the antisymmetric part leads to the rotlet"

f s 1 7~ij -- "~1 (xitj - xjti) dS = -~ ~ eijkT (p) p p

(37.3)


where T(P) is the torque exerted on the particle p, and the summation is over all ~k particles in the volume V. The particle-contributed stress is therefore given by

a ( p ) _ l ( 3 ~ 1 .. T(p) ~ (37.4)

p

The rate of energy dissipation in a large enough volume V to contain all particles is given by

= aijD,j dV - ~ (a,ju,) dV - aiju, nj dS. (37.5)

The second equality comes from the balance of momentum, and the third one from an application of the divergence theorem, assuming that the condition at infinity is quiescent. For a system of rigid particles, the boundary condition on the surface of a particle p is tha t

u = U (p) + fl (p) x x, (37.6)

where U (p) and [~(P) are t he translational and rotation velocities of the particle, which can be taken outside the integral in (37.5). The terms remaining can be identified with the force F (p), and the torque T (p) imparted by the particle to the fluid. Thus the total rate of energy dissipation is

p

Note also that for a system of rigid particles, the integral

~s ( u n + nu) d S - O, p

since

~S U ( P ) n d ~ - - 0 , ~s (N(P) x x n + n n ( P ) x x ) d~--O, P p

by applications of the divergence theorem, and using the fact that the alternating tensor is antisymmetric.

An analysis of the energy dissipation leads to the uniqueness of Stokes solution, some variational statements giving the upper and lower bounds of the rate of dissipation, and a set of lemmas called inclusion monotonicity by HILL and POWER; 21~ the reader is referred to HAPPEL and BRENNER, and KIM and KARRILA 211 for full details. Considerable progress can be made when the concentration of the suspension is small, so that the multi-bodied problem is avoided, or when it is large, so tha t the dominant mechanism for the transfer of momentum between the particles is by the lubrication layer in-between. These cases will be reviewed below.

21~ R. and POWER, G., Q. J. Mech. Appl. Math., 9, 313-319 (1956). 211HAPPEL, J. and BRENNER, H., Low Reynolds Number Hydrodynamics, Noordhoff Interna-

tional Publishing, Leyden, 1973; KIM, S. and KARRILA, S.J., Microhydrodynamics Principles and Selected Applications, Butterworth-Heinemann, Boston, 1991.

37.3 Dilute Suspension of Spheroids We consider now a dilute suspension of force- and torque-free monodispersed spheres in a general homogeneous deformation. The dilute assumption means the volume fraction

47ra 3 r v << 1, (37.8)

3

where v is the number density of the spheres, each of radius a. In this case, in a representaive volume V we expect to find only one sphere. Thus, the microscale problem consists of a single sphere in an unbounded fluid; the superscript p can be omitted, and the coordinate system can be conveniently placed at the origin of the sphere. The boundary conditions for this microscale problem are

u -- U + (D + W ) - x , far from the particle, Ix]--. co, (37.9)

and u = V + w - x , on the particle's surface, I x l - a, (37.10)

where L = D + W is the far-field velocity gradient tensor; D is the strain rate tensor, W is the vorticity tensor, w is the skew-symmetric tensor such that wij -- --e~jkf~k, with t2 being the angular velocity of the particle. The far-field boundary condition must be interpreted to be far away from the particle under consideration, but not far enough so that another sphere can be expected. The solution to this unbounded flow problem is well known 212

a 3 (3 a a 3 ) a5 u - - U + L - x + ~ - - ~ . ( w - W ) - x + ~x+~-ffx 3 ( V - U ) - ~ - f f D - x

and

3 ( V - U ) - x ( a a 3 ) 5 D : x x ( a 3 a 5 ) + X2 X3 X -- ~X ~ X3 x5 x, (37.11)

3 ( V - U ) - x a3 D ' x x (37.12) p - ~7/8a x3 - 5r/~ x5 .

The traction on the surface of the sphere is

t - - o ' ' n l x - a - - 37/8 ( V - U ) - 3r/----~ ( w - W ) - x + 5 ~/--r D �9 x . 2a a a

(37.13)

The force and the torque on the particle can be evaluated:

F - f s a . n dS - - 6 m / s a ( V - U) (37.14)

and

T - L x • naS- (37.1 )

2Z2HAPPEL, J. and BRENNER, H., 1973, op. cit.; see KIM, S. and KARRILA, S.J., 1991, op.

czt., for the multipole solution.

1 where w~ -- ~e~jkWjk is the local vorticity vector. Thus, if the particle is force-free and torque-free, then it will t ranslate with U and spin with an angular velocity of

Returning now to the particle-contributed stress, (37.4),

1

where the stresslet is given, from (37.11), 213 by

. . . . . . l~(x, tj+xjt,) d, 5~--s 5~.(43 a3) D~j, a

and when we recall tha t r - 47ra3u/3, the effective stress will now become

(a> -- -Pl + 2~h (1+ 2~) D- (37.16)

This is the celebrated Einstein's result, 2t4 who arrived at the conclusion from the equality of the dissipation at the microscale and the dissipation at the macroscale as described by an effective Newtonian viscosity. Thus, a dilute suspension of sphere will behave just like a Newtonian fluid, with an enhanced viscosity.

A similar theory has been worked out for a dilute suspension of spheroids by LEAL and HINCH, 215 when the spheroids may be under the influence of Brown- ian motion. Here, if p denotes the unit vector directed along the major axis of the spheroid, then Jeffery's solution 216 yields the rate of change of the particle's orientation:

R 2 - 1 ( D - p - D �9 p p p ) (37.17) I ~ - - W - p + R 2 + 1

where R is the aspect ratio of the particle (length to diameter ratio). The particle- contributed stress may be shown to be

er (p) = 2Wsr {AD �9 <pppp) + B ( D - < p p ) + ( p p ) . D ) + C D + daF (pp)} , (37.18)

where the angular brackets denote the ensemble average with respect to the distribution function of p; A, B, C and F are some shape factors, and dR is the rotational diffusivity. If the particles are large enough so that Brownian motion can be ignored, then the last term, as well as the angular brackets, can be omitted in the previous expression. The asymptotic values of the shape factors are given in Table 37.1.

213 Note that

x d S = O, x x d S = 3

214EINSTEIN, A., inves t iga t ions on the Theory of the Brownzan Movement , Ed. FORTH, R., Transl. A.D. Cowper, Dover, New York, 1956. 215LEAL, L.G. and HINCH, E.J., Rheol. Acta, 12, 127-132 (1973). 216JEFFERY, G.B., Proc. Roy. Soc. Lond., A102, 161-179 (1922).

, , | , i i , | m

Asymptotic R -~ oo R - 1 + 6, 6 << 1

limit (rod-like) (near-sphere) i i

R z 395 62 A 2(ln 2R - 1.5) 147

6 In 2 R - 11 15 ~ _ 395 ~2

C 2 5 2 7 7 6+

3R 2 F !n 2R - 1/2 96 ....

R - - , 0

(disk-like)

10 2O8 37rR + ~ - 2

8 128 - ~ + 1

3~rR 97r 2 8

3~rR 12

7rR

TABLE 37.1. Asymptotic values of the shape factors.

The rheological predictions of this constitutive equation have also been considered by HINCH and LEAL. 217 In essence, the shear viscosity is shear-thinning, the first normal stress difference is positive while the second normal stress difference is negative, but of a smaller magnitude. The precise values depend on the aspect ratio and the strength of the Brownian motion.

When Brownian motion is absent, the complication due to the ensemble averages disappears because of the large size of the particles and (37.17) is solved by

Q (37.19) p = Q ,

where

2 D. (37.20) (~ : s s : L - R2 +------ ~

The effective velocity gradient tensor is s = L - (D, where r -- 2 / (R 2 + 1) is a 'non-affine' parameter. In the start-up of a simple shear flow, where the shear rate is ~, we find that

2 - ( sin Q1 - Qlo cos wt + Q2o ~dt~

Q2 = Q2o cos wt - V 2 ~_ ~Qlo sin wt,

and Q3 - Q3o, where {Qlo, Q2o, Qao} are the initial components of Q, and the frequency of the oscillation is

~R w = ~V/r R 2 + l

From these results, the particle-contributed stress and the viscometric functions can be obtained:

, i , i

217HINCH, E.J. and LEAL, L.G., J. Fluid Mech., 52,683-712 (1972).


20

15

l0

like

- re

5

0

disk-like -5

0 1 2 3 4 5 6 7

Time, o)t

FIGURE 37.1. The reduced viscosity as a function of time. The three cases labelled are: rod-like R = 10, near-sphere R = 1.1, and disk-like R = 0.1.

1. The reduced viscosity:

= 2Apl pl + z (pl + + c ,

2. The reduced first normal stress difference:

(37.21)

NI -- 2AplP2 (p21 - p2) , (37.22)

3. and the reduced second normal stress difference:

N2 = 2pip2 (Ap 2 + B ) . (37.23)

The particles tumble along with the flow, with a period of T -- 27r(R 2 + 1) /~R , spending most of their t ime aligned with the flow. In Figure 37.1, the reduced viscosity is plot ted against the dimensionless t ime wt, for the three asymptot ic cases, rod-like R -- 10, near-sphere R ---- 1.1, and disk-like R -- 0.1. Initially p is kept aligned with the flow, Q0 - { 1, 0, 0}. The first and second normal stress differences are also non-zero and periodic in t ime with the same period. Randomizing the initial configuration of p will smear out the results somewhat, but the basic shape of the viscometric functions will be retained.

In the s tar t -up of an elongational flow with a positive elongational rate ~, we find tha t

Q 1 - Q l o e x p { ( 1 - , ) ~ r t } , Q2 = Q2oexp { - 1 ( 1 - , ) ~ t }

and {1 } Q3 - Q3o exp - ~ ( 1 - (~)~ ,

so that the particle is quickly aligned with the flow in a time scale O(~ -1). At a steady state, the reduced elongational viscosity is

N1 - 3~/s~ R 2 ---- 2 (A + 2B + C) ~ In 2 R - 1.5" (37.24)

Strictly speaking, the dilute assumption means that the volume fraction is low enough, so that a particle can rotate freely without any hindrance from its nearby neighbours. The distance A between any two particles must therefore satisfy l < A, so that a volume of 13 contains only one particle, where l is the length of the particle and d is its diameter. The volume fraction therefore satisfies

d2l C N . A 3 , r <1"

Thus, the reduced elongational viscosity is only O(1) in the dilute limit, not O(R 2) as suggested by the formula. As the concentration increases, we get subsequently into the semi-dilute regime, the isotropic concentrated solution, and the liquid crystalline solution. The reader is referred to DOI and EDWARDS 21s for complete details. Here, we simply note that the concentration region 1 < r 2 < R is called semi-concentrated. Finally, the suspension with r > 1 is called concentrated, where the average distance between fibres is less than a fibre diameter, and therefore fibres cannot rotate independently except around their symmetry axes. Any motion of the fibre must necessarily involve a cooperative motion of surrounding fibres. We shall return to fibre suspensions in a later section.

37.~ L u b r i c a t i o n T h e o r i e s

Suspensions at high concentration exhibit markedly non-Newtonian behaviour: their shear stresses relax under a constant shear rate; they exhibit no recovery when an interrupted shearing is resumed in the same direction, but if the shearing is reversed, a time-dependent hysteresis behaviour is observed. 219 In addition, there is a shear-induced migration of the particles, from regions of high shear rates to regions of low shear rates. 22~ Clearly, there are several parameters that influence the rheology of concentrated suspensions, and analytical progress can only be made for model suspensions.

In the case of an ordered suspension, i.e., a spatially periodic suspension, it is theoretically possible to construct a unit cell for the suspension, and to solve the Stokes equations, obtaining the velocity and stress fields subjected to periodicity constraints, and thus determine the effective rheological properties by an appropriate volume average of the complex stress-strain-rate relation. An example of this

21sDOI, M. and EDWARDS, S.F., The Theory of Polymer Dynamics, Oxford University Press, Oxford, 1988. 219GADALA-MARIA, F. and ACRIVOS, A., J. Rheol., 24, 799-814 (1980); METZNER, A.B.,

J. Rheol., 29, 739-775 (1985); ADLER, P.M., NADIM, A. and BRENNER, H., Adv. Chem. Eng., 15, 1-72 (1990). 22~ R.J., ARMSTRONG, R.C., BROWN, R.A., GRAHAM, A.L. and ABBOTT, J.R.,

Phys. Fluids A, 4, 30-40 (1992).


type of calculation is given by NUNAN and KELLER, 22t who have shown tha t the effective stress is linear in the strain rate, but the viscosity is now a fourth-order tensor, given by

1 ( 2 W,jkl -- ~Ws(1 + f3) 6,kSjl + 5il6jk - ~ i j ~ k l

( 1 ) - - , (37.2 )

in which ~ and ~ are functions of the concentration and the lattice geometry, and 5ijkl is one if all the subscripts are equal, and zero otherwise. The effective shear viscosity is r]s(1 + ~), and the effective elongational viscosity is ~]s(1 + ~). Asymptotic expressions for various parameters at high volume fractions are also given by NUNAN and KELLER; for a simple cubic lattice the leading terms in these expressions are

3 27 1 0(1) 7r 1 0~-- -~71"e -1 +~--~Trlne- + , ~ = -~-lne- + 0 ( 1 ) , (37.26)

where ea is the dimensionless gap between the two spheres. These expressions are found by solving the Stokes equations asymptotically in e-1, and evaluating the relevant integrals tha t contribute to the effective stress tensor. The calculation is incomplete in the sense tha t the t ime evolution of the microstructure cannot be determined within the framework of the method. However, it provides a convenient testing ground for suspension theory. NUNAN and KELLER 222 also consider the problem of the determination of the effective elasticity tensor of a particulate solid consisting of rigid spherical inclusions embedded periodically in an elastic matrix. This problem is more tractable constitutively in the sense tha t the microstructure does not change significantly, at least in the domain of classical elasticity, where only infinitesimal deformation is considered.

In the case where there is no long-range order, bounding techniques can be used to provide information on the effective viscosity of the suspensions. K R I E G E R 223 has introduced the concept of high and low shear rate bounds for the relative viscosity (r/r) and proposed the expression

= r] = 1 (37.27) r/r ~/, (1 - r162 ~ '

where Cm ~ 0.6 is the maximum volume fraction, and a ~ 1.8. This expression has been very popular in correlating experimental data; it has even been extended to cover suspensions of fibres, 224 where Cm is allowed to depend on the aspect ratio of the fibres.

At high volume fractions, where the particles nearly touch each other, the amount of viscous dissipation can be calculated for the array leading to an expression for

221NUNAN, K.C. and KELLER, J.B., J. Fluid Mech., 142, 269-287 (1984). 222NUNAN, K.C. and KELLER, J.B., J. Mech. Phys. Solids, 32, 259-280 (1984). 223KRIEGER, I.M., Adv. Col. Int. Sci., 3, 111-136 (1972). 224KITANO, T., KATAOKA, T. and SHIROTA, T., Rheol. Acta, 20, 207-209 (1981).


the effective viscosity at tha t instant. A notable calculation of this type is tha t of FRANKEL and ACRIVOS, 225 who propose the following form for the reduced viscosity

9 ( r162 1/3 (37.28) - 1 -

ADLER 226 also considers spatially periodic suspensions and points out tha t the singularity in the reduced viscosity as r 1 6 2 --* 1 only occurs instantaneously, and that it should disappear with a time average. In reality, the singularity in the reduced viscosity function is essential for a good fit to experimental data. VAN DEN BRULE and JONGSCHAAP 227 have produced a constitutive equation for the suspensions based on the lubrication theory, which reproduces the correct asymptotic results of NUNAN and KELLER. The missing element in all the theories based on the lubrication argument is again the evolution of the microstructure, which cannot be obtained in any other means short of solving the full problem by a numerical method, for example, by the Stokesian Dynamics Simulation, 228 or the Completed Double Layer Boundary Integral Equation Method. 229 These methods are outside the scope of this chapter, however.

On the other hand, if the microstructure is known completely, then asymptotic methods can be effectively employed to produce the full constitutive equation. We review the technique here, as applied to a concentrated suspension of spheres.

Concentrated Suspensions of Spheres

The evolution of the microstructure can only be derived from a full solution of the multibodied interaction problem. In the absence of this, we need to postulate a reasonable evolution equation for R pq, the centre-to-centre vector between two generic spheres. Based on some limited numerical data, PHAN-THIEN 23~ has proposed the simple model

- - L - R § B(t) , (37.29)

affine fluctuations

where B(t) is the fluctuating component of the motion (in a Lagrangian sense); it is not the Brownian motion, but reflects the spatial fluctuations being experienced by the two generic particles. Here, L is the velocity gradient tensor, and the superscript pq has been dropped for brevity. The simplest statistical model of this fluctuation component is white noise; however, this would require a vanishing time scale for the fluctuations compared with the time scale of the average motion, which is quite

225FRANKEL, N.A., and ACRIVOS, A., Chem. Eng. Sci., 22, 847-853 (1967). 226ADLER, P.M., J. Mdcanzque Th~orzque et Applzqu~e, special issue, 73-100 (1985). 227VAN DEN BRULE, B.H.A.A. and JONGSCHAAP, R.J.J., J. Stat. Phys., 62, 1225-1237

(1991). 228BRADY, J.F. and BOSSIS, G., Ann. Rev. Fluid Mech., 20, 111-157 (1988); DOI, M. and

CHEN, D., J. Chem. Phys., 90, 5271-5279 (1989); CHEN, D. and DOI, M., J. Chem. Phys., 91, 2656-2663 (1989). 229KIM, S. and KARRILA, S.J., M~crohydrodynamics Principles and Selected Applzcatwns,

Butterworth-Heinemann, Boston, 1991. 2a~ N., J. Rheol., 39, 679-695 (1995).

unrealistic. The potential large t ime scale in the fluctuations is taken into account by assuming

(B) -- 0, (B(t + T)B(t)) -- r (T)l , (37.30)

where r (T) is the correlation function of the fluctuations; a realistic functional form for r(T) would be the exponential function, roexp(--T/Tc), where Tc is the correlation time of the fluctuations and r0 -- �89 (B(t) - B(t)). This is clearly a crude model, since it is fairly certain tha t the fluctuations are not isotropic; however, it is a convenient starting point.

The diffusivity of the process R is derived by first defining the total strength of the fluctuations as

V = (aT .a1 )

Then, to O(At2), the incremental form for (3"/.29) is

j!0 At A R - L - R A t + B(t)dt,

leading to

(ARAR) fAt fat / A t /A t

= ] 0 J , ( B ( t ) B ( t ' ) ) d t d t ' - 1 r ( t - t')dtdt' fat~2 0 0

1At J_ At~2 r(T)dT.

The integral of the correlation function can be evaluated if the relative magnitudes of At and T~ are known:

At~2 r(T)dT At~2

f ~ r(T)dT- 2 r 0 T c - 2d~,

f~:12 / = = J-re/2 fAt~2 At

J-At~2 r(T)dT ~__ roAt = d~-~c ,

Thus, the diffusivity in the process R is

if At >> T~,

i f A t "~ ~-~,

if At << To.

ld~ ,

' 2 A t = -~ldR, At ld'

if A t >> T~,

if A t N To,

if At << To.

If we denote the strength of the fluctuations over the period At by

t 2dR -- r(T)dT,

At

then in all cases,

and

d ! R ,

1 i d R - - "~d R,

A t dl

if At >> To,

if At ~ T~,

if At << ~c,

( XRAR (37.32) / -- 1dR . 2 A t

Note that the correlation time scale will be involved in the diffusivity of R only when it is of the same order of magnitude, or considerably greater than At.

The magnitude of dR can be estimated by noting that in a time scale of O(~-1), R is displaced by O(2a), and thus

(2a) 2 = O(a2,~) dR "~ 2~- i

which reduces to zero when the flow ceases. We make a constitutive assumption that

dR -- g a 2 ~ / , (37.33)

where ~ - x/2tr D ~ is the generalised strain rate, and K is an order-one constant. From the assumed motion of the two spheres, and writing R - Rp, where R is the centre-to-centre distance, and p is the unit vector directed along the line of centres, we find that the approach velocity and the rate of change of the relative configuration are given, respectively, by

/~-- RD �9 pp + p - B, (37.34)

and 1

15---- L - p - D " p p p + ~ ( 1 - p p ) - B . (37.35)

At high concentrations, the gap between the two generic spheres is small and the dominant load transfer between the two particles is due to the squeezing mode. Now, the squeezing velocity between the two spheres is Ull - / ~ p , leading to the particle-contributed stress

er (p) = a v f i I ( R D �9 p p p p + p p p - B>, (a7.36)

where • is the number density of the spheres and fll is the squeezing force between the two spheres.

, ,

P r o b l e m 37.A Consider a dynamic quantity Q, whose rate is given by

(~ -- Q1 + Q 2 . B ,


where Q1 is a "deterministic" part, and Q2.B is a "fluctuation" part that is linear in B. Show that

( Q . B ) - (Q(At).B(At)) A t

= ((Q(O) + AtQ~).B(At)) + Q2" / (B(~-)B(At)) dT t J

0

= dRQ2" 1. (37.A1)

Show that the time derivative of ppp consists of a deterministic part, and a fluctuating part that is linear in B. Apply the above result, and show that

( p p p - B ) - - 2dR ( pp - ~ ) , (37.A2)

Thus, the particle-contributed stress is given as

a 2

(o pppp �9

A t high volume fraction, R - 2a + O(e), and the particle-contributed stress can be written as

(r (') -- #(r D : ( p p p p ) + ~ K ~ ( p p ) , (37.37) J

viscous f luc tua t ions ]

where we have set #(r -- apfl lR , defining it empirically rather than using the lubrication formula for fll, although the lubrication formula will lead to a quantita- tively similar result. Allowing a2/R 2 to depend on the volume fraction is somewhat equivalent to K -- K(r if more flexibility is needed in fitting the experimental data.

The particle-contributed stress thus consists of a term linear in the strain rate arising from the squeezing motion between two generic particles, which is appropriately termed viscous contribution, and a term arising from the fluctuations of the generic particles. This should be compared with the constitutive equation for dilute suspensions, (37.18). The terms involving D.pp + pp .D correspond to the shearing terms, if the latter are included. The contribution from the fluctuations is of the same order as the squeezing contribution; but because the strength of the fluctuations is proportional to the strain rate, there is no relaxation of the stress: upon stopping the flow the stress instantaneously reduces to zero.

Next, we need an equation to describe the time evolution of p. From (37.35), we find

d 1 2 ~-~pp- L - p p - p p - i T + 2 D ' p p p p - ~ (pB + Bp) - ~ p p p - B .

An ensemble average is now taken, with the help of (37.32) and the results of Problem 37.A, to obtain

( d } 1 (37.38) A ~-~ (pp) - L - ( p p ) - ( p p ) - L T + 2D" (pppp) + (pp) -- ~1,

where R 2 2

= ~ - - - - (37.39) 6 d R - 3KZy

is the relaxation time. Here, we have already assumed that averages like (pp/R2> can be approximated by

{ pp - ~ ) "" <PP> 4a 2 '

since <R2> N (2a) 2. Closure can be obtained by using the I-lINCH and LEAL approximation: TM

1 (6 (pp>- D. (pp> - D" <pp> <pp> - 2 <pp)2. D1 D �9 (pppp) -

+ 2 (pp>" D 1 ) , (37.40)

which is valid in both weak and strong flows. If the front factor on the right hand side of (37.37) is chosen empirically as

#(r N 8~78 (1 - ~ m ) -~ , (37.41)

then the Krieger viscosity form in a simple shear flow results. The constitutive model has several features, some of which have been observed

in concentrated systems"

�9 an instantaneous response at the inception of the flow;

�9 the stresses instantaneously reduce to zero when the flow is stopped;

if the flow is restarted in the same direction, then the stresses will recover their provious values instantaneously, with the period of rest being of no consequence;

if the shear rate changes from the previous value, the stresses still instantaneously attain the steady state values corresponding to this new shear rate;

if the flow is restarted but in the opposite direction, then the stresses recover partially only, and then relax to their steady state values;

�9 the stresses are linear in the strain rate, which leads to a Newtonian viscosity, and normal stress differences which are linear in the magnitude of the shear rate;

the stress is anisotropic with respect to the strain rate tensor so that the flow resistance will depend on the nature of the flow field;

�9 a universal transient response when the stresses (reduced by/~(r are plotted against ~rt;

�9 a universal response with respect to wt, in small strain oscillatory flows;

2atHINCH, E.J. and LEAL, L.G., J. Fluid Mech., 76, 187-208, (1976).

�9 the dependence of the stresses on the volume fraction is the same in all flows.

The predictions of the model in some simple flow fields have been given by PHAN-THIEN. 232

37. 5 Fibre Suspens ions

As has been mentioned, the concentration of fibre suspensions is usually classified into three regimes: dilute, semi-concentrated (or semi-dilute) and concentrated. The suspension is called dilute if there is only one fibre in a volume of V = l a, where 1 is the length of the fibres; the volume fraction therefore satisfies r < d2l/V, or, CR 2 < 1, where d is the diameter of the fibre and R = lid is its aspect ratio. In dilute suspensions, each fibre can therefore freely rotate. The concentration region 1 < OR 2 < R is called semi-concentrated, where each fibre is confined in the volume d21 < V < dl 2. The spacing between the fibres is greater than the fibre diameter but less than the fibre length. In this regime the fibres have only two rotating degrees of freedom. Finally, the suspension with CaR > 1 is called concentrated, where the average distance between fibres is less than a fibre diameter, and therefore fibres cannot rotate independently except around their symmetry axes. Any motion of the fibre must necessarily involve a cooperative motion of surrounding fibres.

Most of the microstructural models developed by DOI and EDWARDS, 233 HINCH and LEAL, TM DINH and ARMSTRONG, 235 and LIPSCOMB et al., 236

have similar functional forms to those derived from continuum mechanics in the early works of ERICKSEN 237 and HAND. 23s All of these theories have two elements: an evolution equation for the microstructure, represented by a unit vector field, representing the orientation of the fibres, and a stress rule, allowing the stress tensor to be calculated from the unit vector field. For dilute suspensions, it is reasonable to neglect the interactions between the fibres. For non-dilute suspensions, however, we have to consider fibre-fibre interactions, which can affect the flow behaviour. FOLGAR and TUCKER 239 have developed an evolution equation for concentrated fibre suspensions, where the fibre-fibre interactions are taken into account by adding a diffusion term to Jeffery's equation. DINH and ARMSTRONG 24~ discuss the dynamics of non-Brownian particles and derive a constitutive equation

for semi-dilute suspensions; the model takes into account the fibre-fibre interaction and uses a distribution function to describe the orientation state.

, ,

232pHAN-THIEN, N., J. Rheol., 39, 679-695 (1995). 233DOI, M. and EDWARDS, S.F., J. Chem. Soc. Faraday Trans. H, 74, 560-570 (1978); J.

Chem. Soc, Faraday Trans. II, 74, 918-932 (1978). 234HINCH, E.J. and LEAL, L.G., J. Fluid Mech., 52, 683-712 (1972); J. Fluid Mech., 76,

187-208 (1976). 235DINH, S.H. and ARMSTRONG, R.C., J. Rheol., 28, 207-227 (1984). 236LIPSCOMB II, G.G., DENN, M.M., HUR, D.U. and BOGER, D.V., J. Non-Newt. Fluid

Mech., 26, 297-325 (1988). 237ERICKSEN, J.L., Arch. Rational Mech. Anal., 4, 231-237 (1960). 238HAND, G.L., J. Fluid Mech., 13, 33-46 (1962). 239FOLGAR, F.P. and TUCKER III, C.L., J. Reinforced Plastics and Composites, 3, 98-119

(1984). 24~ S.H. and ARMSTRONG, R.C., J. Rheol., 28, 207-227 (1984).


Jeffery-Like Models

The development of the constitutive equation follows a road very similar to tha t of a rigid dumbbell model. First, the fibre orientation is described by Jeffery's orbit,

15 -- W - p + A ( D - p - D �9 ppp) , (37.42)

in which A is given by

A = R2_1 R 2 + i '

and W -- (Vu T - V u ) / 2 is the vorticity tensor, D -- (Vu T + V u ) /2 is the strain rate tensor, and R is the aspect ratio of the microstructure. Note that as 15- p = 0, the magnitude of p is preserved in this time evolution. Thus, if p is initially a unit vector, then it remains a unit vector for all t. There are two physical interpretations for p. Firstly, it can be regarded as the local orientation of an individual fibre. Sec- ondly, in the case where there is some Brownian motion present, then it represents the averaged configuration. The term W - p indicates that p rotates with the fluid, while the term D �9 p represents the component of the strain with the fluid. Since p is of unit length, the stretching component D : p p p must be subtracted, producing the last term. In shear flows of non-interacting fibres, the fibres exhibit a closed periodic rotation known as Jeffery's orbit, which is obvious from (37.58). Note that we can rewrite Jeffery's equation as

15 -- s p - / : " ppp , (37.43)

where the "effective" velocity gradient tensor is s = L - CD, with ~ = 1 - A = 2 / (R 2 + 1). This is reminiscent of the effective velocity gradient tensor that has been used in a number of non-affine network theories.

With Brownian motion, there is a random excitation in (37.43), represented by some noise on the space orthogonal to p. In this case, we write

15 -- s - s + (1 - p p ) - F (b) (t) , (37.44)

where, for example, the Brownian motion can be modelled as white noise of zero mean, and delta correlation function:

( F (b) (t + s)F (b) ( t ) ) -- 2DrS(s)1, (37.45)

in which Dr is the rotational diffusivity of the process, and the angular brackets denote the ensemble average with respect to the probability density function of the process concerned. The factor (1 - pp) in front of F (b) is the statement tha t only rotational Brownian motion is allowed. To complete the description of the micromechanics, the probability density function r must be specified. This quantity satisfies the Liouville equation 241

0r 0 o~ ,[ \ A p A p f/ 2At ) 0 r

_ - - . ~ " _ ( A p -- - ~ } r (37.46)

241CHANDRASEKHAR, S., Rev. Mod. Phys., 15, 1-89 (1943).

With

and

< ~'~Pt > - s - .s " PPP

A p A p > _ Dr (1 - pp) 2At

the Liouville, or the Fokker-Planck, equation for the probability density becomes

0r 0 { 0r } Ot -- " ~ " Dr (1 - p p ) ~ - ( s - s p P P ) r �9 (37.47)

In most of the literature, the diffusivity is usually written as a scalar Dr, while the operator 0 / 0 p is interpreted as the two-dimensional gradient operator on a unit sphere surface, which is essentially equivalent to (37.47).

At high concentrations, Jeffery's evolution equation is no longer valid; in addition, we must also specify the pairwise and higher distributions to account for the multi-particle interactions. However, the dilute theory has been used to approximate the behaviour of suspensions beyond the dilute region. Recently, INGBER and MONDY 242 have reported numerical simulations of three-dimensional Jeffery orbits in shear flows. They have examined wall effects, particle interactions and nonlinear shear flows, and found that the Jeffery theory provides a good approximation of the orientation trajectory for the particle in both linear and nonlinear shear flows, even in close proximity to other particles or walls.

The evolution equation for a dynamical quantity Q(p) can be found by averaging it with respect to r It can be derived, without having to go through the diffusion equation, in a manner similar to that shown in the section on rigid dumbbells - see w Thus, it is possible to derive that the equation of change for Q(p) is given by

< O > = s pm /) - s pmp~,pk

+Dr <(hkt - PkPt) OpkOpt

In particular, with Q - pipj,

OQ Opk = 5~kpj + 5jkp~,

and

leading to

O2Q Op~Opl

= 5ikhjt + 5jkhit

d ~ . (pp) _ ( s + p p . s 2s p p p p ) + 2Dr ( 1 - 3pp)

(37.48)

) 1 A ~-~(pp) + 2 s (pppp) + (pp) -- ~1, (37.49)

where A - 1/6Dr is the relaxation time of the tumbling motion, and A / A t is the upper convected derivative, defined with the effective velocity gradient tensor/2.

242INGBER, M.S. and MONDY, L.A., J. Rheol., 38, 1829-1843 (1994).


This is exactly the expression for a rigid dummbell in a homogeneous flow field (with s replaced by L). The advantages of this derivation are tha t surface integrations on the unit sphere are avoided, and the Brownian motion need not be represented by white noise.

Folgar-Tucker Model

In the FOLGAR and T U C K E R model, 243 the diffusivity Dr is assumed to be of the form Ci~ , where ~/= (2trD2) 1/2 is the generalised strain rate, and the aspect ratio is assumed infinite. The parameter CI is known as the interact ion coefficient, which has been experimentally determined to lie in the range of 1 0 - 2 10-3. YAMANE et al. TM have recently obtained the interaction coefficient in a numerical simulation of semi-dilute suspensions of rod-like particles in a shear flow. However, the predicted values of CI are about two orders of magnitude smaller than those suggested by BAY and TUCKER. 245 It seems tha t this phenomenological constant must be a function of the volume fraction of the fibres, and its aspect ratio; it may even be a tensorial quantity, reflecting the anisotropy of the fluid.

Closure Formulae

A closure approximation is also required to approximate the fourth-order tensor (pppp) in terms of the second order tensor (pp). The simplest approximation is the quadratic closure,

( pppp ) = (pp) (pp) .

This closure is exact in the limit of perfectly aligned fibres. When the P6clet number is small, the fibre orientation tends to be randomized, and the approximation is not recommended. Hinch and Leal's composite closure, (37.40), designed to have the correct limits in both strong (perfectly aligned fibres) and weak flows (perfectly random orientation), is to be preferred here.

More recently, CINTRA and T U C K E R 246 have developed a new family of closure approximations, called orthotropic fitted closure, by transforming the fourth- order tensor in the principal axis system of the second-order tensor, and expressing its three independent components in terms of the second-order principal values. The resulting formula of the closure approximation has been fitted to numerical solutions of the probability density function in a few well-defined flow fields.

A variety of other closure approximations have been proposed. Further details may be found in ADVANI and TUCKER, 247 SZERI and LEAL, 248 and VERLAYE

243FOLGAR, F.P. and TUCKER III, C.L., J. Reinforced Plastics and Composites, 3, 98-119 (1984). 244yAMANE, Y., KANEDA, Y. and DOI, M., J. Non-Newt. Fluid Mech., 54, 405-421 (1994). 245BAY, R.S. and TUCKER III, C.L., Polym. Compos., 13, 317-321 (1992). 246CINTRA Jr., J.S. and TUCKER III, C.L., J. Rheol., 39, 1095-1122 (1995). 247ADVANI, S.G., and TUCKER III, C.L., J. Rheol., 31,751-784 (1987); J. Rheol., 34, 367-386

(1990). 248SZERI, A.J. and LEAL, L.G., J. Fluid Mech., 242, 549-576 (1992); J. Fluid Mech., 262,

171-204 (1994).


and DUPRET. 249 It is known tha t the validity of the closure schemes depends on the type of flow and the degree of alignment of the fibres.

Dinh-Armstrong Model

In the Dinh-Armstrong model, 25~ the fibre aspect ratio is assumed to be infinite, and the bulk stress due to the presence of the fibres in a homogeneous flow field is given by

7r/3v 3 7(P) / LkzpkPlP~Pjr dp (37.50) 'J = r 6 ln (2H/d) j

where L -- ~TuT is the velocity gradient tensor; v is the number of particles per unit volume; l and d are the fibre length and diameter, respectively; and H is the average distance between a fibre and its nearest neighbour, given by

H = (Np/2) -1

for a random orientation, and

H = (Npl) -I12

for a fully aligned orientation. The orientation distribution function r satisfies

0V Ot + V p - [ ( L - p - L" p p p ) r 0, (37.51)

where Vp is the two-dimensional gradient operator on the surface of a unit sphere. The form of the constitutive equation is very similar to tha t of the rigid dumbbell model.

Phan-Thien-Graham Model

In the Phan-Thien-Graham model, TM the particle-contributed stress is given by

tr (p) = 2~/8r162 aR)D �9 ( p p p p ) , (37.52)

where f is a function of the volume fraction and the aspect ratio, which is assumed to take the form

a2R(2 -- r f ( r aR) -- ~i(ln 2aR - 1.5)(1 - C/A) 2' (37.53)

with the parameter A being determined from experimental shear data, using an empirical equation proposed by KITANO et al., 252 i.e.,

1 V/r -- (1 - C/A) 2" (37.54)

249VERLAYE, V. and DUPRET, F., Proc. A S M E Winter Annual Mtg., Nov. 28 - Dec. 3, 1993, New Orleans. 25~ S.H. and ARMSTRONG, R.C., J. Rheol., 28, 207-227 (1984). 251PHAN-THIEN, N. and GRAHAM, A.L., J. Rheol., 30, 44-57 (1991). 252KITANO, T., KATAOKA, T. and SHIROTA, T., Rheol. Acta, 20, 207-209 (1981).

Here r/r ---- r//r/8 is the reduced viscosity of the suspension. The physical interpretation of the parameter A is the maximum allowable volume fraction of the suspension. A linear regression using the data of KITANO et al. leads to

A - - 0 . 5 3 - 0.013aR, 5 < aR < 30. (37.55)

At the aspect ratio aR -- 20, the maximum allowable volume fraction is about 0.27. This equation has been used in a number of flow problems, including the flow past a sphere in a tube. The agreement with experimental data is excellent, if A is chosen to be about 0.46, or aR ~ 5.4.

37. 6 Flow-Induced Migrat ion

Flow-induced migration of particles in suspensions has been observed in a number of experimental studies. One of the first is due to KARNIS et al. 253 In measuring both velocity and concentration of neutrally buoyant spheres in a Newtonian fluid when the suspension is pumped through a tube, they find blunted velocity, but homogeneous concentration profiles for a range of the ratio R/a (R and a being the radii of the tube and spheres, respectively). The first reported experiment of shear- induced migration is that by GADALA-MARIA and ACRIVOS, 254 where steady decrease of the viscosity of a concentrated suspension in a Couette viscometer is noted. Later experiments by LEIGHTON and ACRIVOS 255 show that this is caused by a migration of particles from the Couette gap into the reservoir at the bottom of the device. HOOKHAM 256 uses a modified laser-Doppler technique to measure velocity and concentration profiles for the flow of concentrated suspensions in a rectangular channel. He has found a blunted velocity profile near the centre of the channel; the degree of blunting increases with the bulk particle concentration or with the particle size to gap ratio. However, because of a large degree of scatter in the concentration data, only a qualitative picture of the particle concentration distribution within the flow channel can be provided. KOH et al. 2~7 use a laser Doppler anemometry technique to measure both velocity and concentration profiles. With mean concentrations in the range 0.1 to 0.3, conclusions similar to those of Hookham's about the velocity profiles are obtained. They have found a maximum concentration at the centreline; in some cases, i.e., with the bulk concentration of 30%, the local particle concentration at the centreline approaches a maximum packing value near to 0.65. Non-intrusive NMR technique has also been used to measure concentration and velocity profiles simultaneously. 25s

253KARNIS, A., GOLDSMITH, H.L. and MASON, S.G., J. Colloid Interface Sci., 22,531-553 (1966). 254GADALA-MARIA, F. and ACRIVOS, A., J. Rheol., 24, 799-811 (1980). 255LEIGHTON, D. and ACRIVOS, A., J. Fluid Mech., 181,415-439 (1987). 256HOOKHAM, P.A., Concentration and velocity measurements in suspensions flowing through

a rectangular channel. PhD thesis, California Institute of Technology, 1986. 257KOH, C.J., HOOKHAM, P. and LEAL, L.G., J. Fluid Mech., 266, 1-32 (1993). 25SGRAHAM ALTOBELLI, S.A., FUKUSHIMA, E., MONDY, L.A. and STEPHENS, T.S.,

J. Rheol., 35, 191-201 (1991); ABBOTT, J.R.,'TETLOW, N., GRAHAM, A.L., ALTOBELLI, S.A., FUKUSHIMA, E., MONDY, L.A. and STEPHENS, T.S., J. Rheol., 35, 773-795 (1991); IWAMIYA, J.H., CHOW, A.W. and SINTON, S.W. Rheol. Acts, 33, 267-282 (1994); PHILLIPS,


Based on their circular Couette data, PHILLIPS et al. have proposed a model based on the scaling arguments of LEIGHTON and ACRIVOS. In this model the constitutive equation for the particle flux is considered to be a balance between a contribution due to spatially varying collisions between the particles, and an opposite contribution due to a spatially varying viscosity, which arises due to the migration of the particles. In circular Couette flows, Phillips et al.'s model predicts a migration of particles from the inner wall, where the shear rate is high, to the outer wall where the shear rate is low. The volume fraction reaches a high value at the outer wall, and the velocity decreases rapidly from the inner wall so that there is a relatively large, almost-stagnant region near the outer wall. In circular Couette flows, Phillips et al.'s model does remarkably well in its prediction of the velocity and concentration profiles. Indeed, the two constants in the model have been found by fitting the analytical results to the experimental data. For axisymmetric Poiseuille flows, this model predicts a migration of particles towards the centreline, leading to a cusp-like concentration, and a blunted velocity profile there.

Following earlier studies by JENKINS and McTIGUE, 259 who have proposed conservation equations for mass, momentum and fluctuation energy for the particle phase, NOTT and BRADY 26~ propose a model for concentrated suspensions, in which the concept of a hydrodynamic temperature is used as a measure of the intensity of the velocity fluctuations of the particles. They also conduct a Stokesian Dynamics simulation of a pressure driven flow in a channel for a suspension of monodisperse spherical particles for a range of ratios H / a (where H is the half width of the channel and a is the particle radius), at different bulk concentrations of particles. The main difference between Phillips et al.'s model and that of Nott- Brady's is apparent in channel flow: the former predicts a cusp-like appearance in the concentration profile at the centreline, with the concentration approaching the maximum packing volume fraction era, but in the latter a rounded concentration profile at the centreline is obtained. NOTT and BRADY attribute this to the fact that Phillips et al.'s theory is 'local' in the sense that the diffusivity is allowed to be proportional to the local shear rate, while the theory of NOTT and BRADY is non-local owing to the presence of the energy flux which allows for a non-zero temperature at the centreline and hence a finite particle diffusivity there. LYON and LEAL TM have reported that Nott and Brady's theory seems to provide a better fit to the channel flow data, especially in the velocity profile. The degree of fit to the concentration data is about the same for both theories, judging from the various figures. Nott and Brady's theory is necessarily more complex since it involves an extra equation for the suspension temperature, which is fully coupled with the momentum equations.

| ,

R.J., ARMSTRONG, R.C., BROWN, R.A., GRAHAM, A.L. and ABBOTT, J.R., Phys. Fluzds A, 4, 30-40 (1992). 259JENKINS, J.T. and McTIGUE, D.F., in Two Phase Flows and Waves, Ed. JOSEPH, D.D.

and SCHAEFFER, D.G., Springer-Verlag, New York, 1990, pp. 70-79. 26~ P.R. and BRADY, J.F., J. Fluid Mech., 276, 157-199 (1994). 261LYON, M.K. and LEAL, L.G., I U T A M Symposzum on Concentrated Suspenswns, Denver,

1995.


Phillips et al.'s Model

In Phillips et al.'s model, 262 the stress tensor is given by the generalised Newtonian constitutive equation:

a = - p l + r , r = 2~(r (37.56)

where ~ is the total stress tensor, P is the hydrostatic pressure, r is the extra stress tensor, D -- �89 q- ETu T) is the strain rate tensor, u is the velocity field, is the fluid viscosity, which is assumed to depend on the local volume fraction r of the solid according to Krieger's model:

r/ _ ( r (37.57) i-- .

Here v/8 is the viscosity of the suspending Newtonian liquid, Cm __ 0.68 is the maximum volume fraction, and a ~ 1.82 is a constant.

In addition to the usual conservation equations (mass and momentum), the concentration of the particles is supposed to satisfy the diffusion equation:

0r ~ - + u - V r -- V - N , (37.58)

where N is the flux of the solid phase, given by the constitutive equation

N - Kca2r (~r + Kna2~/r 71) , (37.59)

consisting of a contribution from the collisions between the particles (the first term on the right hand side) and a contribution from the gradient of viscosity (the second term on the right hand side), ~ - x/2trD 2 is the generalised (and positive) strain rate, and Kc and K n are constants. Choosing Kc - 0.41, and K n - 0.62 leads to a good fit to the circular Couette data.

The equation (37.58) is subject to the no-flux boundary condition, N - n = 0, at the solid boundaries, where n is the outward normal unit vector. An average volume fraction over the whole domain is also needed to be specified initially, to fix the level of the volume fraction.

Circular Couette Flow

We now consider a circular Couette flow, where the inner cylinder of radius Ri rotates from rest with an angular velocity ~, and the outer cylinder, of radius Ro, is kept stationary. The particles are uniformly mixed initially.

All length scales are normalised with respect to Ro, velocities to Ro~, and t ime to R2o/a2~. The dimensionless diffusion equation for the volume fraction is

0r + c-2fi . ~7~ -- ET. {Kcr ( ~ ) + Kn~dp2V In r/}, (37.60)

where T is the dimensionless time, the overbars denote a dimensionless quantity and e = a/Ro.

P r o b l e m 37 .B

262PHILLIPS, R.J., ARMSTRONG, R.C., BROWN, R.A., GRAHAM, A.L. and ABBOTT, J.R., Phys. Fluids A, 4, 30-40 (1992).


At steady state, the velocity is unidirectional, and all variables only depend on the radial coordinate. Show tha t the steady state concentration profile is given implicitly by (_~)2 ~_~ ((~m_~) ~o~(Kn/Kc-1 )

(37.B1) -

where r is the concentration at the outer wall. This concentration is determined by the global conservation of mass,

( R o - = i

in which Cb is the initial uniform volume fraction.

Ur r

0.20

~, =0.5,a/h = O. 1, Kc = 0.41, K~ = 0.62 0.6

0.5

0.15 0.4

0.10

0.05

. . . . . . . . . . . . steady velocity and concentration o . - - - - velocity at ,r - - -0.5,1.0,1.5,2.0,5.0,8.0 = fraction at ,r - -0 .5 ,1 .0 ,1 .5 ,2 .0 ,5 .0 ,8 .0

0.3

0.2

0.1

0.0 0.3 0.4 0.5 0.6 0.7 0.8 0.9 dR,,

FIGURE 37.2. The time evolution of the velocity and volume fraction in a circular Couette flow for the case where c = 0.1.

A numerical integration of (37.60) has been done, using a finite volume method, 263 with different values of e. It is found that the dimensionless t ime to reach the steady state concentration profile is about 5. That is, the real t ime needed to reach the steady state is

t , N ~ , (37.61)

which may be considerably long for small sized particles. Some typical velocity and concentration profiles at different times are shown in Figure 37.2, for c -- 0.1.

263pHAN-THIEN, N. and FANG, Z., J. Non-Newt. Fluid Mech., 58, 67-81 (1995).


Note from the figure tha t the approach to the steady state is faster for the velocity than for the concentration. In addition, the steady state concentration at the outer cylinder is considerably higher than tha t at the inner cylinder. Comparison with experimental data in this flow is excellent; in fact, the two constants Kc and K n are found by fitting circular Couette data as mentioned previously.

Channel Flow

U, ~0.7

0.20

0.15

0.10

0.05

~~ =O.5,a/h = 0.1, Kc = 0.41, I~ = 0.62

. . . . . . . . . . . . . . . steady velocity and concentration

o velocity at r =0.5,1.0,1.5,2.0,5.0,8.0 o fraction at '~ =0.5,1.0,1.5,2.0,5.0,8.0

0.6

0.5

0.4

0.3

0.2

0.1

0.3 0.4 0.5 0.6 0.7 0.8 0.9 dR~

FIGURE 37.3. The velocity and the volume fraction profiles at x - 5, 20, 40, 60, 80,and 100, for e = 0.1.

In the plane Poiseuille flow, we define e = a / H , where H is half of the channel height. The results of a numerical integration of (37.60) are shown in Figure 37.3 at different x. We now define the entrance length as the distance from the inlet to the point where the profile of the variable of interest differs from the well- developed profile by a small amount (say 1%). We find tha t the entrance length for the velocity is about 0 . 2 ( H / a ) 2 , whereas the entrance length for the concentration

is about 0 . 5 ( H / a ) 2 , which is within a factor of two from the estimate ( H / a ) 2 of N OTT and BRADY. 264 This entrance length can be quite large for small sized particles.

We now consider the channel flow in detail. Uniaxial flow is admissible, where the velocity is u - {u(y , t) , 0}, in which x is the flow direction, and y is the direction perpendicular to the channel walls. The shear stress, which is zero at the centreline

264NOTT, P.R. and BRADY, J.F., J. Fluid Mech., 275, 157-199 (1994).


y - O, is given by

Ou (37.62) S~ -- - a y -- ~(r

where G = -OP/Ox is the pressure gradient, which can be time-dependent. We shall assume that G is a positive constant here; the modification required for a time dependent G is very minor. The quantity yOu/Oy can be integrated, which leads to the relationship between the flow rate Q (per unit width) and the pressure gradient:

Q - 2G y2r/-1 (r (37.63)

However, this is not a linear relation in the pressure drop, since the concentration profile is also a nonlinear function of the pressure gradient.

From the generalised shear rate, a / = C/y/rl(r the particle diffusion equation becomes

0r a2 0 { ~ _ ~ ( ~ ) c~Ky(r 2 0 r (37.64) = H 2 0-~ g e e + # ( e r a - r '

where r = y / H is the dimensionless y-coordinate , and

HG T = t (37.65)

r/o

is the dimensionless time. The no-flux boundary condition at the wall ( -- 1, and at the centreline (~- 0 is expressed as

0 + aK,r162 0r

. ( era - r or (--0,1 = 0 . (37.66)

At the steady state, the flux is zero everywhere leading to

d-~ In - r . In (era - r -- 0,

which has the implicit solution for the concentration

\ r - r ~(r-~)

(37.67)

with r = r being the concentration at the wall, and F - Kn/Kc. We define

= . ( r - 1). (37.68)

The concentration at the centreline is a maximum only when/~ > 0, or F > 1. In addition, the slope of the concentration is given by

d~ -- d/)w ( dPm -- r )a([ '- l) [ 1 fl ] (37.69) dCb -- -- r Cm -- r -~b + r - r "

Near the centreline, where the concentration approaches the maximum concentration, say Cb -- (1 - - 5 ) e r a , we find tha t

dr dCb

Thus, if/3 < 1, or, equivalently,

( 1 - e ) r m

= O ( ~ f l - 1 ) .

K n i "Kc < 1 +-- 'a (37.70)

then d r -- 0 at ~ -- 0, and a rounded profile for the concentration is obtained at the centreline.

On the other hand, if ~ > 1, then d ~ / d r - 0 at (~ - 0, leading to a cusp profile for the concentrat ion there. Wi th a - 1.82, Kc -- 0.41, and K v -- 0.62, a rounded concentrat ion profile results, for all values of the mean concentration. However, the rounded region is small, and it appears visually as if there is a cusp at the centreline. It is therefore incorrect to s ta te tha t this model yields a cusp concentrat ion profile at the centreline.

P r o b l e m 3 7 . C

Consider the torsional flow between two parallel plates, with a fluid film thickness of H. Show tha t the velocity is given by

uo -- - ~ r z , (37.C1)

where ~ is the angular velocity of the top plate (the bo t tom plate is fixed), and a polar coordinate sys tem (r, 0, z) has been used. Show tha t the s teady s ta te concen- t ra t ion profile r satisfies

r e ~b m -- ~b - constant. (37.C2)

Note tha t the concentrat ion goes to zero as r --. oo. Demonst ra te tha t the above solution consti tutes the exact solution to the finite-

domain torsional flow 0 ~_ r ~_ R, where the outer boundary is frictionless (radial velocity ur and the shear stresses Soz - Sor = 0 there). The concentrat ion profile can be rewrit ten as

-- .... r Cm - r ' (37.C3)

where Cw is the concentrat ion at the wall r -- R. This is a very rare instance where an exact solution can be found for a non-Newtonian flow in a finite domain.

Pulsati le Flows

The behaviour of Phillips et al. 's model in pulsatile flows is quite interesting: it allows for a t ime-periodic velocity field, and yet the concentration profile is driven toward the s teady state. To unders tand this, let us consider a plane Poiseuille flow, in which a pulsatile pressure drop of

Ap -- Ap0 (1 + 5 cos wt) (37.71)


is imposed, where w is the frequency of oscillation, w = 27r/T, T is the period of oscillation, ~ is the dimensionless amplitude and Apo is the averaged pressure drop (corresponding to ~ = 0). There is a large amount of literature on the effects of pulsatile pressure gradients on pipe flows of polymer solutions; for a review, see BIRD eta/. 265 Overall, an increase in the mean flow rate, relative to the steady state value at the averaged pressure drop, is observed. This is due to the effective decrease in the viscosity at the wall because of the oscillations.

In this flow the velocity field is uni-directional, u = {u(y,t), 0}; x is the flow direction, and y is the direction perpendicular to the channel walls. As noted before, the shear stress, which is zero at the centreline y - 0, is given by

r - - z x p ( t ) y = ,7( ) N . (a7.72)

The quantity yOu/o~ can be integrated by parts, which leads to the relationship between the flow rate Q and the pressure gradient:

/o Q(t) - 2Ap(t) y2~-I (r (37.73)

The generalised shear rate is given by ~ = G(t)y/vl(r where G(t) = IAp(t)l. The particle diffusion equation becomes

(9(/) a 2 HG(t) O { ~_~ ( ~~r ) ~Kn~r 2 0 r (37.74) 0t = H 2 ~/0 0r g c r + # ( r 1 6 2 ~ ,

where ( = y /H is the dimensionless y-coordinate , 77 = 7/0/z (r and v/0 is the viscosity of the suspending liquid. We now stretch the time scale to T, where

dT a 2 gG(t) = 0. (s7.75)

dt H2 ~0

Then the volume fraction equation becomes

=~-~ g e e + # ( r 1 6 2 1 6 2 '

which is subject to the no-flux boundary condition at the wall r - 1, and at the centreline (~ = 0, we have

K 0 ( ~ ) ~Ky ' r 0r I - -0 . (37.77)

The preceding two equations are identical to those of a channel flow with a constant pressure drop. They have been shown to have a steady state solution in a t ime scale of the order T, ~_ 5. Thus, the concentration profile will be driven toward a steady state solution in a real t ime scale of ts, where

a 2 /t. v, = G(t)dt. (37.78)

~0H

265BIRD, R.B., ARMSTRONG, R.C. and HASSAGER, O., Dynamics of Polymeric Liquids: Vol. I. Fluid Mechanics, John Wiley & Sons, New York, 1977.

In the case Ap(t) - Ap0 (1 + 6 coswt), where 5 > 0 is arbitrarily, we have G ( t ) -

Ap0 [1 + 5coswt[. Based on the inequality

I1 - 5 Icos ~otl{ < {: + 8 cos ~ot{ < 1 + ~ Icos ~ot{,

it can be established that the time to reach a steady state concentration profile is bounded by

1 r/0H 1 r/0H

independent of the oscillation frequency. Similar results can be established for oscillatory circular Couette and torsional

flOWS.

P r o b l e m 37.D Consider the pulsatile circular Couette flow, where the inner cylinder (of radius

R~) is set to rotate with an angular velocity of ~(t), and the outer cylinder (of radius Ro) is stationary. Show that the torque on either cylinder per unit length is given by //;~ T( t ) - -21r12( t ) , r-3rl-i(dp)dr. (37.D1)

For the case T(t) - To (1 q- ~ cos wt), where ~ is arbitrary, show that the time for the concentration to reach steady state is bounded by

1 2~o/~o 1 2rr/0Ro 4 (37.D2) %'] -b 2~]7r a2T0 <-- ts g Ts [ 1 - 25/~ I a2T0

which is therefore independent of the frequency of the oscillations; it depends only on the magnitude of the oscillating torque.

P r o b l e m 37.E Consider the oscillatory torsional flow between two parallel disks, where the

lower disk is stationary and the upper disk is allowed to rotate at an angular velocity ~(t) = Q0 (1 + 5 cos wt), where 5 is arbitrary. Show that the time for the concentration to reach steady state is bounded by

1 H 2 1 H 2

~ 1 + 2 ~ / . a2a0 < t. < ~ I1 - 2~/-I a m o (ar.E~)

Thus, once again the .velocity field is oscillatory, but the concentration will be driven toward a steady state, and the time taken to reach a steady-state concentration does not depend on the frequency of oscillations; it depends only on the amplitude of the driving force.

. . . . . . . , , , , , , , , . . . . . , , ,

In summary, we have presented the basic elements of microstructural theories, starting with the Langevin description of long chain molecules, and ending with lubrication theories for concentrated suspensions, and the phenomenon of flow- induced migration of particles. For dilute polymer solution, a constitutive equation of the Maxwell-Oldroyd type, with a FENE force law should be adequate in most cases. For polymer melts, the reptation theory is most relevant, not because of the


detailed predictions (some are poor), but because of the broad framework provided by the theory under which the dynamics of entangled polymer chains can be effectively studied. For numerical investigations, given the computing resources available in a typical research environment of today, we believe that constitutive equations of the differential type, similar to Chilcott-Rallison or P T T models, are to be preferred, and we expect to see their increasing use in most commercial packages. The situation with suspensions is less clear; however, there are several numerical tools available to probe the structural evolution. It seems that a constitutive equation of the type proposed by HINCH and LEAL should be relevant; for it to be usable, the correct dependence of the rheological properties of the suspension on the volume fraction should be built into the constitutive equation, as well as the flow-induced migration of particles.

5 The Shape and Nature of General Solutions

In this Chapter, we exhibit many flows which furnish illustrative and, perhaps, instructional, examples from which one may discern the uses of the isotropy restriction on the stress functional of an incompressible simple fluid. The major goal is to discover, if possible, those shear stress components which are zero or those which are interrelated; also, to find if the normal stress differences are invariant under coordinate or flow reversals. The particular instances we have in mind arise from motions with constant stretch history, steady and oscillatory simple shear, extensional flows, a squeezing flow between a wedge when combined with extension, and a squeezing ftow in a cone with an extension superposed on it. These matters form the substance of w and in w we list the equations of motion in Carte- sian, cylindrical and spherical coordinates for use in the subsequent sections and chapters.

Section 40 is confined to a thorough treatfnent of viscometric flows: those per- petual motions from which one may grasp the effects of viscosity and the normal stress differences. We discuss a variety of flows from which these material functions may be measured. They include the Couette flow, the Poiseuille flow in a pipe, and their combination which is the helical flow; there are detailed t rea tments of the torsional and cone-and-plate flows and the axial motion of fanned planes. The underlying principle is to examine when the viscosity function determines the velocity field, for when it does, the apparatus is shown to afford the determination of the viscosity more easily than the two normal stress differences. On the other hand, when the velocity field is specified in its entirety, such as in the torsional flow or the cone-and-plate flow configurations, it is found that these flows provide a better opportunity to measure the normal stress differences, although one may find the viscosity from them as well.

In w the dynamic compliance of rectilinear motions is investigated. It is found tha t as a general rule, rectilinear motions are possible in circular pipes, or in the an-

272 5. The Shape and Nature of General Solutions

nulus between two concentric cylindrical surfaces, or in channels, or as axial motions of fanned planes. It has been known for a long time that in other cross-sections, rectilinear flows occur if the viscosity and the second normal stress difference function of a fluid are interrelated. However, exceptions to this rule do exist and we illustrate a few of them.

In w non-viscometric flows are examined. They include the helical-torsional flow and the helical flow-axial motion of fanned planes. These flows may be considered to be academic curiosities for they are not easily generated in the laboratory.

Later on in w a detailed study of the dynamics of the flow in the eccentric disk rheometer is made when inertia is absent and when inertia does exist. Finally, extensional flows are studied in order to introduce the material property, called the extensional viscosity. Apart from the usual, homogeneous extension of a rod or that of a sheet, the only dynamically possible non-homogeneous extensional flow is mentioned here.

The Chapter closes in w with a list of unsteady flows which are dynamically sustainable in all incompressible simple fluids. These arise quite easily from their steady counterparts in w167 We also include two unsteady flows generated by the squeezing of a wedge or a cone with a superposition of an extensional flow on each. Thus, this Chapter is a compendium of flows whose universal forms are known; the specificity may vary from one fluid to another. It is this last point where the emphasis on generality in this Chapter must be seen as bearing fruit for once the way the general solution depends on the material functions is understood, those particular features affecting a specific fluid may be determined either analytically or numerically.

38 Some Consequences of the Isotropy of the Constitutive Functional

38.1 Path Line and the Constitutive Functional

It is essential to note that except at singular points, such as sources, sinks, saddle points, homoclinic and heteroclinic points and others found in the field of dynamical systems, the path line of a particle is unique. This means that the position ~ of a particle at time T is a unique function of its current position, x, the current time t and T. See w for specific examples.

Thus, the relative strain tensor Ct(T) will be a function of (x, t, T). When this is substituted into the constitutive relation and the extra stress tensor S is calculated over the time interval - o o < T < t, one finds that S is a function of (x, t) in general. However, in steady flows, it is known that ~ is a function of (x, ~ - t) - see (1.25). Equivalently, the position of a particle at time ( t - s), 0 _ s < oo, is a function of (x, s) only. Hence, the extra stress tensor in a steady flow becomes a function of x only, or it too is steady. Thus, in the sequel, these notions of a steady stress tensor being associated with a steady flow and an unsteady one arising in an unsteady flow will be used without further comment. However, the isotropy of the constitutive functional provides additional information about the stress field and we turn to this next.

38 Some Consequences of the Isotropy of the Constitutive Functional 273

38.2 Isotropy of the Constitutive Functional As shown earlier, the constitutive functional (33.3) fSr the extra stress tensor S, viz.,

obeys the restriction

0 < (38.1)

Q(t).Tz(Ct(t - s)) Q(t) T - JZ(Q(t)Ct(t - s)Q(t) T) (38.2)

for all orthogonal tensor functions Q(t). In general, there are six independent components of the stress tensor. However,

because of the nature of the velocity field or the strain history, it is probable that one or more of these components are zero, or they are interrelated. In this section, the isotropy of the constitutive functional will be used time and again to demonstrate how to derive some of these results. For example, it will be seen that in a viscometric flow, there are only four non-zero functions of the shear rate and in an extensional flow, there are only three non-zero functions of the extensional rate. Application of the restriction (38.2) to more complicated flows is possible and some of these are also exhibited here. The main point that needs emphasis is that once the principle of application of (38.2) is understood, it is easy to derive corresponding results for other constitutive functionals.

38.3 Motions with Constant Stretch History We begin with motions with constant stretch history first for they encompass both viscometric and extensional flows. From (38.1)-(38.2), the stress tensor at time 0 is seen to obey

q ( t ) y ( c 0 ( 0 - s))Q(t) ~ - y ( q (t) Co (O - ~) q (t) ~)

= ~e(c,(t- ~)), (38.3)

with the last relation following from the definition (9.1) of a motion with constant stretch history. What (38.3) says is that the stress tensors at times 0 and t experienced by the particle are related through

Q(t)s(o)Q(t) T = s(t) . (38.4)

If a rotating basis orthonormal basis ei(t) is attached to each particle such that

e,(t) = q( t )e , (0) , / = 1,2,3, (38.5)

it follows from (38.4) that

S(t)e~(t) = q(t)S(0)e~(0). (38.6)


The rotating basis has been used earlier in w when discussing the kinematics of motions with constant stretch history and a result similar to (38.6) occurs there.

Now, using (38.5)-(38.6), one obtains

( t ) . -

= ej(o)Tq(t)Tq(t)S(O)ei(O) (38.7)

= �9

The interpretation of (38.7) is tha t the physical components of the stress tensor at times 0 and t are the same. Hence, the particle in a motion with constant stretch history experiences a state of monotonously identical state of stress. Therefore, all memory effects must vanish and the stress tensor at time 0 must be a function of the tensor M which determines the strain history because (cf. (9.3))

c 0 ( o - - ( 3 8 . 8 )

That is, in these motions,

S ( 0 ) - . , ~ ' (C0(0 - s ) ) -- f (M). (38.9)

This function f of M must obey the isotropy condition corresponding to (38.2), viz.,

Q(t)f(M)Q(t) T - f(Q(t)MQ(t)T). (38.10)

We shall examine the consequences of the above in steady simple shear 1 and an extensional flow 2 next.

38.~ Steady Simple Shearing Flow

In steady simple shear flow, the components of the velocity field are described in a Cartesian coordinate system through

u--~y, v----0, w - - 0 , (38.11)

where ~ is the constant shear rate. The tensor M which determines the strain history at t ime 0 is the velocity gradient L of the flow (38.11) because the latter is a constant. Indeed, in this flow, the stress tensor at time t is identical with that at t ime 0 because the basis vectors at t ime t are identical with that at time 0, or the usual triad {l,j, k} is the basis with respect to which the components of the stress tensor are calculated. The matrix of M is given by

[M]-- 0 0 . 0 0

(38.12)

1COLEMAN, B.D., MARKOVITZ, H. and NOLL, W., Vzscometr~c Flows of Non-Newtonian Fluzds, Springer-Verlag, 1966.



Now, in (38.10), choose the orthogonal tensor Q to be the constant diagonal matrix

1 0 O ) [ Q ] - 0 1 0 .

0 0 - 1 (38.13)

Then, the left side of (38.10) gives rise to the following: (lo o)( x o o) 0 1 0 �9 Syv Svz 1 0 0 0 - 1 �9 Szz 0 - 1

( S ~ S.~ -S~) S~ -S~, .

�9 Szz

(38.14)

Note that the symmetry of the stress tensor has been used in replacing the off diagonal elements with dots.

The implication of (38.14) is that reversing the z-axis causes the stresses Szz and S~z to change sign. On the right side of (38.10), we find that

(lo o)(o o)(io o) (i i) 0 1 0 0 0 0 1 0 - 0 . 0 0 - 1 0 0 0 0 - i 0

(38.15)

That is to say, the reversal of the z-direction means that the tensor Q M Q T remains unchanged. Hence, its function must be equal to the function of the tensor M, or the right side of (38.10) must still have the component form

�9 Szz (38.16)

Comparing (38.14) with (38.16) one observes immediately that in a steady simple shearing flow, two of the stress functions Sxz and Suz are both zero. The stress tensor thus has the form

I s ] = . o , �9 S Z Z

(38. 7)

where the four non-zero stresses are functions of the shear rate ~. We may find an additional property of these stresses next by choosing the orthog-

onal tensor Q in (38.10) to be the diagonal matrix [-1, 1, 1]. Physically speaking, this means that the ftow direction is reversed and one finds that the left side of (38.10) takes on the form:

�9 S ~ v 0 ,

�9 Szz

(38.18)


whereas, on the right side, the tensor Q M Q T = - M . Thus the flow reversal causes the shear stress Szu to change sign while the normal stresses S=z, Suu, Szz do not. Hence, the shear stress is an odd function of the shear rate and the normal stresses are even functions of the shear rate. We now introduce the notation:

5'x= -- Suu -- N1, Suu - Szz = N2, S'xu - a , (38.19)

where Nx and N2 are the first and second normal stress differences respectively and a is the shear stress. From what has just been derived, we have

N i ( - ~ ) - Ni(~), i = 1,2; (38.20)

=

It can be seen tha t changing the y-direction to its opposite does not result in any more restrictions on the three material functions - the two normal stress differences and the shear stress function of viscometric flows.

In te rms of the t r iad of shear axes {a, b, c} of a viscometric flow - see w above, (38.19) implies t ha t

Saa - - S b b - -" NI, Sbb -- Scc -- N2, S~b -- a. (38.21)

Quite often, we shall use three other functions: the viscosity ~ and the normal stress difference coefficients ~I and ~2. These are defined through

r / ( ~ ) ~ - a(,~), ~(;y)-~2 -- N~(~/), i - 1, 2. (38.22)

These new functions, r/, ~ l and ~2 are all even functions of the shear rate of course, and for the most part , we shall consider them as functions of ~. However, there are occasions when it is useful to consider them as functions of ,~2. For instance, see w below where it is found tha t the calculations concerning rectilinear flows are simplified if one assumes tha t ~P2 is a function of ~2.

38. 5 Const i tut ive Relat ion in Terms of A] and A2

In a viscometric flow, it is known from w and w tha t only the first two Rivlin- Ericksen are non-zero. Thus, the stress tensor in a viscometric flow is a function of A1 and A2. We shall examine how to obtain a simple form of the consti tut ive relation in viscometric flows in te rms of the three material functions. Since the tensor M is determined by the first two Rivlin-Ericksen tensors in this flow - see (9.29) for an explanation, the consti tut ive relation is an insotropic function of these two tensors. Tha t is,

S -- g(A1, A2), Qg(A1, A2)Q T - g ( Q A 1QT, QA2QT) (38.23)

for all orthogonal tensors Q. The integrity basis for such a function was obtained by RIVLIN 3 and this consists of eight terms, for the ninth term, proport ional to the unit tensor, may be absorbed into the pressure. The eight te rm expansion was reduced to a simpler form 4 and using the notation of this treatise, we may write

, , , ,

3RIVLIN, R.S., J. Rational Mech. Anal., 4, 681-702 (1955). 4CRIMINALE, W.O., Jr., ERICKSEN, J.L. and FILBEY, G.L., Jr., Arch. Rational Mech.

Anal., 1,410-417 (1957).


it as 1

S -- n(~)A1 -b [~1(~)+ ~2(qr - ~ l ( ~ ) A 2 . (38.24)

In checking the above form, recall that in steady simple shear flow, the two Rivlin- Ericksen tensors are represented by (0 o) (! 0 o)

[Al l - - �9 0 , [A21- 2'~ 2 0 . ( 3 8 . 2 5 )

�9 0 �9 0

P r o b l e m 3 8 . A

Let it be given that in a steady simple shearing flow, the heat flux vector q is an isotropic function of the first two Rivlin-Ericksen tensors and the temperature gradient vector g, which has the form

(~ g - - dO/dy , (38.A1) 0

where 0 is the temperature which is a function of y only. Assuming that

Qq(A1, A2, g)QT = q(QA~ QT, QA2QT, Qg) (38.A2)

holds for all orthogonal tensors Q, prove that 5 the component q~ - 0, that q~ is proportional to "~(d~/dy); and that aN is proportional to '~2(dO/dy)2, with the coefficients of proportionality being functions of ~2 and (dS/dy) 2.

38 .6 E x t e n s i o n a l F lows

In an extensional flow, the velocity field has the form(cf. (8.8)):

u - - a x , v - b y , w - c z , (38.26)

where a, b, c are constants. Incompressibility of the fluid is met by imposing the restriction a + b + c -- 0. Because the velocity gradient L is a constant, the extra stress tensor is an isotropic function of this gradient tensor. That is

S - - f(L), Qf (L)Q T - f ( Q L Q T) (38.27)

for all orthogonal tensors Q. Using the latter tensors in the form Q : diag[-1, 1, 1], d iag[1, -1 , 1] and diag[1,1, -1] , it is easily shown 6 that the off-diagonal stresses Sxy,Su~,Sz= are all zero. Also, because the velocity gradient is symmetric, A1 : 2L. Thus, it follows that one may write (38.26) in the form

S = h(A1). (38.28)

5HUILGOL, R.R., PHAN-THIEN, N. and ZHENG, R., J. Non-Newt. Fluid Mech., 43, 83-102 (~992).



Because of (38.27), the expansion of (38.28) may be effected by using the integrity basis in the Appendix to Chapter 2 and one has

S -- a l A 1 + a2A12, (38.29)

where a l , a2 are functions of the invariants trA21, trA~.

38. 7 Unsteady Shear Flow

If one considers the unsteady shear flow

u = u(y,t), v - O , w - O ,

then the shear strain is given by

t

,y(t)= o

The relative shear strain is thus

= N ( y , o ) t

Hence the relative strain tensor has the following form

[Ct(r) ] - - �9 l+72(-r, t) 0 . �9 1

(38.30)

(38.31)

(38.32)

(38.33)

P r o b l e m 3 8 . B

Using (38.2), show that in the unsteady shear flow (38.30), the stresses Sxz and Syz are both zero. Also, show that the shear stress Sx~ is an odd functional of the strain 7(T, t) while the three normal stresses are even functionals of it. 7

38.8 N o n - V i s c o m e t r i c F l o w s

Applications have been made to discover the consequences of the isotropy condition (38.2) to doubly superposed viscometric flows s and the eccentric disk rheometer flow 9 for example. These will not be discussed here.

Instead, attention will be focussed on slight generalisations of two recently discovered motions 1~ that are possible in all incompressible simple fluids provided

7COLEMAN, B.D. and NOLL, W., Ann. New York Acad. Sci., 89, 672-714 (1961). 8HUILGOL, R.R., Quart. Appl. Math., 29, 1-15 (1971). 9HUILGOL, R.R., Trans. Soc. Rheol., 13, 513-526 (1969); GODDARD, J.D., J. Non-Newt.

Fluid Mech., 4, 365-369 (1979). t~ N., J. Non-Newt. Fluid Mech., 16,329-345 (1984); Rheol. Acta, 24, 119-126

(~985).

inertia is ignored. Leaving aside the compatibili ty of the motions with the equations of motion till w below, a brief discussion of the kinematics of these flows and the use of the isotropy of the functional will be given here.

Consider the velocity field in cylindrical coordinates in contravariant form

? -- r [ a + f ' (~, t)], ~ -- - 2 f (O , t), ~ = - 2 a z , (38.34)

where ~ is a constant, f ' - - Of/OO and f(8, t) is to be determined. The above flow is kinematically possible in an incompressible fluid, because the divergence of the velocity field is zero.

Denoting by (~, v], r the coordinates at t ime T of a particle at which is at (r, ~, z) at t ime t, the second and third equations may be integrated, in principle, to yield

~7(T) - f2(~, t , T), ~(T) - ze 2(t-r). (38.35)

It follows tha t ~(T) -- r e f [ (a+f') da ._ r f l (0, t, T). (38.36)

If these pa th lines are used to calculate the covariant components of the relative strain tensor Ct(T) through (3.4), one finds tha t this matrix has the form

( 0.1 0 ) f l 2 r f l OO [Ct(T)],j -- �9 r 2 [ ( - ~ ) 2 + f12 ( - ~ ) 2] 0 , (38.37)

. e4a(t-r~

where we have used the fact tha t the component g22(~) of the metric tensor is given by

g22 -- ~2 _ r2f l 2. (38.38)

The physical components of the strain tensor in (38.37) are easily seen to be functions of (~, t, T) and thus the stresses will be functions of (8, t) when the consti tutive relation is evaluated as a functional over - o o < T _< t. Now, because interchanging the z direction with the - z direction leaves the relative strain invariant, it follows from an application of (38.2) - use Q = diag[1, 1 , -1 ] , tha t the stresses Srz and Soz are zero in the flow (38.34) in all incompressible simple fluids.

P r o b l e m 38 .D

Consider the velocity field in spherical coordinates in contravariant form

? = r [a + f(O, t)], ~ = g(O, t), ~b -- - 3 a r (38.D1)

where a is again a constant. Show tha t incompressibility demands tha t

3 f + g' + g cot 0 -- 0. (38.D2)

where g' = Og/OO. Establish tha t a solution for the pair (f, g) may be found in terms of a new function h - h(#, t), where # = c o s O and 11

Oh 3 f - - - ~ , g - - s i n O h. (38.D3)

11PHAN-THIEN, N., Rheol. Acta, 24, 119-126 (1985).

Calculate the covariant components of the relative strain tensor Ct(~-) and show that its components in the (r, r and (0, r directions are zero. From the covariant form prove that the physical components of the relative strain tensor are functions of (0, t, T) only. Finally, deduce that the stresses Srr and See are again zero.

39 Equations of Motion in Curvilinear Coordinates

The equations of motion for an incompressible simple fluid can be obtained from (15.9), viz.,

div T + pb -" pc, (39.1)

by replacing the total stress tensor T by - p l + S, where p is the pressure. Then, in direct notation, the equations of motion become

- V p + div S + pb = pa, (39.2)

which have the following form:

-P,i + S~j,j + pbi = pai, i , j = 1,2,3, (39.3)

in curvilinear coordinates. In (39.3), Sij ,p,i , bi and ai are the covariant components of the extra stress tensor, the pressure gradient vector, the body force vector and the acceleration vector respectively and the comma (,) denotes the covariant derivative. The set of three equations in (39.3) have to be cast in their respective physical component forms when problems in cylindrical or spherical coordinates have to be solved. Here, the notation of S to represent the physical components of the tensor S is very cumbersome; equally, the notation Sii cannot be accepted because its use is reserved for the covariant components. Hence the following: Phys i ca l C o m p o n e n t N o t a t i o n The physical components of an arbitrary vector v and a second order tensor A will be denoted by va and A~b respectively, where a and b stand for

(i) x , y , z in Cartesian coordinates; (i) r, 0, z in cylindrical coordinates; (iii) r, 0, r in spherical coordinates. The physical components of the velocity vector will be denoted by u, v, w in each

coordinate sytem. For future use, we now list the equations of motion in various coordinates. They can be derived, with some effort, from using the covariant derivative or the dyadic operator X7 because div S = V- S. See Appendix to Chapter 1 on these matters.

Cartes ian Coordinates

O S ~ O S ~ OS~ Op + + + + pbx -o-7 a~ av a~

OS=~ OS~ OS~= Op + + + + pb~ --~ o~' oy o~ or, os:~ os~= os~

- ~ + oz + oy + o; +pb:

f Ou Ou 8u Ou "~ - P ( - X + ' N + l ov av Ov Ov '~

+ u + v + w ,

[Ow Ow 8w Ow "~ + u + v + w

(39.4)

39 Equations of Motion in Curvilinear Coordinates 281

Cylindrical Coordinates

Op OSrr 10Sre OSrz Srr - Soo - ' ~ + or + - ' - - - - + + +pbr r O0 OZ r

YOu Ou y ou O u v 2 , = P~-~ + '~-g + 7 - ~ + ~ ' ~ 7 ) '

10p + OS~o 10Soo OSoz 2 S - 7 0 " 0 Or + 7 00 + Oz + 7 ro+pbo

/Or Ov v Ov Ov uv \

= p ( ~ + ~ g + 7 ~ + ~ g + r ) , Op k OSrz 10Soz OSzz 1S Oz Or H 7 "~ 1- Oz + 7 , ,z+pbz

= p - -g+ ,~ . - -~+- , - -~ + ~ .

Spherical C o o r d i n a t e s

(39.5)

_ _ _ OSrr 10Sro 1 Srr %+ + _ _ _ _ + Or Or r O0 r sin 0 0r

1 + - [ 2 S ~ - See - Sr162 + cot 0 S~o]

r

+pb~ -- p + U or + -r - ~ -t w Ou v 2 + w 2

r sin 0 0r r ) '

10p OSro 10Soo - 7 0 - ~ + o r + - - - - r 00

1 Sor 1 r sin 0 0r b -[3SrOr + cot O(Soo - S,r

(_~_ Ov vOv + pb o - p + U-~r + - r - ~ -t

w Ov uv w 2 cot O) r sin 0 0r I r r

o&o i osoo ,,1 %+ -rs in00- '~ Or r 00

i &~ r sin 0 0r

1 + -[3Srr + 2 cot 0 S0r

r

Ow Ow v o w +pb, - p -~- + ~-~- + 7-b-g +

sinW ~ Owa(p uw vw cot 0 ) ~ _ + + ~ . r r r

(39.6)

39.1 Body Force

Typically, in fluid mechanics, the body force is assumed to be derived from a potential or it is zero. In the former case, b - - V r so that we have to solve the equations

- V p + div S = pa, (39.7)

where p includes the PC term. Of course, when the body force is zero, we have to examine

- V p + div S = pa. (39.8)


39.2 Inertia

When inertia is included, tha t is the te rm pa is included in the equations of motion, it is also assumed tha t the body force is derivable from a potential so tha t (39.7) applies. Quite often, inertia is ignored in the solution of the problem. This means tha t the te rm pa is put equal to zero along with the body force and it is assumed tha t the equations of motion become

- V p + div S = O. (39.9)

39.3 Homogeneous VelocityFields It is known from w tha t in a homogeneous velocity field, the deformation gradient tensor is a function of t only, whether the flow is steady or not. Hence, the relative strain tensor history is a function of t and the t ime lapse and thus, the extra stress tensor in a simple fluid is a function of t only. This means tha t the divergence of this stress tensor is zero. Hence, the equations of motion, in the presence of a conservative body force, are trivially satisfied, because one may ignore inertia if necessary. Of course, when the inertial term is zero or is the gradient of a sclalar, the pressure te rm can be chosen to equilibrate it. In particular, these observations show tha t the unsteady, homogeneous flow (6.6), repeated here for convenience,

= -

- z, (39.10)

-- 0~

is possible in all incompressible simple fluids, for the acceleration field associated with it is zero.

39.4 General Procedure for Solutions of Problems In what follows, except in rare circumstances, we shall assume a framework for the kinematics of the velocity field and determine the components of the extra stress tensor S from it by using a constitutive equation. Sometimes, we shall use the comprehensive category of the simple fluid so tha t we may be restricted to determining tha t S depends on a single coordinate and /o r time. In all of these circumstances, we shall follow the fundamental procedure of RIVLIN 12 and determine the pressure field from the equations (39.7)-(39.9), choosing the relevant form from this set.

In a few circumstances, the pressure gradient may be prescribed, e.g., in pipe flow, along with certain features of the velocity field. Here we have to ascertain tha t the equations of motion are satisfied when the constitutive equation of the fluid, even if it is the simple fluid, is prescribed.

12See the various papers from 1947-1949 as applied to finite elasticity which appeared mainly in the Phil. Trans. Roy. Soc. Lond., Set. A. Also, see the paper in J. Rational Mech. Anal., 5, 179-188 (1956) for the application of this method, sometimes called the 'inverse method', to viscoelastic fluid flow problems.

Numerous examples of these two general procedures appear in the sequel, beginning with their application to viscometric flows next.

40 Viscometric Flows

The kinematics of these flows have been discussed in great detail in w In this section, we shall discuss their dynamics as well as pay at tention to experimental configurations where the three material functions - the viscosity and the two normal stress differences - can be measured.

The dynamic compliance of a kinematically viable viscometric flow may be established in a brute fashion. As stated in w there are only seven classes of such flows and one may examine each and everyone of these classes of flows in turn to determine its dynamic compliance. However, this process does not provide any insight into the two important questions associated with the dynamic feasibility" What are the roles played by the normal stress differences and the viscosity functions? Can the effects of the normal stress differences be separated from tha t of the viscosity? The answers appeared in 1968 - 1970 and we summarise below the results of the two fundamental papers due to PIPKIN 13 and YIN and PIPKIN 14 on these matters, after discussing the case of Couette flow as an illustration.

~0.1 Couette Flow

The velocity field in Couette flow is given by

v ( r ) - rw(r)eo, n l <_ r < R2, (40.1)

in cylindrical coordinates. This azimuthal flow is assumed to occur between two concentric circular cylinders of radii R1 and R2(> R1) respectively and the boundary conditions are tha t the fluid adheres to the cylindrical surfaces; these are not needed at present. Because the shear rate is

dw (40.2) dr '

all the extra stresses are functions of r; also, the stresses Srz and SOz are both zero because the flow, which is locally equivalent to steady simple shearing, is such tha t the vector a is parallel to e0, whereas the vector b is parallel to er. The equations of motion can now be simplified from the general forms in (39.5) and we have

dSrr Srr -- SO0 Opt_ 4- . -O-'-r- dr r

1 019 dSro 4- _ 2 S t ~ -7o'-a + dr' ,"

Op Oz Pg

-- _ffrw 2 '

-- 0, (40.3)

= 0.

13pIPKIN, A.C., Quart. Appl. Math., 26, 87-100 (1968). 14yIN, W.L. and PIPKIN, A.C., Arch. Rational Mech. Anal., 37, 111-135 (1970).


The body force terms in the (r, 0) directions are taken to be zero and that in the z-direction is - g because the z-axis has been assumed to be vertically upwards.

Solving for the pressure function p, it is found that (40.3)3 leads to

p(r, O, z) = -pgz + f (r, 0). (40.4)

However, (40.3)2 demands that Op/OO be independent of 0; if this is not so, p(r, 27r, z) will not be equal to p(r, O, z) as it must, if the pressure field is to be a continuous function of its variables. Thus, we have

o, = -pg + f ( O ,

and the function f ( r ) has to satisfy

(4o.5)

df dSrr Srr - So0 = . . . . . + + (4o.6)

Moreover, (40.3)2 is now simplified to

dSro dr + 2 Sr~ = (40.7)

M Sro - 27rr 2 , (40.8)

and this has the solution

with the constant M being the torque required, per unit height of the fluid column, to maintain the flow. Since Sro is the viscometric shear stress, one has that

M a(~) -- r/(q)~ -- 27rr 2 (40.9)

which is a first order, perhaps nonlinear, ordinary differntial equation for the determination of the angular velocity function w -- w(r). It is clear from the foregoing that the velocity field is determined by the viscosity function and not the normal stress difference functions.

Couette flow is not the only one where the viscosity determines the velocity field. In Poiseuille flow, helical flow and channel flows, the situation is the same; in steady simple shear, torsional flow and cone and plate flow, the velocity field is given and the viscosity plays no part at all. What is equally important is that the normal stress difference functions do not affect these velocity fields, although they play their part in determining the pressure fields. These facts lead to partial controllability and controllability, a pair of terms introduced by PIPKIN 1~ and to their elucidation we turn next.

0.2 Par t ia l Control labi l i ty

The constitutive relation (38.24), which applies in viscometric flows only, can be given the more suggestive dyadic form

S -- r/(~)~/(ab + ba) + ~/2 [~I/1 (~) -~- ~I/2 (~()]aa -}- ~2~P2bb. (40.10)

lSpIPKIN, A.C., Quart. Appl. Math., 26, 87-100 (1968).

As s ta ted in w we shall assume tha t the body forces have ben absorbed into the presure t e rm and so we have to consider

- V p + div S -- pa. (40.11)

Following P IPKIN, we shall call a viscometric flow part ial ly controllable, if for all choices of normal stress difference coefficients ~Pl and ~P2, the divergence of the normal stress te rms in (40.10) is the gradient of a scalar r tha t is,

div{~2[~pl(~) + ~P2(A/)]aa + A/2~2bb } = -V(I). (40.12)

There are only six general forms of such flows and they are included in the seven flows given earlier in w The six are as follows:

(i) Tangential sliding of parallel plane slip surfaces:

v - u(y) i + w(y)k (40.13)

in a Cartesian coordinate system.

(ii) The helical flow, described in cylindrical coordinates through,

v = r w ( r ) e o + u ( r ) e z . (40.14)

(iii) Screw motions of right helicoidal slip surfaces. In cylindrical coordinates,

v -- a ( z - cO)(ree + cez) , (40.15)

where a and c are constants. From a text on differential geometry, 16 we recall t ha t a right helicoidal surface is generated by a straight line which is orthogonal to the axis about which it rotates. Thus, the line z - c O - constant will generate a right helicoidal surface if 2 - c0 and this is reflected in the above velocity field.

If one puts c - 0, then there is no rise in the z-direction as the generating straight line sweeps out a surface and the flow becomes the torsional flow, and this will be investigated later in this section.

(iv) Axial motion of fanned planes. In cylindrical coordinates,

v = C0ez, 0 _< 0 < 27r, (40.16)

where c is a constant.

(v) Rota t ion of conical slip surfaces about a common axis. In spherical coordinates:

v -- -~r sin 0[ln sin 0 - In(1 + cos 0)]er

where -~ is the constant shear rate.

" (40.17)

16WILLMORE, T. J., An Introduction to Differential Geometry, Oxford, 1959, pp. 37- 39.


(vi) The motion with a flexible slip surface:

x (X, t) - r0(1 + ~2 t2 ) - l [ a (a ) - t~b(a)] + z0k. (40.18)

It is interesting to note tha t rectilinear flows do not appear in the above list and the reasons for their absence will be made clear below in w Further, in what follows, we shall not discuss the motion with a flexible slip surface for it is possible over a finite t ime span only. 17

40.3 Roles of the Divergence of the Shear Stress Tensor and the Acceleration Field

Now, it is not necessarily t rue tha t each one of the flows (i)-(v) is compatible with the equations of motion. In order for them to be compliant, one has to solve the equations resulting from (40.10)-(40.12), i.e.,

- V ( p + r + div {~(-~)~(ab + ba)} - pa. (40.19)

A glance at this shows tha t it is desirable to classify the flows in the list (40.13)- (40.17) into addit ional categories, depending on the divergence of the shear stress tensor and the nature of the acceleration field. These aspects will be examined next.

~0.~ Divergence of the Shear Stress Tensor is Irrotational

Let us now examine all the flows in the list (40.13)-(40.17) and seek when the divergence of the shear stress tensor is such tha t for every choice of the viscosity function,

div {~/(-~)~(ab + ba)} -- - V X , (40.20)

where X is a scalar. The following list is exhaustive and contains all such flows:

�9 Steady simple shear.

�9 Couet te flow with a uniform shear rate:

v -- ~/r ln(r/R) eo, (40.21)

where R is a constant. This part icular flow is clearly a special case of the more general form in (40.1).

�9 Screw motion of right helicoidal slip surfaces - see (40.15).

* Axial motion of fanned planes - see (40.16).

t7See YIN, W.L. and PIPKIN, A.C., Arch. Rational Mech. Anal., 37, 111-135 (1970) for a discussion of the dynamics of this motion.


The dynamic compliance of these flows is now reduced to a study of the following:

- V ( p + �9 + x) = (40.22)

It is easily demonstrated that three flows listed as (a), (b) and (d) above have irrotational acceleration fields and hence they are possible in all incompressible simple fluids, regardless of the forms of the viscosity and normal stress differences, i.e., all three viscometric functions. These flows are called controllable viscometric flows. 18 The helicoidal flow, or its special case- the torsional flow, is controllable only if inertia is ignored.

~0.5 Viscosi ty D e t e r m i n e s the Velocity Fie ld

Out of the five velocity fields in (40.13)-(40.17), two flows (40.15) and (40.16) have already been discussed and have been found to be independent of the viscosity. In the remaining three, there are two flows with curl-free acceleration fields. These are:

�9 Tangential sliding of parallel plane slip surfaces - s e e ( 4 0 . 1 3 ) . Indeed, the acceleration vector associated with this velocity field is zero.

* Helical flow (40.14), which has the acceleration field

a = --rw 2 er, w = w(r). (40.23)

Therefore, only in the above two class of flows does the viscosity play a role in determining the velocity field. That is, the velocity field varies from one fluid to another dictated by the viscosity function. Examples will be given later to illustrate the procedure to be followed in using the viscosity function.

0.6 Flow between Ro ta t ing Conical Sur faces

The fifth flow in (40.17) is quite peculiar because it suffers from two drawbacks which are not immediately apparent. The first one is the acceleration field associated with it, which is (cf. (39.6))

w 2 a - ' - - - - - e r --

r

where

w 2 cot 0 ~ e o , (40.24)

r

w = ~/r sin 0[ln sin 0 - ln(1 + cos 0)1. (40.25)

It is quite easily shown, from using the representation of the curl of a vector field in spherical coordinates- see (A1.43), that the acceleration vector a in (40.24) is not curl-free. Hence, we have to ignore inertia to examine the dynamic compliance of the ftow (40.17).

18PIPKIN, A.C., Quart. Appl. Math., 26, 87-100 (1968).


Because the shear rate is a constant, all the extra stresses are constants; also the flow is characterised by conical surfaces slipping past one another while rota t ing about the axis 0 -- 0. Hence, the extra stresses Sro and Srr are both zero. Using all of the above information in (39.6), one finds tha t the equations of motion now become

_ . _ _ 1 Op - s o o - - 0 , Or r

10p cot 0 + (soo-s ) - o, r 00 r

1 0p 2 cot 0 r s in "~vor -}- - - - - - - S 0 r - - O. r

(40.26)

If the pressure t e rm p were to depend on r the pressure function would not be continuous in this variable because p(r, 0, 0) will not equal p(r, O, 2r) . Thus, we have to accept tha t 0 p / 0 r = 0. This means tha t

2 c o t 0 Sor = O. (40.27)

r

The correct solution to this is either tha t cot0 = 0, i.e., O = r / 2 , which means tha t the flow occurs over a fixed plane and not between conical surfaces in relative rotation; or tha t Sor = 0, which is absurd. Hence, strictly speaking, cone and plate flow is not dynamical ly possible in the laboratory.

Nevertheless, cone and plate flow is used extensively in measuring viscometric functions. The justification for this lies in keeping 0 close to ~ / 2 or making the cone angle very small 19 . We shall return to this flow again later for some additional analysis.

~0. 7 Couette F l o w - The Velocity Field

We shall now reexamine the Couet te flow, which is a special case of helical flow, to obtain a general formula for the velocity field in it. Because the measurable quant i ty in the flow is the torque per unit height M and this is related to the shear stress through (40.8), it is preferable express the shear rate as a function of the shear stress in order to solve (40.9). In order to achieve this, we introduce a new function, denoted by r It is called the fluidity function, 2~ for it is the reciprocal of the viscosity, i.e., less the viscosity the more the fluidity and vice versa. Thus, r = 1/~] and we obtain

= o r ( 4 0 . 2 8 )

Turning to (40.8), we are led to the solution of

dw M r - - = a r a = (40.29)

dr 27rr 2 '

19For a discussion of these matters, see WALTERS, K., Rheometry, Chapman and Hall, London, 1975.

2~ A.C. and TANNER, R.I., in Mechanics Today, I, Ed. NEMAT-NASSER, S., Perg- amon Press, 1972, pp. 262-321.

from which it follows quite easily that

dw _1 r (40.30)

This differential equation has the following solution for the angular velocity field

o" 1 1/ ~(~) - ~ ( R , ) =

f f

r dT, (40.31)

where M

a i - 2---~-~12 (40.32)

is the shear stress on the inner cylinder of radius R1. It follows therefore that the magnitude of the difference in angular velocity f~ between the inner and outer cylinders is

o" 1

1 / M (40.33) a - ~ , (~1 e~, o ~ - 2 - 7 ~

i f 2

The solution (40.30) as well as the angular velocity difference (40.33) demonstrate quite clearly the role played by the viscosity function or the fluidity function in Couette flow.

Turning to the problem of finding the fluidity r or the viscosity 7, one has to invert the relation (40.33). This is not easy except when the gap between the cylinders is small, i.e., h -- R 2 - R1 < < R1. In this case, the rate of shear may

_

be assumed to be uniform across the gap and its magnitude is ~ - R~/h, where / ~ - (RI + /52 ) /2 is the average radius. Then, ~(~) - ih/27r~R a.

0.8 Poiseuille Flow

This is another flow which is a special case of helical flow and is assumed to occur in a circular pipe of inner radius R. The fluid flows along the pipe and the velocity field is given by

v = ~(~) ~ , ~(R) = o, (40.34)

in a cylindrical coordinate system. Because the flow occurs in the z-direction and the velocity gradient is in the r-direction, two shear stresses are zero, viz., St0 and Soz, and the remaining extra stresses are functions of r. Since it is known that the viscosity determines the velocity field, the important equation of motion is in the axial direction and this is

dSrz 1 Op ~ + -S~z = 0, (40.35) - 0"~ - d r r

where the body force has been absorbed into the pressure term. It is easily seen that Op/Oz is a constant, - c say, where c is the magnitude of pressure drop per unit length of the pipe. Then, one obtains

Srz(r) = - c r / 2 , (40.36)

where the homogeneous solution to the differential equation (40.35) has been ignored on the assumption tha t the stress Srz is bounded in the pipe. Note tha t the shear stress is negative because the velocity gradient du/dr <_ 0 in the pipe, for u decreases from its maximum at the centre of the pipe to zero on the wall.

Using the fluidity function one may now solve for the velocity distribution in the pipe through

du = - a r a = cr/2. (40.37)

However, this does not lead to a determination of the viscosity function and we turn to the flow rate Q which does. This discharge has the formula

R dr~v

Q - 27r u(r)r d r - --~

o o

(40.38)

where integration by parts and w(R) - 0 has been used along with aw = cR/2 denoting the wall shear stress.

The reduced discharge q -- Q/TrR 3 has the elegant formula

O'w

q(aw) = a~, 3 / T3r dT

0

(40.39)

and the derivative of this with respect to aw is

dq = r 3q daw aw

(40.40)

Because of the definition of the fluidity function, r -- "Yw/aw, and thus

d(ln q) ] (40.41) 9/w = q(a~) 3 + d ( l n a w ) "

This formula leads to the shear rate at the wall being determined from a plot of q versus a,o - cR/2. Hence, the measurements of the flow rate Q which leads to q and the pressure drop c which yields the wall shear stress are sufficient to obtain the wall shear rate and thus the viscosity at any desired value of the (wall) shear rate. When adequate precautions are taken, Poiseuille flow leads to the viscosity function being measured to within an accuracy of less than one per cent 21 .

For power law fluids, the equations (40.37)-(40.38) may be integrated explicitly through the use of the shear stress-shear rate relationship

a(6/) = Kl~YIn-1~, (40.42)

where K > 0 is a constant and n is the power law index, which is unity for Newtonian fluids. Shear thinning fluids have n < 1 with a typical value of n -- 0.5. The fluidity function associated with (40.42) is

0 ( o ) /40

2tTANNER, R. I., Engineering Rheology, Oxford Univ. Press, Oxford, 1985. See pp. 96-97 for the precautions that need to be taken to achieve this level of accuracy.

Thus the velocity profile in power law fluids is given by

u(r): ( 1 ) ( c ~ 1- (R) (n:+l)/n" (40.44)

and the discharge rate is

Q:3n+lTrn (~K)l/nR(3n+l)/n . (40 .45)

Introducing the average velocity ~2 = Q/TrR 2, one may plot the dimensionless velocity profiles u(r)/~ for various values of n. See Figure 40.1 for a representative sample.

3.0

2.O

1.0

1.0 0.5

n=O

0.0 0.2 0.4 0.6 0.8 1.0

r/R

FIGURE 40.1. Dimensionless velocity profiles for various values of the power law index n. Note that n -- 1 corresponds to the Newtonian fluid, n < 1 corresponds to a shear thinning fluid.

0.9 Helical Flow This viscometric flow, which is a superposition of Couette and Poiseuille flows, has the velocity field

v - rw(r) eo + u(r) ez, R I __~ r __~ R 2. (40.46)

This flow occurs in the annular gap between two concentric cylindrical surfaces due to a combination of their rotat ion and the flow along the axis.

Helical flow is also impor tant in tha t it reveals the following nonlinearity" in Couette flow, the shear stresses Srz and Sez are both zero; in Poiseuille flow, Sre and Sez are both zero. If the constitutive relation of the viscometric flow were

linear, then the stress Soz would be zero in helical flow; it will not be, if (40.10) is a nonlinear relation. We shall demonst ra te this next.

The shear direction vector a for the helical flow is given by (cf.(7.17)):

a : c o s a e 0 + s i n a e z , du (40.47) - . "~cosa = r , ~ s i n a = dr

Using this, along with the fact t ha t the shear gradient direction vector b is parallel to er, one finds from (40.10) tha t

~rv - - ~I/2~/2, ~00 - - (II/1 -[- 1I/2) r2 "~r '

( d u ) 2 dw S z z - - (lI/1 q- 1I/2) ~ r ' S r O -- glr '~r '

~ r z = ~l '~r, SOz - - (lI/1 -[- 1I/2)r ~ r ~rr "

Clearly, Soz ~ 0 and all the extra stresses are functions of r. The equations of motion (39.5) take on the simple form

o & ~ & ~ - & e _ Op + + _ _ prw2 or

O&e l o p + + 2 S ~ o -- O,

__.___ OS~z 1 Op }- + - S r z + p b z - O, Oz Or r

(40.48)

(40.49)

where it has been assumed tha t the body force terms br -- 0, bo -- 0. Jus t as in the case of the Poiseuille flow, bz has to be a constant; it is +g, depending on whether the flow along the axis is vertically up or down.

Absorbing this body force into the pressure term, we see tha t p(r, z) -- f ( r ) - cz, and tha t

M b 1 S r o - 27rr 2 , Srz . . . . r 2 cr, (40.50)

where the constant M is again the torque per unit height needed to maintain the rotation; c is the magni tude of the axial pressure drop per unit length and b is a constant which arises because the flow occurs in an annular region. This constant is zero in the Poiseuille flow in order to ensure tha t Srz remains bounded in the flow domain. Whereas one may measure M and c, the inability to determine the constant b experimental ly is the cause of major difficulties in the analysis of helical flOW.

Nevertheless, some progress can be made. One way to est imate 22 the constant b arises from noting tha t the axial velocity u(r) has a maximum in R1 < r < R2, say at r = R. Hence the shear stress Srz = 0 there, or

c R 2 b-- 2 " (40.51)

22HUILGOL, R.R., Proc. Fifth Nat ional Conf. Rheol., pp. 43-46, Melbourne, Australia, 1990.


A second estimate may be obtained when the gap is small. For convenience, let us assume that the inner cylinder rotates with an angular velocity fl and that the outer tube is at rest. Then

w' -- - ~ (40.52) R 2 - R I '

with ~ denoting the derivative with respect to r. We now note from (40.48) and (40.50) that in a helical flow,

U I

rw I

Using (40.52), we get

= = Z ( 2 b - 2) St0 M

(40.53)

du ~2 d-~ -- M(R2 - R1) (cr4 - 2br2)" (40.54)

This can be solved quite easily for the constant b by using the boundary conditions u(R1) -- u(R2) = 0. One derives

b__3c Rh-R51 c ---'10 R~ _ Ra 1 ~ -~RIR2, (40.55)

which is in agreement with (40.51). Turning now to the general case, one has a set of three equations to consider;

again, one may assume that only the inner cylinder rotates. The equations are"

~2 ~2

/ u' (r) d r - O , n+fJ(r)dr-O. R1 R1

There is also the flow rate Q given by

R2 R2

(40.56)

Q - 27r / r u ( r ) dr - -Tr / r 2 u ' (r) dr. (40.57)

R1 R1

Now, it is easy to see that

r = rw', r = u', (40.58)

where the fluidity function has been used and the total shear stress e is given by

a ( r ) 2 _ ( M 2 2 2vr2 ) + [ b - 2 ] . (40.59)

Hence we arrive at the set of equations: 23

R2

,._ o.

RI

23HUILGOL, R.R., Proc. Fifth National Conf. Rheol., pp. 43-46, Melbourne, Australia, 1990. Similar formulae occur in COLEMAN, B.D. and NOLL, W., J. Appl. Phys., 30, 1508-1512 (1959) in terms of the inverse shear rate-shear stress function, viz., ;y -- a - l ( a ( ' ~ ) ) .

R2 / M R1

d r

R2

R1

-- 0, (40.60)

- - 0 .

These equations provide a set of consistency relations among the five parameters: M, b, c, Q, f~. If they are not met, then wall slip or some other wall effects will be present. 24

Practically speaking, helical flow and its special cases, apart from channel flow to be studied next, exhaust all flows where the viscosity determines the velocity field.

O. 10 Channel Flow

Even though it is possible to prove tha t the velocity field (40.13), which describes the tangential sliding of parallel plane slip surfaces, is possible in all incompressible simple fluids, we shall consider the special case of channel flow only. The velocity field in the channel, which is infinitely deep, is given by

v - - u ( y ) 1, u(~:l) = 0. (40.61)

The planes y -- :El are assumed to be at rest and these are the boundary conditions on the velocity field which is driven by a pressure gradient in the flow direction. Thus the important equation of motion is

_ _ _ d S x u _ 0p + -- 0, (40.62) Ox dy

since the acceleration field is zero. Clearly, the pressure gradient must be a constant, say equal to c per unit length of the channel. Then, we have

Sx~ = cy, (40.63)

because the shear stress is zero at the centre of the channel. Employing the fluidity function, one has

d u - - = ( 4 0 . 6 4 ) dy

which can be integrated on using u(1) - 0 to yield

1

y

0 < y < 1. (40.65)

24For an application of the procedure in (40.60), see BOWN, D.J., MIDDELBERG, A.P.J. and NGUYEN, Q.D., Proc. 12th Int. Confce. Slurry Handling and Pipe Transport, 1993, pp. 343-352.


Since, by symmetry, u ( - y ) - u(y) , the solution given above determines the velocity field completely.

The flow rate Q and the reduced discharge q may be computed here as in the case of the Poiseuille flow. One obtains

O'to

q(ew) -- a~2 f T2r dT,

0

a ~ - c, (40.66),

which leads to the analogue of (40.41), viz.,

d(ln q) ] (40.67) " ~ w - q ( a w ) 2 + d ( l n a w ) "

Because channel flows are not easy to produce in the laboratory, the Poiseuille flow formula (40.41) and its application through the capillary rheometer provides the best means of determining the viscosity of a fluid.

We now turn to the determination of the normal stress differences. Here, free surface conditions are important and their violation causes much difficulty in practice.

~0.11 Torsional Flow and the Normal Stress Differences N1 and Nu

The normal stress differences are measured in viscometers, such as the torsional flow apparatus or the cone-and-plate flow system, through total thrust measurements. We shall derive the relevant formulae first for the torsional flow. Here, the flow occurs between two concentric circular disks, which are parallel to one another and are at a distance h apart. The bot tom disk is stationary and the top rotates with an angular velcoity f~ and the velocity field is (cf. (40.15)):

v - a r z eo, (40.68)

where a = Q/h. It is really the change in the angular velocity with z which gives rise to the shearing and thus e0 is parallel to the shear direction a and ez is parallel to the gradient direction b. This leads to the shear rate being given by ~ = ~ r / h - see (A1.45) for a direct verification in terms of the components of the relevant A1.

The shear stresses S~o and S~z are both zero and all the extra stresses are functions of r only. Thus, the equations of motion now become

Op dSr,

-o-7+ + S r r - $00 ~~2z2

r = - p r - - ~ - ,

lop - - ' - - ~ - - O ,

r 00 Op

: O~ Oz

(40.69)

which show clearly why inertia, and hence the body force as well, has to be ignored; thus, one finds that p = p(r) only.


On using the constitutive relation (40.10), one finds tha t Sr~ - 0 and tha t Soo - N1 + N2 which show tha t

p(r) = -f N1 +~ N2 0

+ C, (40.70)

where the constant C is to be found on the assumption tha t the free surface at r -- R is open to the atmosphere or the total radial stress on this surface is zero, modulo the atmospheric contribution. Hence,

means that

Trr(R) = -p(R) -0 , (40.71)

R p(r) = f N1 + N2 . d~, (40.72)

r where it is understood tha t N1 and N2 are both functions of "~ -- f~r/h. This means that the total stress on the bot tom plate is

(40.73) Tzz(r) - N2(f~r/h) + ~f Nl(f~/h) +~ N2(f~/h) d~,

R

where the constitutive relation (40.10) has been used yet again. Hence the total thrust F on the bot tom plate is

(40.74)

R R

f dr, 0 0

where p'(r) = _ N 1 -t- N 2 (40.75)

r is the derivative with respect to r of the pressure function p(r). The final result of thrust per unit area f is

F �9 f "~(N1 - N2) d~

0

(40.76)

which relates the difference N 1 - N2 to the rim shear rate ~/R- A more useful formula derived from this relation is

dln f ) (40.77) N 1 - N 2 = f 2 + d l n , ~ R .

P r o b l e m 40 .A

Let M be the torque needed to turn the rotating plate or hold the bot tom plate steady. Show that m = M/27rR 3 is given by

A/R " - - 3 / m = ~,R V'~ 3 d'~,

0

(40.A1),


where ~ = OR/h is the shear rate at the rim. Hence, deduce tha t the viscosity at the rim shear rate is given by

a(1 ,~ ) ] m 3 + . (40.A2) U(~R) = ~--~ d(ln~R)

O. 12 Cone-and-Plate Flow and N1

The velocity field in this configuration has been discussed before and in spherical coordinates has the form

v = r sin Ow(O) er w(O) = ~[ln sin 0 - ln(1 + cos 0)]. (40.78)

The flow is assumed to occur between a fiat plate at 0 = 7r/2, at rest, and a conical surface at 0 -- r / 2 - a , rotat ing with a constant angular velocity ~. If a is very small, then to O(a) ,

) n -- w ~ - a = ~[ln(cos a) - ln(1 + sin a)] : - ~ a , (40.79)

which shows tha t the shear rate ~ / : - ~ / a , which is a constant as assumed before. The shear rate is negative; because the angular velocity decreases from ~ to 0 as 0 increases from (w/2 - a) to ~/2. Thus, we may take the angular velocity to be given by the simpler form

: _n. (40.80)

The shear direction is along the C-direction and the gradient is parallel to the 0-direction. Hence, the normal stresses are

s ~ = o, Soo = g2, S** = g l + g2. (40.81)

Turn now to the equations of motion (40.26) and observe tha t (40.26)2 implies tha t Op/O0 = 0, because we have put cot 8 = 0, since 0 ~ r / 2 . This means tha t p = p(r) only and the equation (40.26)1 now leads to

p(r) = - ( N 1 + 2N2) ln ( r /R) , (40.82)

where we have used the assumption tha t the total radial stress Trr = -p(r) on the free surface r = R is zero.

The stress Toe, which exerts a force on the fixed plate, is given by

Too = (N1 + 2N2) ln ( r /R ) + N2, (40.83)

leading to the thrus t R

F - -2~ f Toor dr - 2 N i R2, 0

(40.84)

a truly remarkable result. It is thus not surprising tha t the cone-and-plate flow apparatus is used to measure N1.

P r o b l e m 4 0 . B

Let M be the torque needed to turn the rotating cone or hold the bot tom plate steady. Show tha t the torque is related to the shear stress through

a M = 21rRaa(~), - ~ - a / a . (40.B1)

. . . . . .

In the torsional flow, where cylindrical coordinates are used, the free surface is vertical, for the z-direction is vertical. In the cone-and-plate flow, where spherical coordinates are used, the free surface is spherical. However, if the cone angle is around 4 ~ , the error between assuming the latter surface to be spherical or vertical is of no significance. Hence, it ought to be possible to discuss the cone-and-plate flow as a generalised torsional flow, where the surface of constant angular velocity varies continuously from the flat surface at z - 0 through a series of conical surfaces generated by straight lines z / r -- tan/3, where 0 _</3 _< a and a is the cone angle. We turn to a development of the theory of generalised torsional flows next.

40.13 Generalised Torsional Flow

In a cylindrical coordinate system, let it be assumed that a surface of revolution generated by z -- h(r) rotates with a constant angular velocity [~ over a fixed plate si tuated at z ---- 0. It is assumed tha t h(r) is small when compared with the radius R of the free surface and also tha t the slope d h / d r is very small. See Figure 40.2.

Let 0 be the flow direction, r be the direction of the gradient and p be normal to both. We shall assume tha t surfaces of constant angular velocity exist in the fluid so tha t on any such surface, the 0 and p-directions lie on the tangential plane to the surface. The shear rate of such a generalised torsional flow 25 is approximately given by

fir ~/= h ( r ) ' (40.85)

or the shear rate is a function of r. Assuming the flow to be locally a viscometric flow, the non-zero viscometric stresses are Soo,Sr162 and S0r and these are all functions of r only. Using the constitutive relation (40.10), we find tha t

See - N1 + N2, Sr162 - N2,

s p-0, &r (40.86)

Next, the slope of any constant angular velocity surface changes and is approximately given by

z dh tan ~ = h(r) dr ' 0 <_ z <_ h(r). (40.87)

25For two slightly different ways of deriving the results for this class of flows, see PIPKIN, A.C. and TANNER, R.I., in Mechanics Today, 1, Ed. NEMAT-NASSER, S., Pergamon Press, 1972, pp. 262-321, or HUILGOL, R.R., Trans. Soc. Rheol., 18, 191-198 (1974).


I |

I

I |

I R

co t i velocity surface, | -

FIGURE 40.2. Generalised torsional flow, showing the shear axes (0, p, r and their relationship with the coordinate axes (0, r, z).

Because/~ is very small, we take cos/3 = 1,sin/3 -- /3 and ignore terms of order 0(/32). Hence, with h ' = dh/dr, we obtain

0/~ h' Oz h ' Srr - Spp cos 2/3 + Sr162 sin2/3 - 0,

~ Z -- �89 (Spp - Sr162 2 / ~ - -N2/~.

(40.88)

The equation of motion in the r-direction leads to

h I drdP N1 +r N2 _ N2--~ -- O, (40.89)

where inertia has been ignored. This gives the pressure distribution

f [ ~ N2h'(~) d~, (40.90) p(r) -- p(R) - (N1 + N2) + h(~)

R

where the pressure p(R) is to be specified. If this surface is open to the atmosphere, one takes p(R) -- 0; if the apparatus is immersed in a bath, then p(R) has to be determined separately.

Turning to the determinat ion of S~z, one finds that

Szz - Sppsin 2 ~ + Sr162 cos 2 )3 - N2 (40.91)

and thus, the axial stress on the fiat plate is

T~(~) - - p ( ~ ) + N~(~(~)). (40.92)

P r o b l e m 4 0 . C 26

Use the formula (40.92) to solve the following:

1. Obtain the formula (40.73) for torsional flow by using h(r) = h, a constant; the formula (40.83) for the cone and plate flow when h(r) = rc~, since a is very small. Use p(R) = 0 in these two calculations.

2. Assume tha t the cone-and-plate device has an axial separation c at the centre, i.e., h ( r ) - c 4-ra. Find the axial stress distribution on the bot tom plate.

3. Let the cone-and-plate apparatus have an inverted cone, i.e., let h(r) = a ( R - r) + c, so tha t the axial separation decreases from a + c to c. Find Tz~ again on the flat plate.

4. Assume tha t a shallow spherical surface of radius R is the solid body of revoultion rotat ing over the fiat plate. Lett ing h(r) = r2/R, determine Tzz on the flat plate.

5. Obtain the formula relating the torque M and the viscosity r /of the fluid in a generalised torsional flow.

40.1~ M e a s u r e m e n t of N2

Here, we shall begin by considering an indirect way of measuring the second normal stress difference N2. It is clear from the formula (40.84) tha t the thrust in the cone- and-plate apparatus yields the first normal stress difference N1. Knowing NI, one may use the result (40.77), from the torsional flow experiment, to find the second normal stress difference N2. However, it is desirable to examine those flows where N2 appears on its own so tha t its magnitude may be found directly. We shall s tudy these situations next.

First of all, if in the torsional or cone-and-plate flows, the rim pressure p(R) = O, then the axial stress on the fixed p la te - see (40.73) and (40.83) respectively, is equal to N2 at the appropriate shear rate. The difficulty here lies in mounting the relevant pressure transducers close to the edge of the apparatus. Hence we turn to two other flows - the annular flow and the axial motion of fanned planes. However, each of them has its own share of experimental drawbacks which will become apparent shortly.

Annular flow is a special case of helical flow in that the fluid flow occurs in the annular space between two concentric cylindrical surfaces which do not rotate. Thus the velocity field is

V --u(r)ez , R, < r < R2, (40.93)

(R1) = 0, = 0.

26Solutions to these problems may be found in the two papers cited above.


The stress field corresponding to this is available from (40.48) and one finds from (40.49) tha t the radial stress distribution has the form

Trr -- - f N2r dr, (40.94)

which shows tha t R1

- - f dr, (40.9 ) R2

with "~ = du/dr. Despite the elegance of this formula, there are some difficulties: the curved surfaces make it difficult to mount pressure transducers, al though mi- crosensors may overcome this problem; if pressure taps are used, intrinsic errors arise and as will be demonstra ted later in w167 one needs to know N1 or N2 to estimate these errors. To add to this, the shear rate distribution across the gap is not completely known, because the information tha t is available tells us tha t (cf.(40.50)) the shear stress is given by

b 1 Srz = - - - c r , (40.96) r 2

where c is the pressure drop per unit length. The constant b = cR2/2 again and one may take R 2 = RIR for the case of a small annular gap (cf.(40.55)). Of course, if the gap is small, then the radial stress difference in (40.95) is likely to be small. Thus, an extremely elegant formula such as (40.95) seems to be beset with difficulties in delivering N2.

We shall now examine the axial motion of fanned planes, which has the velocity field v = cOez in a cylindrical coordinate system. See Figure 40.3 below.

z

0

/h

FIGURE 40.3. Axial motion of fanned planes.

The bo t tom plate is supposed to be at rest and the top one moves with a constant speed COo in the z-direction. The shear rate is ~ = c/r and thus the extra stresses are functions of r only. They are obtained from (40.10):

S~ = 0 , Soo = N2,

S~ =NI+N2, &~=a. (40.97)

/ / / / / / / / / /

FIGURE 40.4. Plane cone-and-plate flow. The upper "cone" moves axially with a constant speed, while the bottom plate is at rest.

The acceleration field is zero and omitting the body forces, one discovers tha t p depends on r only, and thus

p(r) -- - / N2r dr. (40.98)

This means tha t the difference in the azimuthal stresses Too on the fixed plate at r - R1 and r = R2(> R1) is given by

R2

Too(sh) - Too(Ri) - I N2(cl,')r dr + N2(c/Sh)- N2(c/RI). R1

(40.99)

The difficulty in generating this flow cannot be underestimated, of course. An intriguing way of visualising the above motion as a plane cone-and-plate flow

has been suggested. 27 See Figure 40.4. Here, the bot tom plate is at rest and the top "cone" moves in the axial, z - direction with a constant speed. Thus, in the sector bounded by 0 _< 0 __< 00 or by 7r - 00 _< 0 _< r , the fluid suffers a motion in which the velocity is constant on planes defined by 0 = const. Attractive as this configuration is, it seems unlikely to lead to a direct measurement of N2 at high rates of shear.

The final opportunity for determining N2 arises from rectilinear flows, if the secondary flows are small. Since this topic will be discussed in the next section, we shall omit any consideration of it here.

In conclusion, in this section all dynamically possible viscomertric flows have been discussed and solutions to these velocity fields have been found. Additionally, flows which are approximately viscometric have been described and shown to lead to configurations from which the three viscometric functions may be found, albeit with due care and experimental finesse. In Figure 40.5, we present the viscometric functions of a viscoelastic liquid prepared from dissolving polyisobutylene in cetane. These graphs may be considered to be typical of many polymeric liquids.

41 Rectilinear Motions

Even though rectilinear motions form a special class of kinematically possible viscometric flows, it is desirable to accord them a special s tatus because they are not possible in all fluids; also, they have initiated much research into the geometry of

27We are grateful to Dr. BINDING, D. M. for this observation.

41 Rectilinear Motions 303

10 4

10 3

10 2

Z !

a 10 Z

N

(~

~ _ ~ ~ - N 2 .

I I I

1 10 10 2 10 3

'i'(s-b

FIGURE 40.5. Viscometric functions of 6.8% polyisobutylene in cetane at 24~

slip surfaces, secondary flows and the roles of the viscosity and the second normal stress difference. Some of these aspects will be studied here.

We shall assume that rectilinear flow occurs in the body of an incompressible fluid and that the velocity field is given by

- - 0 , ~l -- O, ~ = w ( x , y ) , (41.1)

where w is the axial velocity component. Using the velocity field (41.1) in the constitutive equation (38.24), it is easy to

show that the extra stress tensor S has the form

S -- ' O2w2,y vlw,y , (41.2)

�9 (lI/1 -[- 1I /2)~2

where the shear rate is given by

Ow Ow ~2 _ w2,x q- w2,y, w,x -- -~x , w,y - Oy . (41.3)

Since the acceleration terms are zero in this problem, the equation of motion in the z-direction becomes

_ O p + (~Tw,~),~ + (vlw,y),y + pb~ -- O. (41.4) Oz


Assuming the body force bz to be a constant and absorbing this into the pressure term, one obtains

p(x, y, z) = cz + h(x, y), (41.5)

where h(x, y) is to be found from the other two equations of motion. Hence, (41.4) gives rise to the partial differential equation

+ = (41.6)

This equation shows that the velocity field is determined by the viscosity function. However, it will be seen shortly that the velocity field found in this manner will not occur in the fluid unless the second normal stress difference function has a special property. That is to say, rectilinear flows are not partially controllable.

To demonstrate this, employ the Greek indices c~, ~ = 1, 2, set the body forces in the (x, y) plane to zero and note that the equations of motion in the x = Xl and y - x2 directions are �9

- h,~ + (r = 0. (41.7)

It follows from the definition of the shear rate that

- ('~2w,~),~w,,~ + ~ 2 w , ~ w , , ~

= (~2w,~) ,~w,~ + ~P2;y;y,~. (41.8)

We shall now assume that ~P2 is a function of ~2, rather than ~; see the discussion following (38.22) above. Using this in (41.8), we derive from (41.7) that

f( ~,~)

h(x, y) -- IP2(~2)~d~ + P(x, y), J o

where the new pressure term P must satisfy

(41.9)

(41.10)

Now, let ~z be the two dimensional gradient operator here and in what follows. Then, it is easily seen that the expression (~P2w,~),f~ is the divergence of a vector. Thus, we define

r V-(qJ2Vw), (41.11)

and find that (41.10) is equivalent to

V P , = CVw. (41.12)

Geometrically speaking, the above equation tells us that the gradient of P must be parallel to that of w, if the rectilinear motion (41.1) were to occur in the given fluid. That is to say, the surfaces on which P = constant must coincide with the surfaces on which the speed w = constant. This means that P must be a function of w, i.e., P = P(w); in turn, this leads to the result tha t

C p - (41.13) dw


Comparing (41.12) with (41.13), one finds that r the two dimensional divergence of (#2~7w), must also be a function of w; 28 an equivalent requirement tha t r = r is the following condition on the Jacobian 29

0(r w) =0. (41.14) (~(Xl ,X2)

In general, neither (41.13) nor (41.14) is satisfied. Hence, rectilinear motion is not possible in all incompressible simple fluids. That is, secondary flows will occur.

Nevertheless, if one assumes that r = r a certain amount of useful information about the role of N2 in rectilinear flows may be obtained. We shall examine this next.

~1.1 Normal Stress on a Pipe Wall and a Free Surface

Assuming that r : r it follows from (41.5) and (41.9) that the pressure field p is given by

[#(=,v) f~(=,v) p(x, y, z) = cz + ~1/2(~2)~d~ + r du + C, JO JO

(41.15),

where C is a constant. Without loss of generality, one may choose C = 0 in pipe flOWS.

Now, at an arbitrary point on the pipe wall, let the unit normal be given by

' 2 2 = 1 . n=nxi+nvj , n=+nv (41.16).

Then, through the constitutive relation (41.2), the normal stress on the wall can be shown to have the following expression:

n-Tn = -p + ~21Vw- nl 2. (41.17)

Using (41.16) and the fact that w : 0 on the boundary of the cross-section of the pipe, this may be converted to

f0 ~ n - T n : ~2~ 2 - ~1/2(~2)~ d~, (41.18)

where ~/ - Ow/On is the shear rate at a point on the wall. This formula shows that measuring the normal stresses on the wall of a pipe yields the second normal stress difference readily, when rectilinear flow does occur.

i i i i i i i

28This was pointed out by ERICKSEN, J.L., Quart. Appl. Math., 14, 318-521 (1956). The development given here follows that in PIPKIN, A.C., Lectures on V=scoelast~city Theory, 2rid Ed., Springer-Verlag, 1986. See also, CRIMINALE, W.O., Jr., ERICKSEN, J.L. and FILBEY. G.L., Jr., Arch. Rational Mech. Anal., 1,410-417 (1957).

29See OLDROYD, J.G., Proc. Roy. Soc. Loud., A283, 115-133 (1965).


Some other results, similar to the above, arise in connection with free surfaces in rectilinear flows. These are contained in the Problems below.

P r o b l e m 41 .A: Free Surface C o n d i t i o n s

Assume tha t the rectilinear flow (41.1) takes place and tha t the flow domain has a free surface. Let the unit external normal n to the free surface be given by

2 2 _ 1 . n - - n ~ i + n v j , n x + n v (41.A1)

Set the unit tangent vector t to the free surface �9

t -- - n v l + n~j. (41.A2)

Establish tha t the zero shear stress condition

t - T n -- 0. (41.A3)

is equivalent to

�9 2(Vw- n)(nxw,v - n v w , x ) - - O.

Verify tha t the above relation may be satisfied in one of three ways �9

(41.A4)

(i) The second normal stress coefficient #2 is zero;

(ii) V w - n -- 0. Tha t is the external normal is orthogonal to the velocity gradient vector.

(iii) n ~ w , y - n v w , x - - O. This is a new free surface condition tha t is congruent with

the demand tha t n x ~ w -- 0, or tha t the external unit normal is parallel to the velocity gradient vector.

P r o b l e m 41 .B: N o r m a l Stress on a Free Surface

Show tha t the normal stress on a surface, in a rectilinear flow, with the unit normal defined by (41.A1) is given by

n - T n - - p + ~ 2 [ ~ w - nl 2. (41.BI)

Prove the following results which are valid on free surfaces"

(i) If Vw- n -- 0, then

n - T n - - p ; (41.B2)

(ii) If n is parallel to ~Tw, then

n - T n - - - p + ~ 2 l ~ w [ 2. (41.B3)


~ 1 . 2 A d d i t i o n a l E x a m p l e s a n d C o u n t e r E x a m p l e s to E r i c k s e n ' s

C o n j e c t u r e

Let us now consider the case when the second normal stress difference coefficient is proportional to the viscosity, i.e., let ~2 -- Kr/, where K is a constant. Then,

(~2w,~),~ = K(r}w,~),~ = Kc, (41.19)

with the last one following from (41.6). This shows tha t r - K c and tha t the pressure p is given by (cf. (41.15))

p -- cz + K c w + ~2(~2)~d~ + C. (41.20)

That is, the rectilinear flow determined by the viscosity is possible in every fluid in which #2 = Kr}.

In general, this proportionality between #2 and 7} does not hold, and so the conflicting requirements of the two separate equations (41.6) and (41.10) cannot be met. The consequence is tha t the rectilinear motion (41.1) does not occur. This observation led ERICKSEN 3~ to conjecture tha t unless ~2 = Kr}, 'these exceptional motions are either rigid or such that the curves of constant speed are circles or straight lines.'

Leaving aside the trivial case of a rigid motion, it has been shown in w tha t Poiseuille flow or annular flow, in which curves of constant speed are circles, and simple shearing motion, channel flow and axial motion of fanned planes, in which the curves of constant speed are straight lines, are possible in all incompressible simple fluids. These demonstrations do not prove that ERICKSEN's conjecture is correct. Indeed, there are exceptions 31 of which two will now be listed. Others, including the rectilinear flow between eccentric, circular, cylinders may be found in the paper by MAYNE. 32

Consider the case of a fluid with constant viscosity and a quadratic second normal stress difference coefficient, i.e.,

r / = 1, 1I/2(@2 ) : KA/2 + L, (41.21)

where K and L are constants. If the pressure gradient c = 0, it is easy to show that

w ( x , y ) = A ( x 2 - y2), (41.22)

where A is also a constant, satisfies (41.6) because the latter reduces to Laplace's equation. Using this velocity field, one finds tha t

r ~Y. (~2~Yw) = 16AK2w. (41.23)

The consequence is tha t the velocity field (41.22) is possible in the fluid with the material functions listed in (41.21). The lines of constant speeds are rectangular hyperbolae rather than straight lines or circles.

3~ J.L., Quart. Appl. Math., 14, 318-321 (1956). 31FOSDICK, R.L. and SERRIN, J., Proc. Roy. Soc. Lond., A332, 311-333 (1973). 32MAYNE, G., Quart. J. Mech. Appl. Math., 42,239-247 (1989).


The second example depends on a non-analytic material function, viz.,

K ~7- 1, ~P2(~ '2) -- V/,~ 2 " b + L, (41.24)

where K, L and b are constants, yet again. Then,

W ( X , y ) - M x 2 + by, (41.25)

where M is a constant, satisfies the equation of motion (41.6) provided 2M = c, the pressure gradient. It is easily shown that in this case

r = 2 L M w (41.26)

and thus the velocity field (41.25) is possible in the fluid defined by the material functions in (41.24). Clearly, the curves of constant speed are parabolae here.

Despite these counterexamples, the basic result that holds for smooth material functions is this : if the flow domain is bounded, the pressure gradient c ~ 0, the speed w = 0 on the boundary of the domain and ~P2 is not proportional to ~, then rectilinear motion will occur in a circular pipe or in the annulus between two concentric circular cylinders. If, instead, the flow domain is unbounded and connected, then simple shear or channel flow, or a combination of the two, or axial motion of fanned planes, will occur.

The proof of the above theorem has beeen given by FOSDICK and SERRIN 33 using the theory of quasilinear elliptic partial differential equations. Subsequently, MAYNE 34 established the same result by using methods from differential geometry.

Using the assumption tha t the number of values of ~2 for which the relation ~(d~P2/d~ 2) - ~P2(d~7/d~ 2) = 0 holds is finite (more precisely, at most countably infinite), McLEOD 3S demonstrated tha t different sets of rectilinear flows arise when the pressure gradient c = 0 and when it is not.

(i) In particular, when c = 0, the flow is of the form w(x ,y ) = Ax + By + C, where A, B, C are constants. Clearly, this flow is a superposition of two simple shearing motions and the condition about the solutions of ~(d~P2/d~/)- ~P2(d~/d'~) = 0 is unnecessary.

(ii) The second case when c # 0, leads to a velocity field whose explicit form is impossible to compute unless six values, say, of w and its derivatives w,x, w,u, w,xz, w,~u and one of the third derivatives of w are specified at a point (x0, y0) in the flow domain. Unfortunately, there do not seem to be any specific examples of this latter result, other than the flow in a circular pipe or the flow in a channel.

We shall now turn briefly to secondary flows which arise in pipes and channels.

33FOSDICK, R.L. and SERRIN, J., Proc. Roy. Soc. Lond., A332, 311 - 323 (1973). 34MAYNE, G., Quart. J. Mech. Appl. Math., 42 ,239 - 247 (1989). 35McLEOD, J.B., in Lecture Notes in Math., 415, 193-204 (1974). A similar assumption appears

in the 1956 paper by ERICKSEN.

42 Non-Viscometric Flows 309

1 . 3 S e c o n d a r y F l o w s

Suppose tha t the chosen geometry has neither a circular nor a plane cross-section and ~2 ~ Krl. Then, one is likely to encounter a secondary flow in addition to the main stream. Tha t is, the velocity field will possess the general form �9

0r 0r

where r = r is the stream function which has to be determined from the equations of motion in the x and y directions. Typically, these secondary flows manifest themselves as closed curves in the cross-section of the flow domain; consequently, the overall flow consists of helices as the fluid particles are driven by the imposed pressure gradient. However, as we have already seen, there are many fluids and an equivalent number of configurations in which the flow is rectilinear and these secondary deviations do not exist.

Secondary flows are rarely observed in the laboratory because the effects induced by the second normal stress difference coefficient are weak relative to the primary flow driven by the pressure gradient and the viscosity of the fluid. Nevertheless, experiments have been done and the influence of secondary flows have been detected in ducts of square and rectangular cross-sections. 36

42 Non-Viscometric Flows

It is trivial to exhibit non-viscometric flows which are dynamically compliant when the fluid is the incompressible simple fluid and inertia is ignored; all one needs is a steady, homogeneous velocity field v = Lx with a velocity gradient L which meets L 2 ~ 0. The flow (2.29) in the eccentric disk rheometer belongs to this class. 37 This flow would have remained a curiosity but for the fact tha t it is a steady flow from which measurements of the dynamic moduli may be made. 38 We shall concentrate here on the dynamics of this homogeneous flow as well as its generalisation (9A.1) only.

A second homogeneous flow, the extensional flow, is possible in every incompressible fluid, 39 even in the presence of inertia. It is also of much significance because it approximates the industrial process of fibre spinning; it is responsible for a polymeric liquid, such as a 1% separan AP-30 in a mixture of 75% glycerin and 25% water, to be drawn upward into a tube whose orifice is not submerged in the liquid. Tha t is, one dips the tube into the liquid, initiates the process of suction and slowly pulls the tube out. It is found tha t the fluid continues to form an as-

36TOWNSEND, P., WALTERS, K. and WATERHOUSE, W. M., J. Non-Newt. Fluid Mech., 1, 107-123 (1976); GERVANG, B. and LARSEN, P.S., J. Non-Newt. Fluid Mech., 39, 217-237 (1991).

37HUILGOL, R.R., Trans. Soc. Rheol., 13,513-526 (1969). 3gGENT, A.N., Brit. J. Appl. Phys., 11, 165-167 (1960); MAXWELL, B. and CHARTOFF,

R.P., Trans. Soc. Rheol., 9, 41-52 (1965). For a thorough discussion, see WALTERS, K., Rheom- etry, Chapman and Hall, London, 1975.



cending syphon and, surprisingly, the process is stable. 4~ The stability can only be explained by a material property, called extensional viscosity, which increases with the extensional rate unlike its viscometric counterpart, which typically decreases with the shear rate in polymeric liquids.

Along with a study of the above two motions, we shall briefly examine the two doubly superposed viscometric flows. These are important because they are some of the dynamically feasible velocity fields which are possible in all incompressible fluids and, from the well known paucity of such examples, deserve a place in this treatise. We shall begin with a study of these superposed flows first in this section.

~2.1 Hel ical - Tor s iona l F low

The velocity field in this flow is a combination of helical flow and torsional flow. The velocity field is (cf. (8.6)) given in cylindrical coordinates by

v = + z).o + (42.1)

where the helical flow, itself a combination of Couette and Poisuille flows, occurs between two concentric cylindrical surfaces in relative rotation. In order to induce the torsional motion, two porous rings lying in the annular space are needed: one of which is at rest and the other one in rotation. See Figure 42.1.

rotating ring uq~

Outer tube . .

rotating cylinder

' I ~ ~ Porous rings

Inner cylinder

FIGURE 42.1. Helical-torsional flow. The inner cylinder and the top porous ring rotate independently of one another.

The idea of inserting the porous rings is due to OLDROYD, 41 who examined the Poiseuille-torsional flow and proved that the latter is indeed possible in all incompressible fluids, when inertia is ignored. The explanation for the dynamic compliance of the flow is easy: firstly, the extra stresses are all functions of the

4~ a photograph of this effect, see PIPKIN, A.C. and TANNER, R.I., Ann. Rev. Fluid Mech., 9, 13-32 (1977).

41OLDROYD, J.G., Proc. Roy. Soc. Lond., A283, 115-133 (1965).


coordinate r, and secondly, inertia has to be set equal to zero because it is absent from the simple case of torsional flow itself.

The generalisation to helical-torsional flow follows quite easily 42 for the same two reasons just mentioned. We shall, however, not pursue the flow in any more detail here.

~2.2 Helical Flow - Axial Motion of Fanned Planes

This flow is a superposition of the axial motion of fanned planes on the helical flow. In cylindrical coordinates, it has the form (cf. (8.7)):

v = + + (42.2)

Because the acceleration field associated with the helical flow is - w 2 r e r and tha t with the fanned plane motion is zero, it it easily shown tha t the velocity field in (42.2) is possible in all simple fluids, 43 even in the presence of inertia. Because the fanned motion is, in itself, very difficult to generate in the laboratory, the flow in (42.2), or its simpler version, viz., the Poiseuille flow - axial motion, have to remain as intellectual curiosities only.

~2.3 Flow in the Ecccentric Disk Rheometer

In Figure 42.2, the configuration of the eccentric disk rheometer is displayed. Each layer of fluid, t rapped between the upper and lower disks, rotates with a constant angular velocity ~t about an axis which is parallel to the z-direction. However, the centre of rotation for each planar surface lies on the straight line y = Cz, where the parameter r = a/h, with a being the displacement of the upper disk from tha t of the lower one in the y-direction and h being the vertical distance between the two disks.

The velocity field is thus given by

u - - - f l ( y - r v = f l x , w = 0 . (42.3)

Because this steady flow is homogeneous, the motion is one with constant stretch history and all the extra stresses are constants. The acceleration field is

a : - - ~ 2 x i - ~2 (y _ ~)z)j, (42.4)

and this is not the gradient of a scalar. Hence, the equations of motion are not satisfied unless inertia is ignored. Assuming this to hold, the flow (42.3) is possible in every incompressible simple fluid. 44 In order to show that the forces Fx and F~ measure the dynamic moduli, a specific linear constitutive relation is needed, a5

42HUILGOL, R.R. Quart. Appl. MaSh., 29, 1-15 (1971). 43HUILGOL, R.R., 1971, Quart. Appl. MaSh., 29, 1 - 15 (1971). 44HUILGOL, R.R., Trans. Soc. Rheol., 13,513-526 (1969). 45For a demonstration, see WALTERS, K., Rheometry, Chapman and Hall, London, 1975.

Fy

Fx / rotation sense

<___

~ ML

FIGURE 42.2. Eccentric disk rheometer.

Turning to a generalisation of the flow (42.3) in order to incorporate inertial effects, 46 let us consider the following: 47

= - a ( y - f ( z ) ) , ~ = ~ ( ~ - g ( ~ ) ) , ~ = o, (42.5)

where the two functions f(z) and g(z) are to be determined. In this flow, each plane z -- constant moves as if it were rigid with an angular velocity ~ about a point, a l though the locus of this point as z varies is not a straight line joining the centres of the two bounding planes, as it is in the flow defined by (42.3). Not only this, a l though the velocity field (42.5) includes the earlier one (42.3) as a special case, the dynamics of the two flows are quite different as shall be seen shortly.

It has been shown earlier in Problem 9.A tha t the flow (42.5) has constant stretch history; it is now easily seen tha t all the extra stresses are functions of z only. The acceleration field being given by

a - -122(x - g ( z ) ) i - ~2(y _ f (z ) ) j , (42.{;)

the equations of motion become (cf. (43.4)):

Op dSxz -o~ ~ d~ - - P ~ ( ~ - g ( ~ ) ) '

_ _ _ dS~ 0V + = _ p ~ 2 ( y _ ](z)), (42.7) Oy dz

46The first successful a t tempt is due to ABBOTT, T.N.G. and WALTERS, K., J. Fluid Mech., 40, 205-213 (1970). They discovered an exact solution of the Navier-Stokes equations using the flow field (42.5).

47RAJAGOPAL, K.R., Arch. Rational Mech. Anal., 79, 39-47 (1982); GODDARD, J.D., Quart. Appl. Math., 41, 107-118 (1983).


_ _ . _ dS~ Op +pg-O. Oz dz

The solution is

p(x,y,z) - S~ + pgz + 2Pft2(x2 + y2), (42.8) , t

provided the shear stresses Sxz and S~z satisfy the equations

dSxz dS~z dz -- Pft2 g(z); dz -- Pft2 f (z)" (42.9)

Hence, the exact forms of the functions .f(z) and g(z) vary from one fluid to another, being determined by the respective constitutive relations. Since the extra stresses are constants in the homogeneous flow field (42.3), the differential equations (42.9) can only be satisfied provided inertia is ignored, i.e., one puts p - 0. This is the reason why the dynamics of the two flows are different as asserted earlier.

~2.~ Simple Extensional Flows

1 0 4

10 3 ~ O i ~ ~

10 2

._o

e - 10 ~ .m

2 1-

10 0 o

10 1

10 .2 �9 �9 | , | . . . . | �9

0 1 2

T ime (s)

FIGURE 42.3. The extensional stress growth data for a high molecular weight polyisobutylene in a mixture of intermediate molecular weight polybutene and another low viscosity Newtonian solvent. The extensional rate k - 2 per sec.

Consider the steady velocity field in Cartesian coordinates

u- -kx , v - - - . ~ y , w- - - -~z , (42.10)

where ~ is a constant, called the extensional rate. The flow in (42.10) describes the extension of a circular bar along its axis, which is in the x-direction, while it suffers a uniform contraction in the plane of the cross-section of the rod. It has already been demonstra ted in (38.26)-(38.29) tha t the stress system corresponding to the flow (42.10) has no shear stresses. In addition, it is easily seen tha t Sxx ~ S~y - Szz as well. Moreover, the diagonal stresses are all constants throughout the rod and depend on the extensional rate k only.

The extensional flow is also dynamically compliant in the presence of inertia and a conservative body force; incorporating the latter into the pressure term, one finds tha t

p(x,y,z) -- - S (4x2 + yU + z2). (42.11)

The important material property in an extensional flow is the extensional viscosity r/E , defined through

Sxx - Suu (42.12) =

From (38.29) it follows quite readily tha t ~E(k) ~ ~E(--k) unless certain conditions are met. The former is given by ~E(k) -- 3(~1 + a2k), while the latter has the form r/E(--k ) -- 3(~1 -- ~2k). Since t r A i - 0, we have that o~ l and ~2 are functions of trA21 and t r A 3. Whether the extensional rate is positive or negative, the former is always 6k 2, while the latter is 10k 3 if k > 0 and - 6 k 3 if k < 0. Thus, if

1. o~ 1 (6k 2, 10k 3) = c~ 1 (6k 2, --6k3);

2. a2(6k 2, 10k 3) -- --a2 (6k 2, --6k3),

then the extensional viscosity in tension is equal to that in compression and not otherwise. Of course, the two are equal in the Newtonian fluid since O~ 1 : 7~ is the constant shear viscosity and a2 - -0 .

It follows readily from the above that the extensional viscosity r/E in an incompressible Newtonian fluid is three times tha t of the shear viscosity r /o f tha t fluid, a result found by TR OUTON. 48 In polymeric liquids, the extensional viscosity is typically much larger than its viscometric counterpart, as Figure 42.3 shows. How- ever, the measurement of extensional viscosity is not as easy as the shear viscosity. It suffices here to note tha t in the most successful set of experiments, 49 the fluid is held between two disks which are moved apart so tha t the extension rate based on the filament diameter is constant. Hence, the stresses Sxx and Syy are functionals of (k, t) and after a certain amount of time, which depends on the magnitude of the extensional rate, the tensile stress growth coefficient defined through

r/+(k, t) = Sxx(k, t) - S,y(k, t) (42.13) k

at tains its steady value r/E. In Figure 42.3, the ratio of r/+/r/0, where r/0 is the zero shear viscosity, is plotted as a function of time. It is seen tha t a steady state value

48TROUTON, F.T., Proc. Roy. Soc. Loud., A77, 942-954 (1906). 49See TIRTAATMADJA, V. and SRIDHAR, T., J. Rheol., 37, 1081-1102 (1993).

43 Dynamically Compatible Unsteady Flows 315

for this ratio is attained after 3 seconds. It is also obvious that the ratio ~7E/~7o is of the order of 103 .

In order to discover whether there exist other feasible experimental configurations to measure the extensional viscosity, PIPKIN 5~ analysed all kinematically admissible steady stretching motions. We summarise the conclusions here.

If all particles are undergoing steady extensional motions, i.e., each particle experiences a strain history of the form

l e- 2as 0 0 I [ C t ( t - s ) ] - 0 e -2b~ 0 , (42.14)

0 0 e -2cs

where a + b + c = 0, then the extension rates for all the particles must be the same. Hence, there is no kinematically admissible, let alone dynamically possible, steady extensional flow in which the condition of adherence to stationary boundaries can be met.

Because all particles experience the same extensional rates, the extra stresses are constants throughout the flow field. Hence, the equations of motion are satisfied if the corresponding acceleration fields are conservative. There are only two such flOWS:

1. The homogeneous velocity field

u = ax, v - - b y , w - - cz, (42.15)

with the three constants satisfying the isochoricity condsition a + b + c - 0. This includes the extensional flow (42.10) as a special case.

2. The inhomogeneous flow described in cylindrical coordinates through

i ' = cr, O = -cO, k = - c z , (42.16)

where c is a constant.

In sum, there are no other flows apart form these two from which the true extensional viscosity may be measued.

43 Dynamically Compatible Unsteady Flows

In w attention was focussed on five steady, viscometric flows which are dynamically compliant; in w steady, rectilinear flow was shown to occur under some restrictions on the material functions; in w four steady flows were placed under scrutiny for their dynamic feasibility. Therefore, it is natural to ask whether the ten steady flows have unsteady counterparts, which satisfy the equations of motion.

5~ A.C., Div. Appl. Math. Report, Brown University, June 1975. Surprisingly, this important research is still unpublished although the inhomogeneous extensional flow is mentioned by PIPKIN, A.C. and TANNER, R.I., Ann. Rev. Fluid Mech., 9, 13-32 (1977).


Here we shall examine four unsteady forms of the viscometric flows which satisfy the equations of motion, for these unsteady flows are locally equivalent to unsteady, simple shearing and their stress functionals are known from (42.30)-(42.33) and Problem 42.B. Full details will not be provided for it is easily seen that in these flows, the extra stresses become functionals of the strain histories involved, which in turn means that the extra stresses are functions of the relevant coordinate(s) and time. As a first example, let us consider unsteady helical flow.

~3 .1 U n s t e a d y H e l i c a l F l o w

The velocity field in cylindrical coordinates is

v = rw(r, t)eo + u(r, t)e~. (43.1)

The extra stresses are functions of (r, t) and the equations of motion now become

_ _ 0 & ~ & ~ - & 0 _ 0p + + _ _prw2, Or Or r

_ 1 ~ _ _ os~o o ~ r O0 t- Or +2Sr~ -- p r ~ , (43.2)

_ . _ OSrz 1 Ou Op+ + - & ~ + p b ~ - p-~. Oz Or r

Clearly, by absorbing the body force in the z-direction which arises from gravity into the pressure term, it is seen that the pressure gradient in this direction must be c(t) per unit length. Hence, one must solve

OS~z 1 Ou Or + - s~ - p-~ + ~( t ) ,

os~o 2 o ~ Or + S,.o = pr , r c~ ~

(43.3)

and the pressure field must be given by

p(r, z, t) -- c( t)z + S ~ + + prw 2 dr. (43.4) r

Thus we conclude that unsteady helical flow is possible in all incompressible simple fluids with the specific form of the velocity field varying from one substance to another.

3 .2 U n s t e a d y C h a n n e l F l o w

Here, the velocity field is assumed to be

v = u(y, t ) I. (43.5)

All the extra stresses are functions of (y, t) and the equations of motion are satisfied provided the body force is conservative and including this in the pressure term, we find that

p(~, y, t) = ~(t)~ + s ~ (43.6)


and OS~ Ou

Oy = P-~ + c(t). (43.7)

The analysis shows, again, that unsteady channel flow is possible in all simple fluids.

~3.3 Unsteady Torsional and Cone-and-Plate Flows

In both of these flows, inertia has already been ignored under steady conditions of opertion; thus the unsteady torsional flow

v -- - - ~ r z e o ,

and the unsteady cone-and-plate flow

(43.8)

are possible in their respective geometries when inertia is ignored.

(43.9)

43.~ Unsteady Ex tens iona l Flow

The next example we consider is unsteady extensional flow, for this is typical of the motion in any spinning operation. Suppose that

v -- a(t)x t + b(t)y j + c(t)z k (43.10)

where a(t)+ b(t)+ c(t) = O. It is easily seen that the shear stresses vanish yet again and that the normal stresses are functions of t. The above flow is compliant with dynamics provided

P [ a x 2 + b y 2 a2(t)x b 2 c 2 z] - - 7 + + + ( t l y + (t) . (43.11)

Having discuused the unsteady counterparts of some viscometric and extensional flows, two unsteady flows, whose kinematics has been examined in w will now be considered for their dynamic compatibility.

~3.5 Flow Generated by Squeezing a Wedge with Ex tens ion

Consider the following velocity field in cylindrical coordinates (cf. (38.34))"

v = r[a + f'(O,t)ler - 2rf(O,t)eo - 2az, (43.12)

where ~ is a constant. As shown by PHAN-THIEN, 51 if this constant were set equal to zero, then the flow can be generated within the confines of a wedge by squeezing

51PHAN-THIEN, N., J. Non-Newt. Fluid Mech., 16, 329-345 (1984).


the sides together; the rate of movement of the walls need not be a constant. Here, we consider the general form in (43.12).

It has already been shown earlier, el. (38.34)-(38.38), that the physical components of the relative strain tensor are functions of (0, T, t) and so the extra stresses are functions of (0,t) only; in addition, the stresses Srz and Soz are both zero. Using this information, neglecting the acceleration terms and the body force, the equations of motion (el. (39.5)) now become

Op 10&o -O'7+r O0

_Op + O0

Srr -- SO0 + - - O ,

r O&e

+ 2&o - o, (43.13) O0

Op ~ ~ O .

Oz

The above equations have a solution provided

o&o + &~ - &o = g(t), (43.14)

where g(t) is a function of time t only. Consequently, it follows that

p(r, O, t) = g(t) In r + Soo + 2 / S~o dO + h(t), (43.15)

where h(t) is some function of t. Hence, the assumed motion (43.12) is possible in all simple fluids, with the function f(0, t) varying from one fluid to another and its determination, subject to boundary and initial conditions, is to be accomplished by using (43.14) in the form

0 [ 0 & 0 ] 0"0 "00 + &~ - See - O. (43.16)

This function f(9, t) and the corresponding velocity field have been determined for the Oldroyd-B fluid by PHAN-THIEN in the work cited above.

~3.6 Flow Generated by Squeezing a Cone with Extension

In a manner similar to the above, one may consider the flow, cf. ( 38 .O1) - (38.03), which has the following form in spherical polar coordinates"

v -- r[a + f(O, t)]er + rg(O, t)eo - 3ar sin 0r162 (43.17)

where a is a constant. When this is zero, it has been shown by PHAN-THIEN 52 that the flow may be generated within a conical shaped region by squeezing its surface. Again, the rate of squeezing need not be a constant.

Consideration of the velocity field (43.17) is the subject of Problem 38.D and it is thus known that the extra stresses are all functions of (0, t); moreover, the

52PHAN-THIEN, N., Rheol. Acta, 24, 119-126 (1985).


stresses S~r and See are both zero. Using these facts, ignoring both the body force and inertia, the equations of motion are (cf. 39.6))

0p Or

1 0 S , o 1 ! b - [2Sr~ - Soo - Sr162 + cot 0 S~0] - 0

r cOO r

1 0 p 10Soo 1 r O0 ~ ~--[3S~o + cot O(Soo - Sr162 - 0 r O0 r

1 Op - - ~ ~ ---- O .

r sin 0 0r

(43.18)

Once again, we find that the above system of equations has a solution provided

oS~o + 2Sr~ - Soo - Sr162 + cot 0 Sro = h( t ) , (43.19)

OO

where h( t ) is a function of t only. If this is so, then

p(r, O, t) -- h( t ) In r + Soo + f ta&o + cot o(s00 - &+)] dO + q(t), (43.20)

where q(t) is an arbitrary function of t. Clearly, the velocity fields f ( O , t ) and g(O,t) are possible in all incompressible

simple fluids, varying from one fluid to another. Since two equations are needed to find these two functions subject to appropriate boundary and initial conditions, we observe that one equation is provided by the incompressibility condition (38.D2), viz.,

3f + g' + g cot 0 = 0. (43.21)

The second arises by differentiating (43.19) with respect to 0. This is:

o [os~o ] - ~ [ O0 + 2Srr - Soo - Sr162 + cot 0 Sro = 0. (43.22)

P r o b l e m 4 3 . A

In a manner similar to the examples shown here, deduce the conditions under which the following five unsteady flows comply with the equations of motion in the relevant coordinate systems:

1. v = u (y , t) i + w(y , t) k.

2. v = c(t)O e~.

3. v = w ( x , y, t) k .

4. v = [r~(r , t ) + ~(t)~] eo + ~(r, t) e~.

5. v = r~(r , t) eo + [~(r, t) + ~(t)O] ~ .

Appendix to Chap te r 5" Strain Jumps and Stress Relaxat ion

Stress Response to Strain Jumps

Consider an elastic solid which has been at rest for a long time. Let a particle of this material be subjected to a sudden jump in the deformation gradient at t ime t - 0, and let it remain in this strained state subsequently. Tha t is, the deformation gradient history has the form �9

F ( r ) - ~ 1, - o o < r < 0, (A5.1) Fo, O<T~_t. L

Now, it is obvious tha t the s tate of stress in the solid will be given by

T(T) = { T0,0' 0-~176 T <<Tt, < 0, (A5.2)

where To is a constant. This shows clearly tha t

lim T ( T ) = 0, lim T ( T ) = To, (A5.3) r---*0- "r---,0+

or there is a jump in the limits at t ime t -- 0. Let us now suppose tha t the same experiment is repeated on an incompressible

Newtonian fluid particle. Here, we find tha t the first Rivlin-Ericksen tensor obeys

0, - o o < T < 0, (A5 .4 ) A 1 - - 0 , 0 < T _< t.

Consequently, the extra stress tensor in such a fluid will be given by

0, -oo <T < 0, (A5.5) S ( ~ ) - 0, 0 < ~ < t .

That is, there is no jump in the limits at t ime t = 0. Hence, one is led to pose the question of determining the reponse of a viscoelastic

fluid in the above situation. To find an answer, we turn to the constitutive relation of linear viscoelasticity mentioned earlier (B3.11) and rewrite it in an amended form :

s(t) = 2C;(0)E(t)+ 2 C(s )E(t - s) as. (As.6)

If one assumes tha t the jump in the deformation gradient is infinitesimal, then the above equation applies and we use (.45.1) to derive

E ( t - s) - ~ 0, t < s < oo, (A5.7) Eo, 0 <_s<t. t

This strain history shows tha t the two sided limits of the stress at t ime t -- 0 will not be the same; indeed, there will be a jump of amount 2G(0)E0. We shall see tha t a jump in stress occurs in the simple fluid as well.

Appendix to Chapter 5: Strain Jumps and Stress Relaxation 321

Let us assume tha t a simple material element has been at rest for an infinitely long time and that , at t ime t = 0, the element has a sudden deformation gradient F0 # 1 imposed on it. Subsequently, the material element remains under this deformation gradient, i.e., (A5.1) applies. The relative deformation gradient may now be calculated and we obtain

Ft(~-) = F(T)F(t) -1 = { F~ ' O<T<_t.--O0 < ~" < 0, (A5.8)

Consequently, the relative strain history has the form

Ct(T) -- Ft(T)TFt(~) -- { B~ 1,

In (A5.9), the tensor B0 is defined through

B0 - F0F T.

- o o < T < 0, (A5.9) O<T<t.

(A5.10)

Let us now assume that the material is a simple fluid. Then, the constitutive relation for the stress tensor in the fluid is given by (cf. (31.11))

S(t) = T/(Ct(T), IF(t)l), (A5.11)

where IF(t)l is the determinant ofF( t ) . Hence, the simple fluid element with a stress response functional depending on the relative strain history will respond to (A5.9) as follows : the fluid particle will experience a state of rest over the time interval (0, t] and a jump in the strain of amount Bo a over the time interval ( -oo , 0). Because B o 1 is a constant tensor, the stress in the fluid element will reduce to a function of this strain tensor with t ime t acting as a parameter. Because of the objectivity condition on the constitutive functional (A5.11), this function will be an isotropic function of Bo 1, i.e.,

S(t) = g(Bo 1, t), (A5.12)

with the restriction �9

Q(t )g (B ~ 1, t)Q(t)T _ g(Q( t )B ~ 1Q(t)T ' t) (A5.13)

for all orthogonal tensors Q(t). The above argument leads to the result that , if a sudden jump in strain is imposed on a simple fluid element which has been at rest for an infinitely long period of time, then the two sided limits of the stress tensor will not be identical. That is, there will be a jump and the fluid element will behave like an isotropic, elastic solid instantaneously. Subsequently, because of the special form of the strain history, the fluid will behave like an isotropic elastic solid, except that the material functions will depend on time; indeed, they will relax to zero as t --~ cr (see below). Be this as it may, those results that hold for isotropic elastic solids must also hold for simple fluids. In particular, if the sudden strain that is imposed is tha t of simple shear, i.e.,

1 7 0 ) F0 = 0 1 0 ,

0 0 1 (A5.14)

where 7 is the constant shear strain, then the following relation must hold:

N1 (t, 7) -- ~cr(t, ~'), (A5.15)

where N1 - - S l l - - S12 is the first normal stress (relaxation) function and a -- S12 is the shear stress (relaxtion) function. Thus, we have derived a universal relation or one tha t applies to all compressible simple fluids.

It is easily shown tha t similar conclusions apply to incompressible fluids as well. The lat ter result is known as the Lodge-Meissner relation (cf. (35.20)). 53

In particular, it must be noted tha t even if the stress in a simple fluid depends explicitly on the Rivlin-Ericksen tensors A1,-- -, AN as well as the difference history G(s) , the above conclusions remain valid. The reason lies in the fact t ha t the Rivlin- Ericksen tensors are zero both before and after the jump (cf. (A5.4)).

Turning to the case of double-step strain, there are no universal relations comparable to (A5.15). However, for a certain class of integral models, consistency relations may be derived for jumps in both shear and extensional strains, sa

Stress Relaxation

In order to offer an explanation of stress relaxation, we need to examine the response of a simple fluid element when a given motion ceases, or when the position occupied by the particle at t ime to is the same as at t ime t, an interval 5 later. Hence,

x (X, to) - x (X, t), t - to - 5. (A5.16)

Thus, the deformation gradients at t ime to and t are identical for t >_ to. Using this, the deformation gradient history following the cessation of motion is given by

F ( t - s) - { F(t0), 0 _ s _< 5, (A5.17) F(to-s+t~), ~ _ < s < o o .

The above history 55 may be called a static continuation of F(t0) through ~. From (A5.17), the relative deformation gradient history may be calculated and the strain history determined. Finally, one obtains the difference history �9

G(s)= {0' O_<s_<5, (A5.18) C t o ( t 0 - s + ~ ) - l , ~ _ < s < c o .

One now observes tha t as the static continuation is preserved indefinitely, i.e., as 5 --~ oo, the difference history is zero over an infintely long time. Consequently, the stresses in a simple material and indeed a simple fluid element must vanish as t ime proceeds from the cessation of motion.

As an example, let us assume tha t following a strain jump, a fluid element is held in tha t configuration for a long time. Then, the stresses will relax to zero.

53For the K-BKZ fluids, see BERNSTEIN, B., Acta Mechanzca, 2, 329-354 (1966); for simple fluids, see RAJAGOPAL, K.R. and WINEMAN, A.S., Rheol. Acta, 27, 555-556 (1988). For viscoelastic solids, see RIVLIN, R.S., Quart. Appl. Math., 14, 83-89 (1956).

54See VRENTAS, J. S. and VRENTAS, C.M., Rheol. Acta, 34, 109-114 (1995) and other references listed therein for a discussion of these matters. In addition, see BROWN, E.F. and BURGHARDT, W.R., J. Rheol.. 40, 37-54 (1996).

55See COLEMAN, B.D., Arch. Rational Mech. Anal., 17, 1-46; 230-254 (1964).

6 Simple Models and Complex Phenomena

In Chapter 5, the emphasis was on finding solutions to problems without assuming any specific forms for the constitutive relations. In this Chapter, we seek expla- nations of fairly complex phenomena by choosing relatively simple constitutive models. Most of these are either the order fluids, notably the second order fluid; or 1-order, 1-integral integral models like the Oldroyd-B fluid; or slightly more complicated, nonlinear relations. The goal is to t ry and understand, if possible, the elastic effects without additional complications arising from the incorporation of inertial or non-Newtonian viscous behaviour; or, to comprehend the role of nonlinear viscosity without the intrusion of elastic effects.

Thus, in w conditions which guarantee tha t a Newtonian velocity field occurs in a second order fluid are sought. These are, for the most part, relevant to creeping flows. It is found tha t while the velocity fields may be identical, the two pressure fields are different. Consequently, the normal stress effects on stationary or free surfaces will exhibit the importance of the elasticity of the fluid. These matters occupy the next two sections w - w In w it is shown that N2 may be measured by the normal stresses on the walls of a pipe of rectangular cross section (cf. w that N2 is responsible for 'pressure slot errors' if a rectilinear flow occurs in the direction of the slot; and, finally, tha t N2 is the main cause of edge fracture.

In w the effects of N1 are studied. It is found that it is responsible for the 'pressure slot errors' if the fluid moves across a slot; it is also found, through a flow in and out of an annulus, tha t N1 may induce a departure from the Newtonian velocity field, if the fluid experiences a finite transit t ime in a given process.

Next, in w experimental observations on the storage and loss moduli arising out of in-line and transverse oscillations superposed on steady shearing are presented. It is shown tha t the theory of nearly viscometric flows provides an adequate explanation of the phenomena observed. This section also describes the crucial test to determine whether a fluid sample being tested is a simple fluid or not.

324 6. Simple Models and Complex Phenomena

In w a theoretical explanation of the experimentally observed increase in the tensile stress growth coefficient ~+ (~, ~, t) due to the imposition of pre-shearing on an extensional flow is given. Here, it is found that both N1 and N2 play a cruical role.

The last two sections deal with the analyses of the stability of a given flow. In w we seek conditions which guarantee the linearised stability of the state of rest of the simple fluid when the fluid model is an integral model or an order fluid. Next, we examine the stability of the torsional flow of an Oldroyd-B fluid in unbounded as well as bounded domains. Both axisymmetric and non-axisymmetric perturbations are considered; steady and unsteady situations are included. After a detailed study of this problem which highlights the methods employed to study such problems arising in other flows, we turn to qualitative dynamics in w Here, no a t tempt is made to solve a given problem. Rather, we examine the constitutive relations and the equations of motion affecting a given problem and, by looking at the structure of these equations, we endeavour to predict the stability of the solution to the problem at hand. Connections between the loss of evolution of a given set of equations and the propagation of acceleration waves and vortex sheets are made. And the role of nonlinear hyperbolic equations in the development of singularities is also discussed.

44 Conditions for Identical Velocity Fields in Newtonian and Some Non-Newtonain Fluids

In this section, we examine when a flow that is possible in an incompressible New- tonian fluid is also possible in an incompressible second order fluid. Some very general applications will also be considered with specific examples to be dealt with in w and w later. Further, we examine conditions under which a Newtonian velocity field occurs in a Maxwell and an Oldroyd-B fluid.

~ . 1 Second Order Fluids

The constitutive equation for the extra stress in the second order fluid has the form (cf. (38.24))

0 2 1 ~OlA2, (44.1) s - oA1 + + -

where ~7o is the constant viscosity and ~p0 and ~p0 are the constant, first and second normal stress difference coefficients respectively.

Assuming a conservative body force and absorbing it into the pressure term, the Newtonian pressure field PN has to obey

VpN -- ~70~7 - A1 - pa. (44.2)

If the same velocity field were to occur in the second order fluid, then an additional pressure term Ps will arise and this has to satisfy

[ 1 ~1~ ] (44.3) Vps - V- (~~ 1 + ~~ - ~ .

44 Conditions for Identical Velocity Fields 325

We shall investigate when the right side of (44.3) is a scalar through four special classes of flows.

Potent ia l Flows

When the velocity field is a potential flow and, consequently v -- Vr incompressibility demands that r - 0 . Since

(A1)~j = 2r (44.4)

it follows that

V - A 1 = 0. (44.5)

Hence, the Newtonian pressure field is given by

1 PN = + 7 v - v l . (44.6)

Next, it is easy to prove that

V. A~ - l v ( t r A~) V. A2 - 3 V ( t r A12) (44.7) 2 ~

This shows clearly that in potential flows, 1 the Newtonian and second order fluid velocity fields are identical, with the second order pressure term given by

1 (r + 4r A12. (44.8) Ps -- "~

P r o b l e m 44 .A

Obtain the following Bernoulli equation 2 for a potential flow in a second order fluid:

P l ( ~ p o + 4 ~ p O ) t r A 2 1 - C , (44.A1) Pr + -~V" v + PN -- "~

where C is a constant.

D i v e r g e n c e o f A2 in the A b s e n c e of Inert ia

Consider a velocity field v such that either the acceleration field is zero or the inertial effects may be ignored. An example of the former which concerns us here arises when the velocity field is that in a rectilinear flow, i.e., v = w ( x , y ) k in Cartesian coordinates; this will be looked at in detail in w In the latter case, the flow is called a creeping flow and we shall study two examples in detail in w

In either case, assuming that the velocity field is possible in a Newtonian fluid, one finds that the Newtonian pressure PN must satisfy the analogue of (44.6), i.e.,

PN -- ?70 V" AI. (44.9)

1This result is due to CASWELL, B. For the citation, see PIPKIN, A.C., in RIVLIN, R.S., Ed., Nonlinear Continuum Mechanics and Physics and their Applications, C.I.M.E. Lectures, Bressanone, 1969, pp. 51-149.

2JOSEPH, D.D., J. Non-Newt. Fluid Mech., 42, 385-389 (1992).


Of course, in creeping flows, the body force is ignored; in a rectilinear flow, the body force term is absorbed into the pressure term, for this simplifies the subsequent argument.

Now, for this Newtonian velocity field to be possible in a second order fluid as well, one has to show that the right side of (44.3) is the gradient of a scalar. Equivalently, it is necessary to find simple formulae for the divergences of A12 and A2 which guarantee this result.

Leaving aside the formula for the divergence of A12, let us turn to the task of proving that V- A2 is the gradient of a scalar. Here, a considerable amount of effort is needed as the independent derivations by GIESEKUS 3 and PIPKIN 4 show. Indeed, if v and PN satisfy (44.9), then their results lead to the following:

l d p N V - A 2 - V - A 1 2 + V ~0 dt + l t r A12]. (44.10)

In (44.10), the material derivative of PN is given by

dPN = v - V p N (44.11) dt

in the present context, for the flow is steady. This result will be useful in w and w

It has thus become clear that V - A 2 is the gradient of a scalar whenever V. A12 is itself a gradient. We shall turn to two examples of this kind in w below.

Plane Creeping Flows

Let the velocity field be that in a steady plane flow, i.e., v -- u ( x , y ) i + v ( x , y ) j . The task of demonstrating that ~7- A12 is the gradient of a scalar is trivial here because the Cayley-Hamilton theorem and incompressibility together show that

A 2 + (det A1)I - 0. (44.12)

Taking the trace of the equation, one finds that

det A1 = - l t r A21 .

Hence, we have the result:

(44.13)

Now, combining (44.10) and (44.14), we arrive at

V ' A 2 - - ~ 7 ~0 dt

(44.14)

(44.15)

3GIESEKUS, H., Rheol. Acta, 3, 59-81 (1963). 4pIPKIN, A.C., in RIVLIN, R.S., Ed., Nonlinear Continuum Mechanics and Physics and their

Applications, C.I.M.E. Lectures, Bressanone, 1969, pp. 51-149. See also PIPKIN, A.C., Lectures on Viscoelasticity Theory, 2rid Ed., Springer-Verlag, 1986.

44 Conditions for Identical Velocity Fields 327

Thus, a plane creeping flow in a Newtonian fluid is also a solution to the plane creeping flow problem in a second order fluid. In w we shall examine the question of uniqueness of this solution after rederiving this result through the use of the stream function.

The extra pressure in the second order fluid may now be shown to be

1 p s - ~ ~ (~P~176 dpN 2?7 0 dt -8 dt -- v . ~ P N . (44.16)

R e c t i l i n e a r F lows

If the velocity field has the form v - w ( x , y ) k , then it is dynamically possible in an incompressible Newtonian fluid, when the pressure gradient in the z-di rec t ion is a constant c, say. The equation of motion reduces to (cf. (41.6)):

02w 02w c + = .--. (44.17)

Ox2 OY 2 ~o

P r o b l e m 4 4 . B Prove that if w obeys (44.17), then s

Here, use the fact tha t the shear rate ~/is given by

4/2 -- w 2,~ + w2,v

and the material derivative of PN by

d p N / d t - v . V p N -- CW.

(44.B1)

(44.B2)

(44.B3)

In sum, it has been shown through (44.10) and (44.B1) that in a rectilinear fl0w, the divergences of A12 and A2 are both gradients of scalars. Hence, a rectilinear flow which occurs in a Newtonian fluid is also sustainable in a second order fluid. The second order pressure field is now given by

(~__~ c ) (44.18) PS -- ~0 + - - - w . rio

This formula is in agreement with that derivable from (41.20).

T h r e e D i m e n s i o n a l C r e e p i n g F lows

Here, we consider a three dimensional creeping flow that is possible in a Newtonian fluid. Combining (44.3) and (44.10), we find that the second order pressure field must satisfy

[ 1 1A1 l ,441 , ~Tps -- V . ( ~ 0 + ~ 0 ) A 2 1 +~7 - - - - - - - + t r . rio dt

5KEARSLEY, E.A., Trans. Soc. Rheol., 14, 419-424 (1970).


Now, it is not always the case that in three dimensions, V- A12 is the gradient of a scalar. However, as GIESEKUS 6 observed, if the normal stress coefficients ~0 and G ~ are such that

~p0 + 1 G0 _ 0 (44.20) 2

then the term containing A12 vanishes and (44.19) can be solved for Ps:

Ps--~01 dpNdt + 4 tr A12" (44.21)

Hence, a three dimensional creeping flow that is possible in an incompressible New- tonian fluid is also possible in a second order fluid if (44.20) holds. If it does not hold, and experimental results do show that the second normal stress difference is not the negative of half of the first normal stress difference in many fluids, it does not mean that the two flows need not be the same. Clearly, what happens when (44.20) does not hold is tha t there is no general result applicable to all three dimensional creeping flows; each flow has to be examined on its own merit.

##.2 Maxwell and Oldroyd-B Fluids

The constitutive relation for a Maxwell fluid may be given the form

(44.22) S M 4- ~1 ~ t - - - ~7~

where A1 is the relaxation t ime and for any symmetric second order tensor S,

7)8 dS = - - - - L S - SL T. (44.23) 7)t dt

The constitutive equation for the Oldroyd-B fluid is the following:

~ S o ( DZ)Atl ) (44.24) S o - ~ ' ) ~ l ~)t = 7 ] 0 AI-I-A2- , ~ 2 _ ~ 1

where A2 is the retardation time. Now, for a fixed velocity field v, it is easily shown that the tensors SM and So are related through

S o - 1 - + 0 AI. (44.25)

The above equations lead to the following:

P r o b l e m 44 .C Let there be three incompressible fluids with the same density and viscosity. Let

one of the three be a Newtonian fluid, the second a Maxwell fluid and the third an Oldroyd-B fluid, with the latter two possessing the same relaxation times. Then, prove that a velocity field that is dynamically possible in any two fluids under the same body force is also possible in the third subjected to the same body force.

, i i , , , , [ , , , ,

6GIESEKUS, H., Rheol. Acta, 3, 59-81 (1963).

45 The Role of the Second Normal Stress Difference in Rectilinear Flows 329

45 The Role of the Second Normal Stress Difference in Rectilinear Flows

In w it has been shown that if the viscosity r/ is proportional to ~2, then a rectilinear flow is possible in all cross sections. In the second order fluid, both of these material functions are constants and hence one is proportional to the other. Consequently, a rectilinear flow will occur in all geometries and in a given domain, the velocity field is identical to that in an incompressible Newtonian fluid, a result established in w It has also been demonstrated in w that the pressure fields in the two fluids are not the same. Not only do these differences affect the stress fields in the respective fluids, the presence of normal stresses also causes the stress disributions in the two fluids to be different. In order to highlight these aspects which lend themselves to the measurement of N2, we shall consider two flows in detail. Lastly, we examine the role played by N2 in edge fracture in rheometry.

~5.1 Flow in a Tube of Rectangular Cross-Section

Consider the rectilinear flow of an incompressible Newtonian fluid in a pipe of rectangular cross-section. Let the length of the rectangle be 2L and let its breadth be 2B, such that 0 < B/L -- h _< 1. The coordinate axes are chosen so that the centroid of the rectangle coincides with the origin in the (x,y) plane and the x - a x i s is parallel to the length of the rectangle. Now, the equation of motion for the velocity field v -- w(x,y)k when the flow is in the z-di rect ion is well known; it is

~0[~-~x 2 § 0y 2 j c,

where ri0 is the viscosity and c is the pressure drop per unit length of the pipe. In this section, we shall follow PIPKIN and RIVLIN 7 and let c - - A P / L , where A P is the pressure drop over the length L. Now, let us introduce some non-dimensional variables through

x* x y, Y w*= 7o (45.2) = T' = -L' LApw"

In terms of these non-dimensional variables, (45.1) now becomes

02W * 02W *

Ox,2. + 0y,2 -- - 1 . (45.3)

In (41.18), a general formula for the normal stress on the stationary wall of a pipe has been derived; in the case of a second order fluid, this reads

n - T n - Tnn - ~ 0 , (45.4)

where ~ is the shear rate at the relevant point on the wall. In the flow Under consideration, the points on the wall that deserve attention are (L, 0) and (0,B).

7pIPKIN, A.C. and RIVLIN, R.S., Zezt. angew. Math. Phys., 14, 738-742 (1963). In this paper, (45.1) has been solved for pipes of elliptical, equilateral triangular and rectangular cross-sections.


Or, in terms of the non-dimensional variables, we are interested in the shear rates at the points (1, 0) and (0, h).

In order to calculate these shear rates, we need to solve for w* through (45.3). This is given by:

1 x. 2 w * ( x * , y * ) - - ~ ( 1 - )

2 x-~ ( - 1 ) ~ c o s h ( n + l / 2 ) ~ y * (n 2_, + 1 / 2 ) ~ x * (45.5)

-n-'~ (n 4- 1/2) 3 cosh(n + 1/2)nh cos n- -0

A fairly simple set of numerical computat ions show tha t the difference between ~2 at (1,0) and (0, h) is a maximum when h ~ 0.603. Hence, reverting to the original

variables, we find tha t , when B ~ 0.603L,

Tn,~(L ,O) - Tn ,~ (O,B) ~ -0 .0432 G ~ (/kV)2 (45.6) V02

This shows tha t if the viscosity of the fluid is known and, if one measures the pressure drop over a length of pipe L, which is the semi-length of cross-section of the rectangle, then the differences in the normal stress measurements at the mid- points of the rectangular walls of the pipe lead to a measurement of the second normal stress coefficient G ~

Of course, for more general fluids, it is t rue tha t a rectilinear flow will not occur in a pipe of non-circular, e.g., square or rectangular, cross-section (cf. w s Nev- ertheless, applying regular per turbat ion theory, it can be shown 9 tha t if the fluid velocity is small, then rectilinear flow will occur in all fluids up to the third order, with the fourth order fluid exhibiting secondary flows.

Returning to the problem at hand, it has just been shown tha t in the second order fluid, the flow in a pipe of rectangular cross-section provides a direct method for the determinat ion of N2. Of course, pipes of different cross-sections may be used for this purpose.

As far as flows in bounded domains are concerned, there seems to be only one other method for the measurement of N2. This is based on the shape of the free surface in the flow of a second order fluid down a til ted trough of semi-circular cross-section. Here, the bulge of the free surface is measured and, from this, the value of N2 is calculated. 1~ It seems to us tha t this experimental set up is more useful as a demonstra t ion of the presence of N2 in a fluid, ra ther than as a tool for the determinat ion of N2, and hence we shall not pursue this mat te r further.

8For experimental results in such pipes, see TOWNSEND, P., WALTERS, K. and WATER- HOUSE, W. M., J. Non-Newt. Fluid Mech., 1, 107-123 (1976); GERVANG, B. and LARSEN, P.S., J. Non-Newt. Fluid Mech., 39, 217-237 (1991).

9LANGLOIS, W. and RIVLIN, R.S., Rend. Mat. Appl. Univ. Roma Inst. Nat. Alta Mat., 22, ~69-185 (1963).

l~ TANNER, R.I., Trans. Soc. Rheol., 14, 483-507 (1970) as well as KEENTOK, M., GEORGES(~U, A.G., SHERWOOD, A.A. and TANNER, R.I., J. Non-Newt. Fluid Mech., 6, 303-324 (1980). For an application of the theory of domain perturbation to the steady flow down a tilted trough, see STURGES, L. and JOSEPH, D.D., Arch. Rational Mech. Anal., 59, 359-387 (~976).


45.2 Flow Along a Slot - An Unbounded Domain

C' B'

A

~y

d P

W

I D

B C

d

A f

FIGURE 45.1. Rectilinear motion with a slot in the direction of the flow.

The rectilinear flow of a second order fluid along a slot - see Figure 45.1 - provides a different method for the measurement of N2 at slow rates. The assumption here is tha t if the slot is deep enough, the shear rate at the bot tom of the slot is zero; and far away from the slot, the flow will be unaffected by the presence of the slot and thus the shear rate will be tha t as if the slot were absent.

To make the argument more transparent, consider the domain depicted in Figure 45.1. Here, the flow along the slot occurs between two parallel plates at a distance D apart; the slot is of width W and of depth d. Three problems worth considering are: the flow driven by the movement of the top plate at a constant velocity V, tha t due to a constant presure gradient in the flow direction, and a combination of the two. Considering the first problem, we see tha t the velocity field w -- w(x, y) satisfies (cf. (45.1))"

02w 02w i - o . ( 4 5 . 7 ) Ox 2 Oy2

The boundary conditions to solve this problem are fairly straightforward. On the top plate, w has a constant value V and on the bot tom plate, including the slot, w - 0 . Nevertheless, this boundary value problem is not easily solved; indeed, one has to employ a Schwarz-Christoffel transformation to map the flow domain from the complex z - p l a n e onto the (~-plane in order to obtain an analytical solution. 11 We shall present this next.

In Figure 45.2, the points in the e - p l a n e corresponding to those in the physical, z - p l a n e are shown. The Schwarz-Christoffel mapping which arises here is given

11KEARSLEY, E.A., Lecture at the 44th Ann. Meeting Soc. Rheol., Montreal, 1973.


- l /k sn a - l /k -1 0 1 I/k 1/k sn a

J

w = O

FIGURE 45.2. Points in the C-plane corresponding to the points in the physical z-plane.

b y 12

dz ( 1 - k2~2/1/2 1 d"~ -- M r , (45.8)

1 - 1 - k2r 2 (~

where M is a constant and k and sn (~ are the modulus and modular sine respectively; the latter arise in the theory of elliptic functions. The boundary values of w in the (~-plane are also shown in Figure 45.2. From these, it is a trivial matter to show that

w(r V V arg ( - - - - C - - - ~ -

71"

1 ) V ( 1 ) + - - arg r + ' (45.9)

ksn ~ 7r ksn

where arg(-) denotes the argument of the complex number. If 4 is the complex potential corresponding to w so that w = ~ m 4, then

V ( l c ~ ) V ( 4(r -- iV +--- log ~ + - - - - l o g ~-- ksn

1 ) . (45.10) ksn

Returning to Figure 45.2, we see that the shear rates at the point O, which corresponds to that at the bottom of the slot, and at C, which is the point far away from the slot, are required. Because w = 0 on the fixed plate, Ow/Ox = 0 at these points in the physical plane. Also, by the Cauchy-Riemann conditions,

- - (45.11)

at the points O and C. Now, using (45.8) and (45.10), one finds that

d~ d4 d~ 2 V k s n ~ ( 1 - r ) 1/2 4-7 = el(, dz = IrM 1 - k2r 2 " (45.12)

The difference in the normal stresses between the points C and O is derivable from (45.4), i.e.,

1 0 ~2 1...0.2 ( A N = ~I/2[~2(C ) - (O)] ~-- ~w2'y (C) 1 "92(0)) (45.13) -y=(c) '

12CARTER, F.W., J. Inst. Elec. Engrs. London, 64, 1115-1138 (1926).


where ":y(C) - V / D . Using (45.11) - (45.12), it is easily shown tha t

1 -~-o �9 2 A N - ~ w 2 " ), (C)G(k~ol)~

1 - k 2 G(k, :

1 - k2sn 2 ~"

(45.14)

Certain remarks are in order here. If t he slot gets deep enough, then k --~ 0 and G -~ 1. On the other hand, for a rb i t rary values of d, D, and W, the relations between D / W , d /W, k, ~ and sn ~ are very complex 13 and a great deal of effort is needed to de termine the function (7. in Figure 45.3, we have drawn the way G --~ 1 for D / W ---- 0.5 as d / W varies from 0 to 0.7. It is seen tha t most of the difference in the normal stresses is evident for a fairly shallow depth of the slot when compared with its width, la

100

90

80

70

60

50

40

30

20

10

D / W = 0.5

i/ I I I I I I 0 10 20 30 40 50 60 70

(xw) %

FIGURE 45.3. The function G(k, ~) plotted against d/W, when D / W --0.5.

taFor example, see RAJAGOPAL, K.R. and HUILGOL, R.R., Rheol. Acta, 18, 456-462 (1979). 14We wish to thank Dr. KEARSLEY, E.A. for this figure.


45.3 Edge Frac ture in R h e o m e t r y

(a) (b)

FIGURE 45.4. Edge fracture in Rheometry.

"Shear or edge fracture" plays an important role in the rheometry of highly elastic fluids, such as melts, because it places an upper limit on the shear rate that can be achieved in a parallel plate or a cone and plate device. This edge effect is noticed as a sudden drop in the torque needed to drive the flow with a concurrent change in the shape of the free surface. Typically, an indentation occurs in the free surface and this moves towards the centre of the apparatus and, as it does, the sheared area is reduced leading to a diminution of the torque. 15 See Figure 45.4. The configuration is fairly stable till the speed is increased further, at which time the fluid flies out of the instrument. 16

A simple explanation of the onset of edge fracture is based on the following: the free surface of the fluid in the rheometer is in equilibrium under the action of the normal stress which is opposed by the surface tension effect. If the free surface is under a compressive normal stress then the free surface will tend to bulge outwards with the surface tension preventing it from moving out; on the other hand, if the free surface is under a tensile normal stress, it will be pulled in while the surface tension will act in opposition to the normal stress. Hence, a first a t tempt at explaining the onset of edge fracture may be made by showing that the free surface, or at least a point on it, is under a tensile normal stress.

To be specific, consider the free surface in a torsional flow apparatus, where the upper and lower plates rotate at equal and opposite speeds. Clearly, the free surface is an even function of the angular velocity f~ of the disks and the first correction to the shape of the free surface, proportional to ~2, may be determined by using the domain perturbation theory. 17 The radial stress at the midpoint of the free surface, proportional to ~2, is also proportional to

sin P~ (1 + sin 2 P , ) + 3H 3 ~ tan2 Pn

n--'l P2

15This observation appears in HUTTON, J.F., Nature, 200,646-648 (1963). 16For example, see CRAWLEY, R.L. and GRAESSLEY, W.W., Trans. Soc. Rheol., 21, 19-49

(1977). 17See w167 in JOSEPH, D.D., Arch. Rational Mech. Anal., 56, 99-157 (1974).

1 ( c o s P ~ - 1 - 3 ] -~" ~-~~=1 p5 ~ tan2 Pn) �9 (45.15)

Here, ~ denotes the real part; if R is the radius of the disk and d is the gap width between the plates, then H0 = 2R/d; finally, Pn,n - 1,2,3 are the eigenvalues listed by JOSEPH. These are:

P1 - 3.748838 + il.384339,

P2 -- 6.949980 + il.676105, (45.16)

P3 - 10.119259 + il.858384.

Using a typical value for H0 -- 20, which corresponds to a radius-gap ratio of 10, one finds that the above expression is 4672 • 10 -3. Hence, the radial stress at the midpoint, over and above the equilibrium hydrostatic value, is tensile.

0w an-0

T Y w=l

J

W---1

h

X

FIGURE 45.5. Rectilinear flow with a semi-circular indentation on the free surface.

In the following, a different derivation of the above result is given, is Basing the argument on the torsional flow configuration which has already been described, it will be assumed that a rectilinear flow occurs between two parallel plates of infinite extent; the orientation of the coordinate axes and the relevant dimensions are shown in Figure 45.5. The free surface, which is open to the atmosphere, has a tiny, semi-circular indentation in it; the origin of this tear in the surface is assumed to be random.

Using the constitutive relation for the second order fluid, the velocity field w = w(x, y) is seen to satisfy Laplace's equation, for the flow is caused by the movement of the two plates. That is, there is no pressure gradient and the boundary conditions are: w(x, h) -- 1, w ( x , - h ) - - 1 , 0 _~ x < oo. On the free surface, the boundary condition is Ow/0n -- 0. Even though the problem is easily posed, it does not possess a simple analytical solution. Hence, we shall use maximum principles to derive bounds on the gradient Vw at the midpoint of the indentation, i.e, at (x, y) =

18For the original idea, see TANNER, R.I. and KEENTOK, M., J. Rheol., 27, 47-57 (1983).

(p, 0). These bounds will be used to confirm tha t the normal stress at this point is indeed tensile. 19

v=O

v - p cosO h

v=O

Y Ov/On=O

Np h F "

x ~ h

avlOn=O

FIGURE 45.6. Boundary values of the conjugate harmonic function v.

We shall replace the function w by u which is defined through

Y u(x, Y) -- w(x, Y) - ~. (45.17)

The purpose of this is to produce a harmonic function u which is zero on the two plates; its normal derivative Ou/On is quite easily calculated from (45.17). If we now consider the function v, which is the harmonic conjugate of u, its boundary values may be determined through Cauchy-Riemann conditions and they are shown in Figure 45.6.

Turning now to the maximum principles for harmonic functions, 2~ we note tha t the maximum and minimum values of a non-constant harmonic function are attained on the boundary; incidentally, this is called Hopf's first maximum principle. Additionally, if the maximum of a non-constant harmonic function v occurs at a smooth point on the boundary, the derivative Ov/0n > 0 there; if the minimum of such a function occurs at a smooth boundary point, then OrlOn < 0 there. This is the second principle of Hopf. The gist of this is tha t the maximum or the minimum of a non-constant harmonic function cannot occur at a smooth point on the boundary where Ov/On -- 0. Applying this reasoning to the problem at hand, we see immediately tha t v cannot have its maximum or minimum on the two parallel plates. The extrema occur on the line segments: x - 0 , - h _< y _< - p , or x - 0, p _ y ___ h, or on the semi-circular indentation. A glance at Figure 45.6 shows tha t the vm== occurs at (x, y) -- (p, 0). Hence, o 1o - -OvlO > 0 at this point.

By Cauchy-Riemann conditions, Ou/Oy = -Ov/Ox, and hence one finds tha t Ou/Oy > 0 at (x,y) -- (p,O) and through (45.17) tha t 0 w / 0 y > 1/h at this point. Since, by symmetry, w -- 0 on the line p g x < oo, y = 0, we see tha t Ow/Ox - 0

z9HUILGOL, R.R., PANIZZA, M. and PAYNE, L.E., J. Non-Newt. Fluid Mech., 50,331-348 (1993); Corrigenda, 55,209-211 (1994).

2~ example, see PROTTER, M.H. and WEINBERGER, H.F., Maximum Principles in Di]- ferential Equations, Prentice-Hall, 1967.

along this line. Hence, the shear rate is given by

0w =

~,y (p,0) (45.18)

Thus, ~(p, 0) > 1/h, i.e., a lower bound to the shear rate at the midpoint of the indentation has now been derived.

~=p2/(x~+f ) < 0

I h an P

FIGURE 45.7. Boundary values of the upper bound solution ~.

To obtain an upper bound, we consider the semi-domian in Figure 45.7. Using the function 21

r y) = g 1 + ~ + y~ , (45.19)

which is also harmonic, we consider a new function r = w - r The boundary values of this new function, which is also harmonic, are shown in Figure 45.7. From the first maximum principle just mentioned, we see that the maximum of r occurs on the line p _ x < oo, y = 0. Hence, as we move away from the point (p, 0) around the arc, r cannot increase. Consequently, 0 r < 0 at the point (p, 0), or

0w

0y (p,0) (p,0)

or 2 = ~. (45.20)

Thus, it is found that 1 2

< ~(p, 0) < ~. (45.21)

Now, from (41.20), we know that the pressure p in a rectilinear flow of a second order fluid, in the absence of a pressure gradient, is given by

1 p = ~ v 0 ~ 2 + c , (45.22)

where C is a constant. In the present context, we may assume that p --, 0 as x --, oo. Using the fact that ~ --, 1/h at infinity, one finds that

1 [ 1] p(~,y) _ ~vo ~(~, y) _ ~ (45.23)

21TANNER, R.I. and KEENTOK, M., J. Rheol., 27, 47-57 (1983).

The normal stress at the point (p, 0) is given by the formula (41.B2), i.e., T,,,, = -p . The inequalities in (45.21) now demonstrate that, at the point (p, 0),

3 0< T~ < ___~o, (as.2a)

where the assumption that the second normal stress coefficient is negative has been used. The foregoing arguments show clearly that the normal stress at the midpoint of the semi-circular indentation is tensile. Hence, the free surface at this point will be sucked inwards as long as this tension exceeds the surface tension effect. Experimental results 22 seem to confirm this explanation of edge fracture and the importance of N2 in it.

W - - I

W----I

h

FIGURE 45.8. A cusp formed in edge fracture in a parallel shear flow.

An intriguing phenomenon that arises is the stable nature of the sheared, hori- zontal cusp after edge fracture has occurred. In order to explain it, it is necessary to note that the velocity gradient is parallel to the unit normal at the cusp - see Figure 45.8. Nevertheless, as shown in Problem 41.A, the surface is free of shear stress at the cusp. Using the result in (41.B3) and the pressure p in (45.23) above, one finds that the normal stress at the cusp is given by

1 0 [ 1 ] T,,,, = g~'2 ~2(~, y )+ g . (45.25)

Since ~ < 0, it follows that the edges of the cusp are under a compressive stress and hence they will be squeezed together, preventing the crack from moving further into the fluid.

46 Plane Creeping Flows and the Relevance of the First Normal Stress Difference

In w it has been shown that the velocity field in a steady plane creeping flow of a Newtonian fluid is also a solution to the steady plane creeping flow problem of a

22See LEE, C.S., TRIPP, B.C. and MAGDA, J.J., Rheol. Acta, 31, 306-308 (1992), who find that it is N2, rather than N1, which determines the onset of edge fracture.

46 Plane Creeping Flows and the Relevance of the First Normal Stress Difference 339

second order fluid. In this section, a condition under which they are identical will be derived and two examples, one of which obeys this criterion and one which does not, will be presented.

Since we are dealing with incompressible fluids in a plane flow situation, there exists a stream function r = r y) such tha t the velocity components u and v are expressible in terms of r as follows:

0r 0r (46.1)

Using these forms for u and v in the constitutive relation (44.1), it is possible to obtain the stresses in a second order fluid in terms of r Substituting the stresses into the equations of motion and eliminating the pressure term through the use of the equality of the mixed partial derivatives, i.e., P,z~ - P,~z, one is led to the following equation 23 for r

1 o ~/0A2r -- ~I/1 V" V(A2~)) : 0, (46.2)

where /k 2 is the two-dimensional biharmonic operator, and ~7 is the corresponding gradient. Now, in a Newtonian fluid, the stream function satisfies the biharmonic equation for r i.e., A2r - 0. From (46.2), we see tha t this is also a solution for the second order fluid, providing a different proof of the result derived earlier in w

Now, it is well known tha t in order to obtain a unique solution to the biharmonic equation A2r = 0, one has to prescribe the velocity v on the boundary. Equiva- lently, the tangential and normal components of the velocity are to be assigned; the former is given by the normal derivative O~p/On, while the latter is described by the derivative along the boundary, i.e., Or where s is the arc length. Integrating the latter, one finds tha t the prescription of the function r to within a constant, on the boundary will do. In sum, the solution of the biharmonic equation A2r = 0 in Newtonian fluids is achieved through both ~ and o ~ / ~ being assigned on the boundary. A question tha t this poses for the second order fluid arises from the fact that the s tream function obeys a fifth order partial differential equation (46.2) and hence these two conditions may not be sufficient. However, there are many flows of interest where the Newtonian solution is the solution to tha t in the second order fluid. We turn to an investigation of this mext.

~6.1 Congruence of the Two Velocity Fields

To examine when the two are identical, s e t / k 2 r --- w for convenience. Then, (46.2) may be rewritten as

w + S v . Vw--O, 5 = - 1 ~Ol. (46.3) 2rio

23TANNER, R.I., Phys. Fluids, 9, 1246-1247 (1966).

Multiply throughout by w and use the fact tha t the velocity field is divergence free. Then, one obtains

w 2 w 2 + 5V �9 (-~-v) = O. (46.4)

Intergrat ing this over the flow domain fl and using the divergence theorem, one has the result

/o w 2 da + 5 ~ - v - n ds = 0. (46.5) fl

Let us observe tha t the value of the first integral is non-negative. If one now accepts tha t the first normal stress coefficient ~ > 0, then 5 < 0, and if on the boundary 0f~ of the domain the value of the line integral

~0 w2 a n 48 < o,

then the second te rm in (46.5) is also non-negative. Hence, (46.5) cannot be satisfied unless w = 0 throughout f / a n d its boundary. Tha t is, the velocity fields in the Newtonian and the second order fluid are identical. 24

~6.2 Flow across a Slot

Y =

FIGURE 46.1. Two-dimensional flow across a slot.

Turning to a specific example, consider the case of a creeping plane flow across a slot - see Figure 46.1. Here, the bo t tom plate is at rest, and the top plate is either at rest or moves with a constant speed in the x-d i rec t ion . Now, on both the bo t tom and top plates in the flow domain, it is obvious tha t v - n -- 0, whether the top plate is moving or not. Since the domain is symmetric about the axis x = 0, we can set r y) = r y). Moreover, far upstream and downstream, the flow is rectilinear; t ha t is, u --+ f (y ) , v -+ 0, as x --+ 4-o0. This means tha t r is a function of y only as x --+ 4-o0, or for such a s tream function, A2r --+ g(y) far away from the slot. Using the properties of v and the the behaviour of w = A2r it is seen tha t the contributions of these two quantities to the line integral in (46.5) cancel each other out as x -+ 4-oo. In sum, the line integral in (46.5) is zero and one thus finds tha t w -- 0 throughout the flow domain. Pu t it another way, the s t ream function in the

| l u l

24See HUILGOL, R.R., SIAM J. Appl. Math., 24, 226-233 (1971).

46 The Role of the First Normal Stress Difference in Plane Creeping Flows 341

second order fluid is also biharmonic and is identical to its Newtonian counterpart in this problem.

Two separate flows may now be considered in the above configuration: the first, when the fluid is driven by the movement of the top plate and the second, when the top plate is at rest and the fluid is pushed across the slot by a pressure gradient. In either case, since the flow is symmetric about the centreline, the velocity component v = 0 along it and, thus, 0 v / 0 y = 0 on x = 0. By the continuity equation, Ou/Ox = 0 as well. More than this occurs; indeed, one can obtain the results contained in the following:

P r o b l e m 4 6 . A Using the properties tha t u(-x, y) = u(x, y), v(-x, y) = -v(x, y) in the flow

domain, show tha t both Ou/Oy and Ov/Ox are even functions of x near x = 0. Also, deduce tha t 0v/cOx = 0 along x = 0.

We shall now use the above result to derive the pressure Ps in the second order fluid with special at tention being paid to the bot tom plate. On this plate, whether the top plate is driven or the flow occcurs due to a pressure gradient, v = 0. Also, along the centreline the only non-zero component of A1 is 0u /0y . In sum, we obtain, through (44.16) tha t

1 -- . (46.7) ps(O, O) ~(~o + 4 ~ )

(0,0)

~=0.75

0.50

=-10 ~ (out~ vortex)

F I G U R E 46.2. Streamlines for pressure driven flow over a rectangular slot (W/D = 1.0, d/D--0.5).


Similarly, in the pressure driven case, 25 v -- 0 on the top plate. Thus,

1 (~)2 ps(0, d + D) -- ~ ( ~ ~ 1 + 4 ~ ~ . (46.8)

(O,d+D)

Moreover, in the pressure driven case, on the top and bo t tom plates, the y y - c o m p o n e n t s of both A12 and A2 are equal to (0u /0y ) 2. Hence, using (46.7)- (46.8) and the const i tutive relation (44.1), we obtain the following result for the difference, Pc, in the normal stresses on the two plates:

P~ - Tyu(O, d + D) - TuN(O , O) - --[pN(O, d + D) - pN(O, 0)1 o0[ ] 0y)(0,~+D) -- (0u/0Y)(0,0) . (46.9) + T 2

The determinat ion of this is aided by the result tha t PN is a constant along the centerline. 2e This follows from Problem 46.A in the manner below:

(i) If both 0 u / 0 y and Ov/Ox are even functions of x near x - 0, then in the Newtonian fluid, the shear stress Txu is an even function of x and thus, OTzu/Ox is an odd function. So, we conclude tha t this derivative is zero along x = 0 .

(ii) Because 0 v / 0 y = 0 on the centreline, we now find tha t the extra stress Suv in the Newtonian fluid is zero and the equation of motion in the y - d i r e c t i o n implies tha t 0 p N / 0 y -- 0. Tha t is, PN is a constant along the y - a x i s .

This simplifies the normal stress difference P~ in (46.9) to the following expression:

2

~1 2 - ( - ~ ) (0,0)] " (46.10)

The above result shows tha t P~ is determined by the first normal stress difference coefficient. The physical interpretat ion is tha t if the flow domain is a channel with a slot in the bo t tom plate, then the normal stress measurement at the bo t tom of the slot is not the same as tha t at the opposite point on the top plate.

Now, far ups t ream or downstream, the velocity is rectilinear with u -- u(y). Let the shear ra te associated with this, at (x, y) -- (:t:oo, d), be ~(d). Then, using NI - V~ find that

( ( 0 u / 0 y - 0 u / 0 y 4P~ (O,d+ D) (0,0) "~1 -~ - ~(d) 2 . (46.11)

In Figure 46.2, we have drawn the streamlines 27 for the pressure driven flow over a slot with the values of W / D - 1.0, d / D - 0.5. As the depth ratio d / D is

| . , , 1

25TROGDON, S.A., and JOSEPH, D.D., J. Non-Newt. Fluid Mech., 10, 185-213 (1982). 26TANNER, R.I. and PIPKIN, A.C., Trans. Soc. Rheol., 13, 471-484 (1969). 27Figures 46.2 - 46.5 are taken from TROGDON, S.A. and JOSEPH, D.D., J. Non-Newt. Fluid

Mech., 10, 185-213 (1982).

46 The Role of the Firs t Normal Stress Difference in Plane Creeping Flows 343

=0.75

0.5

0.01

C)

F I G U R E 46.3. Streamlines for pressure driven flow over a rectangular slot ( W / D = 1.0, d i D - 1.0).

increased, the flow pat tern changes to those in Figs. 46.3 and 46.4. Now, as the slot gets deeper, one would expect tha t the shear rate at the bot tom would diminish and tha t the shear rate at the top would come closer to the value far away from the slot. This should mean tha t 4Pe/N1 --, 1; tha t this is indeed the case is shown in Figure 46.5.

Finally, turning to the problem of the shear driven flow, a formula for Pe, similar to tha t derived here, may be obtained. 2s

~6.3 Radial Flow in and out of an Annulus

Since the flow is steady, (46.4) can be rewritten in terms of the material derivative:

8dw --0. (46.12) w+ dt

Integrating this, one finds tha t

w(X, t) = w(X, 0)e -t /~, (46.13)

where X is the position vector of a fluid particle at t ime t -- 0. The solution (46.13) reveals that , as t --, oo, the function w --, oo because 5 < 0. The only way

28See TANNER, R.I. and PIPKIN, A.C., Trans. Soc. Rheol., 13, 471-484 (1969). Also, s e e

LODGE, A.S., PRITCHARD, W.G. and SCOTT, L.R., IMA J. Appl. Math., 46, 39-66 (1991) for a numerical solution of the slot problem.


u = 0 . 7 5

0.5

0.01

-0 .5 .10 -J

FIGURE 46.4. Streamlines for pressure driven flow over a rectangular slot ( W / D -- 1.0, d / D -- 2.0).

to overcome this si tuation is if w(X, 0) -- 0 to start with. Hence, creeping flows where for each fluid particle, the transit t ime is infinite, w - 0 is the only possible solution. On the other hand, if a fluid element can enter a domain and leave it in a finite period of time, then it is possible to expect tha t the second order creeping flow problem may have a non-biharmonic solution. We turn to an example of this kind next. 29

Let f~ be the concentric annulus with an inner radius R1 > 0 and an outer radius R2. Let the velocity field be

c v -- - e ~ + v ( r ) e o , (46.14)

r

where c is a constant and the function v = v ( r ) has to be determined. The above velocity field is isochoric and hence satisfies the continuity equation.

Ignoring the body force and inertia in a creeping flow sit.uation, the equation satisfied by the function v, in a second order fluid, may be shown to be given by

dr 2 + -~ + "~r -- r- ~,

29HUILGOL, R.R., SIAM J. Appl. Math., 24, 226-233 (1971) exhibited this from FOSDICK, R.L. and BERNSTEIN, B., Int. J. Engng. Sci., 7, 555-569 (1969).

46 The Role of the First Normal Stress Difference in Plane Creeping Flows 345

1.0

0.9

0.8 0.7

Z 0.6

o.5

0.4

0.3

0.2

0.1

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

d ~

FIGURE 46.5. Variation of the hole pressure (pressure driven flow) with d iD for W / D = 0.5 and 1.

where ~ = c5 and, b is a constant of integration. The boundary conditions on the tangential component v are given by v(R1) = v(R2) = 0. It may be shown quite readily that

(i) If b = 0, then v(r) -- 0 and the stream function r = cO to within a constant. This function is biharmonic and is also the Newtonian solution. The physical interpretation is that the fluid motion into and out of the annulus is purely radial.

(ii) If b # O, then

f r 1 ~2/2~ v(r) -- A - r ~ + B ~ e - @, (46 .16 )

where the constants A and B may be used to satisfy the boundary conditions on v. Hence, in the latter case, the fluid will enter the annulus, acquire a tangential velocity within it and exit radially. Clearly, in the second case, we have a non-biharmonic value for the stream function and the velocity fields in the Newtonian and the second order fluids are different.

The reason behind the difference is the following: the shear stress in the second order fluid is given by Sro - b/r 2. Consequently, by putt ing b ~ 0, an extra amount of shear stress is being applied to the fluid which results in the additional motion.


47 Experiments and Theoretical Results to Delineate a Simple Fluid

In this section, we shall describe some experimental results which are typical of many viscoelastic fluids and, thus, we may take them as characterising simple fluids. Hence, it would be desirable to predict the experimental behaviour theoretically and this is the purpose of this section. In turn, the material presented here does answer the following question: Is the fluid a simple fluid or not?

7.1 Exper iments with Smal l Oscillations Supe~Tosed on Simple Shearing

In a series of experiments performed on a cone and plate rheogoniometer, 3~ where the fluid was subjected to a viscometric, simple shear flow with a small in-line oscillatory component superposed on it, measurements of the shear stress Sor and the normal stress Too were recorded. They showed tha t both of the stresses contained their viscometric counterparts along with oscillatory stresses whose frequencies equalled the frequency of oscillation. Using a fairly s tandard notation, one says tha t in these experiments the measured stresses are

Sor w, t) = T(;ym ) + e[G'll (ZYm, W) sin urt + G"ll (;Ym, W) coswt], (47.1)

where ~/ = ~/m + ew coswt (47.2)

is the rate of shear, ~m being the constant mean shear rate, c is the amplitude and w is the frequency of oscillation of the superposed motion. Also, in (47.1), T(#~) is the viscometric shear stress, G'll and G"ll are the parallel, dynamic storage and loss moduli, respectively. The latter depend on the mean shear rate ~m and the frequency of oscillation w. Similarly, one finds tha t

Too(~/, w, t) = Nl (;Ym) + e[N~ll (~m, W)sin wt + N~'ll ('~m, w) coswt].

Here, the symbols are self-explanatory. The situation where a tranverse oscillation was superposed on a viscometric

flow occurred in a Couette device, 31 where the outer cylinder was driven at a constant speed and the inner cylinder was made to oscillate. Again, shear stress measurements were made and it was found tha t the shear stress Sre could be expressed as

Sro(@,w,t) = T(@m)+ e[G'• w) sin ~t + G"x(@m, w) coswt ], (47.4)

Following the practice in linear viscoelasticity, we may associate with each loss modulus G", a dynamic viscosity W' G"/w. Thus, we are led to WII as well as W'

_L �9

3~ H.C., Rheol. Acta, 5,215-221; 222-227 (1966); 7, 202-209 (1968). See also BOOIJ, H.C. and PALMEN, J.H.M., Rheol. Acta, 21,376-387 (1982).

31SIMMONS, J.M., Rheol. Acta, 7, 184-188 (1968).

47 Experiments and Theoretical Results to Delineate a Simple Fluid 347

rJ~

0

I0'

0 0

0

Shear viscosity Tl(u T = 25~

10 o

Orthogonal shear rates (sec -~) �9 0 " 1 7 4 �9 78.5 �9 350

78.5 174 350 I I I

10 ~ 101 102 103

Shear rate, Frequency Ym (secl), r (rad-sec -~)

FIGURE 47.1. Effects of steady orthogonal shear rate "~,~ =, on the real part, ~/'~, of the complex dynamic viscosity for a 4.4% polyisobutylene (MW of 108) in cetane sample at 25~

Turning to specific experimental results, we have exhibited, in Figure 47.1, the lines of the dynamic viscosity r/~ against the frequency w, while in Figure 47.2, the curves of G~II against w are shown. The first graph 32 seems to confirm that , for a fixed mean shear rate ~m, the dynamic viscosity is independent of "Ym in the ultrasonic limit, i.e., as w --~ oo:

~k (r ~) ~ ~k (0, ~). (47.5)

Note that we may simply put 7/~_(0,w) -- if(w), which is the dynamic viscosity from the linear theory.

Secondly, Figure 47.2 implies 33 that the storage modulus has the following behaviour as w --* oo:

OCi(gm, ~) ~ OCi (0, w) (47.6) Ow Ow "

That is to say, the lines of G~_ (~m, w) are parallel to G~ (0,w) for large w. Note that we may put G~_ (0, w ) - G'(w) of the linear theory again.

Below, we shall use the theory of nearly viscometric flows 34 to seek some theoretical justifications for the above experimental results. In turn, our arguments will show when a result is valid for all fluids and when they hold for certain special fluids only.

32This is taken from TANNER, R.I. and WILLIAMS, G., Rheol. Acta, 10, 508-538 (1971), whose experimental apparatus provided accurate measurements of these dynamic moduli.

33SIMMONS, J.M., Rheol. Acta, 7, 184- 188 (1968). a4PIPKIN, A.C. and OWEN, D.R., Phys. Fluids, 10,836-843 (1967).

3000

1000

300

100

30

3 10 30 100 300 1000

FIGURE 47.2. Effect of steady orthogonal shearing of rate ~,, on G'I for 8.54% polyisobutylene in cetane at 25~ and 0.08 oscillatory shear amplitude. Legend: 1: ~m -- 0, 2" "Ym -- 6.36s -I , 3: "~m -- 12-7s-1, 4: '~m -- 25-4s-I, 5: A/m -- 50.9S -1, 6: '~n -- i02s-1, 7: ' ~m --- 2 0 4 S - 1 , 8: ' ~m = 4 0 7 S - 1 .

7.2 Infinitesimal Strain Super'posed on a Large Strain History

We shall suppose tha t we are given a velocity field v ~ (x, t) and we calculate the relative strain tensor ~ ) corresponding to this. If a small per turbat ion be superposed, the new relative strain Ct (~-) can be decomposed as

=o + (47.7)

where Et(T) is the per turbat ion term. In this section, for convenience, we shall adopt the notation C(t , T), ~ ~-), E(t , T) for the above three tensors. The constitutive relation for the extra stress tensor, which is given by

< (47s) S(t)

now takes on the form:

~ ' (C( t , T)) ---- ~ ( ~ T)) 4- 5J~(~ C(t, T)IE(t, T)) 4- O(~), (47.9)

if the norm of the per turbat ion is of order e and this is small. The second term in the above expression is a linear functional of E(t, T) and depends on the base history ~ T). The linearity in E(t , T) is usually expressed as an integral:

(o , 5Jzq(~ ~ijkl C(t ,T) T)Ekl(t,T) dT. (47.10) o o


The above integral shows tha t the per turbat ion stresses depend on the base history and the past in a nonlinear manner through the kernels ~ijkz.

Now, not all per turbat ion histories of small norm can occur, because the new strain history C must have a determinant equal to 1 again. We observe tha t

det C(t , 7.) - det( ~ C(t , 7.) + E(t , 7.))

= de t (~ 7.)) det(1 +o C - l ( t , 7-)E(t, 7))

= det(1 +~ C -1 (t, 7-)E(t, 7.)),

(47.11)

with the latter following from the fact tha t det o C(t , 7.) -- 1. Since E(t , 7.) has small norm, we may expand the last determinant in the expression above and finally arrive at the restriction on E(t , 7.) to be admissible: 35

tr( ~ C - l(t , 7.)E(t, 7.)) -- 0. (47.12)

We shall assume tha t E(t , T) obeys this restriction in what follows.

7.3 Perturbation due to Small Displacements

In w an expression for the addition to a base history when the per turbat ion arises from a small displacement has been derived. Let the displacement gradient be split into its symmetric and skew-symmetric parts:

(47.13) u~,~ = E~r + W~.

Using this, (5.25) can be recast as follows: 36

E(t , 7.) -- - u ( t ) �9 V ~ C(t , 7.) + [W(t) ~ C(t , 7.) - ~ C(t , 7.)W(t)]

-[~ C(t, 7.)E(t) + E(t)~ C(t, 7.) - 2~ (t, 7.)E(7.)~ 7.)]. (47.14)

We shall use this expression to obtain explicit formulae for the stresses arising from the first two terms. If ~ is the stress corresponding to the base history ~ then it follows from the definition of the linear functional in (47.9) tha t

v o s _ )Iv~ (47.15)

and thus the first te rm contributes a stress perturbation"

�9 c ( t , )lv ~ c ( t , (47.16)

since u(t) is not a function of 7.. Next, if the original motion is one of constant stretch history, then we know from (9.23) tha t

- c ( t , dt

(47.17)

35pIPKIN, A.C. and OWEN, D.R., Phys. Fluids, 10,836-843 (1967). 36pIPKIN, A.C., Trans. Soc. Rheol., 12, 397-408 (1968).


where Z(t) is a skew-symmetric tensor. Thus, if ~ is the stress corresponding to a constant stretch history ~ T), then

doS dt - 5~ '(~ T)IZ(t)~ T) _o C(t, T)Z(t))

(47.18) = Z ( t ) o s ( t ) _ o S(t)Z(t),

with the latter following from (34.4). Hence, the second term in (47.14) produces the following stress: 37

W ( t ) o s _o SW(t ) . (47.19)

The contributions of the other two terms do not have such simple representations, although specific forms can be exhibited in three special instances. We turn to them next.

7.~ Per tu rba t ion about a S ta te of Res t

The extra stress arising from a perturbation about the state of rest has been determined earlier in ( B 3 . 1 1 ) - - ( B 3 . 1 2 ) . Here, we shall rederive it by using the general approach developed here. Now, if the base motion is one of a state of rest, then ~ T) ---- 1 and (47.14) reduces to the simple form

E(t, T) -- 2E(T) -- 2E(t), (47.20)

which is precisely (5.9) obtained earlier. The contribution to the stress tensor from the state of rest is zero and, in order to obtain the stresses due to the perturbation, one has to find the most general, fourth order, isotropic tensor. This is given by 3s

ct6,j6kl + ~(6,k6jZ + 5,,Sjk) + 7(5,k6jl -- 6,16jk). (47.21)

Using the above, absorbing the identity term into the pressure term and using the symmetry of the tensor E(t, T), one obtains

s~j(t)- .,~k~(1,r)Ek~(t,~) dr-- , ( t - r )E ,~( t , r ) dr. (47.22)

Thus, in a perturbation about the state of rest, the stress in the fluid is

s(t) - 2 . ( s ) [ E ( t - s ) - E(t)] ds, (47.23)

derived earlier in ( B 3 . 1 1 ) - (B3.12).

7.5 Per tu rba t ion about S imp le Shear and Ex t ens iona l M o t i o n s

In the case when the velocity field is that in simple shear, i.e., x - ~y, y - ~ - 0, or that in the extensional flow, i.e., x -- - k x / 2 , y -- - k y / 2 , ~ -- kz, the respective

37HUILGOL, R.R., J. Non-Newt. Fluid Mech., 5, 219-231 (1979). 3SSee, for example, JEFFREYS, H., Cartesian Tensors, Cambridge Univ. Press, 1969.


strain histories are independent of any spatial coordinates. Hence, we find tha t V ~ C(t , T) = 0. Consequently, (47.14) now leads to

E( t , T) = [W(t) ~ C(t , T) --~ C(t , T)W(t)]

- [~ C(t , T)E(t) + E(t) ~ C(t , T) -- 2~ T (t, T)E(T)~ T)]. (47.24)

Following P IPKIN and OWEN, 39 we call a flow a nearly viscometric flow, if it arises from the superpositions of small perturbat ions on a viscometric flow, and name it a nearly extensional flow if it originates from the superpositions of small per turbat ions on a simple extension, a~

7. 6 T h e N u m b e r o f I n d e p e n d e n t L i n e a r F u n c t i o n a l s

Clearly, there are 81 linear functionals in the set J~jkl- Because of the symmetry of the stress tensor and the strain tensor, we may put

(47.25)

Thus, there are only 36 independent ones left. These can be reduced further if one is willing to assume tha t the extra stress tensor has zero trace; then, one obtains

~ k l = 0. (47.26)

Next, some of the linear functionals turn out to be zero because of the inherent symmetry in the base motion. For example, consider the simple shear flow: x = ~y, y -- ~ -- 0. Here, the following three actions must leave the linear functionals invariant: changing the z - a x i s into its mirror image; or, changing the x - a x i s into its mirror image and replacing ~ by - ~ simultaneously; and finally, changing the y - a x i s into its mirror image and replacing ~ by - ~ simultaneously. While these changes leave the base motion unaffected, they alter the signs of the components of the per turbat ion tensor. Thus, as shown by PIPKIN and OWEN, 41 those linear functionals with an odd number of suscripts equal to 3 are all zero.

In a similar fashion, if the extensional flow is defined through x = - k x / 2 , ~1 - - @ / 2 , ~ = ~z, then rotat ing the (x, y ) - axes through any angle about the z -ax i s , or reflecting them in any plane containing the z -ax i s , or replacing the z - a x i s by its mirror image leave the flow invariant. Hence, some linear functionals turn out to be zero in this case as well. 42

A second set of relations between the linear functionals arises because a nearly viscometric flow may in fact be obtained by a small variation to the original shear rate or a slight rotat ion of the axes. These alterations relate the linear functionals to the viscometric material functions and their derivatives and the resulting conditions are called the compatibility conditions. A similar procedure works for the nearly extensional flows. The details are to be found in the references.

39pIPKIN, A. C. and OWEN, D. R., Phys. Fluids, 10, 836 - 843 (1967). 4~ HUILGOL, R.R., J. Non-Newt. Fluid Mech., 5, 219 - 231 (1979). 41pIPKIN, A. C. and OWEN, D. R., Phys. Fluids, 10, 836 - 843 (1967). 42See HUILGOL, R.R., J. Non-Newt. Fluid Mech., 5, 219 - 231 (1979), for full details.


7. 7 S o m e U n i v e r s a l R e l a t i o n s

A number of universal relations may be derived from the theory of nearly viscometric flows. For example, a3

lim u)--4 0

lim u)---~ 0

lim u)- -a0

- - l i m ~ " - nl(7 , o.,) = n ( # m ) w--*0

Gii(Tm'W) -- lim rlil(~m,w ) -- dT(;'/m) 0.) w--* O d'4f m

Nl ' l l ( ~ m , W) d N l ( '~m)

o) d ~ m '

lim N2'll (X/m' w) -- dN2.(x/m) . (47.27) w --~ O a) d ~ m

These are easily explained because in the first set, the shear rate of the base motion is not affected by the orthogonal oscillation, whereas in the other three, the in-line oscillations change the shear rate.

A result that is of considerable importance is the one which relates the storage modulus of the linear theory with the first normal stress difference. This isaa

= lim Nl(X/) lim _ ~0 (47.28) u)--*0 ~--,0 ,~2 w2 '

where ~0 is the first normal stress coefficient of the second order fluid. Indeed, this result is the one which determines whether the material being tested is a simple fluid or not.

Despite the simplicity of the predictions in (47.27)-(47.28), they have never been verified because of the experimental problems associated with measurements at low frequencies. On the other hand, ultrasonic frequency details are much easier to obtain and, while this is the case, they are not universally valid. We turn to this situation next.

7.8 Ul t rason ic Fea tures

A number of results hold for fluids of the K-BKZ type, which are not universally valid, i.e., they cannot be predicted by using other integral models. An example is the following: using the constitutive relation for K-BKZ fluids and assuming that the relaxation modulus C(s) is such that (~(0) is bounded, it can be proved that 45

lim [W2rl'x (Tin, w)] ~----+ o o

= lim0~-~oo [w2rfll (Tin, w)]

= lim~__,oo[w2~'(w)] = - ( ~ ( 0 ) . (47.29)

43For the derivations, see by PIPKIN, A.C., Trans. Soc. Rheol., 12,397-408 (1968) and HUIL- GOL, R.R., Continuum Mechanics o f Viscoelastic Liquids, Hindustan Pub. Corp., New Delhi, 1975, pp. 210-211.

44COLEMAN, B.D. and MARKOVITZ, H., J. Appl. Phys., 35, 1-9 (1964). 45BERNSTEIN, B. and HUILGOL, R.R., Trans. Soc. Rheol., 15, 731-739 (1971)'

This means tha t for large values of w the lines of r/' r/' , , • fill' and (w) merge. The curves in Figure 47.1 seem to support this contention, i.e., the fluid being tested is tha t of the K-BKZ type. However, it is not possible to ascertain from Figure 47.1 tha t the slope of the graph of In r/'x when plotted against In w is - 2 , which is clearly the content of (47.29). Consequently, it is not clear whether 0(0) is bounded. Thus, it would be desirable to know the form taken by (47.29) if one cannot assume tha t (~(0) is bounded.

Indeed, it may be shown tha t 46

lim [wPrf_L (;Ym, W)] ta)---+ C ~

--lin~oo[wPffll(,~m,W)]

= lin~__+oo[wPrl'(w)] = C, (47.30)

where C is a constant. Here, the index p -- m + 2 is related to the behaviour of G:(s) near s - 0 through the integrability constraint on G(s), which requires tha t G(s) be of order O(sm), where - 1 < m < 0. Tha t is, if m - 0, we recover (47.29), and (47.30) otherwise. A full list of other results which are valid for fluid of the K-BKZ type is also available. 47 Interestingly enough, it can be shown that , for the Doi-Edwards fluid, the ultrasonic limit 4s in (47.30) holds if p - 3/2 or m - - 1 / 2 .

Turning to results tha t affect the storage moduli, it can be proved tha t in K-BKZ fluids

lim [w 30G[l(@m'm) ] w--+oo ~r

-- ~-~oolim [w 3 0G~ (~/m' w)'

= ~-~limoo [wa0G'(w)] - 2 ( ~ ( 0 ) ' 0 w (47.31)

provided (~(0) is bounded. This prediction is in agreement with the experimental results in Figure 47.2 because the latter shows tha t the lines of G'L (~m, w) are parallel to one another for large values of w.

There are a couple of relations which govern the dynamic, normal stress moduli and they are also available in the literature.

7.9 Predic t ions and Per fo rma n ce

Thus, experiments under ultrasonic frequencies seem to indicate tha t the fluids tha t have been tested under laboratory conditions perform as simple fluids, albeit of a special type, which is yet another reason for the s tudy of viscoelastic fluids under this broad canvas.

Of course, in Chapter 4, we have shown tha t fluid models with microstructures predict similar results. Thus, in a given problem, it is quite often a mat ter of convenience on the selection of a model to s tudy the dynamics of a given flow.

46BERNSTEIN, B. and HUILGOL, R.R., Trans. Soc. Rheol., 18,583-590 (1974). 47BERNSTEIN, B. HUILGOL, R.R. and TANNER, R.I., Int. J. Engng. Sci., 10, 263-272

(1972). 4SBERNSTEIN, B. and HUILGOL, R.R., Int. J. Nonlinear Mech., 27, 299-308 (1992).


48

48.1

Cessation of One History and Continuation with Another History

A New Viscometer

Flow in

T

r

r

II

Data acquisition

Vacuum

FIGURE 48.1. A novel viscometer in which a fluid element is subjected to an extensional flow after it has been sheared in the Couette device.

A viscometer which is capable of measuring both the shear viscosity and the first normal stress difference in simple shear as well as the quasi-steady elongational measurements for a large variety of dilute and semi-dilute polymer solutions has been discovered recently. 49 Nun6z Figure 48.1 shows the layout of the instrument.

The arrangement consists of a basic Couette concentric cylinder viscometer in which the torque is measured as a function of the shear rate. The fluid enters the viscometer through the inlet I at a constant flow rate; the rate of shear in the gap is essentially constant because the gap between the cylinders is small when compared with the radius of the inner cylinder. After being sheared, the fluid exits through a small orifice O at the bot tom of the gap. The pressure of the fluid is monitered by the three transducers T as shown. Once a constant flow rate has been established, a suction is applied to the bot tom orifice of diameter D, through a second orifice of diameter d, which is separated from the first by a variable distance h. Both orifices are aligned and as the bot tom plate moves, a filament is formed between both orifices. The tensile stress which causes the extension of the filament is measured by the transducers mentioned above. Finally, the rate of elongation is obtained through a video camera focussed on the filament. Using the information gathered in the manner just described, the tensile stress growth coefficient •+(k, ~, t), which

49See GAMA, J., NUNI~Z, F., ZENIT, R., VON-ZIEGLER, A. and MENA, B., Proc. XIth Int. Cong. Rheol, Vol. 2, 917-919, 1992. Patent pending.

48 Cessation of One History and Continuation with Another History 355

depends on the extensional rate k, the viscometric shear rate ~ and the t ime t of the extensional flow, is calculated.

50

~, 40

o 31)

~ 20

10 ,~ = 6.25 s q.

n o

= 3.14 s-!

3i = 1.26 s "l

0 2 4 6 8 10 12 14 16 18

Axial distance (cm)

FIGURE 48.2. Tensile stress as a function of the axial distance at different pre-shearing rates. Q - 40 ml/min.

r ~ =i 4

~ ' 3 t,-.d

2 =a

+

a" 0

nO

~ - 30 RPM

f~ = 60 RPM

- 1 2 RPM

0 2 4 6 8 10 12 14 16 18

Axial distance (cm)

FIGURE 48.3. Tensile stress growth coefficient versus axial distance at diffferent pre-shearing rates. Q - 40 ml/min.

In Figure 48.2, the values of the tensile stress as a function of the axial distance are shown for both the unsheared and sheared samples of the commercial t rans- mission oil Roshfram's 30. These stresses have been converted to the tensile stress growth coefficient in Figure 48.3. The experiments seem to reveal tha t the effect of superimposing a simple shearing motion upon the system described above is to modify the elongational properties of the fluid as follows:

, the velocity increases along the filament;

�9 the absolute tensile stress increases considerably;

* the values of the tensile stress growth co mci t t) increase with the rotational speed of the Couette viscometer or the shear rate;

�9 the values of the tensile stress growth co mr t) increase along the axis of elongation or with the t ime of extension.

In this section, we seek a theoretical explanation 5~ of the above experimental results through the use of the Oldroyd-B fluid and a fluid of the K-BKZ type. In order to employ these models, it is necessary to understand how a simple shear flow changes into an extensional flow. 51 We turn to this next.

48.2 Smooth Transition from Cessation of Shear Flow to the Initiation of Extensional Flow

We assume that a simple shearing motion occurs over - o o < T _~ 0 and tha t a simple extensional flow follows after that , i.e., over 0 _~ T _~ t. Now, we need to calculate the relative strain history C t ( r ) , - o o < T < t. Let F(7-) be the deformation gradient at t ime T and F(t) be tha t at t ime t. These are calculated with respect to a fixed configuration at t ime 0. The relative strain history is given by Ct(T), --OO < T < t. Using (4.16), it follows tha t

Ct(T) - - ( F ( t ) - I ) T c ( T ) F ( t ) - I . (48.1)

Here, C(T) -- F(T)TF(T) is calculated with respect to the configuration at t ime 0. Let us assume the shearing motion to be the viscometric flow given by

z = o , 9= z, (48.2)

upto t ime 0. Then, we have

1 -~- Af2T2 ~/" O) C ( T ) - - '~T 1 0

0 0 1 , - o o < T < 0. (48.3)

Let the simple extensional flow after t = 0 be

k 9 - - = y , z z

(48.4)

Then, for 0 < 7- < t,

( e - ~ / 2 0 0 / F(T) - 0 e -gr/2 0 .

0 0 e ~ (48.5)

5~ R.R., MENA, B. and PHAN-THIEN, N., preprint. 51PETRIE, C.J.S., J. Non-Newt. Fluid Mech., 4, 137-159 (1978).

48 Cessation of One History and Continuation with Another History 357

This leads to 0 0) C(T) = 0 e -~" 0 , 0 < T < t. (48.6)

0 0 e 2kr

We note tha t the transition from the viscometric to the extensional flow is smooth in the sense tha t the strain history is the identity tensor as T --, 0- or T --, 0 +.

8.3 The Oldroyd-B Fluid In order to obtain a qualitative result which is in accord with the experimental data in Figure 48.3, we assume the constitutive equation for the extra stress tensor S to be tha t of the Oldroyd-B fluid:

)~2 A1 ~1 -- )~2 f_t s - (48.7)

where rl0 is the viscosity, )~1 is the relaxation time, A2 the retardation time, A1 the first Rivlin-Ericksen tensor and Ct(T) -1 is the inverse of Ct(T). Thie above equation may be separated into three parts as follows:

0 ~2 ~1--)~2-t/A1 [ / A 1 ] S(t) = ~/o~-~lAl+r/0 )h 3 e F(t) e "/ C ( T ) - d ~ - F ( t ) T

--OO t

+ rl0 ~13 e F(t) e r/ C(T) 1 0

Thus, at t ime t, the first and the last terms in the above equation give rise to the t ime-dependent stress due to the extensional flow. Because the constitutive equation is linear, this extensional stress tensor is unaffected by the previous viscometric flow history. It is known from w that this extensional stress tensor is diagonal with S=~ (t) : Sy E (t) # S~z (t).

The second term in (48.8) yields the changes in the viscometric stresses due to the subsequent motion. It is easily shown tha t these are given by the matrix

~1 -- ~2 -t/)~l ( Ale- rl~ ~ )h

~ A 2 e - ~ t 0 ) + o .

" )~1 e2~t (48.9)

Now, (48.9) shows tha t the extensional stresses are augmented by an amount due to the modification of the viscometric stresses. Tha t is, in the final analysis, when the extensional flow has been proceeding over a t ime interval t, one finds tha t

Tzz(t) Txz(t) S '~1 -- ~2 --t/~ (e2~t -~t) (48.10) -- -- SZZ (t) -- SEz (t) "4- ?70 ,~21 e 1 -- e .

Now, for the Oldroyd-B fluid, the first normal stress difference N1 is given by

N1 : 2r/0(~X1 - - .~2)"y 2. (48.11)


Thus, we may recast (48.10) in the form

Tz~(t) - T==(t) - sE~(t) - s=E(t) + ~-~12e-t/X~(e2~t - e-~t), 2A1

(48.12)

where ~I/1 - - Nil@ 2 is the first normal stress difference coefficient. The tensile stress growth coefficient r/+(k, ~/, t) is easily obtained from the above

by dividing the stress difference by k. Since the first normal stress difference coefficient lI/1 :> 0 , (48.12) shows clearly that pre-shearing increases the tensile stress growth coefficient; that is to say, pre-shearing increases the "extensional viscosity"; secondly, it is easily seen that

d-t/Al(e2kt e-kt) e-t/)~l [(2~ 1)e2kt.~-(~-~-71)e-kt ] (48.13)

Thus, the tensile stress growth coefficient increases with time or along the axis of elongation if ( 2 k - ~'~1) > 0. Now, from the experimental results, 52 we note that A ~ 1 sec, while k ~ 10/sec. Although the prediction of (48.13) by itself is in agreement with the experimental results in Figure 48.3, these values are inadmissible for the Oldroyd-S fluid, since S~z(t ) - SEx(t) becomes unbounded. Secondly, the experimental results show that the tensile stress growth coefficient is an increasing function of ~, the viscometric shear rate. The Oldroyd-B model used here cannot predict this dependence because N1 is quadratic in the shear rate. Thus, for both reasons, a more complicated model is called for.

~8.~ P r e d i c t i o n s o f a F lu id o f the K - B K Z Type

In order to examine whether the increase in ~+(k,~,t) with ~ is predicted by another model, we examine a constitutive relation of the form

S(t) = _ l ( t - T, I_I , I1 )Ct (T) -1 21- ~[Zl(t- T, I_ I , I1 )Ct (T ) dT, (48.14) o o

where #-1, ~1 are functions of t - T and the invariants:

I_ 1 : t r C t ( - r ) - 1 ' (48.15)

I1 = tr Ct(T).

In the above, tr denotes the trace operator. It is known that in simple shear, the normal stress coefficients are given by

0

1 = / ( ~ - I - -~I) T2 dT, - - 0 0

0

~I/2 = / ~1 T2 dr, (48.16) - - O O

52MENA, B., MORALES-PATINO, A., MOTTA, B., SERRANIA, F. and VON-ZIEGLER, A., Applied Mechanics in the Americas, Proceedings PACAM IV, Vol.I, pp.529-534, Argentina 1995.

49 Linearised Stability and Bifurcation 359

when the two invariants I -1 - - / 1 -- 3 + '~2T2. Because II/1 ~> 0 and ~2 < 0, one may assume tha t #_ 1 >- 0 and #1 -< 0 for k I - / 1 = 3 + ~2T2.

Turning now to the stress at t ime t due to the initiation of an elongational flow after simple shearing, one finds that the kinematics described above lead to the following two separate integrals:

Szz(t) -- Sxx(t) - f~ [~-1 ( e2kt - e-k t ) -1- #1 (e-2~t -- e~t( 1+ ' 272 ) )1 d7

+ ~ot [~_l (e2k(t-r) -- e-g(t-r)) zl- ~tl (e-2g(t-r) -- eg(t-r))] dT.

Here, in the first integral, the invariants are

(48.17)

I-1 "-" e 2kt -t- e-kt(2 -t- ~2T2), (48.18)

I1 -- e -2kt + e~t(2 + ~2T2)-

Let us rewrite the terms in the first integral in the form

(e2et -e-k t ) k ~ t 1 aT ~- (e-2kt--e~t)fOc~.ldT

-- ekt ~ 2 fOoo ~tl T2 dr.

(48

Now, if 0 < t < < 1, then the invariants in (48.18) do not differ too greatly from their viscometric counterparts and thus one may assume tha t their values are approximately given by I_ 1 = I1 = 3 -t-~2T2. Accepting this to be the case and recalling tha t /*-1 >- 0 and /'1 <- 0, we find tha t each component of the above expression (48.19) is positive and an increasing function of the time t. Moreover, because of the presence of ~2, the above expression is an increasing function of the shear rate.

Next, it is interesting to note tha t the contribution from the second integral to the stress difference in (48.17) does not depend on the viscometric flow. This is because, over the t ime interval 0 _< T _< t, the invariants are

I_1 : e 2k(t-r) -t- 2e -~(t-r) ' (48.20)

I1 = e -2~(t-r) + 2e ~(t-r).

In summary, we see tha t chosen model predicts tha t the tensile stress growth coefficient v/+ (k, ~, t), in an extensional flow after simple shearing, depends on k and is an increasing function of both ~/and of the length along the axis of elongation.

49 Linearised Stability and Bifurcation

In this section, we examine a number of problems which are interconnected. These are the stability of a special class of flows of the simple fluid or its special cases


using linearised stability theory. Although two exhaustive reviews of this topic have been published recently, 53 it is vital to understand the major ideas so tha t we can consider the related topics of bifurcation from a given motion of a viscoelastic fluid and its connection with stability, and compare the theoretical predictions with experimental observations where they exist. In addition, the topics considered here have a bearing on the qualitative dynamics of flows which are explicated in the next section.

~ 9.1 Stabi l i ty o f the Res t S ta te

We shall begin by examining the stability of the rest state using linearised stability theory. 54 Let the domain ~t of the flow be a bounded set with a sufficiently smooth boundary O~t and the velocity field be given by

v(~ , t) = ~ ( ~ ) ~ - ~ , v . v = o, (x, t) e a • [0, oo), (49.1)

v = 0, (~, t) e 0 n • [0, oo).

Assuming the pressure field to be given by

p(x, t ) = ~ ( x ) e -at, (49.2)

the linearised stability of the rest s tate is analysed by studying the equations of motion

- p ~ , + V / ~ - V - S L , (49.3)

where SL is the linearised stress, i.e., the extra stress S is linear in the amplitude ~, of the velocity field.

Using (4 .4) - (4 .5) , it is easily shown that

d ~ s s C t ( t - s) -- - F t ( t - s ) T A , ( t - s ) F t ( t - s), (49.4)

where A~ ( t - s ) is the first Rivlin-Ericksen tensor evaluated at t ime ( t - s ) from the velocity field. This is linear in -~, of course, whereas the relative deformation gradient consists of the identity tensor and other terms which are linear and nonlinear in ~,. Hence, to the first order in ~,, we have

d ~ s s C t ( t - s) ~ - A 1 ( t - s). (49.5)

Using the constitutive relation (B3.10) of finite linear viscoelasticity and (B3.12), we find tha t

SL = # ( s )Ct ( t -- s) ds ~ C(s)A1 ( t - s) ds. (49.6)

53See LARSON, R.G., Rheol. Acta, 31, 213-263 (1992) for a wide survey, and SHAQFEH, E.S.G., Ann. Rev. Fluid Mech., 28, 129-185 (1996) for elastic effects in viscometric flows.

54The material in this sub-section is taken from JOSEPH, D.D., Arch. Rational Mech. Anal., 56, 99-157 (1974); 75, 251-256 (1981). For an application of these ideas to fractional derivative models, see PALADE, L.I., HUILGOL, R.R., ATTANt~, P, and MENA, B., J. Rheol. (submitted).

This relation leads to the boundary value problem

- m y + v p - k(o)A ,

k(o) - f o (49.7)

where ~ is zero on the boundary. This boundary value problem leads to an eigenvalue problem with the eigenvalues A1, A2,---, meeting

0 < )~1 -~ "~2 _ ~ ' ' ' , (49.8)

where An --* oo as n --. oo. Moreover, for any given eigenvalue A, a is the root of the equation

a = Ak(a), (49.9)

where, without loss of generality, we have put p = 1. Now, if Na > 0, the rest state is stable and not otherwise. To examine this situation further, let a - ~ + iw, where

and w are real. Then, ~ and ~ satisfy

OO

A / G(s)e ~s cosws ds

0

A G(s)e~" sin ws ds 0

--'r

(49.10)

We observe immediately that

(i) If w -- 0 and G(s) > 0, then (49.10)1 cannot have a root ~ < 0. A glance at (49.1) now shows that all non-oscillatory disturbances must decay with time;

(ii) If w # 0, and G(s) > 0 is a monotonically decreasing function of s, then again the equation (49.10)1 cannot have a root ~ < 0. Hence, even the oscillatory disturbances go to zero as t --. oo.

In conclusion, it follows that the rest state of a simple fluid is stable 55 when the function space of strain histories is the Hilbert space based on the L2 norm (B3.3).

Turning to the order fluids ( B 3 . 1 6 ) - (B3.19), we note from (5.13) that the Rivlin-Ericksen tensors obey

An = ~--~A1, n >__ 2. (49.11)

Hence, the linearised stresses will be given by

S(L 1) = r/oA1 ,

S(L n ) -- S(Ln- 1) _~_ (--1) n -1

1)! n > 2 . (49.12)

55See CRAIK, A.D.D., J. Fluid Mech., 33, 33-38 (1968).

The new constants r are related to those appearing in (B3 .17 ) - (B3 .19 ) of course. For instance,

- r = /3 , r = 2c~2, (49.13)

and so on. More important ly, because of the special relation between the Rivlin- Ericksen tensors (49.11), it follows tha t the Cn are defined in terms of the moments of G(s), i.e.,

r io- a(s) ds,

n _> 2. (49.14)

A second way of deriving (49.14) is to use the fact tha t the velocity field (49.1) imposes the following relation between the tensors A l ( t - s) and A1 (t):

A l ( t - s) ---- A1 (t)e aS. (49.15)

If this is used in (49.5) and the exponential function of s is approximated by a polynomial of desired order, then (49.14) follows. Be tha t as it may, the constants ~0 and Cn, n >__ 2, are all positive because G(s) :> O.

Thus, for a fluid of order n, the function k(a) in (49.9) is

n ~)j(Tj- 1

k((T) - - j 1 E ( j __ 1)!' r -- ~0, (49.16) .=

which leads to the following polynomial equation for a

n ~)j(:Tj- 1 (49.17) ~ ( j - 1)!

j = l

If n -- 1, i.e., the fluid is Newtonian, the root of this equation is given by a = Ar It is well known tha t the eigenvalue A can take on all positive values; indeed by changing the shape and size of the domain ~, one can ensure tha t A --. 0 + and /o r A --, oo. Hence, for all containers, the rest s ta te of the Newtonian fluid is stable since ~ :> 0 always.

If n -- 2, the second order fluid results and here

~(])1 a - (49.18)

(1 - Ar )"

Thus, given the material functions r and r one can find a domain ~ and an eigenvalue A such tha t ~ a < 0. Hence, the rest s ta te of the second order fluid is unstable.

If n = 3, a quadrat ic equation for a is obtained. This is

~(/)3 (72 .~_ ()~)2 -- 1)a +/~)1 - - 0. (49.19) 2

The sum of the roots a l and a2 are such tha t

--(0" 1 -~ 0"2) -- 2 ~ 2 / ~ 3. (49.20)

The right side being positive, one finds that the real parts of both a l and a2 cannot be positive. Hence, at least one is negative which means that the rest state of the third order fluid is unstable as well. Indeed, for any fluid of order n _ 2, one finds from (49.17) that the roots a l , - - - , a n - 1 must satisfy

--((71 -[-' '" ~" ~n-1) -- ( n - 1)r 1

err, (49.21)

Since the right side is positive, it follows that at least one of the roots must have a negative real part and that the rest state of the nth order fluid is unstable. This is an important result showing that the stability properties of the order fluids are not improved by the addition of third and higher order terms; that is, the instability arises at the second order level and persists. On the other hand, the integral model (49.6) leads to a stable situation as seen earlier.

The reason for this "anamoly" is the following: in linearised stability analysis, the amplitude of the spatial disturbance is assumed to be small while the frequency need not be so. Thus, irrespective of the value of the frequency denoted by w in (49.10), the integral model leads to stability because the system (49.10) does not possess a root ~ a < 0, whereas by truncating the exponential function e ~s the opposite results.

P r o b l e m 49 .A

(i) Examine the stability of the rest state of the 1-order, 1-integral fluid

f0 ~176 S - aA1 + G ( s ) A I ( t - s) ds. (49.A1)

(ii) Let the base motion be rigid, i.e., the velocity field has the form v -- w x x, where w is the constant angular velocity. Examine the stability of this motion for the fluid model above.

49.2 Linearised Stability of Fully Developed Flows To begin, let v~ be a dynamically feasible flow in a given fluid. Suppose that one superposes on it a velocity field of the form

fi(x, t) -- f(x) exp[ik �9 x - at], (49.22)

where f may be complex valued, although only the real part has any physical meaning. The constant vector k is real and usually taken to be positive, i.e., each component of k is positive; and, finally, a is a complex number. Clearly, if ~ a > 0, then fi --+ 0 as t --~ oo ensuring that the base motion v ~ is stable; if ~ a < 0, then fi blows up as t --, oo and the original flow is unstable; and if ~ a = 0, then we have a time-periodic motion superposed on v ~ which is, by definition, unstable; and, finally if a - 0, we have a steady motion superimposed on v ~ which is, again, unstable.

In the discussion of flow instabilities, two non-dimensional parameters are important. They are the Deborah number De and the Weissenberg number W i . Here, the usual definitions 56 of these numbers are as follows:

1. De = Characteristic time of a f luid/Characterist ic time of a process.

2. Wi = First normal stress difference/Shear stress in a viscometric flow.

9.3 Torsional Flow of the Oldroyd-B Fluid

To examine the application of the above ideas, we shall look in some detail at the torsional flow between two parallel, infinitely wide disks of the Oldroyd-B fluid and the stability of the base flow. 57 To begin, let the two, parallel coaxial disks be at a distance d apart and let the bottom disk rotate with an angular velocity ~0 and the top one with ~1, which may be in the same or opposite sense to that of ~0. Let r/0 be the viscosity, A~ and Au be the relaxation and retardation times, respectively, of the fluid. Moreover, let ~" and S, respectively, be the velocity and extra stress in physical units, along with ~ and ~ being the radial and axial coordinates. Then, by scaling in the following manner:

-- rd, ~ -- zd,

~'~1 --0t~'~0, --OO < O~ < OO,

- f l 0 v , S = ~0f~0S,

De = ~0A1, 1 - f l - - A2/A1, 0 _ < ~ 1 ,

(49.23)

the constitutive equation for the Oldroyd-B fluid (34.108) takes the form

S d- D e ( S - L S - S L T) = A 1 d - ( 1 - ~ ) D e ( / ~ l - L A 1 - A 1 L T ) , (49.24)

where the superposed dot denotes the material derivative. Note that in defining the Deborah number De, the characteristic time of the fluid is taken to be its relaxation time and that of the process is the reciprocal of the angular velocity of the bottom disk.

In (49.24), L is the velocity gradient corresponding to v and A1 is calculated from L. Incidentally, we note that the Newtonian limit can be obtained from (49.24) in two ways: either by setting ~ = 0 with De arbitrary, or by setting De = 0 and leaving ~ arbitrary. Additionally, the Maxwell model is the consequence of setting /~ -- 1 and De ys O.

We shall now assume that the velocity field is axisymmetric and that it is a function of z and time t only. Thus, the physical components of the velocity field

i

56BARNES, H.A., HUTTON, J.F. and WALTERS, K., A n Introduct ion to Rheology, Elsevier, Amsterdam, 1989.

57For a comprehensive review of this problem, see CREWTHER, I., HUILGOL, R.R. and JOZSA, R., Phil. Trans. Roy. Soc. Lond. A 337, 467-495 (1991).

are expressible in terms of two functions F(z , t) and G(z, t). In order to satisfy the continuity equation, one may follow von K~rmtin and take 5s

(v < r >, v < 0 >, v < z > ) = (rF' , r G , - 2 F ) , (49.25)

where the prime denotes partial differentiation with respect to z. The boundary conditions on the functions are the adherence conditions, i.e.,

F (0 , t ) -- 0, F (1 , t ) = 0, G(0,t) = 1, / (49.26)

f ' ( O , t ) - O, F ' (1 , t ) = 0, G(1,t) - c~. J Given the velocity field (49.25), it can be shown tha t the physical components of the stresses are given by a set of equations, which are polynomials in r, i.e.,

S < rr > - $ 1 0 + r 2 S l l ,

S < r O > = r2S12,

S < rz > = rS13,

S < 0 0 > = S 2 0 + r 2S22,

S < 0 Z > = rS23,

S < zz > = $3o,

(49.27)

(49.29)

where the eight quantities Si0, $11,--- , $30 are functions of z and t. For the sake of brevity, we omit the equations satisfied by the functions Slo and $20 for they do not play any role in the equations to determine the functions F and G. We now list the differential equations satisfied by the remaining six functions:

$11 + D e ( S l l - 2FS~I - 2F"Sa3) = -2 (1 - f l ) D e F ''2,

S12 -~- De(S12 2FS~2 - ' - " - G $13 F $23) = -2 (1 - 13)Def"G' , t, -- F " S13 + De(S13 - 2FS~3 + 2F $ 1 3 - F $3o)

+ ( 1 - ~ ) D e ( F " + 6 F ' F " - 2FF" ' ) ,

$22 + De(S'22 - 2FS~2 - 2G'$23) - -2 (1 - ~ ) D e G '2, (49.28)

$23 4- De(S23 - 2FS~3 4- 2F'$23 - G'S3o) - G '

+ ( i - fl)De((Y' 4-6F'G'- 2FG") $30 + me(S30 - 2FS~o + 4F'$30) = - 4 F '

- 4 (1 - / 3 ) D e ( / ~ ' - 2 F F " + 4F'2),

The equations of motion for the unsteady flow problem are given by:

3S~1 - S~2 + S~'3 = Re(~'" - 2 R E " ' - 2GG') ,

4S12 + S~3 = Re(G + 2 F ' G - 2FG') ,

| m , , ,

58The material here is taken from PHAN-THIEN, N., J. Non-Newt. Fluid Mech., 13, 325-340 (t983).


where R e - p~od2/vlo is the Reynolds number. In the above set (49 .28- 49.29), the superposed dot denotes the partial derivative with respect to t, whereas the prime denotes the partial derivative with respect to z.

We recall from (34.107) that the constitutive equation for the Oldroyd-B fluid may be expressed as a combination of a solvent contribution and a polymer contribution; the latter is really the Upper Convected Maxwell part (cf. (34.105)). It transpires that this procedure 59 eliminates the two time derivatives appearing in (49.24), when we replace S by W, which is defined through

S = W 4- (1 - / 3 ) A 1 . (49.30)

Indeed, the above transformation has the additional effect of removing the time derivatives from the right side of the equations affecting $13, $23, and $30 in the list (49.28), as well as removing all the nonlinear terms from the right side of the constitutive equations for the S 0 in (49.28). It is now a routine matter to recast the constitutive equations and the equations of motion in terms of W~j.

In order to find a solution to this new system, we linearise them about a steady flow. Thus, let V stand for any of the variables (F, G, Wll, .., W30) and assume that

v(~.t.~) = vo(~) + ~ - ~ . ( z ) + o(~) . (49.31)

where V ~ denotes the steady state value and v the perturbation term. Letting (Wl, w2, w3, ..., w8) stand for the perturbations in the variables (F, G, Wll, .., W30) respectively and expanding the new set of equations to the first order in e, the result is:

aReWl"

aRew2

aDew3

aDew4

aDews

aDew6

aDew7

aDew8

- (1 - )~)w 1 (iv) 4- 3w3' - w6 ! 4- w5 ,!

F0mWl t + 2Re[F~ + V~ V ~ w2],

: (1 -- ~)W2" + 4W4 + W7'

4- 2Re[F~ ' - F~ 4- G-~)'w 1 _ G-~)Wl'],

- - ~ + 2D~[F%~' + w~11'~1 + F ~ + W ~ I " ] ,

-- - w 4 4- De[2F~ ' 4- 2W~

4- GO W5 4- WO3 w2 4- FO t tw7 t , 4- WOswl ,,] '

-- - w 5 + ~Wl" + De[2F~ ' 2F ~ - - W 5

4- 2W~13tWl -- 2W~13w I ' 4- F ~ 4- Woowl"], - - ~ + 2D~[F%o' + w ~ ' ~ i + e ~ + w ~ ' l ,

-- --W7 4- ~W2 ! 4- De[2F~ w7 ' - 2 F ~

+ 2 ~ ' ~ , - 2 ~ ~ 1 ' + e ~ + ~ ~ ' ] , = - w s - 4~Wl'

+ e D ~ [ F ~ ' - 2F0'~8 + W ~ _ ewO0~l'].

(49.32)

59See WALSH, W.P., ZeSt. angew. Math. Phys., 38, 495-512 (1987).


The boundary conditions on Wl, W l I and w2 are of course given by

W 1 (0) - - 0 , W 1 (1) -- 0, w2(0) -- 0, ~ (49.33)

Wl'(0) - - O, Wl'(1) -- O, w2(1) = O. J Thus, (49.32) consti tutes an eigenvalue problem for a.

Turning to torsional flow, the basic equation that governs the linearised stability of this flow is obtained as follows. Pu t R e -- 0 in (49.32), for the torsional flow is possible in the absence of inertia only. Thus, the base flow is

F~ -- 0, G~ - 1 + ~z, (49.34)

with the non-dimensional shear rate ~ = a - 1. Next, one finds tha t

Wl~ - - W~)12 - - Wl~ - - W 2 0 = 0 ,

wo2 - 2/~n~q 2,

-

(49.35)

Then the system (49.32) leads to the eigenvalue problem

Wl(iV) + 4~2Wl/! __ 0, (49.36)

where the parameter ~ is given by

~2 _ De2/372[ 3 + 2 /~ - a D e ( 4 - aDe)] (49.37) (1 - a D e ) 2 ( 1 - (1 - f l ) a D e ) 2 "

It now follows tha t (49.36) has non-trivial solutions only if

~--nTr, n - - l , 2 , - - - ,

/ -- tan ~. (49.38)

The smallest root of (49.38) is ~ -- 7r. The implication of the above results is tha t a disturbance, which depends exponentially on t ime t, bifurcates from the torsional flow if, for a set of fixed values of ~, D e and ~ -- 7r, say, there is a solution of (49.37) for a as a function of ~. Note tha t this solution may be complex.

We note further from (49.37) tha t when the fluid is Newtonian, i.e., when D e -- 0 or ~ -- 0, there is no bifurcation from the torsional flow because ~2 = 0 and (49.36) has a trivial solution only. Hence, it is clear tha t the instability in the parallel plate configuration is a purely elastic instabili ty. That is, elastic or normal stress effects cause the instability and, in order for this to occur, it is not necessary for inertia to be present. In addition, it must be noted tha t in the Oldroyd-B fluid, the second normal stress difference N2 is zero and thus, one may assume tha t the first normal stress difference N1 is destabilising.

Condi t ions for a Bi furcat ing Steady Flow

Using the smallest root ~ -- 7r in (49.37) and put t ing a = 0, one finds tha t the torsional flow is unstable if e~

7r 2 = De2 ~.~2 [3 + 2~]. (49.39)

Tha t is, whenever ~ reaches its value specified by (49.39), a steady flow bifurcates from the torsional flow. This secondary flow is in the form of vortices which spiral in an axisymmetr ic fashion between the two parallel plates. The effect of the lat ter is to increase the torque required to maintain the overall motion. Thus, if one converts this torque into an effective viscosity, it is found tha t the bifurcation causes a shear thickening effect.

A b s e n c e o f a S t e a d y B l h t r c ~ t l n g F l o w f r o m a R i g i d B o d y M o t i o n

Although the rest s ta te of the simple fluid has been shown to be stable in w and this a rgument can be extended to tha t of a case of rigid body motion, we shall demonst ra te here in a separate fashion tha t the rigid body motion of the Oldroyd-B fluid is s table within the context of the torsional flow. To this end, we re turn to the equation (49.37) and put ~ = 0 there. Then, ~ = 0 and the boundary conditions on Wl ensure tha t the lat ter is zero as well. This, in turn, means t ha t all the per turbat ions w 2 , - . - , w8 are all zero.

N o n - E x i s t e n c e o f a Bi furcat ing Periodic Flow

In order to examine when a t ime-periodic per turbat ion occurs from the torsional flow, we return to (49.37) and put a = iw there. Then, ~2 may be expressed as ~2 _ ( a - i b ) / ( c - i d ) . Hence, in order for ( a - i b ) / ( c - i d ) to be real and positive, it is necessary t ha t ac + bd > 0 and a d - bc = 0. The lat ter condition leads to the equation

DES(1 - / ~ ) ( 4 - 3/~)w 5 + DES(7 - 7/3 + 5/32 - 2/~3)w 3 (49.40)

+ De(8 + / ~ - 2/32)w ---- 0.

Since all the coefficients are positive when De :> 0 and 0 _< /~ < 1, a simple calculation shows tha t the only possible solution is w -- 0. Hence, a periodic, axisymmetr ic flow does not bifurcate from the torsional flow even if any elasticity is present.

~9.~ Non-Axisymmetric Flows To discuss steady, non-axisymmetr ic flows 61 we turn to Cartesian coordinates (x, Y, 2) and let these be non-dimensionalised through (cf. (49.23))

~: - xd , 9 - yd, ~ = zd. (49.41)

6~ difference between the result (49.24) and that of PHAN-THIEN arises from the way the Deborah number has been defined.

61See CREWTHER, I., HUILGOL, R. R. and JOZSA, R., Phil. Trans. Roy. 8oc. Lond., A337, 46r- 49s (1991).

Let the velocity field "~ be scaled as in (49.23), i.e.,

- - f~oV, (49.42)

where f~0 denotes the steady, angular velocity of the bot tom disk situated at z - 0. Now, let (u, v, w) denote the velocity components of the field v such tha t

u - x F ' - y G + S g ,

v = x G + y F ' - 5 f , (49.43)

w - - 2 F ,

where F, G, f, and g are all functions of the coordinate z, and the prime denotes differentiation with respect to z . The functions F, G are the axisymmetric components, whereas f, g are the non-axisymmetric ones. The boundary conditions (cf.(49.26)) are

F ( 0 ) - 0, F ' ( 0 ) - 0,

F(1) - 0, F ' (1) - 0,

G(0) = 1 , ( 7 ( 1 ) = a , ( 4 9 . 4 4 )

1 S - ~ k + m = O S ~ j k " x k Y m, i----1,2; j = 3 ,

S < 3 3 > = Sa3oo.

There are, in fact, twenty five stress functions in the above list.

f (0) = 0 , f ( 1 / 2 ) = 1 , f ( 1 ) = 0 ,

g(0) = 0, g(1/2) = 0, g(1) = 0.

In (49.44), the boundary conditions on z = 0 and z = I follow from the adherence conditions, whereas on the plane z = 1/2 a measure of non-axisymmetry has been introduced through f (1 /2) . In fact, it is easily seen from (49.43) tha t on the plane z = 1/2 as ( x , y ) --~ (0,0), the in-plane components ( u , v ) --~ ( 0 , - 5 ) . Hence, the non-axisymmetric equations depend on the sole parameter 5 and in order to emphasise this, the constant 5 has been included in (49.43).

If the velocity field (49.43) is substi tuted into the constitutive equation (49.24), it is found tha t the stress tensor S is a quadratic in x and y, with coefficients tha t are functions of z. Indeed, the components of S, denoted by S , have the following representation:

2 S = ~k+m=0 SiJk,,~xkY m, i , j = 1 , 2 ,

(49.45)

The equations of motion for the velocity field (49.43) can be derived quite easily when use is made of the representation of the stresses in (49.45). One obtains �9

- ' - ' - " = 2R(FF'" + GG'), 2Sl120 -{- ~1121 -{- S1310

$1111 -~" 2S1202 "~- S~30I : 2R(FG'- F'G), (49.46)

- ' ~~ -" = R S ( f ~ G + f G ~ + F " g F~ g ~ 2 F g " ) SIIIO "[" ~1201 "[- 81300 -- -- ' ~l "~II + $22o + S2aoo = R (2f'F + f ' g ' - i F " + ga' + g'a).


It is easily seen tha t the first two of the above equations are the axisymmetric, steady flow equations in the new coordinate system, and hence they determine the functions F, G. The next two are to be employed to find the new functions f, g along with the new stress functions. A long, but straightforward calculation shows tha t given the axisymmetric, velocity functions F and G, the non-axisymmetric functions f , g and their stress functions are uniquely determined because the corresponding set of differential equations is linear. Hence for every axisymmetric flow, there is a unique non-axisymmetric flow. Moreover, it may be shown 62 that if

= 0, then all of the non-axisymmetric components vanish; this observation highlights why this parameter appears explicitly in (49.42). To put it another way, a steady non-axisymmetric flow bifurcates from the torsional flow and it depends on the new parameter 5.

We shall now adduce a couple of examples. For a Newtonian fluid, if the base flow is the torsional flow, the non-axisymmetric flow components are easily obtained. Put t ing De = O, Re = 0 in (49.45) one finds tha t

f ( z ) - 4 z ( 1 - z), g(z) : 0. (49.47)

To show tha t in the case of the Oldroyd-B fluid, at least one analytical, non- axisymmetric solution 63 exists, let the underlying axisymmetric flow system be the rigid body motion and introduce a function

N ( z ) = f ( z ) + ig(z). (49.48)

Then, N ( z ) satisfies the following differential equation:

N"'--- i R e ( 1 - i D e ) N ' , (49.49) 1 - i(1 - / ~ ) D e

where Re is the Reynolds number. The above third order equation has the following solution

g ( z ) = Ao + A l e ~'z + A2e"2Z, (49.50)

where #1, #2 are the two roots of

#2 = i r e (1 - iDe) / (1 - i(1 - ~)De). (49.51)

The three constants Ao, A1, A2 in (49.51) are found quite easily from the conditions (49.44) on f and g.

For a variety of non-axisymmetric solutions obtained by numerical means, including the existence of shear layers and the effects of inertia and other parameters, see the review 6a cited earlier. The only question tha t remains is the physical mechanism by which the parameter 5 may be introduced. The obvious one is a misalignment of the parallel disks.

62HUILGOL, R.R. and CREWTHER, I., Proc. Xth Int. Cong. Rheol., Vol. 1, 285-287, 1988. 63HUILGOL, R.R. and RAJAGOPAL, K.R., J. Non-Newt. Fluid Mech., 23,423-434 (1987). 64CREWTHER, I., HUILGOL, R. R. and JOZSA, R., Phil. Trans. Roy. Soc. Lond., A337,

467- 495 (19~1).


~9.5 Non-Axisymmetric Spiral Instabilities

If inertia is ignored, it may be shown by linear stability analysis 65 that non- axisymmetric, spiral solutions of the Oldroyd-B fluid exist in the two-disk configuration. That is, these disturbances possess the general structure

e, t) = Cxp[i. + - o 1. (49.52)

The important feature of these Archimedean spiral motions is that they occur outside a critical radius R, which depends on the radial wavenumber a of the disturbance. Thus, the viscometric, torsional flow is stable in the core region near the centre and unstable outside of it.

This analysis has been extended to the Chilcott-Rallison model by BYARS et al., 66 who, in addition to this derivation, record the data from a series of experiments using two polyisobutylene/polybutene Boger fluids. It is found that the instability occurring at the critical radius (R1) of the 0ZTEKIN and BROWN analysis is subsequently damped by shear thinning and disappears beyond a second critical radius (R2). The advantage of this analysis lies in the fact that the boundary conditions at the outer edge and the symmetry conditions at the centreline can be neglected. Once again, non-axisymmetric disturbances are found to be the most unstable, and this has been borne out by the accompanying experiments. The resulting secondary flow is well described by the analysis, except that the calculated values of the critical radii tend to be smaller than those obtained experimentally. The last point leads us naturally to summarise the knowledge gained from experiments so far.

~ 9. 6 Experimental Results

As indicated, we shall now record some experimental results 6z regarding the flow instabilities in a parallel plate apparatus. These observations lend support to the notion that at a specific critical shear rate, the torsional flow becomes unstable and turns into a time dependent flow. However, the flow ceases to be axisymmetric very quickly. See Figure 49.1. We shall now summarise the observations.

1. The shear stress ~-a(t) is time-averaged and the viscometric stress ~0~ is subtracted. This difference is the ordinate, whereas the abscissa is the shear rate.

2. The flow remains steady upto a critical shear rate of ~c = 42.3 s- 1. Above this shear rate, the flow bifurcates to a time dependent motion. The amplitude of the initial disturbance increases exponentially in time, indicating that the

ii

650ZTEKIN, A. and BROWN, R.A., J. Fluid Mech., 255,473-522 (1993). 66BYARS, J.A., OZTEKIN, A., BROWN, R.A. and McKINLEY, G.H., J. Fluid Mech., 271

173-218, 1994. 67See MAGDA, J.J. and LARSON, R.G., J. Non-Newt. Fluid Mech., 30, 1-19 (1988); McKIN-

LEY, G.H., BYARS, J.A., BROWN, R.A., and ARMSTRONG, R.C., J. Non-Newt. Fluid Mech., 40, 201-229 (1991); BYARS, J.A., 0ZTEKIN, A., BROWN, R.A. and McKINLEY, G.H., J. Fluid Mech., 271, 173-218 (1994).


I

500

Parallel Plates R = 12.5 mm

400 H 1.8 mrrl

300

200

ro = 25oc

~ e

100

Stable base flow 0 oo

0 0

Unstable

- 1 0 0 i I ~ I

0 10 20 30 40 50 60

(see-')

FIGURE 49.1. Torsional flow instability. The base torsional solution becomes unstable at Ale = 4~.38 -1, and the flow becomes time dependent and three dimensional.

linearised stability analysis is effective in determining the critical points on the solution curves. In addition, it is found tha t the flow is initially axisymmetric with roll cells forming at the centreline travelling outwards, and at the outer edge travelling in towards the centre. The disturbance quickly becomes non- axisymmetric with the formation of spiral vortices, and finally develops into a fully nonlinear flow situation.

3. By decreasing the rotation rate, it is possible to follow the t ime-dependent solution below the critical shear rate. This curve does not intercept the base solution; indeed, below the shear rate of ~ -- 33.7 s -1, the t ime-dependent oscillations decay to the steady flow situation. Thus, it seems that there exists a sub-critical Hopf bifurcation in this flow which is not axisymmetric. That is, the purely circumferential, torsional flow becomes unstable and transforms itself into a non-axisymmetric, t ime dependent motion consisting of spiral vortices which travel radially across the disks.

From the foregoing analysis of the instability in an Oldroyd-B fluid undergoing a torsional flow in an unbounded domain, it is clear tha t both steady and un-


steady secondary flows exist. Even if one makes due allowance for the fact that an experiment cannot approximate a truly unbounded flow situation, the experiments suggest that the secondary flows are time-dependent rather than steady, as well as being spiral in character. In order to determine whether these are caused by the finite dimensions of the apparatus, it is necessary to study the bifurcation phenomena in bounded domains and we turn to this next.

49.7 FiniteDomain OLAGUNJU 68 has employed a perturbation scheme to study secondary inertial flows that occur in fluids with arbitrary relaxation times, using a finite geometry in which the fluid is held in by surface tension. Beyond a critical rotation rate the perturbation procedure breaks down, indicating either that a critical rotation rate has been reached or that the scheme is not valid beyond this rate. The conclusion is that a secondary flow exists for all D e and becomes unstable at the critical value

_-- (49.53) De~ V//3 (9 + 2/3)"

where/3 is the retardation parameter. Now, the analyses of OLAGUNJU as well as that of BYARS et al. e9 indicate that

the presence of the free surface is not of extreme importance, since it is perturbed by a small amount only due to secondary flows, which means that the significant factor in the observed difference between the eigenfunctions of PHAN-THIEN's analysis, and the observed disturbance field of McKINLEY et al. is due to the presence of an outer boundary.

AVAGLIANO and PHAN-THIEN 7~ have examined this problem in a different manner. They assume disks of finite extent, with the fluid being held between the plates (a distance d apart and a common radius R) by a frictionless outer bounding shell, and solve a two dimensional linearised problem for the case of axisymmetric disturbances. The radial coordinate is then scaled with respect to R and the axial coordinate with d, so that the domain considered is a unit square. The parameters in the problem are the Weissenberg number De = )~12, where )~ is the Maxwell relaxation time and 12 is the angular velocity of the top disk, the retardation parameter /3 = rip/ri, where rip is the polymer-contributed viscosity, and ri is the total viscosity of the Odroyd-B fluid, and the geometric parameter E = d / R . No-slip boundary conditions are assumed on the top and bot tom disks, symmetry boundary conditions on the symmetry axis, and on the boundary r - 1 no radial flow and zero tangential stress are assumed.

Non-Uniqueness

It is then easily shown, by considering the rate of dissipation, that the assumption of an unbounded Stokes flow leads to a loss of uniqueness of the solution, while that

68OLAGUNJU, D.O., J. Rheol., 38, 151-168 (1994). 69BYARS, J.A., OZTEKIN, A., BROWN, R.A. and McKINLEY, J. Fluid Mech., 271,173-218

(1994). 70AVAGLIANO, A. and PHAN-THIEN, N., J. Fluzd Mech., 312, 279-298 (1996).


for the bounded flow is unique. Hence, the unbounded case cannot be considered as a limiting process of the finite geometry. The findings are summarised in the following exercises.

Problem 49 .B To demonstrate the non-uniqueness of the solution to the unbounded torsional

Stokes flow, consider a v o n K~irm~in solution of the form

~,~ = ,- u(~) + V~(z )~ (no) + u~(~)si~(~) , n - - 0

,,o - ~ v(~) + ~ v~(~) ~ (nO) + V~(z) sin(n0) , n-----'0

~. - w(z) + ~ Wi(z) r + W,(z)~in(~) , n----0

e - Po (z) + ~ 'e , (z) + ~ ( N (z) + ~ ~ (~)) ~ ( n 0 ) n ' - ' 0

+ ~ ( N (~) + ~2e~ (z)) ~in(n0), (49.B1)

and show tha t all the governing equations for Stokes flow are satisfied with u = - �89 U2 = V1, U1 = -V2, W1 = W2 = P ~ 0 - P~ = 0, n - 2. The Stokes flow solution then becomes

vr -- r {P~(z 2 - z ) c o s 2 8 - P~2(z- z2)s in20},

vo -- r { z - / ~ 2 ( z 2 - z ) s i n 2 O - P~(z- z2)cos28},

vz = O, P -- {C + r 2 [P~2 cos 28 + P~2 sin 28] }, (49.B2)

where C is constant, and P~, P~ are the amplitudes of the eigenfunctions. Show tha t the streamlines in the (r, 8) plane are ellipses when �88 (P~2)2+(P~) 2 _ 1,

and hyperbolae otherwise. Show tha t the von K~irm~in form with the functions of z replaced by functions of

z and t also reduces the fully t ime dependent 3D Navier-Stokes equations by two spatial dimensions, and results in a closed system in v, w, U = U1, V -- V1, and P.

With the notation Of~Or ---f, Of/Oz - f', reduce the Navier-Stokes equations to the following system

{ l i z ,+ lw ,2_v2 lww, l } 1 ,, n~ - ~ - ~ + ~ (u 2 + v 2) = - ~ - 2P, (t),

n~ {v - ~ ' ~ + ~ ' } = ~",

4/~2 (t),

{V - w ' r + w , ' } = y" + 4P; (t). (49.B3) Re


Problem 49.C To prove the uniqueness of the solution in the bounded torsional Stokes flow,

consider the volume f~ in three dimensional Euclidean space, whose closure f~= f~US, where S is the bounding Liapunov smooth surface of f~ with outward pointing unit normal n. In Cartesian coordinates, we can express the rate of viscous energy dissipation in the region f~ as

/ r dv= / T,jLji dv= 2rh / Lj,Lj, dv, (49.C1)

' of the which is a positive definite quadratic form. Consider two solutions vi and vi I Stokes' equations satisfying the same boundary conditions on S. Let v~ - - v i - vi.

By applying the divergence theorem, the Stokes equations and the incompressibility condition, show tha t

~ ~ dv

-- ~ ( v * - r * - n - P * v * - n ) da, (49.C2)

which is t rue in any coordinate system. From the boundary conditions for the bounded torsional flow, show tha t the rate

of viscous energy dissipation for the difference solution is zero. Thus, we require L* -- 0 in the entire domain f~. This can only be true if v and v ~ differ by at most a rigid body motion. Since we have defined the velocity field on some part of S, the two solutions must be the same. Hence, the Stokes flow solution is unique in this geometry. This is also t rue in the case of a free surface at r = 1, since the boundary conditions are of a similar form.

Linearised Stability Analysis

We have already noted in w that , in the absence of inertia, the steady torsional flow solution (v = (0, rz, 0)) occurs in all simple fluids and hence in the Oldroyd-S model.

In order to examine the stability of the bounded torsional flow, disturbances of the following form are assumed

v -= Vo + 6 N [u(r ,z) exp -at~De"

I" -- "to A- 6~ [a(r , z) exp -at/De"

P = Po + 6~R [p(r, z) exp -at~De" (49.54)

where ~[-] represents the real part , and Vo, 1"o, and Po form the base torsional solution.

The stability of the torsional flow is determined by studying the linearised equations for the perturbed, by assuming a small parameter 6. The growth rate a determines the stability of the base solution at any set of values of the Deborah

number De, the retardation parameter/3, and the aspect ratio e. If N(a) > 0, the base solution is stable, while ~ (a) < 0 implies the solution is linearly unstable in this region of (De, e, fl) space, and ~ (a) = 0 gives the so called neutral stability curve in the (De, e) or (De,~) plane.

Substitution of the assumed velocity profile into the constitutive equation yields the perturbed stress field as a function of the perturbed velocities. The stress field, when substi tuted into the momentum equations, yields four equations for the perturbed velocity and pressure fields. The pressure term is eliminated, and a stream function introduced so that the continuity equation is identically satisfied. The final set of equations reads

l c ' ~ 1 0 ~ Ur ---- -- - - - - - - Uz - - -- r Oz' r Or'

( 1 - a ) [ 1 - a ( 1 - J 3 ) ] { ~ 2 0 ( lOr~ (ruo)) Ouu~ } +

Oz20r r Oz 2 -~ r = 0 (49.55),

2De~ (1 - a) (2 - a) 02u~ 03 ~ Oz 2 + 2De2B (3 - a) ~ + (1 - a) 2 [1 - a (1 - /3)]

10V (10z,0 ) } 0 Due to the symmetry of the domain, ur and u0 are odd functions of r, while uz

(and hence r is even in r. A mixed Tau-Galerkin procedure is then used to solve the equations, utilising a Fourier series in the axial direction due to the symmetry of the boundary conditions at z -- 0 and z -- 1. In the radial direction, a Chebyshev series is chosen due to its minimax properties and its ability to resolve the solution inside a boundary layer. 7t

Substitution of the spectral approximations into the governing equations yields a cubic eigenvalue problem of the form

[a3B3 (De,/3,e) - a2B2 (De,~,e) +aS1 (De, fl, e) - m (De, Z,e)] v - 0, (49.56)

where A, B1, B2, B3 are all square matrices of size N (N + 1) -- NT, where N is the number of trial functions. For each De there will be 3NT eigenvalues a with corresponding eigenvectors v. The eigenvector is simply the vector of the unknown coefficients of Uog and CN, i.e.,

( )T (49.57) V ---- 04), a l , .., a ~ N T - 1 , b0, b l , " , b~gT -1

Since B3 is a singular matrix of rank NT/2, the number of eigenvalues reduces to 5NT/2. In addition, there is also the solution a -- - 1 , of multiplicity NT/2, which

71G O T T L E I B , D. and ORSZAG, S.A., Numerical Analysis of Spectral Methods: Theory and Applications, SIAM, Phi ladelphia (1977).

0 I m a ~ r

1 . 5 ' ' ~ ' ' ' i . . . . i . . . . i . . . . i . . . . , '

1 . 0

0 . 5

0.0

- 0 . 5

-1.0

- 1 . 5 - 2 . 5

0

0 0

oo [~] 0

0 O0 0 0

0 0 0 0

o o o~ ~ o o~o~ ~ o ~ o o

0 0 0 0 0 0

0 0 0

~ 0 % ~bo o oc t ) Oo o

o o

o o o o o

o

o o ~ 0 0

0

' ' �9 l i i i i I i i i �9 I . . . . I , , , , I ,

- 2 . 0 - 1 . 5 - 1 . 0 - 0 . 5 0.0

FIGURE 49.2. Eigenspectrum for a with N = 12, e -- 0.1 and f~ = 0.41. The critical eigenvalues are marked.

further reduces the number of eigenvalues by NT/2. Hence, we need to solve for 2NT eigenvalues for each De only.

The critical Deborah number (Dec), at which the largest negative, real part of the eigenvalue crosses the imaginary axis determines the onset of instability of the base solution. Let

(A, B1,B2,B3) -- {a" det (a3B3 - a2B2 + aB1 - A) - 0}. (49.58)

As previously observed, the base solution will be linearly stable if Va E )~(A, B1, S2, B3), ~ ( a ) > 0, and unstable if 3 ( a E )~(A, BI,B2,B3)), ~ ( a ) < 0. Neutral stability will occur at Dec when the eigenvalue with largest real part, ac is such that ~ (ac) = 0.

Different techniques have been a t tempted to solve the above problem, with the enhanced initial vector approach discussed by SAAD 72 being the most efficient method. Figure 49.2 shows the spectrum of a, with the critical eigenvalues marked.

When the aspect ratio is 1, there is a dominant roll cell, surrounded by several smaller cells rotat ing in the opposite direction. These secondary roll cells disappear with a decrease in aspect ratio to ~ = 3/4, while a smaller roll cell is developing near the centreline. This cell is fully developed at 6 = 1/2. At a smaller aspect ratio, more roll cells appear, and their dimensions scale approximately with the gapwidth of the disks. The roll cells for the case where 6 = 0.25 are shown in Figure 49.3.

72SAAD, Y., Numerical Methods for Large Eigenvalue problems, Manchester Univ. Press, Manchester, 1992.


FIGURE 49.3. Disturbance streamlines at ac -- -1.47 x 10 -4 + 0.8156i, Dec ---- 2.538 when ~--0.25, and/~----0.41.

Note tha t due to the complex conjugate critical modes, there are two distinct real eigenfunctions at criticality, shown in the figure. The main difference between the two solutions is not readily apparent, when examining the entire region. This is due to the fact tha t no streamlines appear near r = 0, since the ampli tude of the roll cells near r = 0 are several orders of magnitude smaller than the amplitude of the outer cells. Only when the inner region of the disks is examined, it is found tha t one of the eigenfunctions has a single roll cell in the axial direction, while the other has two or more. These multiple vertical roll cells counter rotate in the radial direction, as expected, and corotate in the axial direction.

The form of the disturbance is very similar to tha t observed by McKINLEY et

al. 73 except tha t in the present work, the disturbances are large near the outer edge and small near the centreline. In the linearised stability region the roll cells would first be seen at the outer edge and then appear to travel inwards, as the amplitude of the inner cells increases exponentially in time. In the work of McKINLEY et

al., cells of large ampli tude are observed initially at both the outer edge and at the centreline. A possible reason for this discrepancy is the presence of inertia. Although, as observed earlier, OLAGUNJU has examined the effect of inertia, the perturbat ion scheme falls short of predicting this experimental phenomenon.

i

73McKINLEY, G.H., BYARS, J.A., BROWN, R.A., and ARMSTRONG, R.C., J. Non-Newt. Fluid Mech., 40, 201-229 (1991).

D e 4

' . \ ' , . . . . , ,,. , , , , , . . . . , . . . .

\ ~i.~ . . . . . . . ~ =0.5

' ~ i ' _ - ~ ~ . . . . . . E=O.l

\'~ "-. . , \~ ~ ......................... Phan-Thien 83 : "~ .. o

\~'- ~ ...................... Olagunju '94 \ ' ,

, ~ \ ' : ~ : % .

, ,, :,:,:,:,_, .......... ~ "~"~,~.~i : ' ' '~ ' ' ' -

'~' �9 "~ �9 .... "="-.............

~. ~ . . ~ . . �9 . . . . . . . . . . .

- - - " ' - - ' - ~ . . g 2 " , ~ ' ~ :

0.2 OA o 0.6 0.8 P

|

1 . 0

FIGURE 49.4. The neutral stability curves are plotted in the (De,/}) plane for several aspect ratios./3 : 0 is the stable Stokes' flow,/3 : 1 is the Maxwell model.

Figure 49.4 shows the relationship between the critical Deborah number and the retardation parameter, which is a measure of the elasticity of the fluid, for three aspect ratios. Analysis shows tha t the singularity near/3 -- 0 scales approximately with the square root singularity of both the Phan-Thien and Olagunju analysis, depicted by the dotted and dot-dot-dashed lines respectively.

In Figure 49.5, the critical Deborah number is plotted against the aspect ratio. For fluids with an average to high concentration of viscous solvent (/~ N 0.5), there is an almost linear dependency between Dec and r while for a lower concentration of solvent this relationship is t rue for aspect ratios below about 0.2. In fact, the critical Deborah number can be written as Dec - eK1 - t -K2, where K1,2 depend on/3; for example, when/3 -- 0.41, which is the relevant value in the works of McKINLEY et al., 74 and BYARS et al., 75 one notes tha t K1 - 7.1 and K2 : 1.0 and, the critical dimensionless rim shear rate is A '~c- (K1 + K 2 / r This important fact has been confirmed by the experiments of MAGDA and LARSON, 76 and McKINLEY et al.

Moreover, it is found tha t the critical rim shear rate is inversely proportional to the aspect ratio only when r is small. However, the calculations for the eigenvalues become inaccurate for r _< 0.04, and they are not plotted in Figure 49.5.

74McKINLEY, G. H., BYARS, J. A., BROWN, R. A. and ARMSTRONG, R. C., J. Non-Newt. Fluid Mech., 40, 201-229 (1991).

75BYARS, J.A.,C)ZTEKIN, A., BROWN, R.A., and McKINLEY, G.H., J. Fluid Mech., 271, 173-218 (1994).

76MAGDA, J.J. and LARSON, R.G., J. Non-Newt. Fluid Mech., 80, 1-19 (1988).


6

De 5

1 0.0

. . . . I . . . . I . . . . I . . . . ~ P . . . .

" - . . . . . . . . ~ ! = 0.2 , , " /

. . . . . . . . . ~:0 .41 , " /

................. ~ = 0.7 , "

s S o . . . . . . . . . . . . . . . . . . . . . . . . "

�9 ,, , . . ,""

�9 �9 oS.~. ~~ o . . . . . . . . . . .

s �9 s.s~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

~ ' J . o "

' " " I ' f ' i i " I i i i i i i i i i I , , , ,

0.2 0.4 0.6 0.8 E

FIGURE 49.5. Plot of the neutral stability curves in the (De, e) for a range of/3 values. A linear relationship is apparent for all values of/~ at small aspect ratio.

Shear thinning of ~I/1 has led to appreciable differences between the experimental and theoretical values of Dec for the Oldroyd-B model. McKINLEY et al. 77

suggest a method of comparing their experimental values with the analysis of PHAN-THIEN, by allowing )~, and hence De, to be a function of the applied shear

rate, i.e., ,k = ~(#) and De = De(~) = ,kf~. For the Oldroyd-B model the constant relaxation time is determined by the initial value of II/1 t o be )q -- 0.794. They

use a 4-mode Bird-DeAguir model to determine ~(;/), and have plotted the stability diagrams of the Weissenberg number Wi(= r versus De, for both models. From these diagrams, approximate values of the critical Deborah number may be

determined for several aspect ratios, for both )k I and ~(7). The results are shown in Figure 49.6.

The choice of A = A(X/) results in a significant increase in the correlation between experimental results and the theoretical values obtained with the Oldroyd-B model.

In conclusion, there is a good agreement between the results derived from linearised stability theory and experiments in the parallel plate configuration, when a bounded domain is taken into account, together with a good estimate of the relaxation parameter. However, it must be emphasised that the theoretical results depend on a model with N1 > 0 and N2 - 0. On the other hand, if one employs models with a non-zero N2, it is found that N2 is stabilising. 78 The presence of

i, i ,

77McKINLEY, G.H., BYARS, J.A., BROWN, R.A., and ARMSTRONG, R.C., op. cit, (1991). 78See PHAN-THIEN, N. and HUILGOL, R.R., Rheol. Acta, 24, 551-555 (1985).


De

a

[:] a

a

Oldroyd-B

iL = 0.794

! . . . . I . . . . | , o ,

A N

13 - 0.41

- - I . . I i l ~ i , , I , ' . . I . . . . I , , , ~ I . . . .

0.04 0.06 0.08 F,,, 0.10 0.12 0.14

FIGURE 49.6. Comparison of computed critical modes with experimental values of McKinley et.al, for constant and adjusted relaxation times.

N2 is, ultimately, not all tha t beneficial because large values of N2 lead to edge fracture - see w Thus, torsional flow is but one instance where elasticity alone initiates a very complicated flow regime.

9.8 Cone-and-Plate, Couette and Extrusion Flows

Having considered one flow in detail, we shall record below the stability analyses of various flows; the list is a supplement to the exhaustive summary of the experimental and theoretical results in the review 79 mentioned at the beginning of the section.

1. In the cone-and-plate configuration, the instabilities observed in experiments are similar to those in the parallel plate apparatus, s~ On the theoretical side, a t t empts to explain the non-axisymmetric and time-dependent instabilities have also been made. sl

79LARSON, R. G., Rheol. Acta, 31,213-263 (1992). S~ McKINLEY, G. H., BYARS, J. A., BROWN, R. A. and ARMSTRONG, R. C., J. Non-

Newt. Fluid Mech., 40, 201-229 (1991). slOLAGUNJU, D.O. and COOK, L.P., J. Non-Newt. Fluid Mech., 46, 29-47 (1993); OLA-

GUNJU, D.O., J. Non-Newt. Fluid Mech., 50, 289-303 (1993).


2. As far as a plane Couette or simple shear flow is concerned, linear stability of this flow in a Maxwell fluid has been proven to hold without inertia. 82 If the Oldroyd-B fluid model is used, then linear stability is assured only in the presence of inertia. 83 In the absence of any studies using more complex models, it is not possible to forecast, with a degree of confidence, the effects of inertia and large amount of elasticity on the stability of this class of flows.

3. The Taylor-Couette problem has been studied extensively to determine both the presence of axisymmetric 84 and non-axisymmetric disturbances, s5 In particular, for Maxwell fluids, it has been shown that non-axisymmetric periodic distrubances are more destabilising than axisymmetric ones under certain conditions, when inertia is present. On the other hand, the Taylor-Couette flow problem may be shown to possess a purely elastic instability of the oscillatory flow variety in an Oldroyd-B fluid 8s and this has been confirmed experimentally as well. 8z Again, it is found tha t N1 is destabilising while N 2

is stabilising in this class of flows. 8s

4. It is well known tha t under certain operating conditions or with a combination of the extruding fluids, the interface between two fluid components in an extrusion process is irregular or the interface is unstable. As far as Newtonian fluids are concerned, interface instability may exist in a plane shear flow; this is caused by the difference in the viscosities of the two fluids. 89 Of course, within the context of viscoelastic liquids, it is possible tha t the interface instability may exist solely due to elastic effects. It has indeed been found 9~ tha t when the perfectly smooth interface is perturbed, the jump in N1 of the base flow across the unperturbed interface causes a purely elastic instability.

5. Turning to another class of extrusion problems, it is known 9t tha t the onset of melt fracture is associated with wall slip; inertia is not important here. If the flow regime near the wall is of the stick-slip variety and the slip velocity at the wall is a decreas ing function of the wall shear rate, then it is to be expected

82RENARDY, M., Euro. J. Mech. B., 11,511-516 (1992). 83GUILLOPE, C. and SAUT, J.C., Nonlin. Anal., 15, 849-869 (1990). s4AVGOUSTI, M. and BERIS, A.N., Proc. Roy. Soc. Lond. A 443, 17-37 (1993); AVGOUSTI,

M., LIU, B. and BERIS, A.N., Int. J. Num. Methods Fluids, 17, 49-74 (1993); LARSON, R.G., MULLER, S.J. and SHAQFEH, E.S.G., J. Non-Newt. Fluid Mech., 51, 195-225 (1994).

s5AVGOUSTI, M. and BERIS, A.N., J. Non-Newt. Fluid Mech., 50 ,225-251 (1993). S6LARSON, R.G., SHAQFEH, E.S.G. and MULLER, S.J., J. Fluid Mech., 218,573-600 (1990);

PROCTOR, M.R.E., J. Non-Newt. Fluid Mech., 51,227-230 (1994). sZMULLER, S.J., LARSON, R.G. and SHAQFEH, E.S.G., J. Non-Newt. Fluid Mech., 46,

315-330 (1993). 88SHAQFEH, E.S.G., MULLER, S.J. and LARSON, R.G., J. Fluid Mech., 235,285-317 (1992). 89yIH, C.S., J. Fluid Mech., 27, 337-3XX (1967). 9~ Y. , J. Non-Newt. Fluid Mech., 28, 99-115 (1988); CHEN, K., J. Non-Newt.

Fluid Mech., 40, 261-267 (1991); CHEN, K. and JOSEPH, D.D., J. Non-Newt. Fluid Mech., 42, 189-211 (1992).

91RAMAMURTHY, A.V., J. Rheol., 30, 337-357 (1986); KALIKA, D.S. and DENN, M., J. Rheol., 31, 815-834 (1987).


tha t the flow would be unstable92; the converse is also t rue in the sense tha t if the velocity is an increasing function of the wall shear stress, the flow is stable. W h a t if the velocity is history dependent on the wall shear stress? An answer to this question has been found 9a and it shows tha t instability at the boundary arises in the case of a Maxwell fluid.

6. Elastic instabili ty has also been shown to arise in the flow of a viscoelastic fluid in a curved channel 94 and in the pressure driven flow between two rota t ing circular cylindrical surfaces, called the Taylor-Dean flow. 95

9.9 E n e r g y M e t h o d s and Squ i re 's T h e o r e m

We now tu rn to two ideas which have been very useful in the examinat ion of the stabili ty of the flows of a Newtonain fluid. First of all, it may be shown 9s tha t fi2 = ft . fi of the dis turbance fi(x, t) satisfies the energy equation

1 052

2 0 t = -(a.L~ {(s-s~

(-~ p l ( s - s~ ~ ~v ) + v. (po_p),~§ (49.59)

where (v ~ L ~ S ~ pO) are the base flow values, and (v ---- v ~ 4- fi, S, p) are concerned with the base flow plus the per turbat ion term.

Let f~ denote a spatial domain with a boundary Oft. Thus, if K = fn ~2 2 dr, it follows from the divergence theorem tha t

1 OK

2 0 t = _ f (a. L0a+ !t~p ((s-s~ ~

f~

/o(; 1 ; ) + (v ~ - pla + (s - s o n P ) f i - r . n d a . (49.60)

We shall now limit the choice of the velocity field v considerably by assuming tha t it is spatially periodic in the x3 direction, while the base flow values (v ~ L ~ S ~ pO) are independent of x3. Further, let p and S be periodic in x3. If ~ denotes a volume with its x3 boundaries being apar t by a distance equal to the spatial dependence of v, while the dis turbance u ---- 0 on the remaining parts of 0~ , then it is easily seen tha t the surface integral in (49.60) is identically zero. For instance, such a s i tuat ion arises in the Taylor vortex problem in a Couette flow between concentric circular cylinders.

92PEARSON, J.R.A. and PETRIE, C.J.S., in WETTON, R.E. and WHORLOW, R.W. (Ed), Polymer Systems, Deformatzon and Flow, Macmillan, London, 1968, pp. 163-187.

9SRENARDY, M., J. Non-Newt. Fluzd Mech., 35, 73-76 (1990). 94jOO, Y.L. and SHAQFEH, E.S.G., Phys. Fluids A, 3, 1691-1694 (1991). 95jOO, Y.L. and SHAQFEH, E.S.G., J. Fluid Mech., 262, 27-73 (1994). 96SERRIN, J.B., Arch. Rational Mech. Anal., 3, 1-13 (1959); FEINBERG, M.R. and

SCHOWALTER, W.R., Proc. Vth Int. Cong. Rheol., Vol. 1, pp. 201-206, 1968.


If we assume that the perturbation is steady in time, then OK/Ot -- O. Thus, one finds that fi must satisfy

O= jf (ft. E)~ + tr { ( S - S ~ dv, (49.61)

where we have put p -- 1 without loss of generality, and D O is the rate of deformation tensor associated with the base flow. Now, it is always true that

I~1-I~Ofl ~> ,~min I1" I~1,

where/~rnin is the least eigenvalue of D0. If it can be established that

tr { ( S - S~ >_ a tr{VfiVfi},

(49.62)

(49.63)

where a > 0 depends on the material functions of the fluid, then Serrin's inequality 97 asserts that

t r { V f i V f i } dv > ~ f (x ) dv, (49.64)

where/~ :> 0 is a constant which depends on the domain ~, and f (x ) >__ 0 is scalar valued. Finally, from (49.61)-(49.64), one obtains

f a f i ' f i 0 :> [0~/~ 4- )~minf(X)] f (x ) dr. (49.65)

It follows therefore that fi -- fi(x) cannot exist, i.e., the base flow is stable provided

~/~ 4- Aminf(X) > 0. (49.66) Apart from the various simplifying assumptions made on the domain and the

nature of the velocity fields, the crucial step where the above approach succeeds or fails is the inequality in (49.63). As long as the fluid is described by a finitely linear constitutive relation, the energy method works and one may examine the stability of a given flow. 9s Thus, the energy method has not found a niche in viscoelastic fluid flow problems comparable to the one it occupies in Newtonian fluid mechanics.

Lastly, Squire's theorem in Newtonian fluid mechanics asserts that the three dimensional disturbances of a parallel, steady flow are equivalent, under a set of transformations, to two dimensional perturbations of the base motion. Hence, a considerable amount of labour can be saved here. However, for non-Newtonian fluids, this theorem has been found to hold under very restrictive conditions. For example, in second order fluids, it holds if N2 = 0; for third order fluids, other conditions have to be met. 99 Squire's theorm also holds for the Maxwell model. 1~176 Note that N2 = 0 for this fluid.

In summary, it is seen that two techniques with wide applications in Newtonian fluid mechanics do not achieve the same level of eminence in the flows of viscoelastic fluids.

g7SERRIN,J.B., Arch. Rational Mech. Anal., 3, 1-13 (1959). 98For the Taylor vortex problem, see FEINBERG, M.R. and SCHOWALTER, W.R., Proc.

Vth Int. Cong. Rheol., Vol. 1, pp. 2-01-206, 1968; for the Benard problem, see CARMI, S. and SOKOLOV, M., Phys. Fluids, 17, 544-546 (1974).

99LOCKETT, F.J., Int. J. Engng. Scz., 7, 337- 349 (1969). t~176 G. and BERNSTEIN, B., Phys. Fluids, 13,565-568 (1970).

50 Qualitative Dynamics 385

50 Qualitative Dynamics

In the theory of ordinary differential equations, qualitative theory occupies a central place. By this one means the study of the behaviour of the solution of an initial value problem, without necessarily solving the problem itself. Tha t is, if one has a linear problem x = Lx, where L is an 2 • 2, real matrix and x is a two dimensional vector, then the behaviour of the solution is determined by the eigenvalues of L. It is then seen tha t the origin is a source, or a sink, or a centre, or a saddle point or a node. From this, one may visualise the solution curves in the plane on a short as well as on a long te rm basis. This qualitative study has grown from its simple beginning to higher dimensions as well as into the field of partial and integro-partial differential equations; these are of interest to us here.

Hence, in this section we shall describe some techniques to gain a qualitative understanding of the flows of a viscoelastic fluid. 1~ As before, we shall choose a model problem, viz., the plane flows of the JRS fluid 1~ and examine it in detail, for this problem provides the basis for understanding other developments. To set the scene, we begin with a classification scheme.

50.1 Classification of a Part ial Dif ferential Equat ion

Let us consider a linear differential operator

0 0 P(x , t , -~ , OXl " " ' OXn )' (50.1)

in an n-dimensional Euclidean space and t ime t. Suppose now tha t the order of the highest derivatives in the operator P is m. Then, one may expand P to obtain

P - E . . (x.t)0" + . . (x . t)o". I~l=-~ tal<m

(50.2)

where a = (a0, o~1,---, an) is a multi-index, ]a I = E a , , and

0 ~ = , (50.3)

The polynomial equation

E aa(x't)aa -- O, a - - (O0,al,''" ,an), I"l= (50.4)

o-

is called the characteristic equation for P. Note tha t only the principal part , or the highest terms, of P appear in (50.4). We now consider two examples.

1~ a comprehensive treatment, see JOSEPH, D.D., Fluid Dynamics of Viscoelastic Liquids, Springer-Verlag, New York, 1990. 1~ JOSEPH, D.D., RENARDY, M. and SAUT, J.C., Arch. Rational Mech. Anal., 87,

213-251 (1985).

1. Laplace's equation is given by

j--1

It has the characteristic equation

(50.5)

2 ~o~ - 0 . (50.6) j=l

This has no real zeros; note that a0 = 0 here. A partial differential operator P for which, at every point (x, t), the characteristic equation has no non-trivial real zeros is called elliptic. For elliptic equations, boundary value problems are well posed, i.e., existence, uniqueness and continuous dependence of the solution on the boundary data of such equations can be proved. 1~

On the other hand, initial or Cauchy problems 1~ are not well posed. Indeed, consider the half-plane Q = { (x ,y ) : x > 0 , - c o < y < co} and examine the solution of the Laplace's equation Au = 0 on this domain, with the initial data

Ou(0,u) = f ( u ) , ~(0,u) = 0 , 1 (50.7)

f ( y ) -- - - s inny . n

The solution of this is

1 u ( x , y) -- ~-ff sin n y sinh nx . (50.8)

For large n, the boundary data f ( y ) is very small, while the solution is very large because of the presence of the sinh n x term, which is of exponential order. Hence, a small change in the boundary data leads to large amplitude oscillations of very short wavelength in x > 0. This absence of continuous dependence on initial data is called H a d a m a r d instabili ty.

2. The wave equation is given by

02 u = C 2 n

j--1

It has the characteristic equation

(50.9)

n

o~ - ~ ~ o~ - 0. (50.10) j=l

1~ for example, PROTTER, M.H. and WEINBERGER, H., Maximum Principles in Dif- ferential Equations, Prentice-Hall, 1967. I~ J., Lectures on Cauchy's Problem in Linear Partial Differential Equations,

Dover, New York, 1952.

A real solution is given by a0 - • for all a j , j - 1 , - - - , n , such tha t ~-~j=l ~ a j 2 _ 1. Similar to the elliptic case, an operator P is called hyperbolic if all roots ~0 of the characteristic equation are real for all (~1, '"" , ~ ) E R~\{0}. Unlike the solutions of elliptic equations~ the Cauchy problem is well posed for hyperbolic equations, whereas the boundary value problem is not so.

50 .2 M i x e d E q u a t i o n s

There do exist equations which are hyperbolic on a par t of the domain and elliptic on another. The first example of such an equation which has been studied extensively is the Tricomi equation:

02u 02u OX 2 q- x-~ -- O, (50.11)

which has the characteristic equation

a l + = 0. (50.12)

Clearly, this has non-trivial real solutions when x < 0; hence, it is hyperbolic on this par t of the domain. On x > 0, it is elliptic.

A second example arises from nonlinear elasticity. Consider the one-dimensional motion of a bar 1~ with the stress-strain law T -- T(Ou/Ox), where u is the displacement. The equation of motion is

02u OT �9 (50.13)

We thus obtain

where we have put

02u -- T' 02u (50.14)

dT(v) Ou T'(v) -- . v - - O--x' dv

Clearly, the equation (50.14) is hyperbolic if T'(v) > 0 and elliptic if T'(v) < O.

(50.15)

50 .3 F i r s t O r d e r S y s t e m s

To begin, let us now cast the two-dimensional Laplace's equation as a first order system. Set

Ou Ou - - - = v, - - - -- w. (50.16) Ox Oy

Then

Ov Ow

Oy Ow Ov ox Oy

= O,

0, (50.17)

I~ J.L., J. Elasticity, 5, 191-201 (1975).


with the latter following from the equality of mixed partial derivatives. The two equations in (50.17) may be combined as the matrix-vector set:

(10 1 O) 0 ( v ) + ( O 1 "~x w - 0 "~ w - 0 " 1) 0 ( v ) (0 ) (50.18) To define the characteristic equation, consider the matrices along with the differential operators and replace O/Ox and 0 /0y by ch and a2, respectively, and add; this results in the matrix

The determinant of this matrix is a l 2 + a22, which is the characteristic polynomial of the elliptic equation. In a similar fashion, one may cast the ordinary wave equation in a matrix-vector form and find the corresponding polynomial, which is a02 - c 2 a 2.

50.~ A n Uns teady Shear F low

Turning to a problem in fluid mechanics, l~ consider a motion to occur in the x-direction with the velocity field given by u -- u(y, t). Using the linearised constitutive relation (49.6) and rewriting it as

S = G(t - T)AI(T) dT, (50.20) o o

we find tha t the shear stress a is given by

The derivative of (50.21) with respect to t is

o~ = c(o)~ f_" at ~ + d ( t - -,-) o o

where we have put

O u ( x , - , - )

o'~

( ~ ( t - T) - - dG(s) ds

8 - - - - - t - - r

d~, (50.22)

(50.23)

Next, in the absence of a pressure gradient, the equations of motion reduce to the single equation

Ou Oa P " ~ " - O-"xx" (50.24)

Introducing the notation

(.) (0 ) q = o ' a (0 ) ' (/ 0 ) f - ~ O(t - - ' ~ (50.25)

- - I ) Or dT ' 4 0 0

106See JOSEPH, D.D., RENARDY, M. and SAUT, J.C., Arch. Ratzonal Mech. Anal., 87, 213- 25~ (1985).


we arrive at a first order system

1 - ~ + B - ~ - f, (50.26)

where 1, the identity matrix, has been used to cast the system into a s tandard form. Here, we note tha t the right side is of a lower order than the left side because it contains a derivative with respect to x, which gives an order +1, while the integral with repect to t is or order - 1 ; hence, f is of order 0 as an operator.

Thus, the characteristic equation associated with (50.26) is given by

~176 _ c(o)o 1 = o, ( 5 0 . 2 7 ) P

which means tha t the system (50.26) is hyperbolic. Hence, in this unsteady shear flow, the equations permit waves to propagate with a speed c ---- +y/G(O)/p. This has relevance to the Rayleigh problem which has been studied extensively l~ for the Maxwell fluids as well as those with more general relaxation functions.

50 .5 E i g e n v a l u e s a n d C lass i f i ca t ion

Since matrices and determinants have been introduced, one may as well transfer the problem of ellipticity and hyperbolicity over to the question of eigenvalues and eigenvectors. Hence, consider a system of the form l~

0 u ~ 0 u + - f , ( 5 0 . 2 8 )

j=l

where u is an m-dimensional vector defined over a domain in T~ n and t. Also, A, B1 , - - - , B= are m • m matrix valued functions and f is an m-vector valued function. If these functions depend on (u, x, t) only, then the system is quasilinear. Note tha t (50.26) is an example of such a system.

The system (50.28) is evolutionary in some domain ~ of 7r m • 7r n • 7~ if, for every fixed (u, x, t) in ~ and any n-unit vector D, the eigenvalue problem

- h A -~ i t jBj j-~l

v ----- 0 (50.29)

has only real eigenvalues. Note tha t evolutionary problems involve hyperbolic and non-hyperbolic equations and in order for the system (50.28) to be hyperbolic in the t-direction, (50.29) must have m real eigenvalues, not necessarily distinct, and a set of m linearly independent eigenvectors.

1~ R.I., Zezt. angew. Math. Phys., 13, 573-580 (1962); HUILGOL, R.R., J. Non- Newt. Fluzd Mech., 8,337-347 (1981); ibid, 12,249-251 (1983); NARAIN, A. and JOSEPH, D.D., Rheol. Acta, 21,228-250 (1982); ibid, 22, 519 (1983); ibid, 22, 528-538 (1983); RENARDY, M., ibid, 21,251-254 (1982). I~ D.D. and SAUT, J.C., J. Non-Newt. Fluid Mech., 20,117-141 (1986).

Quasilinear hyperbolic systems are well behaved because they do not permit Hadamard instabili ty to occur. To explain this, consider the homogeneous problem associated with (50.28) and a trial solution vector of the form u - u0 e x p [ i k ( / z - x - At)], where k is real. Subst i tut ion leads to the equation (50.29). Hence, if (50.28) is hyperbolic (evolutionary will do), then A is real and so, the ampl i tude of the solution will not grow without bounds.

Now, as we have seen earlier in connection with a problem arising from elasticity, the equations of motion of a continuous medium are not always hyperbolic. To show tha t this si tuation may arise in fluid mechanics as well, we shall discuss an example from ideal fluid flow, viz., the s teady two-dimensional, irrotational, isentropic flow of an ideal fluid. In te rms of the velocity v - (u, v), the governing equations are

(C2--U2)~X--UV(~+~X ) -.~-(C2--V2)-'~ "- O,

OU OV = O, (50.30)

Oy oz

where c is the sound speed. Let us assume tha t c 2 - u 2 # 0. Then,

0v 0v + B - ~ - - O, (50.31) Ox

where

B - (_2UV/(C2_U2) 0 . (50.32)

Since the x-direction is time-like, the eigenvalue problem is obtained by taking the unit v e c t o r / ~ - j. Then, we have

de t [ -A1 + B] -- 0, (50.33)

which leads to the eigenvalues

~1,2 -- [--UV 4- C(U 2 ~- V 2 -- C2)1/21/(C 2 -- U2). (50.34)

These eigenvalues are real if u 2 + v 2 > c 2 and the equations are hyperbolic if the local speed Iv] is supersonic; if u 2 + v 2 < c 2 the eigenvalues are complex and the local speed is subsonic, with the equations being elliptic. 1~

50.6 Plane Flows of a JRS Fluid

The const i tut ive equation of a JRS 11~ fluid may be writ ten as:

Dt -- 2~/~ § f(S), (50.35)

109JEFFREY, A., Quaszlinear Hyperbolic Systems and Waves, Pitman, London, 1976. 11~ D.D., RENARDY, M. and SAUT, J.C. Arch. Rational Mech. Anal., 87, 213-251

(1985).


where the objective t ime rate of S is defined through

7)8 dS 7:)t = - ~ + S W - W S - ~ (DS + SD). (50.36)

Here, D and W are, respectively, the symmetric and skew-symmetric parts of the velocity gradient, and ~ C [-1,1] is a scalar. On the right side, f is a tensor valued function of the stress tensor S, but not its derivatives. The above equation is thus a generalisation of the Maxwell model; indeed, by choosing a - 1, and f(S) - - S , one recovers the Maxwell equation (34.105).

For convenience, let the stress matrix S in a plane flow have the form 111

Using this, the steady, plane flows of the fluid (50.35) may be written as

Oq Oq _ f/X, (50.37) B I ~ q-B2 N

where q is a column vector with components ( u , v , # , v , a , p ) . The matrices BI and B2 and the vector f depend on q, but not on its derivatives. The eigenvalue problem associated with this is d e t [ - ~ A + B] = 0. This leads to the equation

(1 + ~2)(v -- ~u) 2 [p(v - ~u) 2 + � 8 9 #)(~2 _ 1)

1 + 2 o r (r + + + = 0, (50.38)

where ~ -- r/0/A. This has two imaginary roots: ~ = :t=i; also, it has double real roots ~ -- u / v ; and, finally, the terms in the square brackets lead to two more. These are given by

_ _b q- ( b 2 - ac )1 /2 , (50.39) a a

where

1 ~) �89 ~) a -- /~--pu 2 q - ~ # ( l q - -- -- ,

b = a - p u v , (50.40)

c = ~ _ ~ 2 I -- ~(1 -- ~) q- �89 + ~).

Clearly, these are real when b 2- ac :> O. Whatever the six roots may be, the quasilinear system (50.37) describing the steady, plane flow is neither elliptic nor hyperbolic.

Let us now examine the situation with respect to an unsteady, plane flow of the JRS fluid. This leads to a system of the form

_ ~ 0q 0q A + B I ~ x x q- B 2 ~ " - f/A, (50.41)

111See JOSEPH, D.D. and and SAUT, J.C.J. Non-Newt. Fluid Mech., 20, 117-141 (1986) .

where the first term is the column vector (put, pvt, #t, vt, at, 0), i. e., the matrix A is singular. If one examines the loss of evolution of this system, i.e., when it does not possess real eigenvalues for every unit vector/z in the (x,y) plane, it is found that the conditions are

lu(1 + a)] < O, l u ( l - a ) + ltt(X + c0][/3- 1 # ( 1 - a ) + o2 _ [ /~_ 7

~~,(1 - ~) - 1 ~#(1 + a ) - / 3 < O. (50.42)

Hence, whenever the above inequalities are met, the unsteady plane flows of the JRS fluid lose their evolutionary character and Hadamard instabilities appear.

Surprisingly, there is a connection between this loss of stability and the nature of the equation governing the vorticity. The equation for the vorticity w in a plane, unsteady flow of the JRS fluid is given by

02w 0w 02w 02w 02w p - - ~ + 2 p ( v - V ) - - ~ - a-~x2 - 2b OxOy - c"z-Zay. + l - O, (50.43)

where l is a lower order term. The characteristic polynomial associated with this is

po0 2 + 2 p o o ( ~ , + . o 2 ) - ~ 1 - 2bOlO~ - ~ - o. (50.44)

This is a quadratic equation in a0. For this to possess real roots, the following inequality must hold:

(~ + p~)O~l + 2(b + p ~ . ) o , o ~ + (~ + p ~ ) o ~ > 0 (50.45)

for all a l 2 + a22 = 1. Equivalently, the requirements are

+ pu ~ > o, (b + p~ . )2 _ (~ + pu ~) (~ + p~2) > o. (50.46)

If one uses the expressions for a, b, c from (50.40) in this inequality, it turns out that the resulting inequalities are the same as those in (50.42). One may thus conclude that: the quasilinear system governing the unsteady, plane flow of a JRS fluid is evolutionary if and only if the equation which determines the vorticity is evolutionary.

On the other hand, if the vorticity equation of an inertialess steady flow is hyperbolic, the quasilinear system loses its evolutionary behaviour; conversely, if the vorticity equation of a steady inertialess flow is elliptic and a > O, where a is given by (50.40) with p = O, then the quasilinear system is of evolutionary type.

Now, whether the flow be steady or not, suppose that the vorticity equation loses its stability in the Hadamard sense. Then, the amplitude of the vorticity will oscillate wildly leading to instability of the base flow. If the wavelength is short enough, then one may visualise the transport of vorticity as a moving surface across which the vorticity suffers a jump. Since this aspect may be examined by the theory of acceleration waves, we turn to this next.

50.7 Accelerat ion Waves

The theory of acceleration waves is based on the concept of a propagating surface 112 in a continuous medium across which the acceleration suffers a jump. Since other

kinematical variables suffer jumps as well, let us recall the basic ideas behind such surfaces here. If x -- x ( X , t ) is a motion, then a propagating singular surface of order 2 is one across which:

1. All the first order derivatives are continuous; i.e., the deformation gradient F and the velocity v are continuous.

2. The second order derivatives suffer jumps, i.e., the acceleration a, the velocity gradient L, and the material gradient of F are all discontinuous. Because L has a jump, so does the vorticity vector.

3. It is clear tha t the strain history does not suffer a jump across such a wave because it is the product of deformation gradients whose histories remain intact at the wave front. Hence, a constitutive relation based on the L2 norm of the strain history leads to the fact tha t the stress tensor is continuous across the propagating surface.

4. However, because the acceleration vector suffers a jump, the divergence of the stress tensor must do so.

It may be shown 113 tha t if the speed of propagation of such a wave is U and the unit normal to the wave in the direction of traverse is n, then the jump in acceleration and the material gradient of F are connected through

[a~] - U2A~, (50.47)

[F~,~] - Fja Fk~nj nk A~ ,

where Ai is called the ampli tude vector. In incompressible materials, all waves are transverse and thus A~n~ - O .

Let the constitutive relation for the simple fluid be (cf.(33.3))

<

Then, the gradient of S is

VS = 5~ ' (Ct ( t - s ) iVCt( t - s)). (50.49)

This shows immediately tha t the divergence of the stress tensor is determined by the linear functional 5~ . Hence, if the motion is viscometric, we see tha t the nearly viscometric flow kernels are important; if the motion is an extensional flow, then the nearly extensional flow kernels are important 114 and so on.

112For a thorough treatment, see TRUESDELL, C.A. and TOUPIN, R.A., The Classical Field Theories, Handbuch der Physik, III/1, FLUGGE, S. (ED.), 8pringer-Verlag, 1960. 113HUILGOL, R.R., Arch. Mech. Stos., 25, 365-376 (1973). 114HUILGOL, R.R., J. Non-Newt. Fluid Mech., 5 ,219-231 (1979).


Using the result tha t the jump in the gradient of the pressure field is proportional to the unit normal, it follows that the speed of propagation and the ampli tude are related through an acoustic tensor Q. The exact relation is

where

Q~j(n)Aj - pU2A,, (50.50)

-- 2n, npnknm Jzpmkl(Ct(t- s ) , t - s ) ( C t ( t - s))jl ds

- - ~ ' , m k t ( C t ( t - s), t - s ) ( C t ( t - s))jl ds . (50.51)

As an example, consider the simple shear flow x - ~y, 9 - z - 0. If an acceleration (shear) wave, with a jump in acceleration in the x-direction, moves in the y-direction, then the equation of motion in the x-direction shows that:

1. The jump [Op/Ox]- 0 because there is no pressure gradient in the flow.

2. Hence, [Oa/Oy]- p[al], where a is the shear stress. However

Oa da 04/ Oy dqOy

Put t ing all of the above together, it follows tha t the wave speed is given by pU 2 - da/d~; i.e., the speed of an acceleration wave is determined by the tangential modulus. Of course, this claim may also be proved by using (50.51) and the results from nearly viscometric flows. 115

50.8 Growth of Accelerat ion Waves

The next question tha t arises is this: what happens to the amplitude of the acceleration wave as it moves into the fluid? In order to examine this, it is necessary to take the displacement derivative, i.e., the derivative across the propagating surface, of the jump in acceleration and other essential components. These lead to an evolution equation for the ampli tude of the jump in acceleration which shows tha t there is a critical value for the ampli tude of this quantity. If the initial value exceeds it, the ampli tude gets bigger and eventually reaches infinity in a finite time; if it is less than the critical amplitude initially, it dies to zero exponentially. 116

It may be conjectured tha t if the shear wave does become one of infinite amplitude, then this is equivalent to the velocity suffering a tangential jump. Tha t is, one has a propagating vortex sheet. We shall examine this aspect next.

50.9 Propagating Vortex Sheets

A singular surface of order 1 is one across which both the velocity and the deformation gradient suffer a jump. Since incompressibility demands tha t the normal

115See HUILGOL, R.R., Arch. Mech. Stos., 25,365-376 (1973). 116COLEMAN, B.D. and GURTIN, M.E., J. Fluid Mech., 3, 165-181 (1968).

component of the velocity be continuous, i.e., there is no (normal) shock, one is left to s tudy the propagation of vortex sheets. These are surfaces across which the tangential components of a velocity suffer a jump.

Because the deformation gradient suffers a jump, the stress tensor is also discontinuous. If one assumes the constitutive relation for a simple material to have the form

s ( t ) = 0 < < oo, (50.52)

it follows tha t the history F ( t - s), 0 < s < oo, is the same whether the singular surface is approached from one side or the other. Hence, the jump in the stress arises solely from the discontinuity in F(t) across the surface. Thus, the jump in the stress is given by

[S](t) -- G(F( t - s), F - (t)) - G(F(t - s), F+( t ) ) , (50.53)

where F - and F + are the limiting values from behind and ahead of the wave respectively. Let IF] = F - - F +, and define a secant modulus E S through 117

~ ( F ( t - s), F - ( t ) ) - ~ ( F ( t - s), F + ( t ) ) - ES(t)[F](t). (50.54)

That is, we expect the material to behave instantaneously like an elastic body; see the Appendix to Chapter 5.

Using (50.54), it follows tha t the wave speed of a propagating vortex sheet 11s is given by

pU2(t) = E s (t). (50.55)

Since the secant and tangent moduli are not necessarily identical, it follows tha t if an acceleration wave were to evolve into a propagating vortex sheet, its speed has to change as well. In addition, it is easily seen through the Rayleigh problem for a Maxwell fluid tha t a jump in the velocity at the boundary does move as a vortex sheet through the fluid. This jump, which is described by a Heaviside function~ is accompanied by a jump in the acceleration which behaves like a delta function. 119 Hence, it is not enough tha t the jump in the acceleration become unbounded; it must do so in a very specific fashion for the singular surface to turn into a vortex sheet.

So far there is no proof that a shear, acceleration wave can become unbounded in this manner. In addition, given tha t the growth and decay results have all been derived within the context of the L2 norm based constitutive theories, it is not clear how one may study the growth and decay of kinematical entities which behave like generalised functions. A whole new way of deriving the various jump conditions and their utilisation has to be proved to tackle such matters.

5 0 . 1 0 N o n l i n e a r H y p e r b o l i c E q u a t i o n s

It is obvious tha t singular surface theory has been able to predict the desired wave speeds and the conditions under which certain kinematical quanitities become

117CHEN, P.J. and GURTIN, M.E., Arch. Rational Mech. Anal., 3{}, 33-46 (1970). 11sHUILGOL, R.R., Int. J. Engng. ScL, 11, 75-86 (1973). 119HUILGOL, R.R., J. Mec. Theor. Appl., 4, 725-739 (1985).


unbounded across a propagating singular surface. There is no proof tha t these kinematical variables evolve into more singular forms.

Hence, it becomes necessary to investigate if nonlinear hyperbolic theory provides any answers, t2~ This is because such equations permit the development of nonsmooth solutions even if the initial data is smooth. Indeed, consider a constitutive relation for the shear stress in the form 121

o - f ( ( ) , (50.56)

where f(-) is an odd, nonlinear function of a modified 'shear rate history' (~, defined to be

r t) = _~. o~ N ( y , t - ~) d~. (50.57)

Now, let u -- '~y + ~i, where we consider a superposition of ~i on the steady simple shear flow. Set

C ~

w l (y , t ) -- e - ' ~ s - - ~ ( y , t -- s ) ds ,

o O 0

w (y,t) = f e - ~ ( y , t - s) ds. 0

Then it is easily seen tha t OWl Off - ~ + ~wi = ~ .

Using this in the equation of motion and setting p - 1, one obtains

where

OWl o] -- - ~ W 2 ) - - OtWl, Ot

g J

Ow2 Owx Ot Oy '

(50.58)

(50.59)

(50.60)

(/0 ) / (w2) -- f e-~8[~ + - ~ ( y , t - s)] ds . (50.61)

The system (50.60) is hyperbolic and genuinely nonlinear. It can be shown tha t however smooth the initial data, provided it is large enough, the solution to the pair of equations remain continuously differentiable for a finite t ime only. A glance at (50.58) shows tha t this means tha t cOu/Oy will not be continuously differentiable, or that the vorticity stops being C 1. This does not prove, however, tha t the vorticity is not C ~ Some numerical work 122 on a similar class of problems also suggests loss of C 1 smoothness, but not C ~

It is interesting tha t in the case of the creeping, plane flow of the JRS fluid, loss of stability may occur due to elastic effects only, whereas here, it is entirely due to nonlinearity in the viscosity.

t2~ a review of this area, as well as the role played by singular kernels in the propagation and development of singularities, see RENARDY, M., Ann. R.ev. Fluid Mech., 21, 21-36 (1989).

i2iSLEMROD, M., Arch. Rational Mech. Anal., 68, 211-225 (1978). 122MARKOWICH, P. and RENARDY, M., S I A M J. Numer. Anal., 21, 24-51 (1984).

7 Computational Viscoelastic Fluid Dynamics

Analytic solutions to non-trivial viscoelastic flow problems are rare due to the complexities of the constitutive equations and the nonlinearities of the conservation equations. To make any progress, we have to abandon the search for the exact analytic solution and seek a numerical solution. The theoretical framework for computational fluid mechanics has been well established, especially in the traditional finite difference (FDM), finite volume (FVM), finite element (FEM) and boundary element methods (BEM); for example, see the books by AMES, 1 BEER and

3 FLETCHER,4 JOHNSON 5 WATSON, 2 CROCHET, DAVIES and WALTERS, KIM and KARRILA, 6 PATANKAR, 7 PHAN-THIEN and KIM, s POZRIKIDIS, 9

1 AMES, W.F., Numerical Methods ]or Partial Differential Equations, Barnes and Noble, New York, 1969.

2BEER, G. and WATSON J.O., Introduction to Finite and Boundary Element Methods ]or Engineers, Wiley, New York, 1992.

3CROCHET, M.J., DAVIES, A.R. and WALTERS, K., Numerical Solution o] Non-Newtonian Flow, Elsevier, Amsterdam, 1984.

4FLETCHER, C.A.J., Computational Techniques ]or Fluid Dynamics, Vol. 1 and 2, Springer- Verlag, New York, 1988.

5JOHNSON, C., Finite Element Methods, Cambridge University Press, 1990. 6KIM, S. and KARRILA, S.J., Microhydrodynamics: Principles and Selected Applications,

Butterworth-Heinemann, Boston, MA, 1991. 7PATANKAR, S.V., Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing Corpo-

ration, New York, 1980. 8PHAN-THIEN, N. and KIM, S., Microstructures in Elastic Media: ~ Principles and Compu-

tational Methods, Oxford University Press, New York, 1994. 9POZRIKIDIS, C., Boundary Integral and Singularity Methods ]or Linearized Viscous Flow,

Cambridge Univ. Press, 1992.

398 7. Computational Viscoelastic Fluid Dynamics

REDDY and GARTLING, 1~ and ZIENKIEWICZ and MORGAN. 11 This theoretical background, when combined with continuing advances in computational engineering, both in the hardware and software, especially in the area of parallel and distributed computing, will make it possible in the near future to explore the complex flow behaviour of realistic constitutive models.

In this Chapter, our aim is not to provide a comprehensive reference to computational methods, but rather a guide to some computational techniques dealing with viscoelastic fluids. Hence, this Chapter is not meant to be a replacement of the above standard texts in numerical methods, but rather a discussion of the important issues which arise in numerical viscoelastic fluid mechanics. It starts with an overview, followed by specific sections reviewing standard domain discretisation methods, and their applications. The main causes of the high Weissenberg number problem and the failure of general-purpose codes at a low-to-moderate Weissenberg number have been well understood; and they will be reviewed here.

51 Overview

In any numerical solution procedure, the governing equations are turned into a set of algebraic equations, possibly nonlinear, by a process of discretisation. During the process, the computational domain is divided into a collection of non-overlapping sub-domains, or finite elements, and the search for the exact continuous solution is abandoned in favour of a piecewise solution whose local profile on each finite element will be unimportant in the limit of large number of grid or node points, or elements. A numerical discretisation process is said to be convergent if in the limit of infinite grid points (zero mesh size), the numerical solution converges to the exact solution. It is obvious that it would be difficult to prove convergence, since we do not know the exact solution in the first place. However, it should be possible to establish numerical convergence by successively halving the mesh size, although this may be expensive in computer time, and observing the behaviour of the numerical solutions. A plot of the 'error' with respect to the numerical solution at the finest mesh will establish the degree of convergence of the method. The consistency of the method is also of importance. Here, the term implies that in the limit of zero mesh size, the set of algebraic equations resulting from the discretisation process should reduce to the original governing equations. This seems rather obvious, but there are methods that do not deliver consistency. The concept of numerical stability is also most important in computational mechanics; it is concerned with the question of whether a numerical disturbance, which is always present in a numerical solution, grows or decays. A numerical instability may or may not be related to a flow- induced instability.

Prescribing relevant boundary conditions is also a part of the numerical solution procedure. Here, the type of the governing equations dictates the kind of bound-

I~ J.N. and GARTLING, D.K., The Finite Element Method in Heat Transfer and Fluid Dynamics, CRC Press, Florida, 1994.

tlZIENKIEWICZ, O.C. and MORGAN, K., Finite Elements and Approximations, Wiley, New York, 1983.

51 Overview 399

ary conditions that can be legitimately prescribed, and it would be beneficial at this stage to digress briefly to examine some issues in the classification of partial differential equations, which has been discussed in detail in w

51.1 Class i f icat ion

Consider a second-order PDE in two dimensions

A O2u 02u _02u Ou 0u u) = 0, + S OxOy + C ' - ~ + f ( x , y, u, "~x ' -~ ' (x,y) e D, (51.1)

where A, B, C and F are functions of x, y, u, Ou/Ox, and Ou/Ox. We have the following classification scheme if for (x,y) E D,

B 2 - 4AC < O,

B 2 - 4AC - 0,

B 2 - 4 A C > O,

elliptic PDE,

parabolic PDE,

hyperbolic PDE.

(51.2)

Note that the classification scheme depends only on the coefficients of the highest- order derivatives.

BC needed

BC needed

BC needed

BC needed

v

y~

FIGURE 51.1. Schematic diagram of an elliptic problem.

In an elliptic system, which is primarily a model of a diffusion process at steady state, disturbances are instantaneously felt everywhere, and diffuse outward from the sources. Any discontinuity in the boundary conditions will be smoothed out in the interior of the domain. The important feature of an elliptical problem is that these conditions are needed everywhere on the boundary. Legitimate boundary conditions can be Dirichlet (the unknown is prescribed), or Neumann (the normal gradient of the unknown is prescribed), or Robbins boundary conditions (a combination of the unknown and its normal gradient is prescribed). If only Neumann boundary conditions are specified, then there may be other constraints imposed by the equation that must be met by the boundary conditions for a unique solution.

Steady state heat conduction, Stokes flows, steady state laminar flows, elasticity problems are all elliptic. A schematic diagram of an elliptic problem is illustrated in Figure 51.1.

P r o b l e m 51 .A Show that the Laplace equation"

02u 02u Au = ~ + ~ = 0 (51.A1)

is elliptic in the square domain (0 < x, y < 1). Given the boundary conditions

u(x, 0) -- sin 7rx, u(x, 1) = sin 7rx exp(-Tr), u(0, y) -- 0 - u(1, y),

find the solution by a separation of variables. , , ,

l characteristics

r

u specified

FIGURE 51.2. Schematic diagram of a hyperbolic problem.

Time dependent problems either lead to hyperbolic PDEs (in characteristic problems), or parabolic PDEs (in dissipative problems). The main feature of hyperbolic PDEs is tha t there is a set of characteristics along which the system evolves. The distance along the characteristics can be thought of as time; if this distance is used as an independent variable, then the PDEs can be simplified into total differen- tials. Boundary or initial conditions (the distinction between the two is somewhat blurred for hyperbolic problems) need to be prescribed at the inflow, along a plane that intersects all characteristics. Since there is no dissipative mechanism in hyperbolic PDEs, discontinuity in the boundary conditions will be preserved along characteristics. If these characteristics converge (the variable would have different values on each characteristic), a discontinuity in the normal derivative across these

51 Overview 401

convergent characteristics will result, i.e., a shock wave. In some problems, the solution may only exist for a finite time! Most viscoelastic constitutive equations are hyperbolic in nature, and so are steady state supersonic flows. A schematic diagram of a hyperbolic problem is given in Figure 51.2.

P r o b l e m 51.B Show that the wave propagation equation

02U (~2U = 0 (51.Ul)

Ot 2 Ox 2

is hyperbolic in the strip 0 < t < oo, 0 < x ~ 1. By introducing the characteristics

r ~ - x - t ,

simplify the PDE and show that the solution takes the form

u(x , t) -- f (r + g(~), (51.B2)

where f and g are some arbitrary functions to be determined from the boundary conditions. Given the initial conditions

0~ (~, 0) = ~(~), ~(~,0) = ~(~),

find f and g. Thus, show that the solution is

1 [ f /Lo ( s ) d s ] . (51.B3) ~(~, t) - 7 ~o(~ + t) + ~o(~ - t) + ~+~ J x - - t

Hence, at the point (x, t), only the initial conditions between the two characteristics emanating from ( x - t, 0) and (x + t, 0) determine the solution.

t

BC needed

Propagation in time

T

IC needed

BC needed

r

x

FIGURE 51.3. Schematic diagram of a parabolic problem.

Time dependent problems with a dissipative mechanism lead to parabolic PDEs. The main feature is that the solution propagates forward in time, but diffuses in

space; when the solution is steady, the problem becomes elliptic. Appropriate initial conditions are Dirichlet, and the appropriate boundary conditions are combinations of Dirichlet and Neumann boundary conditions. A schematic diagram of a parabolic problem is given in Figure 51.3.

, ,

P r o b l e m 51 .C Show that the unsteady heat conduction problem

0u o~u - ~ - Ox--- ~ - - 0 (51.C1)

in the strip 0 ~ t < oo, 0 ~ x _ 1 is parabolic. For the initial condition

u(x, O) -- sin lrx,

and the boundary conditions

t ) - 0 = t )

show tha t the solution is

u ( x , t) = sin 7rx exp(-~2t). (51.C2)

Note the decaying nature of the solution in contrast with tha t in hyperbolic problems.

51.2 F o u r i e r M e t h o d

The three-dimensional second-order scalar problem takes the form

02u A " V V u + F = O, A~j Ox~Oxj + F - O, (51.3)

where A = [A~j] is a constant second-order tensor, and F contains all the lower derivatives, including u. Let )~ be an eigenvalue of A. Then

�9 If all the )~'s are non-zero, and of the same sign, the PDE is elliptic.

�9 If all the )~'s are non-zero, and of the same sign except one, the PDE is hyperbolic.

�9 If any of the )~'s is zero, then the PDE is parabolic.

A system of equations in several unknowns can be recast into a set of first-order PDEs:

Ou Ou Ou A - ~ x 4 - B - - ~ 4- C . ~ z = D, (51.4)

where A, B, C, and D are some constants (for a linear system). Then the characteristic determinant is formed:

det [AAx 4- BAu + CAz] = 0, (51.5)

where X -- (Ax, Au, Az) is the direction normal to a surface at point x.

51 Overview 403

�9 This surface is a characteristic surface if all the roots (Ax, Au,A~) are real, and the system is hyperbolic.

�9 If no real root is obtained then the system is elliptic.

�9 If less than n roots are obtained, where n is the order of the characteristic polynomial, and none of the roots is complex, then the system is parabolic.

For a system of equations in several unknowns and in a three-dimensional setting, the Fourier method is convenient for the classification of PDEs. Here, it is more convenient to work in the Fourier transform space, recalling tha t the Fourier transform of a gradient is

( w ) = ka,

where the Fourier transform pair is defined by

fi(k) - (27r) -3/2 i CkXu(x) dV(x),

u(x) -- (2~) -3/2 / e-ikx'f ,(k) dV(k),

and k = (kx,kv, kz) is the Fourier variable. In this approach, any nonlinearity in the system is disregarded and a Fourier transform is taken. This produces a homogeneous system whose determinant is set to zero to obtain the characteristic polynomial. All lower order terms in k are then discarded (since the characteristic polynomial only takes the highest order derivatives into account), and we find that:

�9 If all the roots for (ikz, ikv, ikz) are real, then an elliptic system results.

�9 If there is no real root for (ikx, ikv, ikz), then the system is hyperbolic.

�9 If there are less than n roots for (ikx, ik v, ikz), and none is complex, then the system is parabolic.

An example is needed here to illustrate the idea. Consider Laplace's equation

02u 02u 02u + + = o.

Taking the Fourier transform,

2

we obtain the characteristic polynomial

0 , =

which has no real zeros. The P D E is therefore elliptic.


As a second example, consider the two-dimensional Navier-Stokes equations

Uz --b vy

uux + vu~ -F- Px - v(uxx + u ~ )

uvx + vvu + P~ - v(vx~ + vu~)

After a Fourier transform, we have

- 0,

-- 0,

-- 0.

ikx ik~ 0

2 i(~,k~ + v~,) + ~,(~ + ~) i~

0 --0.

The determinant is set to zero to obtain

(k~ + k~) (i(~k. + ~k,) + ~(k~ + k~)) = 0.

Discarding lower-order terms, we obtain

(k~ + k~12 = O.

Thus the system is elliptic.

P r o b l e m 51 .D Show tha t the unsteady two-dimensional Navier-Stokes equations are parabolic.

P r o b l e m 5 1 . E Show tha t the two-dimensional upper convective Maxwell equations are hyper-

bolic.

51.3 BoundaryConditions

U given

V - 0

U • v - 0

t~=0, t/=0 U ' l l - 0

v

v-O,t~-O w

tx=0

vffi0

FIGURE 51.4. Boundary conditions for an extrusion problem.

We now return to a more detailed description of the boundary conditions in computat ional fluid mechanics.

51 Overview 405

With the Dirichlet boundary condition, the velocity field is prescribed:

u - uo, on Su.

This type of boundary condition is also called an essential boundary condition. Neumann boundary conditions are given in terms of the traction vector:

t - T - n - - t o , on St.

This yields the normal traction,

~ n t~ = n- T - n = - P + 2 w ~ = n- to,

g i n on St,

and the tangential tractions

t - t - n n = t 0 - t 0 - n n , on St.

In FEM or BEM, traction boundary conditions appear naturally, in the process of integration by parts, and are called natural boundary conditions.

Note that we use Su to denote parts of the boundary where velocity is prescribed, and St, parts of the boundary where the traction is prescribed. These surfaces need not be singly connected, but may consist of several patches. Indeed both velocity and traction boundary conditions can be prescribed at a given location, but in different directions, as in the case of the extrusion problem. For example, see Figure 51.4.

Robbins boundary conditions arise in some slip-stick problems, where the slip- page velocity is determined by the tangential traction (shear stress) at the wall. 12

Free Surface Boundary Conditions

For problems with a free surface, e.g., bubbles, extrusion, etc., the location of the free surface is not known and must be found as part of the solution procedure. Here, a kinematic constraint can be used as the condition to locate (implicitly) the free surface. In a steady flow, the kinematic constraint for a free surface is

u - n = 0 . (51.6)

For unsteady flow, if r t) = 0 is the location of the free surface, then r always remains zero on the free surface and its material derivative must also be zero there. The free-surface kinematic constraint thus takes the form

0r --- + u - V ( : - O . (51.7) Ot

Since n = =kV(J IVr is a normal unit vector on the free surface (the sign can be chosen so that n is the outward normal unit vector), this constraint also takes the form

1 0r t - u - n = 0. (51.8) Ivr


AC

t

FIGURE 51.5. A surface element AS, with a bounding curve AC.

In addition to this, the traction on the free surface is known from the physics of the problem. If there is no surface tension, for example, then the traction vector is zero on the free surface. For the case where the surface tension is not negligible, we recall tha t the surface tension 7 is postulated to be the force per unit length acting along the edges of the free surface. The equilibrium condition on an arbi trary surface element AS reads

/ S C

7 q dl = O,

where [ T ] - n - (T + - T - ) - n is the jump in the normal traction, with T + being the stress on the positive side of n, and T - the stress on the negative side of n, and q is the unit vector normal to the boundary curve AC, but tangential to the interface AS; see Figure 51.5. This is the mathematical s ta tement of zero force on AS.

P r o b l e m 51 .F From Stokes' theorem, show tha t

/ACTq dl = /AS V7 dS-- /AST (V" n)n dS. (51.F1)

, , , i , , , ,

Thus, since AS is an arbi trary surface, the constraint on the free surface with a surface tension 7 is

(T + - T - ) - n V 7 - ' 7 ( V - n ) n - O. (51.9)

Since the total stress is usually expressed as - p l + r , where ~" is the extra stress (not necessarily having zero trace), the above condition is equivalent to

(p- - p+) n + (I -+ - ~--) - n + V 7 - 7 (V- n) n - 0. (51.10)

In the limit of no flow, the extra stress is zero, and we have

p - - p + -- 7 V - n, (51.11)

12RAMAMURTHY, A.V., J. Rheol., 30, 337-357 (1986); HATZIKIRIAKOS, S.G. and KALOGERAKIS, N., Rheol. Acta, 33, 38-47 (1994).

51 Overview 407

which is a well-known result in thermostatics, when we recall that the mean curvature of the surface is given by 13

1 1 (51.12) V " n = "~ii 4- R2"

For a spherical bubble of radius R to exist, the jump in the pressure must be at least 27/R. Furthermore, the boundary condition implies that a non-uniform surface tension must necessarily lead to a flow, since a pressure jump alone is generally not sufficient to balance the term ~77; this is referred to as Marangoni flow, a review of which has been given by LEAL. 14

Symmetry Boundary Conditions

Symmetry boundary conditions are very useful to reduce the size of the computation domain and are quite easy to state: a plane of symmetry has no normal velocity component, and no tangential stress (tangential tractions or vorticity are zero), i.e.,

u-n=O, t - t - n n = O , (51.13)

where t is the surface traction.

Inflow and Outflow Boundary Conditions

At the inflow to the solution domain, we know something about the flow, usually the velocity field. For viscoelastic fluids, the stresses are also required there (in essence, these represent the information carried with the fluid from its previous deformation history). The boundary conditions at the inlet are usually Dirichlet boundary conditions. Note that all components of the stress cannot be arbitrarily prescribed. First, the stress components must be consistent with the constitutive equations, otherwise non-physical stress boundary layers will be set up. Secondly, if traction boundary condtions are also given there, then the traction component tangential to the boundary would involve the stress components alone without the pressure terms, and therefore these stress components cannot be prescribed arbitrarily. Finally, RENARDY 15 has pointed out that specifying all the stress components at the inflow may lead to an over-determined mathematical problem.

To have a clear understanding of why this is so, consider the two-dimensional steady flow of the Maxwell model (relaxation time ~ and viscosity v/), where the governing equations can be put in the form

M 0 ~ + N - ~ - f, (51.14)

in which q = p, T = ,

13KELLOGG, O.D., Foundations o/ Potential Theory, Dover, New York, 1953. 14LEAL, L.G., Laminar Flow and Convective Transport Processes. Scaling Principles and As-

ymptotic Analysis, Butterworth-Heinemann, Boston, 1992. 15RENARDY, M., ZAMM, 65,449-451 (1985); J. Non-Newt. Flusd Mech., 36,419-425 (1990).

and

f = r ,

M _ _

N _ ~

0 0 - 1 1 0 0

0 0 0 0 1 0

1 0 0 0 0 0

o o o o

- r ~ - ( r ~ + r ; / ~ ) 0 0 u~ 0

0 -2r~v 0 0 0 u~

0 0 0 0 1 0

0 0 - 1 0 0 1

0 1 0 0 0 0

-2rx~ 0 0 uu 0

- (%~ + r//,X) - r ~ 0 0 u~

o o o o

The characteristics ofthis quasi-linear system are determined by

(y 'ux- u~) 2 [1 + (y,)Z] [A (y,)2 + 2By'+ C] -O,

Uy

where y' -- dy/dx and

(51.15)

A - - T ~ + 7 / / A , B=- -T~v , C--Tvu+~/ /A.

Here, (51.15) consists of three factors. The first factor represents streamlines - they are real characteristics of multiplicity 2. The second factor has two imaginery roots - they represent the elliptic part of the governing equations. The last factor leads to hyperbolicity, if B 2 - AC > 0, o r ellipticity, if B 2 - AC < 0, a s discussed in w

Now, by themselves, the constitutive equations are hyperbolic with streamlines as characteristics of multiplicity 3. Thus, the specification of all the stress components leads to a well-posed problem. Coupled with the equations of balance, however, the streamlines are of multiplicity 2 only, as seen from (51.15). RENARDY 16 has examined the case where the last factor in (51.15) remains elliptic, and concludes that we need to specify two boundary conditions on the whole boundary (one for each elliptic part), and two boundary conditions for the stress at the inflow boundary. Hence, specifying all the stresses will result in an over-determined problem. He further notes that by assuming fully developed flow conditions at the inflow we often inadvertantly satisfy all the requirements on the inflow boundary conditions.

16RENARDY, M., ZAMM, 65,449-451 (1985).

51 Overview 409

It may be prudent in practice to retain the t ime derivative terms in the constitutive equations, even though we may be dealing with steady-state solutions, and integrate the equations in t ime through an initial stress s tate until a steady state solution is reached, including the stress components at the inflow. This ensures tha t a physical solution is obtained.

At the outlet, the flow is usually well developed and arranged so tha t a unidirectional flow results. Outflow boundary conditions therefore usually take the form of no transverse velocity and no axial traction.

Slip or No-Slip Boundary Conditions

The no-slip boundary condition is usually assumed at a solid surface, where the fluid velocity assumes the velocity of the solid surface. This assumption works well for viscous fluids, but there is a large amount of experimental data suggesting tha t it may not be relevant for polymeric liquids in some circumstances. There are extrusion experiments with polymer melts 17 which suggest tha t wall slip may be responsible for melt fracture. In these experiments, the occurence of the extrudate irregularities occurs above a critical wall shear stress, which is accompanied by a fluctuation in the pressure drop. Data on LLDPE suggest a lack of adhesion between the melt and the wall, and the critical shear stress is about 0.14 MPa 18 to 0.26 MPa. 19 LIM and SCHOWALTER 2~ have been able to distinguish four flow regimes in the extrusion of narrow molecular weight polybutadienes. The first flow regime is associated with a smooth, glossy extrudate and a Newtonian flow behaviour. The second flow regime occurs at a wall shear stress of the order 0.1 MPa, from which a loss of gloss occurs. The third flow regime starts at a wall shear stress of the order 0.2 MPa, where the shear stress becomes independent of the shear rate, and a fluctuation in the pressure drop sets in, which is an indication of a slip-stick flow pattern. The fourth flow regime is associated with a non-zero velocity at the wall, but with a reduced frequency and amplitude of the pressure fluctuations. Overall, the extrusion flow becomes unstable at a critical shear stress, or at a critical recoverable strain. The latter is equivalent to a Weissenberg number of the order 1 - 10.

A phenomenological approach to the slip boundary condition has been proposed by PEARSON and PETRIE, 21 where the slip velocity is taken as an empirical function of the wall shear stress. A polymer network model has been proposed recently to account for the dynamic slip velocity. 22 It is likely tha t real progress in this area will be made by a careful consideration of the microstructure near a solid surface.

17BAGLEY, E.B., CABOT, I.M. and WEST, D.C., J. Appl. Phys., 29, 109-110 (1958); KRAYNIK, A.M. and SCHOWALTER, W.R., J. Rheol., 25, 95-114 (1981); RAMAMURTHY, A.V., J. Rheol., 30, 337-357 (1986); KALIKA, D.S. and DENN, M.M., J. Rheol., 31, 815-834 (1987); LIM, F.J. and SCHOWALTER, W.R., g. Rheol., 33, 1359-1382 (1989).

18RAMAMURTHY, A.V., op. cir. (1986). 19KALIKA, D.S. and DENN, M.M., op. cir. (1987). 2~ example, LIM, F.J. and SCHOWALTER, W.R., op. cir. (1989). 21PEARSON, J.R.A. and PETRIE, C.J.S., Proc. 4th Int. Cong. Rheol., Part 3,265-282 (1965). 22HATZIKIRIAKOS, S.G. and KALOGERAKIS, N., Rheol. Acta, 33, 38-47 (1994).


Boundary Conditions for the Pressure

Boundary conditions on the traction vector arise naturally from a weighted residual method. However, in the finite volume method the equation for the pressure is usually solved separately from the velocity field, by using the continuity constraint on the linear momentum equations, and therefore boundary conditions for the pressure are required.

The correct boundary conditions for the pressure can be derived either from the traction boundary condition, or from the momentum equations. Thus, on writing the total stress T - - p l + I", a traction boundary condition will turn into a Dirichlet boundary condition for the pressure:

p = r : n n - t 0 - n , on St. (51.16)

Otherwise, a Neumann boundary condition for the pressure will result from the momentum equation:

{ (0 ) } 0 - ~ = n - --p ~ u - l - u - V u + V - r + p b , onSu . (51.17)

Note that a traction boundary condition at one point on the boundary will implicitly set the pressure. If there is no traction boundary condition, then the pressure can only be determined up to an arbitrary constant. It is important to keep in mind tha t traction boundary condition is not the same as the pressure boundary condition; the former is a physical quanti ty that we can actually impose on the fluid, the latter is a derived quantity, arising only because we are interested in solving the Poisson's equation for p in isolation. If the set of equations, continuity, momentum, are solved jointly, then p would inherit the correct boundary conditions from the boundary conditions for u and t, and there is no need to impose anything on p at all!

P r o b l e m 51.G Consider the flow of a Newtonian fluid on a plane of symmetry, where the bound-

ary condition is u - n - 0. Show tha t

Symmetry also requires

0p (v/Au + pb) (51.G1) ' - " n �9

cOut = 0 , (51.G2)

On where ut is the tangential velocity along the tangential direction t on this plane of symmetry. Using the continuity equation, show that

i)2 un n . / k u -- i)n2 = 0, (51.G3)

and the boundary condition for p is

Op (51.G4) 0--~ -- n - p b .

51 Overview 411

Boundary Conditions for Vorticity

The vorticity-stream function approach is often used in finite volume schemes dealing with Newtonian fluids, especially in two dimensions. The approach is not popular in viscoelastic flow calculations and we only remark tha t if both the stream- function/vorticity equations are solved in a coupled manner, not in isolation, then there is no need for boundary conditions on the vorticity. If boundary conditions are truly needed for the vorticity, then they are derived from either the vorticity t ransport equation or from the continuity equation, which leads t o / k r + w = 0 for two-dimensional flows, where r is the stream function.

Initial Conditions

For t ime-dependent flows, a set of initial conditions is required. The initial velocity prescribed must be divergence free. For viscoelastic fluids, a set of initial values for the stress components is also needed. Note tha t the incompressibility constraint also forbids an impulsive start and stop, normal to the boundary, since

s u . n d S = O.

However, we could prescribe

u - n -- U (1 - exp(-)~t)) .

This would satisfy the continuity condition and allows for a very quick start or quick stop, as required.

Thermal Boundary Conditions

Although most of the viscoelastic flow problems reported in the literature are isothermal, it has been generally acknowledged tha t non-isothermal effects are important in all polymer processes and should be included whenever possible. In these cases we need to specify the boundary conditions for the temperature on all parts of the boundary. These take the form of a Dirichlet boundary condition

T - T0(x, t), on ST, (51.18)

where T0(x, t) is the prescribed temperature, or the form of a Neumann boundary condition

K : V T n + Qc + Qr = Q0(x, t ) , on SQ. (51.19)

Here, K is the thermal conductivity tensor of the fluid; we have tacitly assumed tha t Fourier's law of heat conduction is valid, although there are non-Fourier phenomenological and microstructural theories available. 23 Also, Qc and Qr are the conductive and radiative heat transfers respectively, and Q0(x, t ) is a prescribed function. The convective and radiative heat transfers can be written as

Qc - he(x, T, t) ( T - Ta) , (51.20)

i

23HUILGOL, R.R., PHAN-THIEN, N. and ZHENG, R., J. Non-Newt. Fluid Mech., 48, 83-102 (1992); van den BRULE, B.H.A.A. Rheol. Acta, 28,257-266 (1989).


Q,, = h ~ ( x , T , t ) ( T - T r ) , (51.21)

where hc is the (empirical) heat transfer coefficient, with Ta being the ambient temperature, and hr is the radiative heat transfer coefficient, with Tr being a reference temperature. For a black body enclosure, h~ = aF(T 2 + T~r)(T- T~), where a is the Stefan-Boltzmann constant and F is a form factor. For a finite enclosure, the form of the radiative heat transfer is more complicated. 2a

51.~ Nature of So lu t ions

Consider the following P DE

with the boundary conditions

s = f, in D,

Bu = g, on S.

A strong or classical solution of this system is a function u that satisfies all the ditferentiability requirements imposed by the governing equations, i.e., it is a member of a certain function space, and it also satisfies the governing equations everywhere ~ in D (the closure of D).

Obviously, this concept of a strong solution is too stringent, and in most numerical techniques such as FEM, we abandon the search for the classical solution in favour of the weak solution, which is an element of a different space, say Sh. A function uh is said to be the weak solution if V ~ E Sh,

(q~ -- (r f )

and (v. , B h) = (V., g),

where the angular brackets denote an inner product, for example, the natural inner product

( f ' g) = ]D f " g dV(x),

and integration by parts can be carried out to any required level to lessen the differentiability requirements on the weak solution uh. The subscript h refers usually to the degree of mesh discretisation. The idea is that if the projections of the residuals formed from the governing equations are zero along any direction in the space Sh, then it must be zero everywhere in this space. This space consists of functions that are usually piecewise constant, piecewise linear, and so on. 25 It is clear that a strong solution is also a weak solution; but the converse is not true. We shall return to this in w later.

The problem is said to be well-posed, if

24REDDY, J.N. and GARTLING, D.K., The Finite Element Method in Heat Trans]er and Fluid Dynamics, CRC Press, Florida, 1994.

25See JOHNSON, C., Numerical Solutions of Partial Din'erential Equations by the Finite Ele- ment Method, Cambridge University Press, Cambridge, 1987.

52 Finite Difference Method 413

�9 The solution exists and is unique,

�9 and the solution depends on the data in a continuous fashion.

In computational work we tend not to be too concerned with the questions of existence and uniqueness of the solution, although they are important issues: obviously we would want to know in advance if there is a unique solution to the problem. Wi th the complexity of the consitutive equations, it is expected, however, tha t bifurcations and multiple solutions will exist. Indeed various investigations have shown the occurrence of limit and bifurcation points in numerical solutions, for example, MENDELSON et al., 26 Y E H et al., 27 JOSSE and FINLAYSON, 2s and C R E W T H E R et al. 29 Methods for tracking limit and bifurcated points, and to determine the stability of different solution branches (path tracking, parameter continuation techniques, degree theory, etc.) are quite well-known; for example, see IOOSS and JOSEPH, 3~ KELLER, 3t KUBI(~EK and MAREK. 32 The multiplicity of solutions may be due to the constitutive equations or could be a side effect of the numerical scheme.

The last requirement of a well-posed problem really says tha t if the boundary or initial conditions change by a small amount, then tha t should cause the solution to change by also a small amount. Failure to do this renders the problem i l l -posed

in the sense of HADAMARD; 33 an example of which is given in w above. An ill-posed problem is due mainly to an inappropriate formulation; it could also

be a physical manifestation of the problem itself. In the case of an inappropriate formulation (an example is the first-kind integral equation formulation in Stokes flow for the mobility problem), it may be possible to obtain a good numerical solution provided tha t we do not "push" the solution too far. However, this represents an unsatisfactory solution process in the sense tha t as the mesh is refined, the numerical solution ceases to converge.

52 Finite Difference Method

The finite difference method (FDM) is the most widely known numerical technique for solving partial differential equations, dating back to the relaxation method of

26MENDELSON, M.A., YEH, P.W., BROWN, R.A. and ARMSTRONG, R.C., J. Non-Newt. Fluid Mech., 10, 31-54 (1982).

27yEH, P.W., KIM, M.E., ARMSTRONG, R.C. and BROWN, R.A., J. Non-Newt. Fluid Mech., 16, 173-194 (1984).

28JOSSE, S.L. and FINLAYSON, B.A., J. Non-Newt. Fluid Mech., 16, 13-36 (1984). 29CREWTHER, I., HUILGOL, R. R. and JOZSA, R., Phil. Trans. Roy. Soc. Lond., A337,

467-495 (1991). 3~ G. and JOSEPH, D.D., Elementary Stability and B~]urcation Theory, Springer-

Verlag, New York, 1980. 31KELLER, H.B., in Applications of Bifurcation Theory, pp. 359-384. Ed: RABINOWITZ, P.

H., Academic Press, New York, 1977. 32KUBI(~EK, M. and MAREK, M., Computational Methods in Bifurcation Theory and Dissi-

pative Structures, Springer-Verlag, New York, 1983. 33HADAMARD, J., Lectures on Cauchy Problem in Linear Partial Differential Equations,

Dover, New York, 1952.


SOUTHWELL 34 of the pre-computer era. The basic idea behind the method is that the governing equations are turned into a set of algebraic equations using local expansions of the variables, using a truncated Taylor's series. The final set of equations, which is generally banded, is solved by an iteration method, such as the Newton-Raphson procedure. Although the technique has been widely used in different areas of research, its use in viscoelastic fluid mechanics has not been predominant. Some illustrative problems include the flow past a stationary and rotating cylinder, 35 the flow past a sphere and a bubble, 36 the flows past a protuberance, over a hole and in L-shaped and T-shaped geometries, 37 and the flow through a contraction. 3s

It is fair to say that that the FDM is relatively easy to implement than other techniques, but only for simple geometries, or in those cases that can be reduced to similarity solutions. 39 There are several excellent texts dealing with the subject

34SOUTHWELL, R.V., Relaxation Methods in Theoretical Physics, Oxford University Press, London, 1946.

35TOWNSEND, P., J. Non-Newt. Fluid Mech., 6,219-243 (1980). 36ADACHI, K., YOSHIOKA, N. and SAKAI, K., J. Non-Newt. Fluid Mech., 3, 107-125

(1978); TIEFENBRUCK, G. and LEAL, L.G., J. Non-Newt. Fluid Mech., 10, 115-155 (1982); CHILCOTT, M.D. and RALLISON, J.M., J. Non-Newt. Fluid Mech., 29, 381-432 (1988).

37pERERA, M.G.N. and WALTERS, K., J. Non-Newt. Fluid Mech., 2, 49-81 (1977); 2, 191- 204 (1977); DAVIES, A.R., WALTERS, K. and WEBSTER, M.F., J. Non-Newt. Fluid Mech., 4, 325-344 (1979); PERERA, M.G.N. and STRAUSS, K., J. Non-Newt. Fluid Mech., 5, 269- 283 (1979); TOWNSEND, P., Rheol. Acta, 19, 1-11 (1980); COURT, H., DAVIES, A.R. and WALTERS, K., J. Non-Newt. Fluid Mech., 8, 95-117 (1981); HOLSTEIN, H. and PADDON, D.J., J. Non-Newt. Fluid Mech., 8, 81-93 (1981); COCHRANE, T.C., WALTERS, K. and WEBSTER, M.F., Phil. Trans. Roy. Soc. Lond., A301, 163-181 (1981), J. Non-Newt. Fluid Mech., 10, 95- 114 (1982); CROCHET, M.J., DAVIES, A.R. and WALTERS, K., Numerical Simulation of Non- Newtonian Flow, Elsevier, London, 1984.

3SPERERA, M.G.N. and WALTERS, K., J. Non-Newt. Fluid Mech., 2, 191-204 (1977); GATSKI, T.B. and LUMLEY, J.L., J. Comp. Phys., 27, 42-70 (1978); COCHRANE, T.C., WAL- TERS, K. and WEBSTER, M.F., Phil. Trans. Roy. Soc. Lond., A301, 163-181 (1981); DAVIES, A.R., LEE, S.J. and WEBSTER, M.F., J. Non-Newt. Fluid Mech., 16, 117-139 (1984); SONG, J.H. and YOO, J.Y., J. Non-Newt. Fluid Mech., 24, 221-243 (1987); J. Non-Newt. Fluid Mech., 29, 374-379 (1988).

39PHAN-THIEN, N. and TANNER, R.I., J. Fluid Mech., 129, 265-281 (1983); PHAN-THIEN, N., J. Fluid Mech., 128,427-442 (1983); Rheol. Acta, 22, 127-130 (1983); Rheol. Acta, 23, 172- 176 (1984); J. Non-Newt. Fluid Mech., 16, 329-345 (1984); J. Non-Newt. Fluid Mech., 17, 37-44 (1985); Rheol. Acta, 24, 15-21 (1985); Rheol. Acta, 24, 119-126 (1985); Quart. Appl. Math., XLV, 23-27 (1987); WALSH, W.P., geit. angew. Math. Phys., 38, 495-511 (1987); HUILGOL, R.R. and KELLER, H.B., J. Non-Newt. Fluid Mech., 18, 101-110 (1985); HUILGOL, R.R. and RA- JAGOPAL, K.R., J. Non-Newt. Fluid Mech., 23, 423-434 (1987); MENON, R.K., KIM-E, M.E., ARMSTRONG, R.C., BROWN, R.A. and BRADY, J.F., J. Non-Newt. Fluid Mech., 27,265-297 (1988); LARSON, R.G., J. Non-Newt. Fluid Mech., 28, 349-371 (1988); PHAN-THIEN, N. and ZHENG, R., Zeit. angew. Math. Phys., 41,766-781 (1990); Rheol. Acta, 30, 491-496 (1991); JI, Z., R A J A G O P A L , K.R. and SZERI, A.Z., J. Non-Newt. Fluid Mech., 36, 1-25 (1990); DOR- REPAAL, J.M., J. Fluid Mech., 183, 141-147 (1986); DORREPAAL, J.M. and LABROPULU, F., geit. angew. Math. Phys., 43, 708-714 (1992); PHAN-THIEN, N., ZHENG, R. and TAN- NER, R.I., J. Non-Newt. Fluid Mech., 41, 151-170 (1991); CREWTHER, I., HUILGOL, R.R. and JOZSA, R., Phil. Trans. Roy. Soc. Loud., A337, 467-495 (1991).


of FDM, for example one may consult CROCHET, DAVIES and WALTERS, a~ and R I C H T M Y E R and MORTON. 41

52.1 Path-Tracking

To illustrate some of the important ideas in the FDM, we consider the following one-dimensional problem

s = g,

where/2 is some ordinary differential operator in x E [0, 11, f (x; A) is the unknown, with A being a parameter, and g is a given function (of x and possibly of f) . The discretisation process is carried out by dividing the interval [0, 1] into N sub- intervals, often of equal length 5 = 1 I N (although this is not necessary). Derivatives (of up to fourth order) of f are approximated by the central finite difference scheme, i.e.,

f ; __ f j+l -- f j - I n f j+l -- 2f j + Yj- I + 5 = + ,

{ l l l j j =

f~+2 - 2f~+1 + 2 f r fr 2~3 + O(~2),

f JV __ fj+2 -- 4fj+l -4" 6fj -- 4 f j -1 "4- f j - 2 # "+ 0(~2),

where f j is the nodal value at the node xj -- (j - 1)~; i.e., f j -- f (xj) . Dirichlet boundary conditions for f and g are imposed at the end points, x = 0, and x = 1. Fictitious nodes are invented at x = - 6 and at x = 1 + 5 to satisfy the derivative boundary conditions at the end points. Effectively, the method searches for the function values defined at nodal points.

In general a derivative is approximated by

0 u i _ 1 Oxj -- A x j E cku~ (x + k A x j e j ) + E,j , (no sum in j ) ,

k

where Eij is an error term, and ck are the coefficients to be found either by

�9 expanding the unknown function at point x + k A x j e j about x, then taking a linear combination of ckui (x + k A x j e j ) killing all other terms except Ou~ / Oxj ;

* or, given the nodal points to be used, a polynomial interpolant to ui is constructed using all the available data points and then differentiating the resulting expression.

When the error term E~j = 0 (IAxj ]P), with p > 0, the finite difference scheme is said to be consistent and of order p. The central schemes (using symmetric

4~ M.J., DAVIES, A.R. and WALTERS, K., Numerical Solution of Non-Newtonian Flow, Elsevier, Amsterdam, 1984.

41RICHTMYER, M.D. and MORTON, K.W., Difference Methods for Initial Value Problems, Wiley, New York, 1967.

points about x) noted above are all consistent, and of second order. To keep the approximation at high orders at the boundary, we usually need a one-sided scheme, or fictitious nodes have to be invented with the help of the boundary conditions. A finite difference approximation scheme in graphical form is sometimes referred to as a template, or a computational molecule.

P r o b l e m 52.A Show that the above approximations are accurate to the order indicated. Further,

show that (no sum on i)

Ou, 3ui(x) -- 4 u , ( x - Ax, e,) + u,(x -- 2Axle,) (x) -- 2Ax, + O ([Ax,[2) , (52.A1) Oxj

Ou~ 1 Oxj (x) = 12Ax, { - u , ( x + 2Ax, e,) + 8u,(x + Ax, e,) (52.A2)

- + o �9

The operator

/:DU(X) -- E cku (x + kAxjej) (no sum in j) (52.A3) k

is called a finite difference operator. Show that exp iw-x is an eigenfunction of/:D with eigenvalue

l(w- Ax) -- E ck exp ikw. Ax. (52.A4) k

This shows that the only observable frequencies on the grid are those which satisfy I~-Axl _< ~.

In general, the governing equations may consist of M separate equations, possibly representing different physical laws (e.g., conservation laws, constitutive equations). Following a discretisation process, the governing equations become a set of nonlinear algebraic equations

G,(r = 0; i - 1 , . . . , M , j = 1 , . . . , M N , (52.1)

where N is the number of nodes, excluding those at which the Dirichlet boundary conditions are specified, A is a parameter, and Cj, j -- 1, 2 , . . . , M N represents the solution vector ( M N is the total number of unknowns). This set of equations can be solved by the Newton-Raphson procedure. This is accomplished by first taking a Taylor's approximation about the initial guess r (~ �9

oG~ (~(o) ~), c,(~, ~) - c~(~ (~ ~) + , ~ . - -~ ,

where A~b- ~b- ~(0); this increment is obtained by solving a linear system

j . ~ = _a(V(0), ~) (52.2)


in which J -- OG(~5 (~ , A)/O~ is the Jacobian of the system. Techniques for solving a set of linear equations include LU decomposition (direct method), frontal solvers, 42 iterative strategies (matrix splitting methods including Jacobi, Gauss-Seidel, relaxation methods, conjugate gradient methods including ORTHODIR, ORTHOMIN, ORTHORES, 43 and GMRESa4). With large scale and three dimensional problems, iterative solution strategies are possibly the most efficient ways to obtain a solution, and we provide a brief review of some popular methods in the Appendix.

Returning to the linear system (52.2), we recall that the condition det (J) ~ 0 implies a unique solution A~5, and the point p - {rg, A} is said to be a regular point. In the neighbourhood of a regular point the Implicit Function Theorem guarantees the existence of a unique solution ~ = ~()~) which is also continuous in A. Otherwise (det (J) = 0), the point p is said to be a singular point; it may be a limit point (the solution curve in A doubles up), or a bifurcation point (multiple solutions). To explore the behaviour of the solution for a range of parameter values, possibly including bifurcation and limit points, a parameter continuation technique may be used, i.e., given a solution point P0 = {~0,A0}, a nearby solution point Pl = {(~1 , ~ 1 } is generated, where )~1 = )~0 -{- A~. To negotiate possible limit and bifurcation points, the list of the independent variables is extended to include the parameter )~, and the discretised system (52.1) is augmented with an arc length equation 45

N ( r = + - 1 -- 0, (52.3)

where s is the t rue arc length along the solution curve. The unknown vector ~5 and the parameter A are considered as functions of s along the arc length.

To generate the solution point Pl at sl -- so + As from the solution point P0 at s -- so, we use the Newton iteration scheme:

(/)1 : (~0 -~- i ( ~ , ~1 -- )~0 "~- i )~ , (52.4)

where

A ( s 0 ) - - , A)~ N(~0, A0)

(52.5)

in which /

_ G (V0, 0) A(s0)

N~ (~50, A0)

\ (V0,

) N~(~0, A0)

(52.6)

and J - G4, - OG/O~, G ~ - OG / Oh, N~ - ONIOn, and N~ -- ON/Oh. To classify the singular point, we recall tha t the linear system J x -- y has a

solution if and only if n - y - 0, where n is the null solution of jT , ie., J T n -- 0

42HOOD, P., Int. J. Numer. Methods Eng., 10,379-399 (1976). 43JEA, K.C. and YOUNG, D.M., Lm. Alg. Appl., 52, 399-417 (1983). 4aSAAD, Y. and SCHULTZ, M.H., SIAM J. Sci. Stat. Comp., 7,856-869 (1986). 45KELLER, H.B., in Applications of Bifurcation Theory, pp. 359-384, Ed: RABINOWITZ. P.

H., Academic Press, New York, 1977.


(note tha t det (J) = 0). Now suppose tha t P0 = (r is a solution point, i.e. det (J) -- 0 at this point, and tha t n is the null solution of j T . We take the dot product of n and the first of (52.5) to obtain

n-G~AA = O.

If G~ is in the range of 2, ie., there is a solution of J x - Gx, then n is perpendicular to G~, and AA can be non-zero, and point P0 is a l i m i t po in t . Otherwise, if G w i is

not in the range of J then the point P0 is a b i f u r c a t i o n po in t . Techniques for jumping onto a bifurcated branch are outlined in KELLER, 46 KUBI(~EK and MAREK, 47 and R H E I N B O L D T and B U R K H A R D T . 48

The method has been used in a number of similarity flow problems with the Oldroyd-B fluid. 49

52.2 Two-Dimensional Problems

Two-dimensional viscoelastic flow problems solved by the FDM tend to have simple geometries and boundary conditions, so tha t a s t ructured mesh can be constructed allowing simple finite difference approximations to be employed. In almost all cases, a vort ici ty-stream function formulation is used (resulting from the experience of solving Navier-Stokes equations), where the s t ream function ~p and the vorticity w are defined, respectively, by

0 r 0 r (52.7) Ux -- Oy ' Uy -- -- OX ,

Ou~ Ou~ - = _ A r (52.8)

0x 0y

where /k is the Laplacian operator. Wi th these derived variables, the continuity equation is satisfied identically, and the pressure can be eliminated from the momen tum equations, leading to, with an appropriate normalisation,

46KELLER, H.B., in Applications of Bifurcatwn Theory, pp. 359-384, Ed: RABINOWITZ. P. H., Academic Press, New York, 1977.

47KUBICEK, M. and MAREK, M., Computational Methods in Bifurcatwn Theory and Dissi- pative Structures, Springer-Verlag, New York, 1983.

48RHEINBOLDT, W.C. and BURKHARDT, J.V., A C M Trans. Math. Software, 9, 215-235 (1983).

49WALSH, W.P., Zeit. angew. Math. Phys., 38,495-511 (1987); HUILGOL, R.R. and KELLER, H.B., J. Non-Newt. Fluid Mech., 18, 101-110 (1985); HUILGOL, R.R. and RAJAGOPAL, K.R., J. Non-Newt. Fluid Mech., 23, 423-434 (1987); MENON, R.K., KIM-E, M.E., ARMSTRONG, R.C., BROWN, R.A. and BRADY, J.F., J. Non-Newt. Fluid Mech., 27, 265-297 (1988); LAR- SON, R.G., J. Non-Newt. Fluid Mech., 28, 349-371 (1988); PHAN-THIEN, N. and ZHENG, R., Zeit. angew. Math. Phys., 41, 766-781 (1990), Rheol. Acta, 30,491-496 (1991); JI, Z., RA- JAGOPAL, K.R. and SZERI, A.Z., J. Non-Newt. Fluid Mech., 36, 1-25 (1990); DORREPAAL, J.M., J. Fluid Mech., 163, 141-147 (1986); PHAN-THIEN, N., ZHENG, R. and TANNER, R.I., J. Non-Newt. Fluid Mech., 41, 151-170 (1991); CREWTHER, I., HUILGOL, R.R. and JOZSA, R., Phil. Trans. Roy. Soc. Lond., A337, 467-495 (1991).


where Re = p U L / r I is the Reynolds number, p is the fluid density, and r is the normalised extra stress (with respect to ~?U/L, in which r / is a scale viscosity, U, a velocity scale, and L is a length scale). In addition to these, one has a set of constitutive equations; for the Maxwell fluid,

( ) Tzz + W i u . VTxx -- 2"-~-x rXZ -- 2 - ~ ' r z u -- 2 0 x '

( Ou Ou~ ) Ou~ Ou~ r zy + W i u . V r z y - O : rxx - Oy "ruu -- Oy ~ Oz '

_ Ouy Ou~ ~ Ou~ + u - - = 2 (59,.10)

A second-order central finite difference scheme is usually adopted, with the boundary conditions for the vorticity implied from the boundary conditions for the stream function. This scheme tends to be quite unstable, or restricted to a low grid Reynolds number, due to the absence of the diffusion terms in the vorticity t ransport equation. The remedy for this is either to adopt the Oldroyd-B model, which is more realistic for polymer solutions, or by an explicit introduction of the viscous term, ~" = 2r/sD + S, where r/s is chosen arbitrarily. 5~ Alternatively, upwinding can be introduced into the finite difference scheme. This simply means tha t the first-order gradients of the vorticity are approximated by one-sided first-order schemes, depending on the sign of the local velocity. If the local convective velocity is positive (information is coming from the left hand side of the nodal point), then a backward finite difference scheme is used; otherwise a forward finite difference scheme is used if this velocity is negative. The O(h) error in this approximation is (h is a mesh spacing)

1 R e ( lu l l O~ w 02 w ~

which vanishes when h --, 0, but may still be important at a high Reynolds number, or in a region where abrupt changes in w occur. One can a t tempt to restore the second-order nature of the approximation scheme by introducing upwinding only when needed by a more continuous switching method, sl The numerical problems associated with solving the vorticity equation are, in fact, less serious than those associated with the constitutive equations, since the Reynolds number is small in all cases of interest. We shall return to a discussion of upwinding in the finite volume method later.

Succesful finite difference approximations on the constitutive equations are all first-order accurate in space, with an embedded upwinding scheme, which is associated with a false stress diffusion of the order

2 W i h JuxJ~x2+Juu l~- - ~ ~-ij,

5~ M.G.N. and WALTERS, K., J. Non-Newt. Fluid Mech., 2, 49-81 (1977), 2, 191- 204 (1977); TIEFENBRUCK, G. and LEAL, L.G., J. Non-Newt. Fluid Mech., 10,115-155 (1982).

51pATANKAR, S.V., Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing Corpo- ration, New York, 1980.


1.0

Re=3 Wi=O

0.3

R e = 3 ~ ~ ) Wi-0.38

0.3 0 . . . . .

1.0 1.0

Re =6 Wi=O

0.3

Re=6 Wi=0.38

1. 5

F I G U R E 52.1. Flow of a 4-cons tan t Oldroyd fluid over a deep hole. A1 = 0.45, )~2 -- 0.075, /z : 0.0007.

which can be large when the Weissenberg number is high, or in problems where there are steep stress boundary layers. A example of a finite difference calculation of a flow past a deep slot for a four-constant Oldroyd fluid is shown in Figure 52.1, after COCHRANE et al., S2 where the agreement with experimental data is good. The Maxwell model, however, fails to converge at a Weissenberg number of 0.38. The main feature of the ftow is that the vortex retreats further into the slot.

52.3 Time-Dependent Problems

Time derivatives bring some other complications to finite difference schemes: the phenomena of dissipation and dispersion. This may be easily understood by considering the Cauchy problem

Ou Ou _ 02u 03u = + +

a '~x3 , O x z u(z, o) = ~ ,

52COCHRANE, T.C., WALTERS, K. and WEBSTER, M.F., Phil. Trans. Roy. Soc. Lond., A301, 163-181 (1981).

which has the solution

u(x, t) -- exp (iw(x + at) -- bw2t -icwat).

In a purely convective problem (b = c - 0), the initial profile translates along the time axis with a speed of lal independent of w. When b ~> 0, the amplitude of the profile decreases in time, which typifies a diffusion process. And when c ~ 0 the profile translates along the t ime axis with a frequency-dependent speed - this is a dispersion phenomenon. We can investigate a finite difference scheme in the same manner, noting tha t exp iwx is an eigenfunction of any finite difference operator with eigenvalue l(wAx), see Problem 51.G. Thus, if s is a finite difference approximation for s then the approximate solution is exp (iwx + l(wAx)t) ; dissipation or dispersion behaviour of the numerical scheme can be found by expanding l(wAx) for small wAx.

P r o b l e m 52.B Consider a purely convective problem 0u /0 t + Ou/Ox = 0. Show that a second-

order central finite difference scheme for Ou/Ox is purely dispersive. Show tha t a first-order backward finite difference scheme for Ou/Ox has a false diffusion of 1Ax(O2u/Ox2). 2

Now, having applied spatial discretisation to the governing equations, we invariably end up with a set of first-order differential equations in t ime to be integrated, subject to a set of relevant initial conditions. Time integration schemes can either be single-step methods, where the variables at the next t ime step depend only on the previous t ime step (implicit Euler's, or implicit backward schemes), or multistep methods (predictor-corrector methods). Here, an incorrect choice for the t ime step may result in numerical instability. The existence of such an instability can be investigated for a general single-step method, in which the discretised equations are

t + a t ) - t ) ,

where ,So and L D a r e some finite difference operators. We seek a solution of the form (separation of variables)

u(x, mat) = y~ [Ak(~)l ~ Uk(~)exp ico- x. k,to

Substi tuting a single Fourier mode into the discretised equations yields a generalised eigenvalue problem

Ak(w)SDUk(w) --l(r Ax)Uk(w) ,

which can be solved for the amplication factor Ak. If IAkl < 1 then the method is stable; otherwise it will be unstable. In general, if the method is stable and consistent, it is also convergent to the exact solution in the limit of Ax, At --, 0. This stability analysis is called the von Neumann analysis.

P r o b l e m 52.C

For the parabolic problem 0 u / 0 t - 02u/Ox 2, show that the explicit scheme

At u(x, t + At) - u(x, t) + ~ (u (x + Ax, t) - 2u(x, t) + u (x - Ax, t)) (52.C1)

leads to the amplification factor

4At ( w A x ) A - 1 - A x i s i n 2 - - - - . (52.C2)

Deduce tha t the scheme is stable if 2At/Ax 2 < 1. This is usually referred to as the Courant-Priedrichs-Lewy (CFL) condition.

The stability of multistep methods can also be investigated by a yon Neumann analysis, although one has to solve a much more complicated eigenvalue problem.

We close this discussion with a v o n Neumann analysis of a numerical scheme for the creeping flow of a second-order fluid. 53

P r o b l e m 5 2 . D The stream function in a two-dimensional flow for the second-order fluid satisfies

(cf. (46.2)): ~0

~ 2 ~ ) _ ~1 U" V (/~2~)) __ 0, (52.O1) 2~?0

where 70 and ~0 are the constant viscosity and first normal stress coefficient for the fluid, and u is the •uid velocity. Construct a central finite difference scheme for f -- /k2r using a uniform grid. Assume tha t u - Uex is constant, perform a von Neumann analysis and show tha t the flow is stable only if

~lV v 2r/0 h = A ~ < 1. (52.02)

53 Finite Volume Method

The finite volume method (FVM) has been very popular in computational fluid dynamics dealing with Euler and Navier-Stokes flow problems. There are two main approaches in the finite volume method.

53.1 Chorin- Type Methods

The first approach uses an artificial compressibility condition to satisfy the continuity equation. A pseudo-transient formulation is then adopted in the momentum equations and the steady state solution is considered as the asymptotic solution of a t ime-dependent problem with time-independent boundary conditions (if need be), and these are computed by a time-marching scheme. The method is due to

53TANNER, R.I., J. Non-Newt. Fluid Mech., 10, 169-174 (1982).

53 Finite Volume Method 423

CHORIN, s4 and is very commonly used in computational fluid dynamics dealing with high Reynolds number flows. Integration in time can be carried out either implicitly or explicitly. In implicit schemes, it is necessary to solve a system of equations at each time step, making the scheme very expensive for large three- dimensional problems. However, such schemes can be made unconditionally stable and relatively large time steps can be used. On the other hand, in explicit schemes, it may be necessary to solve only a simple mass matrix, or it may not be necessary to solve any system of equations at all. The main disadvantage of explicit schemes is tha t the time step is restricted by the CFL stability condition. However, there are several ways to accelerate convergence such as residual averaging, local time-stepping and multigrid schemes. PHELAN et al. ss use a similar scheme, but with a modification to suit hyperbolic types of constitutive equations, for solving cavity-driven flow problems for the UCM model. The same method has also been used to solve the slip-stick problem of Maxwell-type constitutive equations. 56 We briefly review the method here.

We first recall that if the fluid is compressible, then the density can be written, using the definition of the isothermal compressibility, as

p - p0[1 + ( P - P0)/~], (53.i)

where P0, P0 are some reference density and pressure, and/3 is the inverse of the isothermal compressibility. Under practical conditions ( p - p o ) / / 3 << 1. The conservation of mass would then take the form

dp + [~ + (p _ P0)]V" u = 0, (53.2) dt

where d ( . ) / d t denotes the material t ime derivative. The false-transient governing equations for an incompressible fluid at zero Reynolds

number flow can thus be rewritten as

0p + / 3 V - u = 0, (53.3) 0t

0u (53.4) P 0 - ~ = V - r .

Since we are interested in the steady state, inertialess solution, 3 and P0 can be both set to unity. Note that (53.3) can be regarded as a penalty-type method for finding the pressure field.

The method is easily adapted to differential constitutive equations. We illustrate the methodology with Chilcott and Rallison's model, s7 where the stress is written in terms of a structural tensor A as

-- -~- f (R) A, (53.5) 7" A

54CHORIN, A.J., J. Comput. Phys., 2, 12-26 (1967). 55pHELAN, F.R., Jr., MALONE, M.F. and WINTER, H.H., J. Non-Newt. Fluid Mech., 32,

197-224 (1989). 56 JIN, H., PHAN-THIEN, N. and TANNER, R.I., Comp. Mech., 13,443-457 (1994). 57CHILCOTT, M.D. and RALLISON, J.M., J. Non-Newt. Fluid Mech., 29, 381-432 (1988).


in which f(R) represents the FENE spring force law,

1 R 2 = trA, (53.6) f(R) - 1 - R2/L 2'

and A evolves according to (cf. (34.100)):

OA )~-~-- -- f ( R ) ( 1 - A ) - R ( u - V A - L A - ALT). (53.7)

In these equations, R represents the extension of the microstructure (modelled as a dumbbell), L represents the ratio of the length of a fully extended dumbbell to its equilibrium length (in the UCM model L is infinite), )~ is the relaxation time, % is the 'solvent-contributed' viscosity, and ~p is the 'polymer-contributed' viscosity.

The computational domain f~ is discretized into a set Th of non-overlapping triangular cells Ki (the choice is rather arbitrary here; for unstructured mesh, triangular cells are probably the best). The governing equations, (53.3-53.7) are now integrated over a finite "volume" K 5s with boundary OK, using the divergence theorem,

jfK ~ .-~ d~

/K Op �9 -~ df~

~ an

-- ~o n-o" dF, K

-- L n - u d F , K

- - /Ir [ . f (1-A) - )~(LA+ALT)] d~-- )~~K n. (uA) aT '

where all the coefficients for the time derivatives have been put to unity and n is the outward normal to OK~ at a point in dF.

In the simplest form of the finite volume method (but by far most commonly used in CFD problems), the integral equations for each cell are evaluated by assuming that the integrands in the volume integrations are constant in each-cell and equal to their values at the center of the cell. A set of ordinary differential equations in time is obtained. For example, we have, for K~ E Th,

-- n-~r dF

.... = --- n- u dF, 0t A~ K,

0A~ : rL|f(1 -- A ) - )~(LA + ALT)|,lj

0t - n - ( u A ) dF,

where the subscript i indicates the variables for the cell Ki and Ai is the area of Ki. To calculate the integration over edge j of Ki, the values Pedge, Ue~ge and Aedge on the edge can be computed by taking the average of the values in the cells on either side of the edge. This simple scheme is in fact second-order accurate in space on a grid patch consisting of uniform equilateral triangles. 59

58In two-dimensional problems, the volume of a cell is actually its area. 59JAMESON, A. and MAVRIPLIS, D., AIAA Journal, 24, 611-618 (1986).


The set of ordinary differential equations resulting from this discretisation process can be integrated in time using a Runge-Kutta scheme, for example. The time steps will be restricted by the CFL condition. But the convergence to the steady state solution can be accelerated by using a local time step Ati which reflects the size of the local element:

Ai At~ - ~ A t ~ ,

where At~ and A~ are, respectively, the time step and area of cell K~, At~ and A~ are the corresponding time step and area of the smallest cell. In this way disturbances are expected to sweep a large cell in about the same number of iterations as with a small cell, and yet the numerical stability is maintained.

The method has been used succesfully in the slip-stick problem, where convergent solutions were obtained at a Weissenberg number of O(10). 6~ The same problem has also been solved by finite element methods, sl where convergence up to a Weissenberg number of the order 30 has been reported. The main feature of the solution, different from that of the Maxwell fluid, is the appearance of an overshoot in the velocity along the wall, shortly after the exit from the stick boundary. At W i - 10, the amount of overshoot is about 10%. Furthermore, it takes a longer exit length for the development of a full plug flow. This is because of the quantity R 2 -- t rA, representing the amount of extension of the microstructure, increases to a near maximum value L 2 on approaching the singularity, and then decays monotonically to its equilibrium value. At W i - 10, the decay rate of R 2 is low, so that it takes virtually the whole exit length (10 units) for R 2 to reach its equilibrium value. In contrast, there is little microstructural extension along the centreline of the flow. As L --, oo, one approaches the Oldroyd-Maxwell models; these are more difficult to compute and need longer exit regions.

5 3 . 2 S I M P L E R - T y p e M e t h o d s

The second approach in finite volume method is much more popular and forms the basis of some commercial packages dealing with Navier-Stokes equations. The resulting algorithms are known as SIMPLE (Semi-Implicit Method for Pressure- Linked Equations Revised), SIMPLER (SIMPLE Revised), 62 SIMPLEC (SIMPLE Consistent), 63 or PISO (Pressure-Implicit with Splitting of Operators). 64 In these schemes, the momentum equations are solved sequentially, and the pressure is corrected from the continuity equation and the linearised momentum equations.

Applications of the SIMPLER algorithm to viscoelastic flows include the 44o- 1 entry flow and the die entry problems, 65 and the non-isothermal flow past a

6~ H., PHAN-THIEN, N. and TANNER, R.I., Comp. Mech., 13,443-457 (1994). 61APELIAN, M.S., ARMSTRONG, R.C. and BROWN, R.A., J. Non-Newt. Fluzd Mech., 27,

299-321 (1988); KING, R.C., APELIAN, M.R.; ARMSTRONG, R.C. and BROWN, R.A., J. Non-Newt. Fluid Mech., 29, 147-216 (1988).

62pATANKAR, S.V., Numerical Heat Transfer and Fluid Flow, McGraw-Hill, New York (1980). 6aVAN DOORMAAL, J.P. and RAITHBY, G.D., Num. Heat Transfer, 7, 147-163 (1984). 64ISSA, R.I., J. Comput. Phys., 62, 40-65 (1985). 65YOO, J.Y. and NA, Y., J. Non-Newt. Fluid Mech., 39, 89-106 (1991); NA, Y. and YOO,

J.Y., Comp. Mech., 8, 43-55 (1991).


cylinder. 66 The PISO method has been applied to the non-circular pipe flow of the Criminale-Ericksen-Filbey model (38.24) to study the effects of the weak secondary flow on the pressure drop. 67 A variant of the SIMPLER method, termed SIMPLEST (SIMPLE with Splitting Technique) has been used recently to study secondary flows in non-circular pipe flows. 6s There are several good texts for the SIMPLER method and its variants, and it is sufficient to review the method via generic convective, diffusive and convective-diffusive problems, followed by an application of the SIMPLEST method to the secondary flows in a pipe of rectangular cross-section for a version of the Phan-Thien-Tanner model.

I i,j,k+l

i-Ij, k ~ ~ij~: i+Ij, k " ~

i

i,j~r

FIGURE 53.1. A 3-D finite volume cell. The filled circles denote a grid point, where the unknown value is sought; and the open circles denote the points where the grid lines intersect with the faces of the control, or finite volume.

Convective Problem

The essential idea behind the finite volume method is the idea of conservative discretisation. This can be illustrated by the following convective problem

0r ~ - + V - F - Q , x e D , (53.8)

where F = (Fx, F v, F , ) is the flux of ~, i.e., F = u( , where u is the velocity field, and Q is a source term. The computational domain is divided into finite volume cells, which need not be regular, or structured. However, in most finite volume codes, the grid lines are regular and are parallel with the Cartesian axes x, y, z, as shown in Figure 53.1. Here, the control volume for the grid point i , j , k is a regular prism of volume V~jk = A x A y A z , and bounding surface Sijk. By integrating (53.78) over a

66HU, H.H. and JOSEPH, D.D., J. Non-Newt. Fluid Mech., 37, 347-377 (1990). 67GERVANG, B. and LARSEN, P.S., J. Non-Newt. Fluid Mech., 39,217-237 (1991). 68XUE, S.-C., PHAN-THIEN, N. and TANNER, R.I., J. Non-Newt. Mech., 59,191-213 (1995).


cell, one has

where ~ijk, Qijk are the mean values of ( and Q over the cell. This is a mathematical statement of the conservation of the quantity (~ over the finite volume V~jk. Any discretisation method that has this property is said to be a conservative discretisation. Indeed, the starting equations of any conservative discretisation scheme can be regarded as conservation statements on a finite volume level; there is no need to make a reference to the set of partial differential equations obtained in a limiting process of allowing the arbitrary finite volume shrinking to zero.

The flux calculations can be approximated by

f s - F~S~ + F~S~ - F ,S , + F~S~ - FdSd, F . IldS F~S~

where the lower case subscripts e, w, n, s, u, and d refer to the east, west, north, south, up and down faces with respect to the central grid point i, j , k (these cor-

1 1 respond to the nodes i T �89 i - ~ , j , k , i , j + �89 i , j - �89 i , j , k + ~, and 1 respectively). i , j , k - ~ ,

Different ways of approximating the fluxes and the average values result in different finite volume schemes. In the most straightforward scheme,

and the fluxes are taken to be the averages of neighbouring nodes; for example,

Fe-- ~l (Fi+ijk + Fijk)

In the case where F -- u(~, where u -- {u, v, w} is a constant velocity, and the finite volume V/jk is a rectangular prism, this leads to the discretised equation

r __ 2Vii kSeAt (Ur -- UCn-ljk) 2~/~jk (Vr -- Vr

S~At ~ -- 2Vij--- ~ (wr -- wr ~- AtQ,~ k + r 1,

where ~i~k -- ~ (xijk, n a t ) , and we have approximated the time derivative by the implicit scheme

~ in __ ~ijk -- ~i3-k 1 ,~jk At + O(At2)"

In the standard notation of FVM, we write

a p ( p -- E a~br + b, (53.9)

where the subscript P denotes the current node, and nb denotes all the neighbouring nodes (upper case for a node, and lower case for a face), and b represents all the source terms:

&At s,~At ap -- 1, aE ---- aw -- --------u, aN ---- as = ---------v,

2V~k 2t~j k

S ~ A t au -- aD -- - -2Vi jkw, b - AtQ~j k + ~i3-k 1.

Diffusive Problem

i=l W

a i-1

a control volume

W

i-1/2 i+1/2

E i=N v l w ,

i+l b

FIGURE 53.2. A 1-D finite volume discretisation. The filled circles represent the nodes, and the vertical lines represent the "faces" of the control, or finite volume.

Next, we consider the unsteady s ta te heat conduction (parabolic) problem in the interval a < x < b"

~

o t - ~ ~ +~ with some relevant initial and boundary conditions. The finite volume diseretisation is illustrated in Figure 53.2. Integrating this equation over the control volume, both in space and in time, we obtain

PCAxi 0T~ f t i " -~-~= + A t

07' _ K(T/'

/ tn+At + Ax~adt.

Jtn

The flux out of and into the volume are est imated as

-~x e - K e A x e ' ~ ~ , = K w Axw '"

Furthermore, the t ime integration may be expressed in terms of a low-order quadrature such as

( t t"+AtT(s)ds - [ fT( tn+l) + (1 - f )T( t~)] A t ,

where f is just a weighting factor. This leads to

_ i x i [ke~E+l _ ~+1 - k~

~+1 _ ~+1 ] AXw _t_ Axian-4-1

+(1 - 1) [k~ ~ ~ - ~ - k~ T~ - ~ w + Ax~a~] Axw ]

This can be rearranged, remembering that the subscript P refers to the current grid point i, to obtain

apT~p +1 -- a E [fT~E +I -+- (1 -- f)Tr~E] -+-aw [f~w +1 -~-(1 -- f)~]

+ [a ~ --(1 -- f ) a w --(1 -- f)aE] T~e + Ax, [ fa ~+1 + (1 - f )a~] ,

where

(53.11)

_Axi K~ K. a~ -- pC'--~-, aw -- A x e ' a s - - Axe ' ap -- a~ + faE + faw . (53.12)

The case f = 0 corresponds to a forward finite difference scheme for the time derivative; it leads to the explicit scheme:

apT~p +1 -- aET~E Jr- a w ~ w + [6~ -- a w -- aE] V~p -]- A x i o rn, (53.13)

which may be applied repeatedly until a required time span is reached. This scheme corresponds to the assumption that the temperature during the current time step remains at the old level until t -- tn + At, when it asssumes the new value; i.e., the interpolation function is piecewise constant.

P r o b l e m 53 .A Carry out a von Neumann stability analysis for the above explicit scheme by

ignoring the source terms and considering just one single Fourier mode. Show that , when the mesh size is uniform and the properties are constant, the amplitude of the error is given by

A - 1 - 4 p V A x 2 sin 2

Thus deduce that the scheme is stable only if

A t < p C A x 2 (53.A1) - 2 K "

Any scheme with non-zero f is said to be implicit, in the sense that a system of equations results. Of these, we are interested in the Crank-Nicholson scheme, which is obtained by setting f -- 0.5, and the fully implicit scheme, which corresponds to f = l .

The Crank-Nicholson scheme corresponds to the temperature varying linearly with time, whereas the fully implicit corresponds to a piecewise constant temperature, but the temperature in the current time interval is in fact the value of the temperature at the next time step.

P r o b l e m 53.B By a von Neumann analysis, show that the amplitude of the error is given by

1 - 4s(1 - f)sin2(O/2) (~.~1~ A

1 + 4s f sin2(0/2) '

where

8 --- K A t

~ A x 2 "

Furthermore, show tha t this function is monotonic in sin2(O/2), and thus the ex- t reme values of A occur at the end points of sin 2 (0/2). Deduce that as long as s > 0 and tha t

2(1 - 2f ) s < 1, (53.B2)

then [A] < 1.

From the previous problem, it is clear tha t any implicit scheme with f _> 1/2 is unconditionally stable for any value of At and Ax. Tha t does not mean tha t any solution obtained with a stable scheme is a reasonable solution. We also need to consider the accuracy with respect to a given mesh size. From a physical point of view, the fully implicit scheme seems to be a better model for the diffusion problem than the Crank-Nicholson scheme and usually preferred in FVM (the exact profile decays exponentially rather than linearly).

In the fully implicit scheme, we have

ap~p +1 -~ aET~E +1 -{- a w ~ w +1 -~- a~ + Axib n+l , (53.14)

where the coefficients are given in (53.12). This results in a tri-diagonal system, which can be solved in an order O(N) operations; we return to this later.

The extension to 3-D diffusion equation is straightforward. For example, consider the problem:

pC~ = V . ( K V T ) + a. (53.15)

A finite volume such as the one illustrated in Figure 53.1 can be chosen and integration over this volume yields

pCVijk dt = K V T . ndS clt + Vijkaijk dt. d t~t C~ d t~t Mr

The surface integral is in fact the net flux into the volume V~jk through the surface S~jk. If the cell is a regular prism aligned with the axes x, y, z, of size Ax, Ay, Az, then assuming the fully implicit scheme, this will lead to,

ap~~p + 1 - aE~I~E+I + a w ~ + I + a N ~ N + I + a s T ~ s + I +auT~u+I + a D ~ + I + s , (53.16)

where K . A y A z K w A y A z

aE = a w = Axe ' Axw ' K , , A x A z K, A x A z

-- , a s "- aN Ay~ K = A x ~ y

a u = aD = Azn

A y s ' K d A X A y

A Z d

(53.17)

- n+l 0 = AxAy/Xzai j k + apT~ijk , (53.18)

a p = aE + a w + aN + a s + a u + aD + a~ (53.19)

Before considering the convective-diffusive problem, let us discuss some common issues in implementing the FVM.


J

E w

P

A x e

AXe Ax +

A V

E

FIGURE 53.3. Interfacial properties.

Interface Properties

In the diffusion problem above, the thermal conductivity can be a function of the temperature, making the problem nonlinear. The question is: how should the interface thermal conductivity be chosen in a manner that is consistent with the degree of interpolation within a volume cell?

To illustrate this, we return to the 1-D diffusion problem and consider the east face relative to the current point P, as shown in Figure 53.3. Consistent with the piecewise constant temperature profile chosen in the numerical approximation, the thermal conductivity is also considered to be piecewise constant. Over the current cell centred at P, K = g p (to the left of the east face), and over the cell centred at E, K = K s (to the right of the east face). A straightforward scheme would be taking the interface thermal conductivity to be the interpolated value:

Ke - f K p + (1 - f ) K E ,

where f -- A x + / A x ~ . By considering the limit of K E ---* O, one finds that this approximation has a serious problem: this limit corresponds to an insulator at the east node, which would be associated with no heat flux through the boundary.

A better approximation would be to consider a composite slab of two layers side- by-side, and an elementary heat transfer calculation show that the heat flux across the interface is

Tp - Te T e - TE q = = R - R + '

where Te is the interface temperature, and R - = A x e / K p , R + - A x + / K E are the corresponding thermal resistances. This leads to

T p - TE Tp - TE q-- = K ~ ~ ,

R - + R + Ax~

giving 1 - f f ) - 1

Ke - K p + ~ ' (53.20)

which, for f -- 0.5 is a harmonic mean rather than the average of the two thermal conductivities. In the limit of K E ---* O, Ke "* 0 as required by the physics of the situation. The analysis applies equally to the viscous dissipation problem.

Boundary Conditions

qb

. l r I

Ax e <

�9 e w

r 1 Ax

w

E

FIGURE 53.4. A boundary half cell.

Dirichlet or Neumann boundary conditions can be conveniently incorporated into a FV scheme, although the end cells may need to be considered separately from the internal cells. If a Dirichlet boundary condition is prescribed at the end, then this tempera ture will enter the discretised equations; and if a Neumann boundary condition is given, then the flux which enters through the end face is known, say qb

- refer to Figure 53.4 for the half cell of the 1-D problem. The discretised equation for this half cell is

A x pC-~ (~+' Ke (T~E+I _ T~p+ 1) ..~. o.n+l Ax -- ~ ) - - qb + --~X e

In the case where the heat flux enters the computational domain through a heat transfer coefficient,

qb = h(T,~ - T p ) ,

we find tha t the relevant discretised equation at this end point is

a p T~p +1 -- a E r i E +1 -}- S ,

where

and

K~ A x

a p - - aE + a ~ + h

S - a ~ + A x b + hTa .

Extensions to 3-D diffusion problems are straightforward.

Desirable Proper ty of anb

For large-scale problems, such as those arising from 3-D calculations, an iterative method is used to obtain a solution. The condition for convergence of an iterative method, such as the Gauss-Seidel method, should serve as a 'desirable property ' of the system matrix.

Let us recall tha t a sufficient condition for the convergence of the Gauss-Seidel method is (see Appendix)

laP[ < 1,

for all equations,

for at least one equation.

(53.21)

which is usually referred to as the Scarborough condition. If ap - ~ anb, as is often the case, then a sufficient condition for the convergence of the Gauss-Seidel method is tha t all the coefficients should be positive. This is so, since any negative coefficient will lead to

a violation of the Scarborough criterion. A "nice" FV formulation therefore should have all the coefficients in the discretised equations positive.

Source Term Linearisation

The source te rm in the diffusion equations can be a nonlinear function of the dependent variable T. In this case, a linearisation of the source term is preferable than treat ing it as a constant term within a control cell. That is, by expressing a as

• - a c + g p T p .

This will lead to the same discretised equation as before, but with

- A x A y A z a c i j k + k,

ap - aE + a w + aN + as -~ au ~- aD + a 0 -- AxAyAza~p+lk .

If the coefficients are to be positive and the system matrix is as diagonal-dominant as possible, then we want a p ( 0. Physically reasonable systems will behave in this manner, for example, some constitutive equations for viscoelastic fluids. If this is not the case, then one has to create artificially a linearisation tha t has a p < O.

Relaxation

To help smoothing out any wild oscillation at the start of the calculation, we may use a relaxation factor. Consider the generic algorithm

a p T p - E anbTnb T S, nb

and denote by TOp its solution at the previous iteration. Thus,

T p -- ~p T A (~-~nb anbTnbap + S

where A is any constant. The quanti ty inside the brackets can be considered to be the change in the variable Tp between successive iterations, and therefore with

< 1, this change can be controlled. Rewriting this equation, we obtain

ap A TP- -EanbTnb+S+(1- -A)aP -~T~ (53.22)

nb

This give us the rule to introduce the relaxation factor. When A ~ l, it is said to be under relaxation, otherwise it is over relaxation. Typically~ an under relaxation factor of the order 0.7 is acceptable. Too low an under relaxation may induce a false convergence.

Another way to introduce a relaxation factor is by a false transient term, i.e., through

(ap + m)Tp -- E anbT,~b + S + roT~ nb

where m is the 'inertia'. This has the effect of increasing the t ime scale of the problem.

Convective-Diffusive Problems

Convective-diffusive problems are at the heart of CFD, but a universal and agreed t rea tment of these problems is not available at present. We start by assuming tha t the velocity field is known. Later on, the fully coupled problem will be discussed.

Consider the 1-D convective-diffusive problem

p ~ + ~ - ~ n ~ �9

To mimic the balance of mass, we also need Ou/Ox - 0, i.e., u is a function of t ime only. Let us assume tha t u is a constant for now. The FV formulation leads to

(0,) (0,) where the three grid points W, P, E have been considered, with the east and west faces denoted by e and w, respectively. Furthermore, assume tha t the mesh is uniform, and points e and w are half way in between points P and E, and W and P, respectively. Thus, one could take

1 1 r - ~ (r + r r - ~ (@ + Cw),

(o<) n ~ ~ - ~ ~(r162 ~ ~ - ~ ~ ( r 1 6 2

which results in

O@ I OUh pAx---~-

(53.24)

P r o b l e m 5 3 . C

Show that, if a forward difference in time is adopted, this will lead to the explicit FV scheme:

a O ~ o +1 -- aE~nE + a w ~ v 4 - a p ~ , (53.C1)

where, with the notation D - ~7/Ax, F - pu, v - ~?/p,

a E - D~ F~ F~ A x - ' ~ - , a w - D ~ , + - - ~ , a ~ t ,

[ aO a p = - D ~ + --ff- + D ,o - +

In finite difference notation, show that this corresponds to

u A t ~ ~ v a t ~ ~n+l _ ~i "~- 2AX (~i+1 -- ~ i -1 ) -- A x 2 (~i+1 -- 2 ~ t "~- ~ i -1 ) , (53.C2)

Perform a von Neumann stability analysis, and show that (53.C2) is stable if

2 2C < Rezxx < U ' (53.C3)

where R e A x -- u A x / v is the cell Reynolds number and C - u A t / A x is the Courant number.

, , ,

To discover the physical reason for this conditional stability, let us start with the discretised equation (53.C2), and find out in the limit of zero mesh size (both in time and space) the type of equation that we are faced with. Now, expanding the terms in a Taylor polynomial

~+1 __ ~i -~- At (~t)i "~- ~ At2 (~tt)i + O ( A t 3 ) ,

~i+1 -- ~i + i X (~x)i + ~ i x 2 (~xx)i + u i x 3 (~xxx)i + O ( i x 4 ) ,

r "-- r -- i x (~z)i "4- ~ A x 2 (~xz)i -- ~ i x 3 (r + O(~kx4) �9

Since everything will be referred to grid point (i, n), we can drop the subscript i and the superscript n. Then the discretised equation becomes

1 r + ~Atr + ~r = re= + o(At2, A~2),

which is quite different from the original equation. To discover how different it is, we must find the term ~tt accurate to at least O(At). By a process of differentiation, we have

1 ~tt "4-~AtCttt + UCz t ---Vr + O ( A t 2 , A x 2)

and 1

r + -~Atr + ur = v;~= + O(At 2, ~z2).

Then

Next

1 ~ t t - - "~uAtr

1 + u2r - v u r + vCX., - ~ ate,,, + O(At 2, Az2).

1 - ur + ur + O(At 2, Ax2).

Thus

2At~tt 1 2 1 2 - ~u Atr - ~uAtr + ~v A t r + O(At 2, Ax2),

and the equation the discretised system solves is

, t + u , z - - ( v - l u 2 A t ) ( ~ , z + v u A t ~ z x z - lv2At~ 'zzz + z O(At2' Ax2)" (53.25)

Herein lies the conditional stability of the scheme: this equation has a kinematic viscosity of v - �89 the scheme has introduced a negative artificial viscosity of va -- -�89 For stability, the net viscosity must be positive

lu2At

leading to one of the important criteria already noted. Not only does the discretised equation introduces artificial viscosity, it also in-

troduces numerical dispersion where there should be none.

P r o b l e m 53.D Show that the backward difference scheme leads to the following discretised equa-

tion apr -- aEr + awr + a~162 -1 (53.D1)

o r n n - u i t (~in+i _ r 1) __ v A t n n

~i - - ~ i 1 + 2Ax Ax 2 ( ~ i + 1 - - 2 r + ~ i - 1 ) �9

Furthermore, show that this scheme solves the equation

~, + ur - ~, + 7u At r - vuzXtr + -~v Z X t r + O(zXt 2, Az2).

It is therefore unconditionally stable, but it involves an artificial viscosity of va = �89 and a numerical dispersion.

Upwinding and Other Schemes

We have mentioned upwinding before, in the finite difference section. In its simplest form, upwinding approximates the convection term by a backward differencing using only upstream information, i.e.,

~i - - ~ i - 1 u if u > 0, 0~ Ax '

u T x x -

r - r if u < 0. U A X

(53.26)

In the FV notation, we write, for example,

( -- (pu~)e -- ~ FeaR, if u > 0,

(Fr ( FeCE, i f u < 0 .

With the notation [a, b] = max{a,b},

we can rewrite the above in the following compact form

(F(~)e - (~p [Fe, 0] - (~E r-Fe, 01 (53.27)

with a similar expression for (F~)~. The discretised equation for the FV (53.26) results, but with the coefficients given by

aE -- De + a w = D,,, + r. .m . i X

a ~ = P A t ' (53.28)

Up -- De + [Fe, O] + D~ + [ - F ~ , O] + a ~

- a E + a w + F e _ F w + a ~

(53.29)

Prob lem 53.E The upwinding scheme discretised the convective equation into (with u > 0)

n u A t n r _ r -4---~X ( r r --" O.

Show tha t this solves the equation

0~ 0r 1 02~ Ot + U-~x - ~uLXx(1 - C) - O ( A t 2, Ax 2) (53.E1) --- ~ x 2 ,

where C = u A t / A x is the Courant number. Deduce tha t the scheme is stable if

C ~ 1, (53.E2)

which is the CFL condition. It merely states tha t disturbances should not be allowed to travel more than an element width in one time step.

The leap frog scheme uses central differencing in both t ime and space, so tha t the discretised equation for the convective problem becomes

u A t n ~n+i _ ~ n - i _~_ " ~ X (~in-I-i -- ~i-1) -- 0.

This scheme employs two levels in t ime and space; it solves the equation

(53.30)

or or + = o ( a t a.=),

and is therefore neutrally stable. It is not too popular due the possibility of a checkerboard solution.

The L a x - W e n d r o f f method relies on the fact tha t

so tha t

n 2 n ~n+l __ ~n .4_ A t (~t) i -~- A t 2 (~t t ) i -~- O(At3),

~nq-1 _ ~n 0A ! 02 A t = cot + 2 A t u 2 ( Ox 2"

Thus, the t ime derivative is discretised as

~n4-1 _ ~n _ 1 A t u u (~in+l __ 2~ n -4- ~in_l) At 2 Ax 2

leading to the discretised equation for the convective equation

1 n ,~ 1C2 (r 2r n -{- r (53.31) r __ r -- ~ C (r - ( i - l ) -{- 7

It can be shown tha t the Lax-Wendroff scheme is stable if the Courant number is less than one.

There are a number of popular schemes in FVM to deal with the convective- diffusive equation, and we review some of them here.

First of all consider the steady state problem

0, 0(0,)

on the interval 0 < x _< L, where u is constant and

(53.32)

(~(0)- ~0, r eL-

The exact solution to this is

- ~o e x p ( R e x / L ) - 1

(~L -- (~0 exp(Re) -- 1 (53.33)

where Re = puL/v l is the Reynolds number (or Peclet number). Note tha t when the Reynolds number is large in magnitude, the value of ~ at a point inside the interval (0, L), i.e., on the interface, is nearly the same as the value of (~ from the upstream boundary. This is precisely the idea behind upwinding.

Now, let us identify x = 0 with our current node P, and x = L = A x e with the east node E (refer to Figure53.5). The east face of the control volume is in between, at x -- 5Xe. The exact solution is rewrit ten as

- ~p e x p ( R ~ x / A x ~ ) - 1 , ,

(~E -- (~P exp(R~) - 1

with the cell Reynolds number defined by Re = puAxe / r l . The convective-diffusive equation is in fact

d J ~- - " - - ' 0 , dx


i=l W , d h , , ~

W e

Axw ~r" Ax e

E i=N A v

FIGURE 53.5. A control volume for the convective-diffusive problem.

where 1 2 -

= p u ~ - ~?a~ J

is the total flux. We shall continue to denote pu by F, and r//Ax by D, so that

Now

But

and

Thus,

In a similar fashion

F~ Re-- De"

e

(e -- ~P + (~E -- ~P) e x p ( R e S x e / A x e ) - 1 exp(Re)- 1

( d~ ) e x p ( R e S x e / A x e ) ~7 "~x e -- F e ( ( E - - r e x p ( R e ) - i

J e - F e (~p-[-

/' J~ - F~ Jew +

and the discretised equation becomes

exp(Re)- 1J "

Cw-r ) e x p ( R ~ ) - 1 '

J~- J ~ - o

or

leading to

where

F~ ( ~ + ~xp(Ro)- 1 ( W - - ( P ) - -0 ,

~xp(R~)- 1

a p ( p -- aE~E "4- a w ~ w ,

F~ a E ~ -

exp(Re) - 1' aw Fw exp(Rw) exp(Rw)- 1'

a p -- a E 4 - a w + F~ - F,,, .

(53.34)

(53.35)

(53.36)


S c h e m e F u n c t i o n f(R)

central difference

upwind 1

1 - 0 . 5 1 R I

hybrid [0,1 - 0.5[R[]

power law [0, (1 - 0.1 [R[) 5]

exponential [ R I / [ e x p ( [ R [ ) - 11

TABLE 53. i. Functional form for f (R) for various schemes.

Note the relations like J~ - F ~ p = aE (r - (E) , (53.37)

which follow readily. This scheme is called the exponential scheme, and it produces the exact solution

to the 1-D convective-diffusive equation. The main feature of the scheme is tha t

aE _ Re a w _ Rw exp(Rw) = Rw + f (Rw). De -- e x p ( R e ) - 1 = f ( R e ) , D~v -- e x p ( R w ) - 1

Based on this and various approximations of the expected profiles, it is proposed that 69

aE = D ~ I ( R e ) + [ - F e , 0] , a w = D ~ I ( R ~ ) + [F~, 0] ,

ap = aE + aw + R e - Fw,

where various forms for the function f are shown in the Table 53.1.

(53.38)

(53.39)

2-D Convective-Diffusive Problems

Extension of the 1-D FV scheme to higher dimension is quite straightforward. For the 2-D case, we deal with

+ + = + + s , (53.40)

where we have used the 'conservation' form, S is a source term which may be nonlinear, and (53.118) is subject to some relevant initial (Dirichlet) and boundary conditions (Dirichlet and/or Neumann). We assume that S can be linearised, so that

S = S c + Spr (53.41)

where ,.-qc contains all the nonlinear effects. This equation can be rewritten as

0 - .~(p() + V . J - S c + Sp~, (53.42)

69PATANKAR, S.V., Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing Corpo- ration, New York, 1980.


FIGURE 53.6. A control volume for the 2-D convective-diffusive problem.

where the 'flux' J is defined as

0r J = (pur p v r (53.43)

We assume that the domain is regular Cartesian, so that the mesh can be discretised in a straightforward manner. A domain cell centred about point P with neighbouring points E (east), W (west), N (no r th ) and S (south) is sketched in Figure 53.6. The east, west, north and south faces of the control volume are denoted by lower cases e, w, n, and s. If one uses the fully-implicit scheme, i.e., backward time difference, then this will lead to the discretised equation

V (p.r _ por + g~ _ g~ + j~ _ g, = (so + s~r At

where V - AxAy is the 'volume' of the control cell, and variables with superscript 0 denote their values at the previous time step. Although we have not discussed the fully coupled flow problem, the conservation of mass leads to

V _ ( p e _ pO) + F~ - F = + F . - F , = 0 , At

where

F~ - - (pu)~y , F~ = ( p ~ ) ~ y , F~ = ( g o ) ~ , F, = ( ~ ) ~ (53.44)

are the net mass fluxes across the control volume faces. We now multiply the conservation of mass equation by ~p and take the difference with (53.44) to obtain

p~ ( r _ ~ o ) + (Je - FeaR) - (J~ - F w r + (J,~ - FrieR) - (Ja - F s r At

= (sc + sp@) v.


We now make use of the 1-D relations (53.37)"

Je - F e ( p -- aE (~p -- ( E ) , Jw - F w ( p -- a w ( ( w - ~P) ,

J n - F n ( p - - a N ( ~ p - ( , N ) ,

with the coefficients given by

J , - F , r = a s ( r - CP) ,

aE -- D e f (Re) + r-Fo, ol ,

aN -- Dn.f (P~) + r-F..,o], in which

aw -- D , , , I ( R ~ ) + [F~, 0] ,

a s -- D s I ( R s ) + FF.,Ol,

(53.45)

(53.46)

~/~Ax ~hAx De = rle AY D~o = rlw AY On = . 08 = .... �9 (53.47) AXe ' A.T.~ ' Ayn ' Ay 8

Note tha t we have not demanded tha t ~ be constant; it is allowed to take on different values on the control faces - an interpolation like (53.20) may be used in this case.

The fully discretised equation now becomes

a p ~ p -- a E ~ E ..+- a w ~ w -b. a N O N -.b a s ~ s -.b b, (~3.48)

where ~P = ~ + ~w + ~ + ~s + ~o _ s p v , (53.49)

P ~ o o ~ o = "-dY' b = s ~ v + ~pr (53.50)

with the function f ( R ) to be chosen from Table 53.1. Extension of this scheme to 3-D situation is quite straightforward for the case of

a regular geometry. All tha t is required is to keep track of the extra fluxes coming in through the top and bot tom control surfaces.

Newtonian Fluid Problems

In the u - p formulation, the governing equations are (in conservation form)

Op + V - ( p u ) - -0 , (53.51)

- ( p u ) + V - ( p u u ) -- - V p + V - ( ~ V u ) + pb, (53.52)

0 ~ ( p C T ) + V . ( p u C T ) = r + V . ( K V T ) + ,or. (53.53)

One now needs a way to discretise the pressure gradient and the means to satisfy the balance of mass.

Consider the 1-D gradient Op/i)x. A straightforward way to discretise this, in the case of a uniform mesh, would be

P~ - P~ PE -- PW Ax 2Ax "


0 velocity nodes

�9 pressure, temperature, stresses,...

FIGURE 53.7. 2-D staggered grid and control volumes for u, v, and p.

This involves pressures at al ternate grid points, not adjacent ones. The implication is tha t checkerboard-type of solutions are allowed to set up. Consider, for instance, the sequence of pressure values along a grid line (100, 200, 100,200, 100). The pressure gradient taken from alternate grid points is zero, but the pressure field is highly oscillatory.

The balance of mass also contains first-order derivatives of the velocity, and the same consideration will show that a straightforward discretisation will not be adequate.

The remedy for this is quite simple: we use different control volumes for the velocities and the pressure, and other variables such as temperature. Such a staggered grid has been used in the Marker and Cell method, and is illustrated in Figure 53.7 for a 2-D problem, together with the control volume for u, v, and p. The extension to a 3-D staggered mesh is the same, but lack the clarity of the 2-D mesh.

Consider one component of the balance of momentum equation:

Ot ( pu ) O Ou 0 Ou 0 Ou 019 + ( p u u - n ~ l + N ( ~ u - n N l + ~ ( ~ u - n ~ ) = - - - + S ~ o ~ , (53.541

which is already in the form of the 3-D convective-diffusive equation, except for the pressure gradient term, the volume integral of which can be reduced to

- fv Vp dV -- - ~ pn dS.


Thus, if the current velocity grid point is e, with the east and west faces being E and P respectively as shown in Figure 53.7, then (53.54) will be discretised to

a~u~ -- E a~bu,,,b 4- b= 4- (pp - pE)Ae, (53.55)

where the coefficients are given exactly as in the convective-diffusive problem, bz is the source term, including any possible transient terms, and the pressure force is the pressure drop from the adjacent grid points, with Ae being the cross sectional area of the control volume perpendicular to the x-direction. In a similar fashion, the other two discretised equations for v and w are

anvn = E anbVnb 4- bu 4- (pp - PN)An, (53.56)

auwu = E anbW,~b + bz + (pp -- pu)Au. (53.57)

These equations can be solved sequentially by the method of line-by-line, for example, to obtain the velocity field with a given pressure field. The equations for the temperature and other variables can be discretised in the same manner. However, one needs a mean to update the pressure field, and this is where different forms of FV schemes come in.

The most popular method in FV is the so-called SIMPLER scheme, which has been thoroughly tested and implemented in a number of commercial packages, and we discuss this scheme here. In essence, the scheme uses the continuity equation

0 0 0 0 N(p) + ~ ( ~ ) + N ( ~ ) + ~ ( ~ 1 - 0, (53.5s)

to correct the pressure, as follows. With a control volume centred at node P, the discretised equation for the balance

of mass becomes

Y (pp _ pO) + [(pu)~ - ( ~ ) ~ ] A y A z + [ ( ~ ) , - (p~) , ]AxAz At

+[(pw)= - (pw)d]AxAy = 0. (53.59)

Now, from the velocity equations, (53.55)-(53.57),

u~ ~ a~,bu~,b + b= A~ = + (vP - w ) a'-:

ae

and two similar relations for v and w. These can be written as

u~ -" ~t~+d~(pp--pE),

where

and

E anbUnb 4- bx ~te

ae

v,~ = 0,, +dn(pp--PN), w~, - -~v , ,+d~,(pp-pu) , (53.60)

~,, = E ~,,~,,,~ + b~ , ~,,, = E ~ , ~ , , ~ + b= (53.62) an au

A~ A,~ A~, ~ ~,, 4 , (5a.6~)

ae an au


The discretised relations (53.60) are inserted into the discretised conservation of mass equation to obtain

a p p p -- aEPE + a w P w + aNpN + a s p s + auPu + aDPD A- B , (53.63)

where aE - p e d ~ A y A z , a w -- p e d e A y A z ,

aN -- p n d n A x A z , as -- p s d s A x A z ,

au = p~,d~,AxAy, aD = PdddAXAY,

ap -- aE + a w + aN �9 as �9 a~ + aD,

(~3.64) (53.6~) (~3.66) (~3.67)

B = - ~--TV (p~ _ p o ) _ [ (p~) , _ ( p ~ ) ~ l ~ y ~ + [ ( ~ ) = _ ( ~ ) ~ 1 ~ ~

_ [ (p~ )~ _ ( p ~ ) ~ ] ~ ~ y . (~3.68)

In essence, it is the lack of conservation of mass that drives the pressure correction. In sum, the SIMPLER algorithm consists of the following steps:

�9 Start with a guessed velocity field;

�9 Calculate ~, 0, ~ from the discretised momentum equations according to (53.62);

�9 Calculate the pressure, according to (53.63);

�9 Solve for the velocity, according to (53.60);

�9 Use this velocity vector instead of fi in the mass source terms (53.68), and solve for p using (53.63);

�9 Correct the velocity field using (53.60) and the previous velocity profile in place of fi;

�9 Solve for the temperature and any other variables,

�9 Repeat the cycle until convergence (or divergence) occurs.

5 3 . 3 S e c o n d a r y F l o w i n P i p e s o f R e c t a n g u l a r C r o s s - S e c t i o n

A variance of the SIMPLER method, called SIMPLEST (SIMPLER with Splitting Technique) has been applied successfully to the three-dimensional, incompressible, isothermal flow in a pipe of rectangular cross-section of the P T T model, including the UCM model. 7~

The method adds an additional step to the SIMPLER scheme to ensure tha t both the velocity and pressure satisfy the same momentum equations. The method has been found to perform well, and some typical results for the flow in a pipe of rectangular cross-section are shown in Figure 53.8, where only one positive quadrant of the pipe is shown. In the figure, the cross sectional areas of the pipes are kept

7~ S.-C., PHAN-THIEN, N. and TANNER, R.I., J. Non-Newt. Fluid Mech., 59, 191-213 (1995).


~176176176176

J

FIGURE 53.8. Velocity vector field and streamlines in one quarter of the cross section of pipe with aspect ratio of (a) 1, (b) 1.56, (c) 4, and (d) 6.25.

54 Finite Element Method 447

at unity, but their aspect ratios are chosen from 1 to 16 (only the first four aspect ratio cases are shown, the last two, at aspect ratios 9 and 16, are not because of the lack of graphic clarity). The wall shear rate in these flows is of O(102), allowing a significant second normal stress difference to be developed, which is out of proportion to the viscosity function and which drives the secondary flow in this model fluid. We have verified that no secondary flow is present, i.e., of magnitude less than 10 -6 compared to the primary flow, when

1. both ~2(~) and 7/(~) are constant;

2. ~2(~) = n~(~),

as dictated by the theoretical results in w Otherwise, the magnitude of the secondary flow is of the order 10 -2 to 10 -4 compared with that of the primary flow.

Figure 53.8 clearly shows eight vortices in all cases, two vortices in each quadrant. They are completely symmetrical about the diagonal of the quadrant in pipes of square cross-section; while with increasing aspect ratio r, the vortex near the long wall expands, squeezing out the vortex near the short wall.

The main disadantage of the FVM is its inflexibility in accommodating complex geometry and traction boundary conditions. To address the issue of complex geometry, unstructured mesh can be used, but the nice tri-diagonal form of the system matrix is lost. Nevertheless, unstructured FVM has been found very robust in the flow of the UCM between eccectrically rotating cylinders. 71 At low eccentricity ratios, convergent solutions are obtained up to a Deborah number O(100), higher than the pseudo-spectral finite element method, which fails to obtain a convergent result at De -- 95. The traditional finite element method also fails to obtain a convergent solution at De : 0(3). 72 It is clear that the FVM will be a serious competitor to the more traditional FEM.

54 Finite Element Method

The finite element method (FEM) is undoubtedly the most powerful numerical technique for solving differential and integral equations in various fields of engineering and science. The method is a generalisation of the classical method of calculus of variations, which is associated with Rayleigh and Ritz, and can be regarded as a weighted residuals method. 73 As with FD and FV methods, the computational domain is sub-divided into a collection of non-overlaping finite elements of simple shapes, usually generalised triangles or quadrilaterals. The search for the numerical solutions is then restricted to a sub-class of piecewise varying functions, by approximating the unknown functions with a linear combination of simple interpolation

71HUANG, X.-F., PHAN-THIEN, N. and TANNER, R.I., J. Non-Newt. Fluid. Mech., 64, 71-92 (1996).

72BERIS, A.N., ARMSTRONG, R.C. and BROWN, R.A., J. Non-Newt. Fluzd Mech., 16, 147-172 (1984); J. Non-Newt. Fluzd Mech., 22, 129-167, (1987); J. Non-Newt. Fluid Mech., 13, 109-14s, (1983).

7aFINLAYSON, B.A., The Method of Wezghted Residuals and Variatwnal Principles, Academic Press, New York, 1972.


functions, usually polynomials of less than cubic in their orders. These interpolation functions are chosen so tha t they assume a value of one at a certain point in the element, called the node, and zero elsewhere. The coefficients of the interpolation functions in the approximation can then be directly linked to the values of the unknown functions at the nodes, which are usually located at the vertices, the mid-points of the element sides, and the centroids of the elements. Since the unknown functions are simply linear combinations of the interpolation functions, these latter functions also serve as a basis spanning the linear space in which the numerical solutions reside; this is the finite element space. The finite dimension of this space is directly linked to the order of the approximation, and the number of finite elements used. The agreement with the governing equations is enforced by computing the residuals, and requiring the latter to be nil, i.e., orthogonal to the basis functions, in this finite dimensional space. This results in a set of algebraic equations on which the boundary conditions can be imposed. This is then followed by a solution procedure, which is usually of an iterative nature. The FEM thus yields a weak-form solution. Different choices of the interpolation functions, i.e., the finite element space, and the methods of forcing the residuals to zero give rise to different names of the method. The detailed implementation and the theoretical justifications of the method can be found in several texts, for example, see the texts by CROCHET, DAVIES and WALTERS, 74 JOHNSON,75 REDDY and GARTLING, 76 and ZIENKIEWICZ and MORGAN. 77 Here, we briefly review the method, and outline some of the usual schemes tha t have been used successfully in viscoelastic fluid mechanics.

5~. 1 Fini te E l e m e n t Formula t ion

The main idea in the FEM can be discussed via a nonlinear steady s ta te heat conduction problem:

V . (KVT) -- - a , x e ~, (54.1)

where K(T) is the thermal conductivity and a represents the heat source. This is an elliptic problem, demanding a combination of Dirichlet and Neumann boundary conditions on the boundary. We assume

T - T o o n S T

and

q= n . K V T = qo onSq,

where To and q0 are some prescribed functions on the boundary ST O Sq, with non-overlapping ST and Sq; qo may depend on T for a convective problem with a

74CROCHET, M.J., DAVIES, A.R. and WALTERS, K., Numerical Solution of Non-Newtonian Flow, Elsevier, Amsterdam, 1984.

75jOHNSON, C., Finite E lement Methods, CUP, Cambridge, 1990. 76REDDY, J.N. and GARTLING, D.K., The Finite Element Method in Heat Transfer and

Fluid Dynamics, CRC Press, Florida, 1994. 77ZIENKIEWICZ, O.C. and MORGAN, K., Finite Elements and Approximations, Wiley, New

York, 1983.


known heat transfer coefficient. A strong, or classical solution to the problem is a function T(x) , satisfying the smoothness requirements imposed by the governing equations and the associated data. This concept of solution is too restrictive to be useful in numerical work, which often involves simple approximations tha t may not satisfy the differentiability requirements imposed by the governing equations.

We now discretise the domain ~t into a set of non-overlapping finite elements, for example a collection of triangles ~te:

~t - (.J gt~, e

and search for a weak solution Th, where h denotes the size of an element. Tha t is, for all test functions w in some test space, we t ry to satisfy

( V - ( K V T ) , w) + (a, w) - 0,

where (a, b) denotes the natural inner product f~ ab d~. In other words, we force the residual to become zero in this test space, in the sense of the norm induced by the inner product. To lessen the smoothness demand on the numerical solution T, we integrate the above equation by parts, with the help of Green's theorem, to obtain

- ~ V w . K V T d ~ + ~ s w K V T . n d S + ~ a w d ~ - O . (54.2)

The weak formulation therefore demands tha t both w and T be at least linear in the spatial coordinates. Since the von Neumann boundary condition on Sq can be directly incorporated in the weak formulation, it is sometimes called the natural boundary condition, as opposed to the Dirichlet boundary condition, which is referred to as an essential boundary condition.

Next, on each of these elements ~k, the temperature is approximated by a simple piecewise varying function; for example, using piecewise polynomials, we obtain

M

T(x) T. (x) - E T?)N?)(x), j = l

(54.3)

where T (~) denote the unknown values at M points on the element e, at x - Xk,

k : 1 , . . . , M, called the element nodes, and N~ e) (x) are the shape functions for the element e; the order of the shape functions dictates how many nodes are required. They are derived from the property

N (~) (xk) - 5,k, (54.4)

which is a Lagrangian interpolation procedure. Indeed, we can also ensure tha t

N~ ~) (x) have compact support in ~t~. Thus, a global approximation for T(x) is

N M

e--1 j - -1

(54.5)


Thus, if N (e) (x) are appropriately orthonormalised, then Th (x) belongs to the space

spanned by (N(e) (x )} , the finite element space. In the Rayleigh-Ritz-Galerkin finite element method, the test space for w is

chosen to coincide with the finite element space Lh so that the residual is minimized in this space. By substituting the above approximation for the temperature into the weak formulation (54.2) and taking w to be N (I), we arrive at

which can be rearranged in the form

[K] {T} = {b},

where [K] is the system stiffness matrix, resulting from the integration of K V N (f). V N (~) and N~I)VN (e) on the Dirichlet boundary ST, {T} is the global solution vector, and {b} is the right hand side vector, resulting from the integration of the source term and the von Neumann boundary condition on Sq. If both K and a are constant, then the above is simply a linear, symmetric and banded system of algebraic equations, which can be solved by several standard techniques, of which some iterative methods are reviewed in the Appendix. Otherwise, it represents a set of nonlinear algebraic equations which can be solved by an iterative technique such as Newton-Raphson's, a brief review of which has been given in the section dealing with FDM above. The assembly process, i.e., forming the entries of the stiffness matrix, is by far the most time consuming part of a finite element solution. This process mainly involves numerical integration, which can be efficiently carried out by a suitable numerical quadrature, such as the Gauss-Legendre's rule reviewed in the Appendix.

It is clear that in order to have a consistent finite element formulation, the shape functions must satisfy all the smoothness requirements demanded by the

weak formulation, and that { N (e) (x)} must be complete and linearly independent.

Libraries of different types of elements are tabulated in different texts, and the . t

most useful class of elements is perhaps the so-called parametric elements, where the geometry of the element is also modelled by the same type of shape functions. For example, the coordinates of a point in the element ~e are given by

X (~1' "2) -- Z N(~) (71,712)X~ e) ' J

where x~. e) are the position vectors of node j on element e, and r/j are the homogeneous coordinates, varying between - 1 and 1. If the degree of interpolation in the geometry is the same as that of the unknowns, then the element is said to be isoparametric; superparametric if it is greater, and subparametric, if it is less. We have recorded the linear and quadratic shape functions for triangular and quadrilateral elements in the Appendix.


The major task in the assembly process is integration. In most cases the integrands are regular, and a standard quadrature coupled with a coordinate transformation would be sufficient. For example, two-dimensional integrals over an element ~te can be appropriately converted into

_/11/11 fa~ f(x) aS(x) f (X(~l,~2))IJ(~1,~2)1 d~l&?2

j

where r and w~ are the quadrature points and weights, J(r/1,772 ) is the Jacobian of the coordinate transformation, and f can be either a scalar or a component of a tensor.

To evaluate the Jacobian of the transformation for the element e, we define the tangential vectors along the direction r/1 and r/2 as p and q:

P - - 0,1 = E 071 (71, '2) , q = 0~72 - - E 071 J J

The sides of the differential area are given by p d~l and q d~2 , and the differential area at the point x is given by

dS(x) = IP • ql dVldV2.

Thus the Jacobian at the point x is precisely the magnitude of the vector p • q noted above.

Integration schemes for triangular elements can be developed in a like manner; see for example, ZIENKIEWICZ and MORGAN. 7s Alternatively, we can treat triangular elements as degenerate quadrilaterals, with one side shrunk to a point; similarly, for a quadratic quadrilateral element, three of the nodes become one. The integrations for both triangular and quadrilateral elements can now be handled by one single subroutine. It is worthwhile to note that, for a degenerate quadrilateral, the Jacobian is O(r), where r is the distance from the degenerate point. Integrations in three dimensions can be developed in a similar manner.

5~.2 Viscoelastic Fluids

With viscous fluids, we now have the pressure field arising from the incompressibility constraint, and it is fully coupled with the velocity field. In the weak formulation, in which the gradients on the pressure (first order) and the velocities (second order) are transferred to their shape functions, the continuity requirement of the pressure interpolant need not be the same as that of the velocity interpolant. In fact, it has been shown that the order of the pressure interpolant should be one

78ZIENKIEWICZ, O.C. and MORGAN, K., Finite Elements and Approximations, Wiley, New York, 1983.


fewer than that of the velocity interpolant, if an overdetermined discretised system is to be avoided. 79 A more general consideration of the consistency of the finite element formulation leads to the L a d y z h e n s k a y a - B a b u s k a - B r e z z i condition on the shape functions, s~ which must be satisfied if a convergent finite element solution is to be obtained.

In the Galerkin method, the residual of the continuity equation is weighted by the interpolation functions of the pressure field, and the residuals from the momentum equations are weighted with the interpolation functions of the velocity field. This gives rise to the so-called m i x e d f o r m u l a t i o n , since the velocity variables are mixed with force-like variables, e.g., the pressure term. The alternative is to use a pena l t y

method , sl where the pressure is given by

P = - V V " u,

in which 7 is a constant. This is to be constrasted with Chorin's method, where the time rate of change of the pressure is proportional to V- u. The mixed formulation seems to be the preferred method, in all the viscoelastic flow simulations, although the penalty formulation may have more physical justification. As an example, consider the compressible Hookean solid, where the pressure can be identified with 2#v~7- u / ( 1 - 2v), with # being the shear modulus and v the Poisson's ratio.

Early viscoelastic finite element formulations are basically extensions of the New- tonian formulations, with the stress components interpolated in the same way as the pressure field; these approaches have been reviewed by CROCHET and WAL- TERS. 82 Notably, they all suffer from a lack of convergence at a Weissenberg number of the order one. This high Weisenberg number problem may be due to a combination of different reasons: the presence of extremly thin stress boundary layers, even for seemingly regular flows; 83 the singular nature of the stress components near a geometric singularity or a boundary condition singularity; 84 the lack of a unique solution in the discretised equations; 85 and the strong hyperbolicity of the constitutive equations, which may induce a change of type in the governing equations. 86 An instability due to a bad choice of the constitutive equation, or due

79TAYLOR, C. and HOOD, P., Computers and Fluids, 1, 73-100 (1973); SANI, R.L., GRESHO, P.M., LEE, R.L. and GRIFFITHS, D.F., Int. J. Num. Methods Fluids, 1, 17-43 (1981); 1,171-204 (1981).

8~ J.T. and CAREY, G.F., Finite Elements, Mathematical Aspects, Vol IV, Prentice- Hall, Englewood Cliffs, New Jersey, 1983; BREZZI, F. and FORTIN, M., Mixed and Hybrid Finite Element Methods, Springer-Verlag, Berlin, 1991.

stHUGHES, T.J.R., LIU, W.K. and BROOKS, A., J. Comp. Phys., 30, 1-60 (1979). 82CROCHET, M.J. and WALTERS, K., Ann. Rev. Fluid Mech., 15,241-260 (1983). 83BERIS, A.N., ARMTRONG, R.C. and BROWN, R.C., J. Non-Newt. Fluid Mech., 22,129-

167 (1987). S4KEUNINGS, R. and CROCHET, M.J., J. Non-Newt. Fluid Mech., 14, 279--299 (1984);

CROCHET, M.J. and KEUNINGS, R., 7, 199-212 (1980). 85yEH, P.W., KIM, M.E., ARMSTRONG, R.C. and BROWN, R.A., J. Non-Newt. Fluzd Mech.,

16, 173-194 (1984); DEBBAUT, B. and CROCHET, M.J., J. Non-Newt. Fluid Mech., 20, 173- 185 (1986); DUPRET, F., MARCHAL, J.M. and CROCHET, M.J., J. Non-Newt. Fluid Mech., 18, 173-186 (t985).

8SjOSEPH, D.D., RENARDY, M. and SAUT, J.-C., Arch. Rational Mech. Anal., 87, 213-251 (1985). See also w above.


to a badly implemented numerical scheme can also occur. All of these contributing factors have been much discussed. 87

Extensive work on FEM points to the need of a higher-order approximation for the stresses, and a better treatment of the constitutive equations. This leads to the so-called Streamline Upwind Petrov-Galerkin method (SUPG), 88 which is very stable, although we need to be careful that residual stress diffusion introduced by upwinding does not alter the true convergent solution. In the SUPG implementation of MARCHAL and CROCHET, a bilinear element for the stress is used (by sub- dividing the parent element into 4 x 4), satisfying Ladyzhenskaya-Babuska-Brezzi condition, whereas an upwinding scheme is used in the integration of the stress. This implementation, known as SUPG4 • 4, has been used in calculating the flow past a sphere, and in a corrugated tube. 89 The results compare well with other techniques, and convergence with mesh refinement has been obtained at a high Weissenberg number of O(10).

A different approach used by FORTIN and FORTIN, 9~ employs a discontinuous interpolation function for the stresses, in conjunction with an upwinding scheme. This is also very effective in obtaining high quality solution at high Weissenberg numbers.

A notable reformulation to emphasize the ellipticity of the governing equations is the Explicitly Elliptic Momentum Equation method (EEME), which has been proposed by RENARDY 91 for the Maxwell model. Here, the operator (1 + Au- ~7) is applied to the momentum equation, where ,k is the Maxwell relaxation time, to obtain

v . ( x - w ) + Vu. ( v . = Vq, (54.6)

where X - r/1 +)~r, q -- p~- Au- Vp, r/is the viscosity and r is the extra stress. This equation is always elliptic for the velocity field, and a standard Galerkin technique, which is designed for elliptic problems, should be adequate. KING et al. 92 have implemented the method, using a Galerkin technique on the continuity and the elliptic momentum equations, and an upwinding Petrov-Galerkin technique for the constitutive equations, and have shown that the method is stable and convergent with mesh refinement for the flow between eccentrically rotating cylinders, and the slip-stick problem. Convergence up to a Weissenberg number of about 1.6 is also obtained for the flow past a sphere at the centre of a tube. 93 The EEME method

87CROCHET, M.J., Rubber Rev., 62,426-455 (1989); KEUNINGS, R., Rheol. Acta, 29, 556- 570 (1990); BERIS, A.N., KEUNINGS, R. and BAAIJENS, F.P.T., J. Non-Newt. Fluid Mech., in press (1996).

SSMARCHAL, J.M. and CROCHET, M.J., J. Non-Newt. Fluid Mech., 26, 77-114 (1987). sgCROCHET, M.J. and LEGAT, V., J. Non-Newt. FLuid Mech., 42 ,283-199 (1992). 9~ M. and FORTIN, A., J. Non-Newt. Fluid Mech., 32, 295-310 (1989). 91RENARDY, M., Zeit. angew. Math. Mech., 65, 449-451 (1985). 92KING, R.C., APELIAN, M.R., ARMSTRONG, R.C. and BROWN, R.A., J. Non-Newt. Fluid

Mech., 29, 147-216 (1988). 93LUNSMANN, W.J., GENIESER, L., ARMSTRONG, R.C. and BROWN, R.A., J. Non-Newt.

Fluid Mech., 48, 63-99 (1993).

has been extended subsequently to apply to Maxwell-type constitutive equations, 94 and to multimode models of Maxwell-type. 95

The EEME method, being designed for Maxwell-type models, cannot be extended to models which have a viscous component, such as the Oldroyd-B model. The Elastic Viscous Split Stress formulation (EVSS) is designed to handle a New- tonian component in the constitutive equations. 96 In the formulation, the extra stress is split into a viscous component, i.e., the Newtonian-contributed stress, and a polymer-contributed stress. The viscous component is substituted into the momentum equation to obtain an elliptic operator, and a Galerkin finite element technique is applied to the resulting equation. This method has been used to solve viscoelastic free surface problems, 97 and the flow past a sphere in a tube, 98 for both the Maxwell and the Oldroyd-B models. In the latter flow for the Maxwell model, the agreement with the results obtained by the EEME method is good.

An extension of the EVSS has been implemented by SUN et al. 99 The scheme, called Adaptive Viscous Split Stress (AVSS), is different from the EVSS in that the local Newtonian component is allowed to depend adaptively on the magnitude of the local elastic stress. The method is extremely robust, with no upper limiting Weissenberg number found for the Poiseuille flow problem. For the flow past a sphere in a tube, convergent results up to a Weissenberg number of 3.2 are obtained, even with decoupled schemes; without the adaptive scheme, the limiting Weissenberg number for the decoupled method is about 0.3.

Figure 54.1 shows a comparison between different computational methods in predicting the drag force on a stationary sphere of radius a at the centreline of a tube of radius R -- 2a, as a function of the Weissenberg number W i -- )~U/a, where U is the tube velocity. With W i _< 1, an agreement to three significant figures is obtained between different methods, using a mesh with about 105 unknowns. The results for 1.0 < W i <_ 2.2 are in agreement to two significant figures, t~176 Convergence with mesh size has not been clearly demonstrated for 2.2 < W i < 3.2 in all of the published results for this problem. 1~

So far, we have discussed only techniques applied to differential models. With integral models, we have the extra complication of particle tracking and deformation

94JIN, H., PHAN-THIEN, N. and TANNER. R.I., Comp. Mech., 8, 409-422 (1991). 95RAJAGOPALAN, D., ARMSTRONG, R.C. and BROWN, R.A., J. Non-Newt. Fluid Mech.,

36, 159-192 (1990); NORTHEY, P.J., ARMSTRONG, R.C. and BROWN, R.A., J. Non-Newt. Fluid Mech., 36, 109-133 (1990).

96RAJAGOPALAN, D., ARMSTRONG, R.C. and BROWN, R.A., J. Non-Newt. Fluid Mech., 3e, 159-192 (1990).

97RAJAGOPALAN, et al., op. cit. (1990). 9SLUNSMANN, W.J., GENIESER, L., ARMSTRONG, R.C. and BROWN, R.A., J. Non-Newt.

Fluid Mech., 48, 63-99 (1993). 99SUN, J., PHAN-THIEN, N. and TANNER, R.I., J. Non-Newt. Fluid Mech., 65, 75-91 (1996).

I~176 W.J., GENIESER, L., ARMSTRONG, R.C. and BROWN, R.A., J. Non-Newt. Fluid Mech., 48, 63-99 (1993); TALWAR, K.K. and KHOMAMI, B., J. Rheol., 36, 1377-1416 (1992); JIN, H., PHAN-THIEN, N. and TANNER, R.I., Comput. Mech., 8,409-422 (1991); FAN, Y. and CROCHET, M.J., J. Non-Newt. Fluid Mech., 57, 283-311 (1995). 1~ the review by WALTERS, K. and TANNER, R.I., in Transport Processes in Bubbles,

Drops, and Particles, Eds. CHHABRA, R.P., DE KEE, D., Hemisphere Publishing Corporation, New York, 1992, pp. 73-86.


6"5 1 ~ ,

~o 6.0

~ .5

~ 5.0

AVSS/SI

AVSS/SUPG Luramann et al. [7]

Jin et al. [8] Khomami et al. [10]

---0--- Crochet and Legat [9] Fan and Crochet [11]

Luo [13]

]

4.5

4.0

3.5

0 1 2 3

Wi

FIGURE 54.1. The dimensionless drag force predicted by different methods. Typical meshes used by the computations are shown in the following figure.

calculations. In the early phase of this research, this has been accomplished by a Lagrangian formulation, 1~ which suffers from badly distorted elements. This can be avoided by periodically remeshing the computational domain, t~ An improved technique due to BERNSTEIN, KADIVAR and MALKUS t~ involves using linear triangular element for the velocity field with a penalty function for the pressure field. The velocity gradient is thus piecewise constant and the deformation gradient can be found analytically within each element.

A more general approach has been subsequently developed by DUPONT and CROCHET, 1~ by constructing an upstream streamline passing through a particle. The deformation history can then be computed by traversing along this streamline, using the current velocity field. LUO and TANNER t~ have further developed the method, using streamline elements; note that the interpolation function for two

I~ M. and CASWELL, B., J. Non-Newt. Fluid Mech., 6, 245-267 (1980). I~ O. and BISGAARD, C., J. Non-Newt. Fluid Mech., 12,153-164 (1983). I~ B., KADIVAR, M.K. and MALKUS, D., Comp. Meth. Appl. Mech. Eng., 27,

279-302 (19Sl). I~ and CROCHET, M.J., J. Non-Newt. Flu{d Mech., 29, 81-91 (1988). I~ X.-L. and TANNER, R.I., J. Non-Newt. Fluid Mech., 21, 179-199 (1986); 22, 61-89

(19s6).


opposite sides of the elements is also tha t of the stream function. The method has been used to calculate the die-swell and other problems for a K-BKZ fluid up to a high Weissenberg number. 1~ In essence, the pathline of a particle presently at location x(t) , and previously at t ime t ' = t - s at location r(s) is calculated by integrating the equation

O r ( s ) = - u ( s ) , u(0) = u(t) , r ( 0 ) = x(t) (54.7)

by a predictor-correetor method, such as Runge-Kutta integration, with an adaptive step size control. The deformation gradient can be obtained by solving (of.(1.17))

~sF(S) = - L ( s ) F ( s ) , F ( 0 ) = 1 (54.8)

in the same manner. Stress integration along the pathline is then done with a numerical quadrature, such as Gauss-Leguerre.

The method has been applied successfully to the problem of the flow past a sphere: the results compare well with those obtained by other methods, as shown in Figure 54.1.

The extrusion flow out of a long die using a K - B K Z constitutive equation with a discrete spectrum of relaxation times has been simulated by the method. Here, the Weissenberg number is defined to be W i = Aa'Ya, where the apparent shear rate is "~a = 4Q/7rR2 , where Q is the flow rate, and R is the radius of the pipe. A typical result is shown in Figure 54.2: at a Weissenberg number of 587, the amount of die swell is predicted to be about 218%.

The question of relevant boundary conditions at the exit lip of the die becomes very important, especially at a high Weissenberg number. This is clearly illustrated in the extrusion from an orifice, shown in Figure 54.3. The amount die swell relative to the orifice size is quite large (about 360% at a Weissenberg number of about 37), and a redefinition of the at tached point to the die wall has to be considered. A further progress in numerical analysis must await a bet ter understanding of the boundary conditions at a solid interface near a geometric singularity.

55 Other Methods

Other methods have also been used in computational viscoelastic fluid mechanics, among which are the spectral and boundary element methods, and the s t ream-tube approach.

|

l~ X.-L., Finite Streamline Element Simulation of non-Newtonian Flow, PhD Thesis, Department of Mechanical Engineering, University of Sydney, 1987; LUO, X.-L. and MITSOULIS, E., J. Rheol., 33, 1307-1327 (1989); Int. J. Num. Meth. Fluids, 11, 1015-1031 (1990); PARK, H.J. and MITSOULIS, E., J. Non-Newt. Fluid Mech., 42,301-314 (1992); BARAKOS, G. and MITSOULIS, E., J. Rheol., 39, 1279--1292 (1995); SUN, J., LUO, X.-L. and TANNER, R.I., Proc. XIth Int. Gong. Rheology, Vol. I, pp. 394-396, 1992; SUN, J. and TANNER, R.I., J. Non-Newt. Fluid Mech., 54, 379-403 (1994); SUN, J., Some Developments and Applications of the Mixed Finite Element and Streamline Integration Method for Non-Newtonian Fluid Flows, PhD Thesis, Department of Mechanical & Mechatronic Engineering, University of Sydney, 1995.

55 Other Methods 457

MI

M2

M3

FIGURE 54.2. Extrusion from a long die showing the meshes, and convergence with mesh refinement.


M1

. . . . . . _ _ _ _ m _

M2

M3

FIGURE 54.3. Extrusion from an orificed die showing meshes and convergence with mesh refinement.

55 Other Methods 459

In the hybrid spectral/finite element method designed specifically to deal with the flow in the annular region between two eccentric cylinders, 1~ the stream function and the stresses are approximated by Fourier series in the azimuthal direction, and a finite element approximation in the radial direction. Solutions have been obtained up to a Weissenberg number of 90, for the case of small eccentricity ratio (0.1), and up to 7.3, for the case of a moderate eccentricity ratio (0.4). In the latter case, the computation is terminated because of a limit point near W i = 7.24. Other problems solved by the technique include the ftow through an undulating tube, 1~ and the evolution of Taylor-Couette flow instabilities. 11~

In the boundary element method (BEM), an iterative decoupled technique is used, breaking the problem into the boundary element solution of the elliptic field equations with the nonlinear terms treated as pseudo-body forces, and the integration of the constitutive model using the fixed velocity profile obtained in the previous iteration. The inclusion of free surfaces can also be treated iteratively, by up-dating the geometry using the previous fixed velocity field. Separate solution of the momentum, constitutive, energy (if needed), and free surface update allows different techniques to be used in each stage of the iterative procedure, which permits some optimization in the solution process. The main disadvantage of the method is that the iterations, usually of the Picard-type, converge at best linearly, and the radius of convergence is often limited. Despite this, the flexibility of the decoupled boundary element method makes it a good vehicle for testing constitutive behaviour in different flow problems. 111

There are a large number of non-Newtonian flow problems that have been solved by the BEM, including the axi-symmetric extrusion problem, 112 the three-dimensional extrusion and die-design problem, 113 and the finite deformation of a rubber-like material. 114

An example of a BEM calculation is given in Figure 55.1, showing the extrudate shape from a square die ( W i = 0.9), and from a triangular die ( W i = 1.6) for the Maxwell model. The amount of swelling is of the order 30%, at W i = O(1), to be compared with 13% for a Newtonian circular extrudate. The Weissenberg number shown is based on the wall shear rate at the centre of the die far upstream.

In the die-design problem, we wish to find the shape of the die for a given extrudate profile. In this inverse problem, we update the free surface from the

l , ,

I~ A., ARMSTRONG, R.C. and BROWN, R.A., J. Non-Newt. Fluid Mech., 22,129-167 (1987). I~ S. and BERIS, A.N., J. Non-Newt. Fluid Mech., 31, 231-287 (1989). 11~ M., LIU, B. and BERIS, A.N., Int. J. Numer. Methods Fluids, 17, 49-74 (1993). 111BUSH, M.B. and TANNER, R.I, Int. J. Numer. Meth. Fluids, 3, 71-92 (1983); TRAN-CONG,

T. and PHAN-THIEN, N., Rheol. Acta, 27, 639-650 (1988); ZHENG, R., PHAN-THIEN, N. and TANNER, R.I., Rheol. Acta, 30, 499-510 (1991); ZHENG, R., PHAN-THIEN, N. and COLE- MAN, C.J., Comp. Mech., 7, 79-88 (1991); 8, 71-86 (1991); TULLOCK, D.L., New Development and Applications of the Boundary Element Method for some Problems in Elasticity and Viscous Flows, PhD Thesis, Department of Mechanical Engineering, University of Sydney, 1993. 112BUSH, M.B., TANNER, R.I. and PHAN-THIEN, N., J. Non-Newt. Fluid Mech., 18, 143-162

(1985). 113TRAN-CONG, T. and PHAN-THIEN, N., Rheol. Acta, 27, 639-648 (1988); J. Non-Newt.

Fluzd Mech., 30, 37-46 (1988). 114TRAN-CONG, T., ZHENG, R. and PHAN-THIEN, N., Comp. Mechanscs, 6, 205-219 (1990).


1~,~ ~lJJ.~ 11

~adate udate

FIGURE 55.1. Extrudate shape from a square die ( W i = 0.9), and from a triangular die ( W i -- 1.6) for the Maxwell model. The Weissenberg number is based on the wall shear rate far upstream.

given cross-sectional profile of the extrudate, and follow a particle path until we get to the die exit station. The die profile is then updated and the cycle of iterations continues. In practice, extrusion dies usually have a gradual transition from a simple cross-section, usually circular, to the final profile forming section. This is also the adopted method in the numerical simulation; it makes the task of specifying the inflow boundary conditions much simpler, and the algorithm much more robust, since the inflow boundary conditions are fixed. To date, only simple die designs have been attempted, e.g., square and triangular extrudates. Our die profile to produce a square extrudate compares well with existing dies in use, but there is no information on a die profile that produces a triangular extrudate. At low W i , we find that the die profiles for the P T T model are similar in shape to those designed for the Newtonian fluid. Thus, the Newtonian die design could form an excellent starting point for the corresponding viscoelastic die design. Figure 55.2 shows a die design for a square extrudate, for the P T T model at W i = 0.34, based on the shear rate at the wall far upstream. The amount of shrinkage, relative to the extrudate, is about 24%.

Finally, we would like to mention the stream-tube approach, 115 in which the entire domain, including the free surface when it exists, is mapped on to a computational domain where the streamlines are rectilinear. Solving the relevant problem in the new domain, which requires the determination of the mapping function and its inverse, the results in the physical setting are recovered. The above procedure has been used to solve the extrudate problem for a K-BKZ fluid exiting from a duct of square cross-section, as well as entry and contraction flows for a variety of fluids.

tlSNORMANDIN, M. and CLERMONT, J.-R., Proc. IMACS-COST Conf. Comp. Fluid Dy- namics, Lausanne, 1995, 195-201.

56 Epilogue 461

Die Wall

Die Exit

Extrudate

FIGURE 55.2. Square extrudate die design ( W i = 0.34) for the PTT fluid.

56 Epilogue

Although most of the important issues in computational viscoelastic fluid mechanics are now well understood, and the curse of the high Weissenberg number problem has been considerably lessened, there still remain several outstanding problems which require considerable progress in constitutive, asymptotic and numerical modelling.

The first is the nature of the stress singularity at either a geometric or a boundary condition discontinuity. There has been some work done on this, 116 but useful results can only come if an asymptotic element can be designed for a general constitutive equation, in a manner similar to the crack-tip element in fracture mechanics. The problem is compounded by extremely thin stress boundary layers in seemingly harmless flow fields where there are neither geometric nor boundary condition discontinuities; for example, the stress boundary layers near the stagna- tion point behind a sphere, 117 near the walls in a Couette geometry with a small eccentricity ratio, where the velocity profile is not much different from tha t in the

116DAVIES, A.R. and DEVLIN, J., J. Non-Newt. Fluid Mech., 50, 173-191 (1993); HINCH, E.J., J. Non-Newt. Fluid Mech., 50, 161-171 (1993); RENARDY, M., J. Non-Newt. Fluid Mech., 58, 83-89 (1995); J. Non-Newt. Fluid Mech., 52, 91-95 (1994). 117CHILCOTT, M.D. and RALLISON, J.M., J. Non-Newt. Fluid Mech., 29, 381-432 (1988);

HARLEN, O.G., HINCH, E.J. and RALLISON, J.M., J. Non-Newt. Fluid Mech., 44, 229-265 (1992).


circular Couette geometry, tt8 near the wall in a corrugated tube. 119 It may be that more understanding of the constitutive behaviour near a wall is required before the problem of steep stress boundary layers can be resolved. The progress in wall phenomena will help to clear up the nature of the slip boundary conditions at a solid wall, which have been linked to shark skin and melt fracture in the extrusion problem.

The second issue that needs better understanding is the area of viscoelastic instability, which has been examined in w - w Most of the theoretical work done in this area is confined to Oldroyd-type constitutive equations in simple flow geometries where a similarity solution exists. This is necessary partly because a path tracking method is invariably adopted, which would require large computing resources in multi-dimensional flows, and partly because of the need of the base solution, which is not known analytically and must be computed in complex flow geometry. Exceptions are the numerical works of KEUNINGS, 12~ and DEBBAUT and CROCHET, 121 who provide at least two complex viscoelastic flows with limit and bifurcation points, using a path continuation technique as discussed in the FDM section, and a theoretical work concerning the torsional flow of the Oldroyd- B fluid in a bounded geometry, summarised in w which has yielded information in good agreement with experimental observation. The main point here is that we need a concurrent experimental understanding of flow instabilities. The information can then be recycled in building useful constitutive equations that mimic real features. Chasing fictitious limit or bifurcation points, due to the adoption of an inappropriate constitutive model, is also a worthwhile venture, in designing better and more robust algorithms, but sometimes the effort far outweighs the return. Finally, complex three dimensional time dependent flow simulations are now being attempted.

The ingredients for a successful large scale viscoelastic computation will be a combination of automatic mesh generation facility in three dimensions, and a method that combines the flexibility of the finite element technique, and the relatively modest demand on computing resources in a distributing environment, consisting several loosely connected super workstations. Several results in suspension simulations have been published using this approach. 122 The similar need to extend parallel and distributed computations to viscoelastic fluid mechanics cannot be overestimated, considering the fast evolution of computer architecture in these direction.

11SBERIS, A.N., ARMTRONG, R.C. and BROWN, R.C., J. Non-Newt. Fluid Mech., 22, 129- 167 (19S7). 119PILITSIS, G. and BERIS, A.N., J. Non-Newt. Fluid Mech., 39, 375-475 (1991). 12~ R., in Proc. Fourth Int. Con]. Num. Meth. Laminar and Turbulent Flow, Ed.

TAYLOR, C., Pineridge Press, Swanaea, Vol. 2, 1773-1782 (1985). 121DEBBAUT, B. and CROCHET, M.J., J. Non-Newt. Fluid Mech., 20,173-185 (1986). 122KIM, S. and KARRILA, S.J., Microhydrodynamics: Principles and Selected Applications,

Butterworth-Heinemann, Boston, 1991; PHAN-THIEN, N. and KIM, S., Microstructures in Elas- tic Media: Principles and Computational Methods, Oxford University Press, New York, 1994.

Appendix to Chapter 7 463

Appendix to Chapter 7

L i n e a r So l ve r s

TDMA Solver

In the 1-D finite volume formulation, the following discretised equation results,

~i -- bi~i+l q- vi i i -1 q- di, i = 2 , . . . , N - 1. (A7.1)

This can be extended to include the boundary points i = 1 and i = N, by using the boundary conditions. For example, if Dirichlet boundary condition is known at i = 1, then r = dl, and bl = Cl = 0. If the derivative of r is known at i = N, for example, then Cg and dN can be chosen appropriately, and bN = 0. This extended equation can be written in the tri-diagonal matrix form:

1 - b 1

-c2 1 -b2

--CN_ 1 1 --bN-1

--CN 1

~1 dl !

(~2 d2 ,

(N- I ] dN-1

r dN .

The algorithm to solve this system is surprisingly simple and effective; it is called the Thomas algorithm, or the Tri-Diagonal Matrix Algorithm (TDMA). In essence, (AT.l) states that ~1 is known in terms of ~2, and so on. If we carry out this symbolic substitution, then when we get to the end, (~g will be given by all the previous r and the boundary condition ~g+l" At this point a back substitution is carried out. Mathematically, the forward substitution process corresponds to

r : /~r -[- Qi. (A7.2) Thus, (AT.l) becomes

r -- bir + ci(Pi- lr q- Qi-1) + di.

Rearranging, r (1 - c i P i - 1 ) - - bi~i+l 4- ciQi-1 T di.

If we identify this with the forward substitution process (A7.2), then the following recurrent relations for/~ and Qi are obtained:

p ~ _ b~ , Q~_C~Q~_l+d~. (A7.3) 1 - c~P~_l 1 - c~P~_l

With Pl "- bl, Q1 - dl, (A7.4)

these recurrent relations can be used to obtain P~ and Qi. It is noted that PN = O, and thus

CN = QN. (A7.5)


The relation (A7.2) can now be used to find all the values r recurrently. The TDMA is computationally very efficient, being of O(N) in operation counts.

In particular, it can be generalised to a tri-diagonal block matrix, and other banded matrices. The penta-diagonal solver, for example, requires two forward sweeps and one backward sweep.

For multi-dimensional problems, the line-by-line is possibly the best method for generating the solution without requiring a large computational resource. It combines the Gauss-Seidel iterative method with the TDMA. In essence a grid line is chosen, say in the x-direction. The values of the unknowns along this chosen line are solved using the TDMA assuming that all other values are known, from the latest iteration. This is repeated for other lines along the same or different directions until the process converges. For convergence, it is sufficient to require that all the coefficients in the FV formulation be positive. The line-by-line method has several advantages in large-scale simulations:

�9 the storage requirement is of O(N); For example, a viscoelastic problem involving 2 • 10 4 grid points, with 2 x 10 5 unknowns requires only a core of 10 MB;

�9 the CPU time requirement is modest, the operation count is of O(MN), where M is the number of sweeps;

�9 convergence is relatively fast, since sweeping quickly brings the boundary conditions into the computational domain.

Matrix Splitting Methods

Unless the matrix A is well behaved, the iterative solution of

A x = b

requires some form of preconditioning, that is, a non-singular matrix M is sought for, such that the equivalent system

M A x = M b

can be solved iteratively. Most classical methods involve matrix splitting, i.e.,

A = M + N

and the algorithm reads

xk -- xk- 1 + M - irk, (A7.6)

where rk is the residual

rk = b - (M + N)xk-1 . (A7.7)

Different choices of M, the preconditioning matrix, and N lead to different methods. In the Jacobi method, one writes

A = D + L + U ,

where D is the diagonal, L the lower triangular, and U the upper triangular part of A. The preconditioning matrix is taken to be D, leading to the following algorithm for the solution vector x, and the error ek = x - xk:

Xk -- D - l b - D -1 (L + U) Xk_l, (A7.8)

ek = - D - 1 (L 4- U ) e k - 1.

Convergence of the method is guaranteed by the fixed point theorem, if

(A7.9)

p (D -1 (L + U)) < 1,

where p (M) denotes the spectral radius of matrix M, its largest eigenvalue in magnitude. Matrix A meets this condition if it is irreducibly diagonally dominant, that is, its components aij satisfy, for all i,

la"l >- E la'Jl (no sum on i), (A7.10) J

with strict inequality for some of the i's. The method usually converges algebraically in its spectral radius, i.e., as (p (D -1 (L + U ) ) ) a , for some constant a.

In the Gauss-Seidel method, the preconditioning matrix is taken to be

M = D + L ,

and the solution algorithm can be written as,

(D + L)xk = b - U x k - t , (A7.11)

(D + L)ek = Uek-1. (A7.12)

Solutions to these equations are obtained by a forward substitution process. Again, convergence occurs if A is irreducibly diagonally dominant. For other preconditioning methods, including relaxation and incomplete factorisation methods, we recommend the development in EVANS. t23

Conjugate Gradient Methods

Conjugate gradient methods were introduced by HESTENES and STIEFEL as direct methods. 12a In essence, we search for an iterative solution of the form

Xk+l -- x0 + dk,

where dk e Kk+l(ro, A) -- span { ro ,Aro , . . . , A k r o ) ,

r0 = b - Axo being the initial residual and Kk+l(ro ,A) the Krylov subspace of dimensions at most k + 1. To have a more precise understanding of the "correct"

t23EVANS, D.J., Computer J., 4, 73-78 (1961). 124HESTENES, M.R. and STIEFEL, E.L., NBS J. Research, 40, 409-436 (1952).


choice of dk, we star t with a definition. Let B be a positive definite Hermitian matrix. Then the B-norm of x is defined as

IIxIl - ( B x , X) 1/2.

A congugate gradient method C G ( B , A ) is one tha t chooses dk to minimize the error in B - n o r m . Mathematically, we wish to impose on the error ek+l the condition:

(Bek+l, Y) -- O, Vy E Kk+l.

Since K k C. Kk+l, this lea~ to

(Be/c+l,Z) - 0, Vz E Kk.

But ek+l = ek - dk, and (Bek,z) = 0. Thus

(Bdk, z)-0, V z E K k ,

i.e., dk E g k + l and dk _L Kk in B - n o r m . We can seek dk in the form

dk -- akPk (no sum in k),

where Pk is a conveniently chosen vector, and ak is a scalar. Again, use of the orthogonality condition (Bek+l , Pk) = 0 leads to

(Bek, pk)

(Bpk,pk)"

For pi, we can use a B - o r t h o n o r m a l basis for g k + 1 "

P0 -- r0 , i

Pi+I -- A p ~ - ~ ~ i j p j , j-O

(BApi , p j ) ~Tij-- (Bp~,pj ) "

There are two important optimal properties of such conjugate schemes: the error is monotonically decreasing in the B - n o r m ; and the process converges in a finite number of steps, of the order of the number of distinct eigenvalues of A. We outline the algorithm for a conjugate gradient method known as ORTHOMIN: 125

1. For a known initial solution x0, set

r 0 = b - A x 0 , p 0 - r 0 , r 0 - P 0 = r 0 ,

(ro, ~o) Wo=O, ~ o = (Apo ,Fo), x l - - x o + ~ o p o ;

125jEA, K.C. and YOUNG, D.M., Lm. Alg. Appl., 52, 399-417 (1983).


2. For each iteration m - 1 , . . . , do

r m -- r m - 1 -- ~ m - 1 A P m - 1, r m -- r m - 1 -- O l i n - l A T e ) m - I ,

( r m , r m )

~ T m - ( r m - i , r m - 1 ) ' 0 ~ m -

P m -- r m + ~TmPm- 1,

( rm, r m )

(APrn, e,~)

~)m -- r m "~- ~ m ~ ) m - 1,

Xm+ 1 = Xm + ~ m P m ;

3. Check for convergence

The GMRES (preconditioning Generalised Minimal Residual) 126 is a conjugate- gradient-like method, periodically restar ted using the current i terate as the new initial guess, with B -- AT A. We outline the GMRES algori thm below. 127

1. For a known initial solution x0, set

x0 ro -- b - AXo, Vl - - [] ]] ,,Xotl ;

2. For each i teration m - 1 , . . . , k do

hi,m = ( A v m , v i } , i - 1 , . . . , m m

Vm+l -- A r m - Z h i , m V i ' i -1

Vm+l = V m + l / h m + l , m ,

Construct the N x k matr ix V - - { Y l , . . . , Y k } , the upper k x k Heissenberg mat r ix H formed from h,,m, and the vector e = {llx011,0, . . . , 0}.

3. Find y to minimize I[Hy - ell and form the approximate solution

xk = x0 + Vy .

4. If convergence is satisfied then the process is terminated, otherwise take x0 = xk as the new initial es t imate and go back to step 1.

126SAAD, Y. and SCHULTZ, M.H., S I A M J. Sci. Stat. Comp., 7, 856-869 (1986). 127REDDY, J.N. and GARTLING, D.K., The Finite Element Method in Heat Transfer and

Fluid Dynamics, CRC Press, Florida, 1994.


Numer ica l Quadrature

There are several excellent texts dealing with numerical quadra tu re , e.g., DAVIS and RABINOWITZ, t2s and STROIYD and SECREST. 129 We provide a brief summary of the Gauss-Legendre integration rules here.

Denote by

I ( f ) -- f (x)dx (B7.1)

the exact definite integral of f (x) in the interval [a, b]. The quadrature Q(f) is just any rule tha t provides an approximation of (B.1). In general, any information on the integrand may be used to construct I ( f ) ; one often has a set of values f(r at a set of discrete points ~=, n -- 1 , . . . , N, called the quadrature points; and the quadrature rule takes the form

N

Q(f) - Z w,~f(~,,,) + E, (B7.2) n--1

where w=, n = 1 , . . . , N, are called the quadrature weights, E is the error in the quadrature. Sometimes the integrand is written as a product of a known function, w(x), called the weight function, and h(x), to emphasize the special form of the integrand. Obviously the smaller the value of the error, the better is the quadrature, for a given integrand. Of the different methods for choosing quadrature points and weights, the most practical one would be the error annihilation rules.

Suppose tha t {hi(x)}, i = 1 , . . . , g , is a finite basis tha t spans the class of integrands chosen. Then any of these integrands can be represented by a linear combination of hi(x). It is clear then tha t the quadrature rule will be exact for the chosen class of integrands, if it is exact for these basis functions. We thus write, for an N-point quadrature,

N b

E w,~hi(r -- mi -- ~a hi(x) dx, i -- O, 1,. . . ,K. (B7.3) n--1

This represents a set of K equations, in the 2N unknowns {w=, ~n}" One can choose the points beforehand (say evenly spaced in the interval [a, b]), set K -- N - 1, and determine the weights by solving a set of linear equations (B.3), provided tha t all the mi exist. A choice of hi = x i, i = 0 , 1 , . . . , K will lead to Newton-Cotes rules; the lowest order ones are the trapezoid and Simpson's rules . Thus, an N-point Newton-Cotes quadrature is exact for polynomials of degree at most N - 1.

The most important class of numerical integration is undoubtedly Gauss quadratures. Here, both the weights and the points are unknowns, so tha t by setting K = 2 N - 1, one hopes to annihilate the error for a larger class of functions than before. Equation (B.3) can now be solved for {w=,~=}, by taking hi(x) -- x',

128DAVIS, P.J and RABINOWITZ, P. Methods of Numerical Integration, Academic Press, New York, 1975.

129STROUD, A.H. and SECREST, D., Gausszan Quadrature Formulas, Prentice Hall, New York, 1966.


i - - 0 , 1 , . . . , 2 N - 1 �9 N bi+l _ ai+l

Z w n ( ~ = i + 1 "

n----I The solution to these nonlinear equations provides us with the quadrature points and weights. However, we can make a great deal of progress without having to solve them, as will be shown below. First, let us rescale the argument x with the aid of the transformation

b - a a + b 2 2 '

so that we are dealing with an integral between • Suppose hi(x) are normalized so that

=

1

and let Cn, n -- 1 , . . . , N be the distinct and real zeroes of hN(x) between •

hg( (n ) -- O, n - - 1 , . . . , N .

Such an orthonormal basis is the normalized Legendre polynomial:

h i ( x ) - - i ( 2 i + l ) 2 P~(x),

where /~(x) is Legendre polynomial of order i.t30 For any polynomial f ( x ) of degree 2 N - 1 at most, we can write by a process of long division:

f (x) -- q(x)hn (x) + r(x), (B7.4)

where q(x) is the quotient and r(x) the remainder, and both are polynomials of degree at most N - 1. These can be expressed as linear combinations of hi, i - 0, 1 , . . . , N - 1. Thus

f I ( f ) -- I(r) = r(x)dx, (B7.5) 1

due to the orthogonal condition of the normalized Legendre polynomials. We can now express the remainder polynomial as a Lagrangian interpolation of the values

N r(x) -- Z r((n)l(N)(x) ' (B7.6)

rim1

where l(N)(x) are the Lagrangian interpolation polynomials: TM

l(nN)(X) __ (X -- (:I)""" (X -- (n_,,l,)(X -- (n-l-l)''" (X -- (:N)

(('n -- ('I)""" ((:n -- ~'n-l)((:n -- ('n-i-l)''" ((:n -- (~N) N

n jC~n - r "

130 ABRAMOWITZ, M. and STEGUN, I.A., Handbook of Mathematical Functions with Formu- las, Graphs and Mathematical Tables, 9th Edition, Dover, New York, 1972. :3:RALSTON, A., A First Course in Numerical Analysis, McGraw-Hill, New York, 1965.

From ( B . 4 - B.6) and the fact that (n are the zeros of hn(x), we have

N N

,(:) - =

n = l n = l

(B7.7)

where

' z N) -

1

(B7.8)

It is clear that, once the points are known, the weights can be found by a simple integration. It may be shown from the properties of Legendre polynomials that

W l ~ "---

2

(N + 1)PN+I((n)P~((n)"

Furthermore, if f(x) is not a polynomial of, at most, degree 2 N - 1, then the error involved in the quadrature (B.7) is

22N+l(Nt) 4 E = (2N + 1)[(2N)[] 3f(2N)(~)' --I < C, < i.

The rule (B.7) is most popular and is known as Gauss-Legendre rule; it is exact for polynomials of order 2 N - 1 or less. Extensive tables of weights and points are given in STROUD and SECRF_~T; 132

Different Gauss rules can be derived (Jacobi, Chebyshev, Radau, Lobatto, etc.), depending on the types of integrands and the integration interval (finite, semi- infinite or infinite); a good review of these can be found in RALSTON 133 .

For integrals of the form

~ f(x) dS(x),

where fie is a two-dimensional region, a suitable transformation can be devised (via a parametric element, for example) such that. one now deals with

i l f l g((,~) d(d~, 1 1

where the integrand now involves both f(x) and the Jacobian of the transformation. Such an integral can be evaluated by the Gauss-Legendre rule

1 1 N

1 1 m,n=l

132STROUD, A.H. and SECREST, D., Gaussian Quadrature Formulas, Prentice Hall, New York, 1966. 133RALSTON, A., A First Course in Numerical Analysis, McGraw-Hill, New York, 1965.

Linear and Quadratic Shape Functions

The number of nodes per element depends on the order of the interpolant and the shape of the element. A linear triangular element has three nodes located at the vertices of the element. Its nodal shape functions are given by

N1 - 711 ,

N2 -- 7]2, 0_< 7]~,712 -< 1. (C7.1)

N 3 -- 1 - - 711 - - 7]2 ,

The triangle defined by this interpolation has straight sides, due to the degree of interpolation. To model a curved triangular element accurately, a quadratic interpolation is employed, with the following nodal shape functions:

N1 - - 711 - - �89 Na + N 6 ) ,

N2 -- 712- � 8 9

N 3 -- 1 - 711 - T]2 - I ( N 5 + N 6) ,

N4 = 4711 7]2 ,

N5 -- 4712(1 -7 ] : - 7]2) ,

N e -- 47] 1(1 - 711 - 7 ] 2 ) ,

0 ~ 7 1 1 , 7 ] 2 --~ 1. (C7.2)

Note that there are six nodes per element, the three extra nodes being located at the midsides of the triangle.

One can also have a linear quadrilateral, i.e., with straight sides, having four nodes located at the vertices with associated nodal shape functions:

i I _ 1 - - ~ ( 1 - - ~71)(1 - - ~72),

N 2 -- 1(Iq-711)(1-712) 4

N 3 _ 1 - - ~ ( 1 + 711)(1 + 712),

i 4 __ 1 -- ~ ( i - 711)(1 q- 712),

- 1 _< 7]1 , T]2 _~ i . (C7 .3)


Quadrilaterals of curved sides can be modeled by a quadratic quadrilateral with eight nodes and nodal shape functions:

N1 - 1( 1 - "1)( 1 - ?]2) - �89 N5 + N8)

N 2 = �88 + . 1 ) ( 1 - . 2 ) - �89 N5 + N 6 ) ,

N a = �88 + . 1 ) ( 1 + . 2 ) - �89 Ne + N T ) ,

N 4 - - �88 - .1)(1 -[- " 2 ) - ~( N7 + N s ) ,

N 5 - - � 8 9

N e - �89 - , i ) ( 1 + , 1 ) ,

N 7 -- � 8 9

N s -- � 8 9

with - 1 _~ "1, "2 - 1. It should be noted that

Z N ~ = 1,

which is a property of the Lagrangian interpolation function.

(cT.a)

(c7.5)

Index

Abbott, J.R., 249, 261, 263, 312 Abbott~ T.N.G., 21~ 34 Abramowitz, 469 acceleration

field, 3 gradient, 26, 63 material description, 4 motion with zero, 38 spatial description, 4 waves, 393

Acierno, 229 Acrivos, 241, 242, 249, 251,261 Adachi, 414 Adler, 240, 249, 251 Advani, 259 affine motion, 221 Airy stress function, 126 Alexandrov, 100 Altobelli~ 261 Ames, 397 angular momemtum, 95 annular flow

axial, 43, 300 helical, 43, 291 second normal stress difference in,

301 annulus

radial flow, 343 Antman, 142 Apelian, 425, 453 approximate constitutive equations, 148 approximations to strain history

infinitesimal strain, 29 infinitesimal velocity, 30 small displacements added to a

large motion, 31 small velocity added to a large

velocity, 33 approximators to regions with fractal

boundaries, 114 Armstrong, 156, 177, 180, 191, 199-

201~ 204, 207, 219, 221,238, 249, 256, 260, 261, 263, 268, 371,378-380, 413, 414, 418, 425, 447, 452-454, 459, 462

Arnold, 122 artificial viscosity, 436 assembly process, 450 asymptotic relations, 353 Atkinson, 174, 199 Attan6, 143, 360 autonomous system, 5

as a steady velocity field, 6 Avagliano, 373

474 Index

Avgousti, 382, 459 axial motion

fanned planes, 44, 301 plane cone-and-plate flow, 302 rectilinear flow, 302

axisymmetric flow, 364

B-norm, 466 Babuska, 452 Bagley, 409 balance equations

for angular momentum, 95 for energy, 97 for linear momentum, 89 for mass, 88

Barakos, 456 Barnes, 364 Barnsley, 108 basis vectors

natural, 75 reciprocal, 75

Batchelor, 178, 242 Bay, 259 bead-spring model, 166 Beavers, 239 Beer, 397 Bergen, 41 Beris, 382, 447, 452, 459, 462 Bernstein, 7, 234, 322, 344, 352, 353,

384, 455 Bhatia, 7 Bianchi, 189 bifurcation, 359

point, 417 limit point, 417

Binnington, 187 Bird, 156,177, 180, 191,199-201,204,

207, 218, 219, 221,227, 228, 234, 238, 239, 268

Bird-Curtiss model, 238 Bisgaard, 455 Blankenship, 174 Boger, 187, 190, 256 Boger fluid, 187 Boltzmann temperature, 160 bond probability, 160 Booij, 199, 346

Bossis, 241,251 boundary condition

essential, 405, 449 finite volume, 432 free surface, 405 inflow-outflow, 407 initial, 411 natural, 405, 449 pressure, 410 slip, no-slip, 409 symmetry, 407 thermal, 411 vorticity, 411

boundary element method, 459 Brady, 241,251, 262, 265, 414, 418 Brauer, 5, 12 Brenner, 240, 241, 244, 245, 249 Brezzi, 452 Brooks, 452 Brown, E.F., 322 Brown, R.A., 249, 261,263, 371,373,

378-380, 413, 414, 418, 425, 447, 452-454, 459, 462

Brownian forces, 168 Burghardt, W.R., 322 Burkhardt, J.V., 418 Bush, 459 Byars, 371, 373, 378-380

Cabot, 409 Cantor set, 104 Carey, 452 Carlson, 10 Carmi, 384 Carter, 332 Caswell, 325, 455 Cauchy stress tensor, 91 Cauchy-Green strain tensor, 9

relative, 11 three-dimensional, 10

Cauchy's first law of motion, 91 Cauchy's reciprocal theorem, 92 Cauchy's second law of motion, 96 Cayley-Hamilton theorem, 10 CFL condition, 422 CG(B, A), 466 Chandrasekhar, 172, 174, 257

channel flow, 294 characteristic equation, 385 Chartoff, 15, 309 Chaur~, 58, 64 Chen, 241,251,395 Chilcott, 194, 414, 423, 461 Chilcott-Rallison model, 194, 423 Chirinos, 190 Chorin, 423 Chorin-type methods, 422 Chow, 261 Christoffel symbols, 77 Cintra, 259 circular Couette flow, 43 classical solution, 449 classification, 399

eigenvalues, 389 first order systems, 387 PDE, 385

Clermont, 460 closure

composite, 259 orthotropic fitted, 259 quadratic, 259

Cochrane, 414, 420 Coleman, B.D., 14, 30, 41, 51, 74, 133,

148, 149, 152, 274, 277, 278, 293, 309, 322, 352, 394

Coleman, C.J., 459 components of tensors

contravariant, 75, 76 covariant, 75, 76 mixed, 76 physical, 76

compressibility, 423 cone-and-plate flow, 297

viscomteric functions from, 297 configuration, 2

change of local, 70 change of reference, 73 spatial, 3

conical surfaces, 287 conjugate gradient methods, 465 conservation

angular momentum, 95 energy, 97 linear momentum, 89

Index 475

mass, 88 conservative discretisation, 426 constant stretch history, 51 constitutive equation, 130, 138

approximations to, 148 Bird-Curtiss model, 238 elastic fluid, 132 elastic material, 130 fading memory, 148 finite linear, 150 first order fluid, 151 fourth order fluid, 151 Fr~het differential, 149 general principles, 128 Hilbert space, 149 isotropic solid, 131 isotropy of a functional, 143 Johnson-Segalman model, 228 K-BKZ fluid, 358 linear viscoelasticity, 150 Maxwell model, 186 Oldroyd B-fluid, 186 path lines, 272 PTT model, 221 reflection, 129 restrictions due to frame indiffer-

ence, 134, 140 restrictions due to isotropy, 136 restrictions due to incompressibil-

ity, 142 resrictions due to objectivity, 134,

140 restrictions due to symmetry group,

72, 130 restrictions due to unimodular group,

72, 74, 132, 138 retardation of a process, 152 Riesz representation theorem, 150 second order fluid, 151 second order viscoelasticity, 150 simple fluid, 138 simple material, 138 third order fluid, 151 uses of integrity bases, 144

continuation by one history after cessation of the first, 354

continuity equation, 88

476 Index

contravariant metric tensor, 75 controllability, 284

partial, 284 Cook, 381 copolymer, 157 Couette flow, 283, 288 Courant number, 435 Court, 414 covariant

derivative, 76 metric tensor, 74

Craik, 361 Crain, 190 Crank-Nicholson, 429 Crawley, 334 creeping flows

three-dimensional, 327 two-dimensional, 326

Crewther, 364, 368,370, 413, 414, 418 Criminale, 276, 305 Crochet, 397, 414, 415, 448, 452-455,

462 curl of a vector

Cartesian coordinates, 79 cylindrical polar coordinates, 79 spherical polar coordinates, 80

Currie, 234 Curtiss, 156, 177, 180, 191, 199-201,

204, 207, 219, 221,234, 238, 239

Davies, 397, 414, 415, 448, 461 Davis, 468 de Bats, 219 de Gennes, 156, 230, 231 Debbaut, 452, 462 Deborah number, 364 Debye's relation, 166 definition of

extensional flow, 14, 48, 59 extensional viscosity, 314 fluidity function r 288 non-viscometric flow, 47 normal stress difference coefficients

kI/1 and ~P2,276 normal stress differences N1 and

N2,276

shear stress function T, 276 viscomteric flow, 40 viscosity 7, 276

deformation gradient, 5 evolution, 5

del operator Cartesian coordinates, 81 cylindrical polar coordinates, 81 spherical polar coordinates, 82

Denn, 256, 382, 409 derivative

material, 4 Devlin, 461 Diagonally dominant, 465 dilute polymer solutions

diffusion equation, 164 dumbbell model, 166 Gaussian bond probability, 159 general characteristics, 157 random walk model, 159 Rouse model, 166 time scales, 170

Dinh, 256, 260 Dinh-Armstrong model, 260 dispersion, 420 displacement gradient, 29 dissipation, 420 divergence of a vector

Cartesian coordinates, 79 cylindrical polar coordinates, 79 spherical polar coordinates, 79

divergence theorem regions with fractal boundaries,

114 regular regions, 113

Doi, 156, 230, 231,233, 241,249, 251, 256, 259

Doi-Edwards model, 231 Currie potential, 234 differential form, 235 Larson's approximation, 235

Dorrepaal, 414 dumbbell model, 166, 180 Dupont, 455 Dupret, 260, 452 dyadic analysis, 74 dynamic moduli

loss, 346 storage, 346

dynamical system, 6

eccentric cylinder, 21 eccentric disk rheometer, 3.11 edge fracture, 334 Edwards, 156,213,230, 231,233,249,

256 EEME, 453 Einstein, 246 elastic fluids, 132

integrity basis, 137 elastic materials, 130

objectivity restrictions, 134 elliptic, 386, 399 elongational rate, 193 end-to-end vector, 158, 182 energy balance energy method, 383 entropic spring force, 167 Epstein, 212 equations of motion

body force, 281 Cartesian coordinates, 280 cylindrical polar coordinates, 281 inertia, 282 spherical polar coordinates, 281

equipartition of energy, 170 Ericksen, 25, 41, 63, 256, 276, 305,

307, 387 Ericksen's conjecture, 307

counter-examples, 307~ 308 Eringen, 147 error annihilation, 468 Eulerian description, 4 Evans, 199~ 465 evolutionary, 389 EVSS, 454 excluded volume, 197 experimental results

extensional flow, 313 extensional flow after simple shear,

355 viscomteric flow, 302

exponential scheme, 440 extensional flow, 48, 277, 313

Index 477

inhomogeous, 315 extensional viscosity, 314 exponential function of a matrix, 12 Putzer's method~ 15 extra stress tensor, 142

fading memory, 148 Fahy, 137, 147 Falconer, 102, 104, 105, 108, 109, 111,

113 Fan, X.-J., 191 Fan, Y., 454 fanned plane motion, 44, 301 Feder, 99 Federer, 112 Feinberg, 383, 384 FENE model, 191,424

elongational viscosity, 194 viscometric functions, 192

Ferry, 213, 215, 216 Filbey, 276, 305 Finger tensor, 223 finite difference method, 413 finite difference operator, 416 finite elasticity, 130 finite element method, 448 finite norm, 148 finite volume methods, 422 Finlayson, 413, 447 Finzi, 127 Fixman, 196, 197 Fletcher, 397 Flory, 156, 158, 164, 166, 167, 213,

22O flow across a deep slot, 340 flow along a deep slot, 331 flows in tubes, 302

rectangular cross-section, 329,445 fluctuation-dissipation theorem, 169,

170 Fokker-Planck equation, 172 Folgar, 256, 259 Folgar-Tucker model, 259 force on a chain, 166 Fortin, 452, 453 Fosdick, 129, 307, 308, 344 Fourier

478 Index

method, 402 transform, 403

fractal curves box dimension, 107 dimension, 101 divergence theorem, 113 Hausdorff dimension, 103 length, 99 Minkowski dimension, 107 snow flakes Stokes' theorem, 115 unit normal, 109 unit tangent, 109

frame indifference, 129 frames of reference in relative motion Frankel, 251 free-draining, 167 free surface

edge fracture, 334 trough flow, 330

Freed, 213 Frenkel, 240 Fujimoto, 214 Fukushima, 261 function basis, 147 fundamental matrix, 12

Gadala-Maria, 241,242, 249, 261 Gama, 354 Gartling, 398, 412, 448 Gatski, 414 Gauss quadratures, 468 Gauss-Legendre rules, 468, 470 Gauss-Seidel method, 465 generalised torsional flow, 298 Genieser, 453, 454 Gent, 15, 309 Georges~u, 190, 330 Gervang, 309, 330, 426 Geurts, 239 Giesekus, 50, 187, 199, 230, 326, 328 Giesekus form, 187, 199 Gleissle, 239 GMRES, 467 Goddard, 278, 312 Goldsmith, 261 Gordon, 221,229

Gottleib, 376 gradient

acceleration, 26, 63 deformation, 5 displacement, 29 relative deformation, 23 velocity, 4

Graessley, 334 Graham, 249, 260, 261,263 Green, 128, 156, 219 Green-Cauchy tensor

relative, 11 two-dimensional, 10

Greenberg, 9 Gresho, 452 Griffiths, 452 Guillope, 382 Gurtin, 91,117, 121,394, 395 Guth, 219, 221

Hadamard, 386, 413 Hadamard instability, 386 Hand, 256 Happel, 241,244, 245 Harrison, 113, 115 Hassager, 156, 177, 180, 191,199-201,

204, 207, 219, 221, 238, 268, 455

Hatzikiriakos, 405, 409 Hausdorff

dimension, 103 measure, 103

helical flow, 43, 291, 316 steady, 43 unsteady, 316

helical-axial fanned flow, 48 helical-torsional fanned flow, 311 helical-torsional flow, 48, 310 helicoidal flow, 43 Herstein, 72, 74, 131 Hestenes, 465 Heuer, 214 Hilbert space, 149 Hill, 244 Hinch, 167, 169, 182, 197, 205, 237,

241,246, 247, 255, 256, 461 Hoger, 10

Holstein, 414 homogeneous material, 130 homopolymer, 157 Hood, 417, 452 Hookham, 261 Huang, 447 Hughes, 452 Hur, 256 Hutton, 364 hydrodynamic interaction, 194 hyperbolic, 399

equations, 395

ill-posed, 413 incompressibility, 142 independent alignment assumption, 231 infinitesimal

strain, 29 velocity, 30

Ingber, 258 integrrity bases, 144 interface properties, 431 internal energy, 97 internal viscosity, 198 interpolation, 449 invariance restrictions, 133 Iooss, 413 isochoric motion, 88 isoparametric, 450 isotropic function, 136 isotropic functional, 143 isotropic solids, 131

objectivity restrictions, 136 Issa, 425 Iterative methods, 464 Ito, 163 Iwamiya, 261

Jacobi method, 464 Jacobian, 451 James, 156, 219-221 Jameson, 424 Jameux, 51 Jea, 417, 466 Jeffery, 246 Jeffery's solution, 246 Jeffrey, 390 Jeff-reys, 350

Index 479

Jenkins, 262 Ji, 414, 418 Jin, 423, 425, 454 Johnson, 201,213, 221,397, 412, 448 Johnson-Segalman model, 228 Jones, 19, 20, 34, 161 Jongschaap, 229, 251 Joseph, 151, 152, 239, 325, 330, 334,

342, 360, 385, 388-391,413, 452

Josse, 413 Jozsa, 364, 413, 414, 418 JRS model, 390

plane flows, 390 jump in strain, 320 jump in stress, 321

K-BKZ model, 234 ultrasonic limits, 352

Kadivar, 455 Kalika, 382, 409 Kalogerakis, 405, 409 Kamal, 234 Kaneda, 259 Karnis, 261 Karrila, 241,244, 245, 251,397, 462 Kataoka, 250, 260 Kato, 214 Kaye, 234 Kearsley, 234, 327, 331,333 Keentok, 190, 330, 335, 337 Keller, H.B., 413, 414, 417, 418 Keller, J.B., 240, 250 Kellogg, 407 Kepes apparatus, 19 Keunings, 452, 462 Khomami, 454 Kim, M.E., 413, 452 Kim, S., 241,244, 245, 251,397, 462 Kim-E, 414, 418 kinematics, 3 kinetic energy, 97 King, 425, 453 Kirkwood, 167, 194 Kitano, 250, 260 Koch curve, 104, 105 Koh, 261

480 Index

Kramers, 209 form, 180, 200 stress tensor, 204

Kraynik, 409 Krieger, 241,250, 263 Krylov subspace, 465 Kubfc(~k, 413, 418 Kubo, 132 Kuhn, 156, 159, 166, 180, 198 Kuhn segment, 159

La Mantia, 229 Labropulu, 414 Ladyzhenskaya, 452 Lagrangian

description, 3 interpolation, 469

Lamb, 165, 241 Landau, 169, 178, 242 Landel, 216 Langevin

equation, 169 function, 168

Langlois, 142, 330 Lapidus, 115 Laplace equation, 400 large strain history

superposed infinitesimal motion, 348

Larsen, 309, 330, 426 Larson, 156, 177, 178, 234-238, 360,

371,379, 381,382, 414, 418 Laun, 215, 218 Lax-Wendroff, 438 Leal, 167, 197, 198,205, 237, 241,246,

247, 255, 256, 259, 261,262, 407, 414, 419

leap-frog, 437 Lee, 191,217, 338, 414, 452 Legat, 453 Legendre polynomial, 469 Leighton, 261 length of a curve, 100 Leonov, 230 Levy, 240 Lifshitz, 169, 178, 242 Lira, 409

limit point, 418 Lin, 169, 171 line by line, 464 linear viscoelasticity

finite, 150 infinitesimal, 150

linearisation from a large motion31, 348 from a large velocity field, 33,363 from a state of rest, 360 from a viscometric flow, 351 from an extensional flow, 351

linearised stability from a fully developed flow, 363 from a rigid body motion, 368 from a state of rest, 360 from a torsional flow, 364 in bounded domain, 374 in unbounded domain, 364 of axisymmetric torsional flow, 364 of non-axisymmetric flow, 369 spiral motions, 371 steady bifurcations, 368 unsteady bifurcations, 368

Liouville equation, 222, 257 Lipscomb, 256 Liu, 382, 452, 459 Lockett, 384 Lodge, 156, 187, 190, 212, 214, 219,

225, 229, 234, 343 Lodge model, 187 Lodge-Meissner rule, 229, 322 Lumley, 414 Lunsmann, 453, 454 Luo, 455, 456 Lyon, 262

Mackay, 190 Magda, 338, 371,379 Majumdar, 234 Malkus, 455 Malone, 423 Mandelbrot, 102 Manero, 167, 197, 198 Marchal, 452, 453 Marek, 413, 418 Marin, 19

Markovian process, 164 Markovitz, 274 Markowich, 396 Marlow, 142 Marrucci, 229, 234, 235 Martin, 9 Martins, 121, 122 Mashelkar, 234 Mason, 261 matrix

exponential funcion, 13 nilpotent, 14 Putzer's method, 15 splitting, 464

Mavriplis, 424 maximum principle for elliptic equa-

tions, 336 Maxwell equation

classification, 407 Maxwell model, 186

start-up elongational flow, 188 start-up shear flow, 188

Maxwell, B., 15, 309 Maxwell, J.C., 186 Mayne, 307, 308 McKean, 163 McKinley, 371,373, 378-381 McLeod, 308 McTigue, 262 mean curvature, 407 measurement of N1

cone-and-plate flow, 297 flow across a slot, 342 torsional flow, 296

measurement of N2 annular flow, 301 cone-and-plate flow, 300 flow along a slot, 331 flow down a tilted trough, 330 flow in pipes of non-circular cross-

section, 305 generalised torsional flow, 299 motion of fanned planes, 301 plane cone-and-plate flow, 300 torsional flow, 296

measurement of viscosity r/ Couette flow, 289

Index 481

Poiseuille flow, 289 torsional flow, 296

Meissner, 229 memory

fading, 148 integral expansions, 150, 151,152,

153 Mena, 143, 354, 356, 358, 360 Mendelson, 413 Menon, 414, 418 Metzner, 230, 239, 241,249 microtructures, 155 Minkowski, 107 Mitsoulis, 456 mixed formulation, 452 Mizel, 9, 121,149 Mizutani, 214 modulus

loss, 346 storage, 346

Moller, 15 Mondy, 258, 261 Morales-Patino, 358 Morgan, 398, 448, 451 Morland, 217 Morton, 415 motions with constant stretch history

definition, 51 eccentric disk rheometer, 15, 53 extensional flow, 14, 48, 59 helical-axial fanned motion, 48 helical-torsional flow, 48 necessary conditions, 54 Poiseuille-torsional flow, 48 strong flow, 60 sufficient conditions, 56 viscometric flow, 41 weak flow, 41

motions with An = 0 for n odd, 35 motions with zero acceleration, 38

flows with A3 --0, 38 Motta, 358 Muller, 382

N1,276 N2, 276 Na, 425

482 Index

Nadim, 249 Nagasawa, 214 Narain, 389 natural inner product, 412 nearly viscometric flows, 351

number of linearly independent functionals, 351

small oscillations on simple shear, 346, 352

stresses, 348 ultrasonic limits, 352

neo-Hookean solid network theories, 219 Newton-Cotes, 468 Nguyen, 190 Nijenhuis, 190 N oda, 214 Noetinger, 241 Nohel, 5, 12 Noll, 5, 14, 30, 48, 51, 62, 70, 74, 117,

123, 128, 130, 132, 133, 138, 141,148, 152, 274, 277, 278, 293, 309

non-atone deformation, 198 non-autonomous system, 6

as an unsteady velocity field, 6 non-extensional flows, 50 non-isothermal problems

Fourier law, 411 non-Fourier law, 411

non-viscometric flows, 47, 278, 309 normal stress

on a free surface, 305 on a pipe wall, 305

normal stress difference coefficients qs 1 and ~2,276

normal stress differences N1 and N2, 276

Normandin, 460 Northey, 454 Norton, 113, 115 Nott, 262, 265 Nunan, 240, 250 Nunez, 354

objectivity, 129 of a scalar, tensor or a vector, 62

principle, 129 restrictions on constitutive rela-

tions, 134, 140 observer, 60 Oden, 452 Ohl, 239 Olagunju, 373, 381 Oldroyd, 21, 26, 48, 73, 128, 140, 186,

305, 310 recursive formulae, 26

Oldroyd-B model, 186, 187 elongational viscosity, 188 oscillatory flow, 187 viscometric functions, 188

order fluids, 151 Orszag, 376 orthogonal group, 131 orthogonal tensor, 9, 18 Orthomin, 466 Oseen tensor, 195 Ottinger, 156, 196, 218, 239 Owen, 347, 349, 351 0ztekin, 371,373, 379

Paddon, 414 Palade, 143, 360 Palmen, 346 Panizza, 336 Papanicolaou, 174 parabolic, 399 parallel-plate flow parametric element, 450 Park, 456 partial differential equation

elliptic, 385, 402 evolutionary, 389 heat equation, 385, 402 hyperbolic, 385, 389, 402 Laplace's equation, 385 maximum principle, 336 mixed, 387 nonlinear hyperbolic, 395 parabolic, 402 quasilinear, 390 wave equation, 402

Patankar, 397, 419, 425, 440 path lines, 11

Oldroyd's method, 20 pert urbation, 19

path tracking, 417 Payne, 336 Pearson, 383, 409 penalty method, 423, 452 Perera, 414, 419 perturbation

about a state of rest, 350 about extensional flow, 350 about simple shear, 350 small displacements, 349

Peterlin, 189, 191 Peterlin approximation, 191 Petit, 241 Petrie, 356, 383, 409 Phan-Thien-Graham model, 260 Phelan, 423 Phillips, 249, 261,263 Phillips et al.'s model, 263 physical components, 76 Piola-Kirchoff stress tensor, 98 Pipkin, 31, 41, 44, 49, 64, 133, 146,

147, 150, 283, 284, 286-288, 298, 305, 310, 315, 325, 326, 329, 342, 343, 347, 349, 351, 352

plane creeping flows, 326 role of N1,338

Poiseuille flow, 289 polar decomposition, 9 polymer contour length, 159 polymerisation index, 157 position vector, 3 potential flows, 325 Power, 244 Pozrikidis, 398 Prager, 234 principle of frame indifference, 129 Pritchard, 343 Proctor, 382 Protter, 386 PTT model

elongational viscosity, 227 integral form, 223 oscillatory flow, 226 step strain, 228

Index 483

viscometric functions, 226 Putzer, 15 Putzer's method

quadrature, 468 points, 468 weights, 468

qualitative dynamics, 385 plane flows of a JRS fluid, 390

Rabinowitz, 468 radius of gyration, 165 Raithby, 425 Rajagopal, 53, 312, 322, 333, 370, 414,

418 Rajagopalan, 454 Rallison, 194, 241,414, 423, 461 Ralston, 469, 470 Ramamurthy, 382, 405, 409 random walk model, 159 Rayleigh problem, 389, 395 Rayleigh-Rit z-Galerkin, 450 rectilinear flow, 42, 327

role of N2, 329 rectilinear motions, 302

Ericksen's conjecture Reddy, 398, 412, 448 reference configuration

change of, 70 relative deformation gradient, 23 relaxation function, 150 relaxation method, 434 Renardy, 152, 382, 383, 385, 388-390,

396, 407, 408, 452, 453, 461 reptation model, 230 Reshetnyak, 100 retardation of a history, 252 retardation time, 186 Reynolds, 85 Reynolds' transport theorem, 85 Rheinboldt, 418 Richtmyer, 415 rigid body motion, 18 rigid dumbell, 199 Riseman, 167, 194 Rivlin, 25, 40, 63, 74, 128, 132, 146,

276, 282, 322, 325, 329, 330 Rivlin-Ericksen tensors, 25, 276

484 Index

general formulae, 25 in motions with constant stretch

history, 54 Oldroyd's formulae, 27

Rizzo, 229 Roe, 212 Rosenberg, 142 rotlet, 243 Rotstein, 234 Rouse, 166, 208 Rouse matrix, 208 Rouse model, 166, 207

complex viscosity, 211 non-isothermal viscometric func-

tions, 217 relaxation times, 210

Rouse relaxation time, 182 Ryskin, 241

Saad, 234, 238, 239, 377, 417, 467 Sakai, 414 Sanchez-Palencia, 239, 240 Sani, 452 Saut, 151,152, 382, 385, 388-391,452 Scarborough, 433 Schaeffer, 234 Schieber, 239 Schowalter, 221,229, 383, 384, 409 Schrag, 190, 213 Schultz, 417, 467 Schur norm, 29, 148 Scott, 343 secant modulus, 395

in propagating vortex sheets, 395 second-order fluids, 324

edge fracture, 334 flow across a slot, 340 flow along a slot, 331 free surface, 335, 337 Giesekus-Pipkin theorem, 325 Huilgol's theorems on plane creep-

ing flows, 339 pipe flows, 308 plane creeping flows, 326 radial flow across an annulus, 343 role of N1,340 role of N2,329

Tanner's theorem, 339 three dimensional creeping flows,

327 uniqueness theorem, 339

secondary flow, 309 Ericksen's conjecture, 307 pipes of rectangular cross-section,

329 pipes of square cross-section, 330

Secrest, 468, 470 Segalman, 221 Serrania, 358 Serrin, 129, 307, 308, 383 sets of finite perimeter, 122 shape functions, 449 shape of a body, 123 Shaqfeh, 360, 382, 383 shear rate shear stress function % 276

importance in viscometric flow, 283

Sherwood, 190, 330 shift factor, 214 Shirota, 250, 260 ~ilhavp, 120, 123 Simmons, 346, 347 simple fluid, 130

incompressibility, 142 objectivity restrict_ions, 140 symmetry group, 138

simple shearing flow steady, 274 superposed oscillations, 346 unsteady, 278

SIMPLER, 444 Simpson's rules, 468 Sinton, 261 Slemrod, 396 slot

flow across, 340 flow along, 331

small displacement on a large motion, 31

small oscillations on simple shear, 346 asymptotic relations, 352 nearly viscometric flow, 352

small velocity on a large velocity, 33

Smith, 74, 137, 147 Smoluchowski equation, 174 smoothed-out Brownian force, 177 Sokolov, 384 Song, 414 Southwell, 414 spectral radius, 465 Spencer, 128, 147 Squire's theorem, 383 Sridhar, 190, 314 stability, 359

cone-and-plate, 381 Couette, 381 extrusion, 381 rest state, 360 torsional, 364, 368, 371, 373, 374 von Neumann, 421,429, 435

staggered grid, 443 static continuation, 322 Staudinger, 214 Staudinger's rule, 214 steady flow, 7 Stegun, 469 Stephens, 261 Stiefel, 465 stiffness matrix, 450 Stippes, 120 stochastic differential equation, 169 Stokes-Einstein relation, 171 Stokes' theorem

regions with fractal boundaries, 116

regular regions, 115 storage modulus, 346 strain jump, 320 strain tensors, 23

calculation of, 24 Strauss, 414 stream-tube approach, 460 stress jump

in double step strain jumps, 322 in single step strain jumps, 322 Lodge-Meissner rule, 229, 322

stress power stress relaxation, 322 stress tensor

Cauchy, 9

Index 485

continuity, 120 existence, 95, 117 Piola-Kirchhoff, 98

stress vector, 92 stresslet, 243 strong and weak flows, 59 strong flow, 182 strong solution, 412, 449 Stroud, 468, 470 Sturges, 330 subparametric, 450 ' Sun, 454, 456 superparametric, 450 SUPG, 453 surface tension, 406 suspensions, 239

channel flow, 265 circular Couette flow, 263 concentrated, 249, 251 dilute suspension of spheroids, 245 effective properties, 242 fibres, 256 flow-induced migration, 261 Jeffery's orbit, 257 pulsatile flows, 267 torsional flow, 267

symmetry group, 72 of a fluid, 72, 132 of a solid, 72 of an isotropic solid, 131

symmetry restrictions, 130 system

autonomous, 6 evolutionary, 389 hyperbolic, 386 non-autonomous, 6 nonlinear hyperbolic, 395

Szego, 7 Szeri, 259, 414, 418

Tait, 1 Talwar, 454 tangential modulus, 394

propagating acceleration waves, 393

tangential sliding, 42 Tanner, 44, 49, 60, 156, 177, 182, 183,

190, 197, 199, 221,225, 288,

486 Index

298, 310, 315, 330, 335, 337, 339, 342, 343, 353, 389,414, 422, 423, 425, 426, 445,447, 454-456, 459

Taylor, 452 tensile stress growth coefficient, 314,

358 tensor analysis, 74 Tetlow, 261 thermorheologically simple, 217 0 temperature, 158 third order fluid, 151 Thomson, 1 Thurston, 214 Tiefenbruck, 414, 419 time-temperature superposition prin-

ciple, 214 Tirtaatmadja, 314 Titomanlio, 229 Tiver, 36, 38 Tlapa, 384 Tobolski, 156, 219 torsional flow, 43, 295

generalised, 298 non-uniqueness, 374 uniqueness, 375

torsional instability, 364, 368,371,373, 374

bifurcating, 368 experimental results, 371 finite domain, 373 finite domain, 373 non-axisymmetric, 368 spiral instabilities, 371

Toupin, 64, 127, 393 Townsend, 309, 414 traction vector, 92 Tran-Cong, 459 transition probability, 164 Treloar, 157 Tripp, 338 Trogdon, 342 trough flow Trouton, 314 Truesdell, 64, 127, 393 Tschoegl, 150 Tucker, 256, 259

Tullock, 459

ultrasonic limits, 352 unimodular group, 74 uniqueness theorem in plane creeping

flow, 339 universal relations, 352 unsteady flows, 315

channel, 316 cone-and-plate, 317 dynamical system, 6 extensional, 317 helical, 316 squeezing of a cone, 318 squeezing of a wedge, 317 torsional, 317 with A 3 - 0, 36

upwinding, 419, 436

van den Brule, 251,411 van Doormaal, 425 van Loan, 15 van Wiechen, 199 velocity field, 3

linear autonomous, 12 non-autonomous, 16 steady, 4, 8 Zorawski, 64

velocity gradient, 4 Verlaye, 260 Virga, 123 Viriyayuthakorn, 455 viscometric flows, 40, 47, 283

axial motion of fanned planes, 44 between conical surfaces, 287 channel flow, 42 cone-and-plate flow, 297 controllability, 285 Couette flow, 43 definition, 41 divergence of the stress, 286 helical flow, 43 intrinsically unsteady, 45, 68 nearly, 351 plane cone-and-plate flow, 302 Poiseuille flow, 43 rectilinear flow, 42

screw motion of helicoidal slip surfaces, 43

simple shear flow, 42 stresses, 276 unsteady, 45, 58 with a flexible slip surface, 45

viscometric functions, 276 von Neumann stability analysis, 421,

422 von-Ziegler, 354, 358 vortex sheets, 394 Vrentas, 322

Wagner, 228, 230, 234 Walsh, 366, 414, 418 Walters, 19-21, 34, 288,309,311,312,

397, 414, 415, 419, 420, 448, 452, 454

Walters, K., 364 Wang, 55, 131 Warner, 168, 191,199 Warner spring, 168 Waterhouse, 309 Watson, 397 weak flow, 182 weak solution, 412, 449 Webster, 414, 420 Weese, 213 Weinberger, 336, 386 Weissenberg number, 364 well-posed, 412 West, 409 Wetton, 383 Weyl, 145 White, 230 Whitlock, 241 Wiegel, 219 Wiener measure, 163 Wiest, 218, 239 Williams, 121,216, 347 Willmore, 43 Wineman, 147, 322 Winter, 423 WLF function, 216 Wu, 212, 214

Xue, 426, 445

Index 487

Yamakawa, 156, 177 Yamamoto, 219, 222 Yamane, 259 Yarit z, 190 Yeh, 413, 452 Yih, 382 Yin, 41, 64, 283, 286 Yoo, 383, 414, 425 Yoshioka, 414 Young, 417, 466

Zapas, 234 Zenit, 354 Zheng, 107, 277, 411,414, 418, 459 Ziemer, 112, 122 Zienkiewicz, 398, 448, 451 Zimm, 167, 196, 211,212 Zimm relaxation times, 214 Zorawski, 64 Zorawski problem, 64 Zorawski velocity field, 65 Zuzovsky, 240

[r.r. huilgol n. thien fluid mechanics of org

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