Low Reynolds number hydrodynamics.

Mechanics of fluids and transport processes

editor: RJ. Moreau

Low Reynolds number hydrodynamics with special applications to particulate media

John Happel

Columbia University Department of Chemical Engineering and Applied Chemistry New York, New York USA

Howard Brenner

Department of Chemical Engineering Cambridge, Massachusetts USA

1983 MARTINUS NIJHOFF PUBLISHERS a member of the KLUWER ACADEMIC PUBLISHERS GROUP

THE HAGUE / BOSTON / LANCASTER

Distributors

for the United States and Canada: Kluwer Boston, Inc., 190 Old Derby Street, Hingham, MA 02043, USA for all other countries: Kluwer Academic Publishers Group, Distribution Center, P.O.Box 322, 3300 AH Dordrecht, The Netherlands

Library of Congress Catalogue Card Number 72-97238

First edition 1963 Second revised edition 1973 First paperback edition 1983

ISBN-13: 978-90-247-2877-0 e-ISBN-13: 978-94-009-8352-6 DOl: 10.1007/978-94-009-8352-6

Copyright

© 1983 by Martinus Nijhoff Publishers, The Hague.

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publishers, Martinus Nijhoff Publishers, P.O. Box 566, 2501 CN The Hague, The Netherlands.

Contents

Preface

Symbols

1. Introduction

1-1 Definition and purpose, 1. 1-2 Historical review, 8. 1-3 Application in science and technology, 13.

2. The Behavior of Fluids in Slow Motion

2-1 The equations of change for a viscous fluid, 23. 2-2 Mechanical energy dissipation in a viscous fluid, 29. 2-3 Force and couple acting on a body moving in a viscous fluid, 30. 2-4 Exact solutions of the equations of motion for a viscous fluid, 31. 2-5 Laminar flow in ducts, 33. 2-6 Simplifications of the Navier-Stokes equations, especially for slow motion, 40. 2-7 Paradoxes in the solution of the creeping motion equations, 47. 2-8 Molecular effects in fluid dynamics, 49. 2-9 Non-newtonian flow, 51. 2-10 Unsteady creeping flows, 52.

3. Some General Solutions and Theorems Pertaining to the Creeping Motion Equations

3-1 Introduction, 58. 3-2 Spherical coordinates, 62. 3-3 Cylindrical co­ordinates, 71. 3-4 Integral representations, 79. 3-5 Generalized reciprocal ;heorem, 85. 3-6 Energy dissipation, 88.

23

58

4. Axisymmetrical Flow

4-1 Introduction, 96. 4-2 Stream function, 96. 4-3 Relation between stream function and local velocity, 98. 4-4 Stream function in various co­ordinate systems, 99. 4-5 Intrinsic coordinates, 100. 4-6 Properties of the stream function, 102. 4-7 Dynamic equation satisfied by the stream function, 103. 4-8 Uniform flow, 106. 4-9 Point source or sink, 106. 4-10 Source and sink of equal strength, 107. 4-11 Finite line source, 108. 4-12 Point force, 110. 4-13 Boundary conditions satisfied by the stream function, 111. 4-14 Drag on a body, 113. 4-15 Pressure, 116. 4-16 Separable coordinate systems, 117. 4-17 Translation of a sphere, 119. 4-18 Flow past a sphere, 123. 4-19 Terminal settling velocity, 124. 4-20 Slip at the surface of a sphere, 125. 4-21 Fluid sphere, 127. 4-22 Concentric spheres, 130. 4-23 General solution in spherical coordinates, 133. 4-24 Flow through a conical diffuser, 138. 4-25 Flow past an approximate sphere, 141. 4-26 Oblate spheroid, 145. 4-27 Circular disk, 149. 4-28 Flow in a venturi tube, 150. 4-29 Flow through a circular aperture, 153. 4-30 Prolate spheroid, 154. 4-31 Elongated rod, 156. 4-32 Axisymmetric flow past a spherical cap, 157.

5. The Motion of a Rigid Particle of Arbitrary Shape

in an Unbounded Fluid

5-1' Introduction, 159. 5-2 Translational motions, 163. 5-3 Rotational mo­tions, 169. 5-4 Combined translation and rotation, 173. 5-5 Symmetrical particles, 183. 5-6 Nonskew bodies, 192. 5-7 Terminal settling velocity of an arbitrary particle, 197. 5-8 Average resistance to translation, 205. 5-9 The resistance of a slightly deformed sphere, 207. 5-10 The settling of spher­ically isotropic bodies, 219. 5-11 The settling of orthotropic bodies, 220.

6. Interaction between Two or More Particles

6-1 Introduction, 235. 6-2 Two widely spaced spherically isotropic particles, 240: 6-3 Two spheres by the method of reflections and similar techniques, 249. 6-4 Exact solution for two spheres falling along their line of centers, 270. 6-5 Comparison of theories with experimental data for two spheres, 273. 6-6 More than two spheres, 276. 6-7 Two spheroids in a viscous liquid, 278. 6-8 Limitations of creeping motion equations, 281.

7. Wall Effects on the Motion of a Single Particle

7-1 Introduction, 286. 7-2 The translation of a particle in proximity to container walls, 288. 7-3 Sphere moving in an axial direction in a circular cylindrical tube, 298. 7-4 Sphere moving relative to plane walls, 322. 7-5 Spheroid moving relative to cylindrical and plane walls, 331. 7-6 k-coeffi­cients for typical boundaries, 340. 7-7 One- and two-dimensional problems, 341. 7-8 Solid of revolution rotating symmetrically in a bounded fluid, 346. 7-9 Unsteady motion of a sphere in the presence of a plane wall, 354.

Contents

96

159

235

286

Contents

8. Flow Relative to Assemblages of Particles

8-1 Introduction, 358. 8-2 Dilute systems-no interaction effects, 360. 8-3 Dilute systems-first-order interaction effects, 371. 8-4 Concentrated systems, 387. 8-5 Systems with complex geometry, 400. 8-6 Particulate suspensions, 410. 8-7 Packed beds, 417. 8-8 Fluidization, 422.

9. The Viscosity of Particulate Systems

9-1 Introduction, 431. 9-2 Dilute systems of spheres-no interaction effects, 438. 9-3 Dilute systems-first-order interaction effects, 443. 9-4 Concen­trated systems, 448. 9-5 Nonspherical and nonrigidparticles, 456. 9-6 Com­parison with data, 462. 9-7 Non-newtonian behavior, 469.

Appendix A. Orthogonal Curvilinear Coordinate Systems

A-I Curvilinear coordinates, 474. A-2 Orthogonal curvili,[ear coordinates, 477. A-3 Geometrical properties, 480. A-4 Differenfiation of unit vectors, 481. A-5 Vector differential invariants, 483. A-6 Relations between carte­sian and orthogonal curvilinear coordinates, 486. A-7 Dyadics in orthogonal curvilinear coordinates, 488. A-8 Cylindrical coordinate systems, 490. A-9 Circular cylindrical coordinates, 490. A-I0 Conjugate cylindrical coordinate systems, 494. A-ll Elliptic cylinder coordinates, 495. A-12 Bipolar cylinder coordinates, 497. A-13 Parabolic cylinder coordinates, 500. A-14 Coordinate systems of revolution, 501. A-15 Spherical Coordinates, 504. A-16 Conjugate coordinate systems of revolution, 508. A-17 Prolate spheroidal coordinates, 509. A-18 Oblate spheroidal coordinates, 512. A-19 Bipolar coordinates, 516. A-20 Toroidal coordinates, 519. A-21 Paraboloidal Coordinates, 521.

Appendix B. Summary of Notation and Brief Review of Polyadic Algebra

Name Index

Subject Index

358

431

474

524

537

543

Preface

One studying the motion of fluids relative to particulate systems is soon impressed by the dichotomy which exists between books covering theoretical and practical aspects. Classical hydrodynamics is largely concerned with perfect fluids which unfortunately exert no forces on the particles past which they move. Practical approaches to subjects like fluidization, sedimentation, and flow through porous media abound in much useful but uncorrelated empirical information. The present book represents an attempt to bridge this gap by providing at least the beginnings of a rational approach to fluid­particle dynamics, based on first principles.

From the pedagogic viewpoint it seems worthwhile to show that the Navier-Stokes equations, which form the basis of all systematic texts, can be employed for useful practical applications beyond the elementary problems of laminar flow in pipes and Stokes law for the motion of a single particle. Although a suspension may often be viewed as a continuum for practical purposes, it really consists of a discrete collection of particles immersed in an essentially continuous fluid. Consideration of the actual detailed boundary­value problems posed by this viewpoint may serve to call attention to the limitation of idealizations which apply to the overall transport properties of a mixture of fluid and solid particles.

It is hoped that the research worker in this and related fields will be stimulated by noting that not only does the hydrodynamic viewpoint lead to a clearer correlation of much existing work, but that, at every turn, there exists a host of new problems awaiting solution. Among those which seem especially intriguing are the effect of variation in particle size and arrange­ment on the dynamic behavior of particulate systems, and the possibilities of extending the present treatments to higher Reynolds numbers. Engineers may be interested in the availability of fluid dynamic models which can serve as the framework for more extended investigations involving other transport processes, coupled perhaps with chemical reactions.

Preface

The treatment developed here is based almost entirely on the linearized form of the equations of motion which results from omitting the inertial terms from the Navier-Stokes equations, giving the so-called creeping motion or Stokes equations. This is tantamount to assuming that the particle Reynolds numbers are very small. Many systems which involve bulk flow relative to external boundaries at high Reynolds numbers are still characterized by low Reynolds numbers as regards the movement of particles relative to fluid. Also, inertial effects are less important for systems consisting of a number of particles in a bounded fluid medium than they are for the motion of a single particle in an unbounded fluid.

The subject matter is largely confined to a development of the macro­scopic properties of fluid-particle systems from first principles. General hydrodynamic and mathematical concepts are not treated in detail beyond what is required for further development. Most of the experimental data reported are confined to critical experiments aimed at demonstrating the applicability of the theoretical results to actual physical systems. Following the first few introductory chapters, subsequent material is organized on the basis of the class of boundary-value problems involved, similar to the ap­proach used by C. W. Oseen in his classical "Hydrodynamik." Starting with the motion of a single particle in an unbounded medium, the problem of the motion of several particles interacting with each other, of particles moving in the presence of bounding walls and finally of combinations of these factors are considered successively. Final chapters deal with the movement of fluids relative to particulate assemblages and with the viscosity of suspensions of particles. The latter treatment is more provisional and includes comparison of theory with empirical equations and data.

Much of the material presented is based on original investigations of the authors and their students, especially Jack Famularo. We were also fortunate in having the advice of o. H. Faxen, who provided us with many early papers by himself and C. W. Oseen, and who also carefully read the entire manu­script. S. Wakiya was also kind enough to review the manuscript and to provide additional useful suggestions. Support in the form of grants and fellowships from the Petroleum Research Fund of the American Chemical Society, The National Science Foundation, The Institute of Paper Chemistry, The Pulp and Paper Research Institute of Canada, and The Texas Company is gratefully acknowledged. The Courant Institute of Mathematical Sciences of New York University graciously provided considerable computer time.

The authors are convinced that the foundation of a scientifically sound development of the fluid dynamics of particulate media at low Reynolds numbers is now available, which should serve as a sound basis for future investigations. They hope that this book will serve to illustrate the versatility of this field of study.

JOHN HAPPEL

HOWARD BRENNER

SYMBOLS

The following is a list of the most frequently occurring sym­bols used in the book. Symbols not defined here are defined at their first place of use. A few of these symbols are oc­casionally used in other contexts.

a A b B

Sphere radius

Cross-sectional area

Distance to cylinder axis

Center of buoyancy

c Particle dimension

Cn Drag coefficient

C ; (C); C ij Coupling dyadic; matrix; tensor

Cjk Coupling dyadic in multiparticle system

d Diameter

D/ Dt Material (convected) time derivative

(eh e2, e,l) == e j Triad of right-handed orthonormal eigenvectors

£ Total energy dissipation rate

£2 Stokes stream function operator

fk«(J, <1» Surface spherical harmonic

F Force

F, Fj Vector force

(fF) Force matrix

g Local acceleration of gravity vector

(h], h." kj) c" hj Metrical coefficients

Yf", I" . .f;, Gegenbauer functions i =,..j-=1 Unit imaginary number

(i], i 2 , i;l) ecCe i j Unit vectors

i, j, k, or i.e, i y , iz Cartesian unit vectors

/", K" Modified Bessel functions

I Dyadic idemfactor

k Wall-effect constant; Kozeny constant

k Wall-effect dyadic

K Translational resistance coefficient for isotropic particle; Darcy permeability constant

K Average translational resistance coefficient

K], K 2 , K.l Eigenvalues of translation dyadic

(.]f') Resistance matrix

K; (K) ; Kij Translation resistance dyadic; matrix; tensor

Kjk Translation dyadic in multiparticle

I

m

system Length; distance to boundary; distance between particles Hydraulic radius

m f ; mp Mass of displaced fluid; mass of particle

"'1 Center of mass

on Distance ,measured normal to surface

n Unit normal vector

Nile Reynolds number

o Arbitrary point fixed in particle

P Local pressure

PI/ Solid spherical harmonic

PI/(cosf)); P::'(cosf}) Legendre function; associated Legendre function

!1P or!1p Pressure drop

P Intrinsic vector "pressure" field

!1P Vector pressure drop

[ijJ Intrinsic triadic "stress" field

(ql> q2' q3) = qj Curvilinear coordinates Q Volumetric 'flow rate or volume

(r, f), <1» Spherical polar coordinates

r Position vector

r() Position vector relative to origin at 0

rop Vector from 0 to P

R Center of reaction

R. Radius of circular cylinder

R Position vector

RR Vector defined in Eq. (5-7.14)

os Distance measured along a surface

s Unit tangent vector in intrinsic coordi-nates

S Surface area

S p Particle surface

dS Directed element of surface area

t Time

t Unit tangent vector

T Absolute temperature or torque com­ponent

T, T, Vector torque

u, v, W Component velocities in cartesian coordinates

(u l , uh u3) = Uj Components of vector u

U Local fluid velocity

U Particle speed or superficial fluid velocity

Uo Settling velocity of a single particle, or centerline velocity

U MF Mean fluid velocity

UOl' Centerline velocity of fluid

UTS Terminal settling velocity of sphere

('11) Velocity matrix

U, UI Particle velocity vector

Urn ; Urn Superficial velocity vector; speed

(VI' V 2• Va) = Vj Components of vector v

V Local fluid velocity vector

v~ Local fluid velocity vector at infinity

V Volume or velocity

V rn Mean velocity of flow

V Intrinsic dyadic "velocity" field

W Rate of doing work

x, y, z Cartesian coordinates

(XI' x 2, Xa) Xj Cartesian coordinates

Xn «(), cf», Yn(IJ, cf», ZII(IJ, cf» Surface spherical harmonics

z' Complex variable

f3 Slip coefficient

Ojk Kronecker delta

oCr - rn) Dirac delta function

A, Atj Rate of strain dyadic. tensor

f Small deformation parameter or frac-tional void volume

fjkl Permutation symbol

IE Alternating isotropic triadic

, Vorticity vector

/C Bulk viscosity

/-L (Shear) viscosity

/-Lr Relative viscosity

/-Lsp Specific viscosity

[/-L] Intrinsic viscosity

v Kinematic viscosity

II, IIij Pressure (stress) dyadic, tensor

lIn Stress vector

p, p f Fluid density

PP' p' Mean particle density

Ap Density difference

(p, cf>, z) Circular cylindrical coordinates

II Internal-external viscosity ratio

T Volume

cf> Fractional volume of solids

cf>~ Dimensionless translation dyadic in un· bounded fluid

cI> Local energy dissipation rate

cI>n, cf>,. Solid spherical harmonic

X,. Solid spherical harmonic

"'" Stream function CU, w Angular velocity vector, speed

o Angular velocity vector

0; (0); n'j Rotational resistance dyadic; matrix; tensor

Ojk Rotational resistance dyadic in multi­particle system

w Radial cylindrical coordinate

V Nabla operator V2 Laplace operator

Transposition operator

-1 Reciprocal dyadic

Dot product

Double-dot product

X Cross product

X Double-cross product X

To our wives and parents

" ... e tutto il frutto ricolto del girar di queste spere. "

Dante Alighiere