effects of variable viscosity and variable permeability on ... · effects of variable viscosity and...

Effects of Variable Viscosity and Variable Permeability on Fluid Flow

Through Porous Media

by

Sayer Obaid B Alharbi

Master of Mathematics, University of New Brunswick, 2012

Bachelor of Science, Qassim University, 2007

A Dissertation Submitted in Partial Fulfillment

of the Requirements for the Degree of

Doctor of Philosophy

in the Graduate Academic Unit of Mathematics and Statistics

Supervisor(s): M. Hamdan, Ph.D., Mathematics and Statistics

T. Alderson, Ph.D., Mathematics and Statistics

Examining Board: J. Watmough, Ph.D., Mathematics and Statistics, Chair

R. McKay, Ph.D., Mathematics and Statistics

I. Gadoura, Ph.D., Engineering (Electrical)

External Examiner: A. M. Siddiqui, Ph.D., Department of Mathematics, Penn State York

This dissertation is accepted by the

Dean of Graduate Studies

THE UNIVERSITY OF NEW BRUNSWICK

October, 2016

©Sayer Obaid B Alharbi, 2017

ii

Abstract

In this work, we study the effects of variable viscosity and variable permeability on single-

phase fluid flow through porous structures. This is accomplished by first deriving the

equations governing fluid flow through porous structures in which porosity (hence

permeability) is a function of position and viscosity of the fluid is pressure-dependent. The

governing equations are derived using intrinsic volume averaging, and viscous effects are

accounted for through Brinkman’s viscous shear term.

When the Darcy resistance, Brinkman’s viscous shear effects and Lapwood’s macroscopic

inertial terms are accounted for, the governing equation is known as the Darcy-Lapwood-

Brinkman equation, and it governs the flow through a mushy zone undergoing rapid

freezing, and is important in slurry transport. Three exact solutions to the Darcy-Lapwood-

Brinkman equation with variable permeability are obtained in this work. Solutions are

obtained for a given vorticity distribution, taken as a function of the streamfunction.

Classification of the flow field is provided and comparison is made with the solutions

obtained when permeability is constant. Interdependence of Reynolds number and the

variable permeability is emphasized. Exact solutions are also obtained for this equation

when the vorticity is proportional to the streamfunction, and a derivation of the

permeability function that satisfies the governing equations is provided.

The problem of laminar flow through a porous medium of variable permeability, behind a

two-dimensional grid is considered in this work to further shed some light of the effects of

permeability variations. Expressions for the permeability profiles are derived when the

model equations are linearized and permeability is calculated at the stagnation points of the

flow. Conditions on the parameters involved in the exact solution are analyzed and stated

iii

and the flow is classified and compared with the case of flow through constant permeability

media. This work might be of interest in the stability analysis of flow through variable

permeability media.

In studying the effects of pressure-dependent viscosity on fluid flow, this work provided

analysis involving viscosity stratification. Coupled parallel flow of fluids with viscosity

stratification through two porous layers is initiated in this work. Conditions at the interface

are discussed and appropriate viscosity stratification functions are selected in such a way

that viscosity is highest at the bounding walls and decreases to reach its minimum at the

interface. Velocity and shear stress at the interface are computed for different permeability

and driving pressure gradient.

Consideration is given to two-dimensional flow of a fluid with pressure-dependent

viscosity through a variable permeability porous structure. Exact solutions are obtained for

a Riabouchinsky type flow using a procedure that is based on an existing methodology that

is implemented in the study of Navier-Stokes flow with pressure-dependent viscosity.

Viscosity is considered proportional to fluid pressure due to the importance and uniqueness

of validity of this type of relation in the study of Poiseuille flow. The effects of changing

the proportionality constant on the pressure distribution are discussed.

Since a variable permeability introduces an additional variable in the flow equations and

renders the governing equations under-determined, the current work devises a

methodology to determine the permeability function through satisfaction of a condition

derived from the specified streamfunction. Illustrative examples are used to demonstrate

how the variable permeability is determined, and how the arising parameters are

determined. Although the current work considers flow in an infinite domain and does not

iv

handle a particular engineering problem, it nevertheless initiates the study of flow of fluids

with pressure-dependent viscosity through variable-permeability media and sets the stage

for future work in stability analysis of this type of flow. It is expected that the current work

will be of value in transition layer analysis and the determination of variable permeability

functions suitable for such analysis.

v

Dedication

This dissertation is dedicated to my loving mother Deghaima, the loving memory of my

father, Obaid and my loving wife, Mona.

vi

Acknowledgments

I would like to express my deep and sincere gratitude to my supervisor Dr. M. Hamdan for

all his time, kind assistance and advice over entire length of this work, without his guidance

I would not have learned as much as I have. It was a great privilege and honor to work and

study under his guidance. I am extremely grateful for what he has offered me. I am also so

thankful to my Co-supervisor Dr. T. Alderson for his valuable support and suggestions.

I would also like to thank my thesis committee members for spending their time on careful

reading of my thesis as well as for their valuable comments.

I would like to express my heart-felt appreciation to my mother Deghaima for her never-

ending support while I spent most of my time away from home. Thank you very much for

your continuous support in my life. Many sincere thanks to my brothers Badar, Bander,

Abdullah and my sisters Badriah, Hailah, Modi, your encouragement and support are

always there. I gratefully acknowledge to my deeply-loved wife Mona, thank you for being

next to my side, your constant understanding, support and encouragement.

vii

Table of Contents

Abstract .............................................................................................................................. ii

Dedication .......................................................................................................................... v

Acknowledgments ............................................................................................................ vi

Table of Contents ............................................................................................................ vii

List of Tables .................................................................................................................... ix

List of Figures .................................................................................................................... x

List of Symbols .............................................................................................................. xvii

1. Introduction ................................................................................................................... 1 1.1 Basic Definitions: Porous Matter, Porosity, Permeability, Viscosity .................. 1

1.2 Fluid Flow through Porous Media ....................................................................... 4

1.3 Porous Media with Variable Permeability ........................................................... 6 1.4 Flow of Fluids with Pressure-Dependent Viscosity ............................................. 9 1.5 Scope of the Current Work................................................................................. 12

2. Flow of a Fluid with Pressure-Dependent Viscosity through Porous Media ........ 15 2.1 Chapter Introduction .......................................................................................... 15

2.2 Governing Equations .......................................................................................... 16 2.3 Averaging the Governing Equations .................................................................. 19 2.4 Analysis of the Deviation Terms and Surface Integrals ..................................... 21

2.5 Final Form of Governing Equations................................................................... 22 2.6 Chapter Conclusion ............................................................................................ 24

3. Riabouchinsky Flow of a Pressure-Dependent Viscosity Fluid in Porous Media . 25 3.1 Chapter Introduction .......................................................................................... 25

3.2 Governing Equations .......................................................................................... 27 3.3 Method of Solution............................................................................................. 30

3.4 Results and Analysis .......................................................................................... 33

3.4.1 Example 1 ................................................................................................. 33 3.4.2 Example 2 ................................................................................................. 37

3.4.3 Example 3 ................................................................................................. 39

3.5 Chapter Conclusion ............................................................................................ 44

4. Coupled Parallel Flow of Fluids with Viscosity Stratification through Composite

Porous Layers .............................................................................................................. 45

4.1 Chapter Introduction .......................................................................................... 45

4.2 Problem Formulation.......................................................................................... 47 4.3 Solution Methodology ........................................................................................ 49 4.4 Results and Discussion ....................................................................................... 53 4.5 Chapter Conclusion ............................................................................................ 62

5. Exact Solution of Fluid Flow through Porous Media with Variable Permeability for

a Given Vorticity Distribution ................................................................................... 63

viii

5.1 Chapter Introduction .......................................................................................... 63

5.2 Governing Equations .......................................................................................... 64 5.3 Method of Solution............................................................................................. 67

5.3.1 Integrability Condition and Permeability Equation .......................................... 67

5.3.2 Determination of Streamfunction, Vorticity and Velocity Components ... 69 5.3.3 Determination of the Permeability Function .................................................. 71 5.3.4 Determination of Pressure ................................................................................ 74 5.3.5 Summary of Solution ........................................................................................ 75

5.4 Chapter Conclusion ............................................................................................. 94

6. Analytic Solutions to the Darcy-Lapwood-Brinkman Equation with Variable

Permeability................................................................................................................ 95

6.1 Chapter Introduction .......................................................................................... 95 6.2 Governing Equations .......................................................................................... 97

6.3 Solution Methodology ........................................................................................ 99 6.4 Sub-Classification of Flow ............................................................................... 104

6.4.1 Determining the values of ........................................................................ 104

6.4.2 Stagnation Points ............................................................................................ 106 6.4.3 Comparison with Constant Permeability Solutions ................................... 109

6.5 Chapter Conclusion .......................................................................................... 118

7. Permeability Variations in Laminar Flow through a Porous Medium Behind a

Two-dimensional Grid .............................................................................................. 119

7.1 Chapter Introduction ........................................................................................ 119 7.2 Governing Equations ........................................................................................ 121 7.3 Solution Methodology ...................................................................................... 123

7.4 Results and Discussion ..................................................................................... 128

7.4.1 Total Flow ....................................................................................................... 128 7.4.2 Stagnation Points ............................................................................................ 129 7.4.3 Determination of Constants .......................................................................... 130

7.4.4 Comparison with the Case of Constant Permeability and Kovasnay’s

Solution .......................................................................................................... 131

7.4.5 Graphical Representation of Solutions ........................................................ 134

7.5 Chapter Conclusion ................................................................................... 144

8. Conclusions and Recommendations ........................................................................ 145

Bibliography .................................................................................................................. 149

Appendix ........................................................................................................................ 159 Appendix A. Streamsurface Figures ........................................................................... 159

Appendix B. Vorticity Figures .................................................................................... 168

Curriculum Vitae

ix

List of Tables

4.1: Velocity and shear stress at interface for different 𝐶

𝜇0 and permeabilities 121 bb ,

121 , 11 L 12 L , 11 a 12 a . ................................................................. 56

4.2: Velocity and shear stress at interface for different 1L ,

2L , 1a and

2a , 121 bb , 1

12 , 121 kk ,C

μ0= -1. ....................................................................................... 56

7.1: Ranges of Values of 𝛼 at the Stagnation Points ...................................................... 133 7.2: Choice of 𝛼 used in Case 1. .................................................................................... 134 7.3: Choice of 𝛼 used in Case 2. .................................................................................... 138

x

List of Figures

3.1: Effect of parameter a on0P

P,fixed permeability and parameter A ............................. 35

3.2: Effect of parameter combinations on0P

P .................................................................... 36

3.3: 0P

P for A=1, and a=1 ................................................................................................. 38

3.4: 0P

P for A=1, and a=10, .............................................................................................. 39

3.5: Permeability function, for 1,12 aAC ............................................................... 41

3.6: Permeability function, for 1,22 aAC .............................................................. 41

3.7: Permeability function, for 1,52 aAC .............................................................. 42

3.8: Permeability function, for 1,102 aAC ............................................................ 42

3.9: 0P

P for 1,12 aAC ............................................................................................... 42

3.10: 0P

P for 1,102 aAC ......................................................................................... 43

4.1 : Representative Sketch ............................................................................................... 47

4.2: Viscosity Profile in the Lower Layer ......................................................................... 59

4.3: Viscosity Profile in the Upper Layer ......................................................................... 60

4.4: Velocity profiles when 11 L , 22 L 5.0,1 21 aa and 10

C ...................... 60

4.5: Velocity profiles when 21 L , 12 L , 2,1 21 aa and 10

C ........................ 61

4.6: Velocity profiles when 11 L , 12 L , 1,1 21 aa and 100

C ........................ 61

4.7: Velocity profiles when, 21 L , 12 L , 2,1 21 aa and 100

C .................... 62

5.1: Permeability Distribution, 1 , 1)(1 D , 1n , 3/ , 1)(2 C ............ 78

5.2: Permeability Distribution, 1 , 1)(1 D , 1n , 6/ , 1)(2 C ........... 78

5.3: Permeability Distribution, 1 , 1)(1 D , 1n , 3/ , 1)(2 C ......... 78

5.4: Permeability Distribution, 1 , 1)(1 D , 1n , 6/ , 1)(2 C .......... 79

5.5: Permeability Distribution, 1 , 4)(1 D , 5.0n , 6/ , 1)(2 C ....... 79

5.6: Permeability Distribution, 10 , 1)(1 D , 1n , 4/ , 1)(2 C ......... 79

xi

5.7: Pressure Distribution, 1 , 1)(1 C , 2n , 3/ , 1)(2 C , 2)(4 C 80

5.8: Pressure Distribution, 1 , 1)(1 C , 2n , 6/ , 1)(2 C , 2)(4 C 80

5.9: Pressure Distribution, 2 , 7)(1 C , 1n , 3/ , 1)(2 C , 2)(4 C

........................................................................................................................................... 80


........................................................................................................................................... 81

5.11: Pressure Distribution, 1 , 4)(1 C , 5.0n , 6/ , ,1)(2 C 2)(4 C

........................................................................................................................................... 81


........................................................................................................................................... 81

5.13: Streamsurfaces, 1 , 1)(1 C , 1n , 3/ , 1)(2 C ......................... 82

5.14: Streamsurfaces, 1 , 1)(1 C , 1n , 6/ , 1)(2 C ......................... 82

5.15: Streamsurfaces, 1 , 7)(1 C , 1n , 3/ , 1)(2 C ...................... 82

5.16: Streamsurfaces, 1 , 3)(1 C , 1n , 6/ , 1)(2 C ..................... 83

5.17: Streamsurfaces, 1 , 4)(1 C , 5.0n , 6/ , 1)(2 C ..................... 83

5.18: Streamsurfaces, 10 , 1)(1 C , 1n , 4/ , 1)(2 C ....................... 83

5.19: Vorticity Distribution, 1 , 1)(1 C , 1n , 3/ , 1)(2 C ............... 84

5.20: Vorticity Distribution, 1 , 1)(1 C , 1n , 6/ , 1)(2 C ............... 84

5.21: Vorticity Distribution, 1 , 7)(1 C , 1n , 3/ , 1)(2 C .......... 84

5.22: Vorticity Distribution, 1 , 3)(1 C , 1n , 6/ , 1)(2 C ........... 85

5.23: Vorticity Distribution, 1 , 4)(1 C , 5.0n , 6/ , 1)(2 C .......... 85

5.24: Vorticity Distribution, 10 , 1)(1 C , 1n , 4/ , 1)(2 C ............. 85

5.25: Permeability Distribution, 3 , 5.0)(1 B , 5.0)(2 B , 1m , 36/7 ,

1)(2 D ........................................................................................................................... 86

5.26: Permeability Distribution, 5 , 1)(1 B , 1)(2 B , 9.0m , 3/ , 1)(2 D

........................................................................................................................................... 86

5.27: Permeability Distribution, 4 , 5.0)(1 B , 5.0)(2 B , 1m , 6/ , 1)(2 D

........................................................................................................................................... 86

5.28: Permeability Distribution, 1 , 5.0)(1 B , 5.0)(2 B , 6.0m , 4/ , 1)(2 D

........................................................................................................................................... 87

5.29: Permeability Distribution, 1 , 5.0)(1 B , 5.0)(2 B , 1m , 3/ , 1)(2 D

........................................................................................................................................... 87

5.30: Permeability Distribution, 5.0 , 5.0)(1 B , 5.0)(2 B , 1m , 3/ , 1)(2 D

........................................................................................................................................... 87

5.31: Pressure Distribution, 3 , 5.0)(1 B , 5.0)(2 B , 1m , 36/7 , 1)(2 D

........................................................................................................................................... 88

5.32: Pressure Distribution, 5 , 1)(1 B , 1)(2 B , 9.0m , 3/ , 1)(4 C

........................................................................................................................................... 88

5.33: Pressure Distribution, 4 , 5.0)(1 B , 5.0)(2 B , 1m , 6/ , 1)(4 C

........................................................................................................................................... 88

xii

5.34: Pressure Distribution, 1 , 5.0)(1 B , 5.0)(2 B , 6.0m , 4/ , 1)(4 C

........................................................................................................................................... 89

5.35: Pressure Distribution, 1 , 5.0)(1 B , 5.0)(2 B , 1m , 3/ , 1)(4 C

........................................................................................................................................... 89

5.38: Streamsurfaces, 5 , 1)(1 B , 1)(2 B , 9.0m , 3/ ................... 90

5.39: Streamsurfaces, 4 , 5.0)(1 B , 5.0)(2 B , 1m , 6/ ................... 90

5.40: Streamsurfaces, 1 , 5.0)(1 B , 5.0)(2 B , 6.0m , 4/ ............... 91

5.41: Streamsurfaces, 1 , 5.0)(1 B , 5.0)(2 B , 1m , 3/ ................... 91

5.42: Streamsurfaces, 5.0 , 5.0)(1 B , 5.0)(2 B , 1m , 3/ .................. 91

5.43: Vorticity Distribution, 3 , 5.0)(1 B , 5.0)(2 B , 1m , 36/7 .... 92

5.44: Vorticity Distribution, 5 , 1)(1 B , 1)(2 B , 9.0m , 3/ ......... 92

5.45: Vorticity Distribution, 5 , 1)(1 B , 1)(2 B , 9.0m , 3/ ......... 92

5.46: Vorticity Distribution, 1 , 5.0)(1 B , 5.0)(2 B , 6.0m , 4/ .... 93

5.47: Vorticity Distribution, 1 , 5.0)(1 B , 5.0)(2 B , 1m , 3/ ......... 93

5.48: Vorticity Distribution, 5.0 , 5.0)(1 B , 5.0)(2 B , 1m , 3/ ....... 93

6.1: Permeability when

1Re ..................................................................................... 111

6.2: Permeability when

3Re ..................................................................................... 111

6.3: Velocity )(yu when

1Re , 1,1,1 21 cc ................................................. 112


3Re , 1,1,1 21 cc .................................................. 112

6.5: Velocity )(xv when

1Re , 5.021 cc ............................................................. 113


3Re , 5.021 cc ............................................................. 113


1Re , 12 c ....................................................................... 114


3Re , 12 c ...................................................................... 114


1Re , 5.021 cc ............................................................ 115


3Re , 5.021 cc .......................................................... 115


1Re , 1,1 21 cc , 1 .............................................. 116


3Re , 1,1 21 cc , 1 ............................................... 116

xiii


1Re , 1,1 21 cc ......................................................... 117


3Re , 1,1 21 cc ......................................................... 117

7.1: )(YK when 12 C ................................................................................................. 135

7.2: )(YK when 22 C ................................................................................................. 135

7.3: )(YK when 52 C .................................................................................................. 136

7.4: )(YK when 102 C ............................................................................................... 136

7.5: )(YK when 202 C ............................................................................................... 137

7.6: )(YK when 12 C , 10Re .................................................................................. 137

7.7: )(XK when 11 C ............................................................................................... 139

7.8: )(XK when 11 C , 10Re ................................................................................ 139

7.9: Streamsurface when 1 ....................................................................................... 140

7.10: Streamsurface when 2 ..................................................................................... 140

7.11: Streamsurface when 5 ..................................................................................... 141

7.12: Streamsurface when 10 ................................................................................... 141

7.13: Vorticity Distribution when 1 ......................................................................... 142

7.14: Vorticity Distribution when 2 ....................................................................... 142

7.15: Vorticity Distribution when 5 ......................................................................... 143

7.16: Vorticity Distribution when 10 ....................................................................... 143

A.1: Streamsurface, , , , ......................................... 159

A.2: Streamsurface, , , ......................................... 159

A.3: Streamsurface, , , , ................................................ 160

A.4: Streamsurface, , , , .............................................. 160

A.5: Streamsurface, , , , ................................................ 160

A.6: Streamsurface, , , , .............................................. 160

A.7: Streamsurface, , , , ........................................... 160

A.8: Streamsurface, , , , ......................................... 160

A.9: Streamsurface, , , , ........................................... 161

A.10: Streamsurface, , , , ....................................... 161

A.11: Streamsurface, , , , ....................................... 161

A.12: Streamsurface, , , , .................................... 161

A.13: Streamsurface, , , , ........................................... 161

A.14: Streamsurface, , , , ......................................... 161

A.15: Streamsurface, , , , ........................................... 162

A.16: Streamsurface, , , , ............................................ 162

121 cc 1.1 1/1.1Re 1

121 cc 1.1 1/1.1Re 10

121 cc 2 1/2Re 1

121 cc 5 1/5Re 10

121 cc 5 1/5Re 1

121 cc 5 1/5Re 10

121 cc 10 1/10Re 1

121 cc 10 1/10Re 10

121 cc 20 1/20Re 1

121 cc 20 1/20Re 10

121 cc 1.1 3/1.1Re 1

121 cc 1.1 3/1.1Re 10

121 cc 2 3/2Re 1

121 cc 2 3/2Re 10

121 cc 5 3/5Re 1

121 cc 5 3/5Re 10

xiv

A.17: Streamsurface, , 10 , 3/10Re , 1 ........................................ 162

A.18: Streamsurface, , , , .................................... 162

A.19: Streamsurface, , , , ....................................... 162

A.20: Streamsurface, , , , ..................................... 162

A.21: Streamsurface, , , ................................................... 163

A.22: Streamsurface, , , .......................................................... 163

A.23: Streamsurface, , , .......................................................... 163

A.24: Streamsurface, , , ..................................................... 163

A.25: Streamsurface, , , ..................................................... 163

A.26: Streamsurface, , , .................................................. 163

A.27: Streamsurface, , , .......................................................... 164

A.28: Streamsurface, , , .......................................................... 164

A.29: Streamsurface, , , ..................................................... 164

A.30: Streamsurface, , 20 , 3/20Re ................................................... 164

A.31: Streamsurface, , , , ....................................... 164

A.32: Streamsurface, , , , .................................... 164

A.33: Streamsurface, , , , .............................................. 165

A.34: Streamsurface, , , , ............................................ 165

A.35: Streamsurface, , , , .............................................. 165

A.36: Streamsurface, , , , ............................................ 165

A.37: Streamsurface, , , , ......................................... 165

A.38: Streamsurface, , , , ....................................... 165

A.39: Streamsurface, , , , ......................................... 166

A.40: Streamsurface, , , , ....................................... 166

A.41: Streamsurface, , , , ....................................... 166

A.42: Streamsurface, , , , .................................... 166

A.43: Streamsurface, , , , ........................................... 166

A.44: Streamsurface, , , , ......................................... 166

A.45: Streamsurface, , , , ........................................... 167

A.46: Streamsurface, , , , ......................................... 167

A.47: Streamsurface, , , , ....................................... 167

A.48: Streamsurface, , , , .................................... 167

A.49: Streamsurface, , , , ....................................... 167

A.50: Streamsurface, , , , .................................... 167

A.51: Vorticity, , , , ............................................... 168

A.52: Vorticity, , , , .............................................. 168

A.53: Vorticity, , , , ...................................................... 168

121 cc

121 cc 10 3/10Re 10

121 cc 20 3/20Re 1

121 cc 20 3/20Re 10

121 cc 1.1 1/1.1Re

121 cc 2 1/2Re

121 cc 5 1/5Re

121 cc 10 1/10Re

121 cc 20 1/20Re

121 cc 1.1 3/1.1Re

121 cc 2 3/2Re

121 cc 5 3/5Re

121 cc 10 3/10Re

121 cc

121 cc 1.1 1/1.1Re 1

121 cc 1.1 1/1.1Re 10

121 cc 2 1/2Re 1

121 cc 2 1/2Re 10

121 cc 5 1/5Re 1

121 cc 5 1/5Re 10

121 cc 10 1/10Re 1

121 cc 10 1/10Re 10

121 cc 20 1/20Re 1

121 cc 20 1/20Re 10

121 cc 1.1 3/1.1Re 1

121 cc 1.1 3/1.1Re 10

121 cc 2 3/2Re 1

121 cc 2 3/2Re 10

121 cc 5 3/5Re 1

121 cc 5 3/5Re 10

121 cc 10 3/10Re 1

121 cc 10 3/10Re 10

121 cc 20 3/20Re 1

121 cc 20 3/20Re 10

121 cc 1.1 1/1.1Re 1

121 cc 1.1 1/1.1Re 10

121 cc 2 1/2Re 1

xv

A.54: Vorticity, , , , ..................................................... 168

A.55: Vorticity, , , , ....................................................... 169

A.56: Vorticity, , , , .................................................... 169

A.57: Vorticity, , , , ................................................. 169

A.58: Vorticity, , , , ............................................... 169

A.59: Vorticity, , , , ................................................. 169

A.60: Vorticity, , , , ............................................... 169

A.61: Vorticity, , , , ............................................... 170

A.62: Vorticity, , , , .............................................. 170

A.63: Vorticity, , , , .................................................... 170

A.64: Vorticity, , , , ................................................. 170

A.65: Vorticity, , , , .................................................... 170

A.66: Vorticity, , , , .................................................... 170

A.67: Vorticity, , , , ............................................... 171

A.68: Vorticity, , , , ............................................. 171

A.69: Vorticity, , , , ............................................... 171

A.70: Vorticity, , , , ............................................. 171

A.71: Vorticity, , , ........................................................... 171

A.72: Vorticity, , , .................................................................. 171

A.73: Vorticity, , , .................................................................. 172

A.74: Vorticity, , , ............................................................. 172

A.75: Vorticity, , , ............................................................. 172

A.76: Vorticity, , , ........................................................... 172

A.77: Vorticity, , , .................................................................. 172

A.78: Vorticity, , , .................................................................. 172

A.79: Vorticity, , , ............................................................. 173

A.80: Vorticity, , , ............................................................. 173

A.81: Vorticity, , , , ............................................... 173

A.82: Vorticity, , , , ............................................. 173

A.83: Vorticity, , , , ...................................................... 173

A.84: Vorticity, , , , .................................................... 173

A.85: Vorticity, , , , ...................................................... 174

A.86: Vorticity, , , , .................................................... 174

A.87: Vorticity, , , , ................................................. 174

A.88: Vorticity, , , , ............................................... 174

A.89: Vorticity, , , , ................................................. 174

121 cc 2 1/2Re 10

121 cc 5 1/5Re 1

121 cc 5 1/5Re 10

121 cc 10 1/10Re 1

121 cc 10 1/10Re 10

121 cc 20 1/20Re 1

121 cc 20 1/20Re 10

121 cc 1.1 3/1.1Re 1

121 cc 1.1 3/1.1Re 10

121 cc 2 3/2Re 1

121 cc 2 3/2Re 10

121 cc 5 3/5Re 1

121 cc 5 3/5Re 10

121 cc 10 3/10Re 1

121 cc 10 3/10Re 10

121 cc 20 3/20Re 1

121 cc 20 3/20Re 10

121 cc 1.1 1/1.1Re

121 cc 2 1/2Re

121 cc 5 1/5Re

121 cc 10 1/10Re

121 cc 20 1/20Re

121 cc 1.1 3/1.1Re

121 cc 2 3/2Re

121 cc 5 3/5Re

121 cc 10 3/10Re

121 cc 20 3/20Re

121 cc 1.1 1/1.1Re 1

121 cc 1.1 1/1.1Re 10

121 cc 2 1/2Re 1

121 cc 2 1/2Re 10

121 cc 5 1/5Re 1

121 cc 5 1/5Re 10

121 cc 10 1/10Re 1

121 cc 10 1/10Re 10

121 cc 20 1/20Re 1

xvi

A.90: Vorticity, , , , ............................................... 174

A.91: Vorticity, , , , ............................................... 175

A.92: Vorticity, , , , ............................................. 175

A.93: Vorticity, , , , .................................................... 175

A.94: Vorticity, , , , ................................................. 175

A.95: Vorticity, , , , .................................................... 175

A.96: Vorticity, , , , ................................................. 175

A.97: Vorticity, , , , ............................................... 176

A.98: Vorticity, , , , ............................................. 176

A.99: Vorticity, , , , ............................................... 176

A.100: Vorticity, , , , ........................................... 176

121 cc 20 1/20Re 10

121 cc 1.1 3/1.1Re 1

121 cc 1.1 3/1.1Re 10

121 cc 2 3/2Re 1

121 cc 2 3/2Re 10

121 cc 5 3/5Re 1

121 cc 5 3/5Re 10

121 cc 10 3/10Re 1

121 cc 10 3/10Re 10

121 cc 20 3/20Re 1

121 cc 20 3/20Re 10

xvii

List of Symbols

CBAa ,,, parameters used in chapter 3

2121 ,,, bbaa parameters used in chapter 4

dC Forchheimer drag coefficient

212121 ,,,,, BBDDCC parameters used in chapter 5

dc, constants

pd average pore diameter

F the deviation of an averaged quantity

F , H fluid quantities used in chapter 2

g

gravitational acceleration

I.F. integration factor

K permeability distribution, dimensionless permeability in chapter 7

1K transition layer permeability

K permeability of ambient medium, permeability at the edge of

boundary layer, both used in chapter 1

k dimensional permeability

1k permeability function in layer 1

2k permeability function in layer 2

*k dimensionless permeability used in chapter 6

hK hydraulic conductivity

wK permeability at the wall

0k , c ,n curve fitting parameters

L characteristic length, macroscopic length used in chapter 2,

generalized pressure function used in chapter 5

1L boundary layer 1 thickness

2L boundary layer 2 thickness

l microscopic length

nm, parameters used in chapter 5

rnnmm ,,,, 2121 characteristic roots

n unit normal vector pointing into the solid

p fluid pressure

xviii

*p dimensionless pressure

refp reference pressure

xp1 pressure gradient in layer 1

xp2 pressure gradient in layer 1

q mean filter velocity (or Darcy velocity)

Re Reynolds number

S the surface area of the solid matrix

U characteristic velocity

0u average velocity

1u velocity across layer 1

2u velocity across layer 2

VU , dimensionless velocity components used in chapter 7

u, v velocity vector fields

** ,vu dimensionless velocity components used in chapter 6

V bulk volume of the porous medium

pV volume of pores

sV volume of the solid matrix

V the effective pore space

permeability-adjustment parameter, parameter used in chapters 5

21, parameters used in chapter 4

positive parameter used in chapter 1

parameter used in chapter 1

parameter used in chapter 5

, viscosity functions of the fluid

1 viscosity in layer 1

2 viscosity in layer 2

0 reference viscosity

eff , * effective viscosity of the fluid

0 , a positive constants used in chapter 1

kinematic viscosity

vorticity function, dimensionless vorticity used in chapter 6

xix

gradient operator

2 Laplacian operator

fluid density

porosity

fluid-phase function

dimensionless streamfunction

streamfunction, dimensionless streamfunction used in chapter 6

vorticity distribution

dimensionless vorticity

1

Chapter 1

Introduction

Chapter Summary

In this Chapter, an overview of the basics of flow through porous media is presented

together with a review of the available literature on key points that related to this thesis.

The chapter ends with objectives and plan of the current work, together with organization

of the chapters.

1.1 Basic Definitions: Porous Matter, Porosity, Permeability, Viscosity

A porous medium may be defined as a solid containing voids (pores), either connected or

not, dispersed within it in a regular or random manner. A porous medium therefore consists

of a solid matrix and pores. If pV is the volume of pores and sV is the volume of the solid

matrix, then sp VVV is the bulk volume of the porous medium. Porosity, , of a porous

medium is a dimensionless quantity that represents the void fraction and is defined as the

ratio of pore volume to the bulk volume of the medium, namely V

V . It is clear that

10 , with 0 representing a complete solid (an absence of pores) and 1

representing free-space (or the absence of a solid matrix).

2

Depending on the compaction of the solid matrix in a given porous medium, pores may be

connected or disconnected. Disconnected or isolated pores are referred to as dead-end pores

and do not support fluid flow in the medium. Connected pores on the other hand can form

a flow network through which fluid flows. Volume fraction of connected pores is referred

to as effective porosity. In this work, any reference to porosity implies reference to effective

porosity.

Typical examples of porous media include earth layers, rock, soil, human and animal

tissues, biological systems, most metals, and most plastics. Porosity of a medium can serve

as a tool to distinguish one porous medium from another, and can be (locally) either a

constant quantity or a function of position and time. If the porous medium is non-

sedimenting with time (that is, there is no collapse in any part of the solid matrix, or

corrosion), then porosity is time-independent, but may still depend on position. In addition,

if porosity is independent of position then it must be constant.

Hydraulic conductivity, hK , is a property of soil or rock that describes the ease with which

water can move through pore spaces or fractures. It depends on the intrinsic permeability

of the material and on the degree of saturation (where if all pores are filled with fluid the

material is fully-saturated, otherwise, it is partially-saturated). Hydraulic conductivity is

specific to the flow of a certain fluid (typically water, sometimes oil or air).

Intrinsic permeability (k) is a parameter of a porous media which is independent of the

fluid. It is a measure of conductivity of the porous material to fluid flowing through it.

3

This means that, for example, hK will go up if the water in a porous medium is heated

(reducing the viscosity of the water), but k will remain constant.

The two are related through the following equation

kKh (1.1)

where

hK is the hydraulic conductivity [LT-1 or ms-1];

k is the intrinsic permeability of the material [L2 or m2];

is the specific weight of water [ML-2T-2 or Nm-3], and;

is the dynamic viscosity of water [ML-1T-1 or kgm-1s-1].

Intrinsic permeability (or simply permeability) is the coefficient that appears in Darcy’s

law, namely

pk

q

(1.2)

where q

is the mean filter velocity (or Darcy velocity), and p is the driving pressure

gradient. Permeability has the dimension of 2L and is measured in units of Darcy (D), (cf.

[57] and the references therein).

Permeability is used to distinguish between two types of porous media: isotropic and

anisotropic media, defined as follows:

(a) Isotropic: Permeability is not direction dependent. It may be treated as a constant

or a scalar function of position.

4

(b) Anisotropic: Permeability is direction dependent (usually a tensor). Anisotropic

media are in general referred to as variable permeability media.

Variations in permeability are in general dependent on the porous microstructure (pore

size, grain size distribution, and shape of grains, and the packing of grains). These

variations impact velocity distributions, heat and mass transfer in porous sediments.

1.2 Fluid Flow through Porous Media

The discussion in Section 1.1 suggests that fluid can flow in a porous medium if at least

some of the pores are interconnected. Now, the flow of a viscous fluid is governed by the

well-known Navier-Stokes equations, written here as:

gvpvvvt

2)( (1.3)

where v

is the velocity vector field, is the fluid density, is the gradient operator, p is

the fluid pressure, g

is the gravitational acceleration, and is the viscosity of the fluid.

When the flow domain is a porous structure, the Navier-Stokes equations describe the

microscopic, local flow in the pore space. Complexity of the geometry of pore space

restricts one’s ability to realistically solve boundary value problems while using the

Navier-Stokes equations. This has given rise to modeling flow through porous media using

a macroscopic approach based on averaging the Navier-Stokes equations over a

representative elementary volume (that is, a control volume who properties are the same as

those of the porous medium).

5

Available models of flow through porous media have been reviewed and discussed in detail

by Hamdan [31] and include the experimental Darcy’s law (equation (1.2)), which is valid

in cases when changes in flow quantities at the microscopic level are insignificant at a

macroscopic length scale. Darcy’s law ignores viscous shear effects that arise in the

presence of solid boundary, and both macroscopic and microscopic inertial effects that

arise due to convective accelerations and tortuosity of the flow path, respectively.

The most general model of flow through porous media is the Darcy-Lapwood-Forchheimer

-Brinkman model, which takes the following form:

gk

uuCu

kupuuu defft

2])([ (1.4)

where u

is the macroscopic (averaged) velocity vector field, k is the permeability, is

the fluid density, is the gradient operator, p is the fluid pressure, 55.0dC is the

Forchheimer drag coefficient, is the viscosity of the base fluid, and eff is the effective

viscosity of the fluid in the porous medium. There is no agreement in the literature on

porous media on the relationship between and eff (cf. Almalki and Hamdan [10] and

the references therein).

In equation (1.4), the term uu

)( is the macroscopic inertia that is due to convective

acceleration of fluid particles, k

uuCd

is the Forchheimer microscopic inertial term that is

due to tortuosity of the flow path and the converging-diverging pore structure, uk

is the

6

Darcy resistance, and ueff

2 is the Brinkman viscous shear term that is important in the

presence of a solid boundary.

1.3 Porous Media with Variable Permeability

In anisotropic porous media, permeability is direction-dependent and is represented as a

tensor. Modelling and simulation of groundwater flow in three dimensions is an example

of flow through anisotropic porous media with variable permeability (cf. [57] and the

references therein). In isotropic porous media, permeability can be treated as possibly

constant or a function of position. While studies involving equation (1.4) with constant

permeability are abundant, there are situations involving flow through and over porous

layers that require considerations of flow through variable permeability media. These are

discussed in what follows.

1) Ergun’s equation, [57] gives the following relationship between porosity and

permeability:

2

23

)1(150

pdk (1.5)

where pd is the average pore diameter in a channel-like porous material.

If porosity in (1.5) is a variable function of position, then so is permeability. This

occurs in inhomogeneous porous media such as earth layers. Clearly, either porosity

or permeability must be specified as a function of position.

2) Equation (1.4) encompasses the popular models of flow through porous media.

In particular, in the presence of macroscopic solid boundary, viscous shear effects are

dominant in a thin layer near the solid boundary, while the flow remains Darcy-like away

7

from the boundary. Nield [56] argued that the use of Brinkman’s viscous shear term

requires a redefinition of the porosity near a solid boundary, while some authors [42,69]

emphasized the need for a variable permeability Brinkman model. Parvazinia et al [59]

concluded that when Brinkman’s equation is used, three distinct flow regimes arise,

depending on Darcy number, namely: a free flow regime (for a Darcy number greater than

unity); a Brinkman regime (for a Darcy number less than unity and greater than 610 ); and

a Darcy regime (for a Darcy number less than 610 ). Their investigation [59] emphasized

that “the Brinkman regime is a transition zone between the free and the Darcy flows”.

Hamdan and Kamel [32] argue that in the presence of a solid boundary, there is a need to

redefine porosity and permeability due to the increase in permeability and porosity near a

solid wall, thus giving rise to channeling effects. Channeling effects have an impact on

shear stress and heat transfer near the wall, (cf. [42,44,66]).

The above situations gave rise to the need for variable permeability models to be used in

connection with models of flow through porous media which employ a viscous shear term

(such as (1.4)).

Fluid flow through porous media of variable porosity and permeability, and its

applications, has been discussed by a number of investigators (cf. [66,69,82]), with

applications ranging from cooling and heating systems to irrigation of agricultural land and

the recovery of oil and gas from reservoirs, and groundwater recovery. Further applications

can be found in the design of artificial porous media, artificial skin layers and bone (cf.

Hamdan and Kamel [32] and the references therein). Permeability is usually modeled in a

number of ways depending on the porous structure and the application sought. Hamdan

and Kamel introduced a permeability function for Brinkman’s equation in their solution of

8

Poiseuille flow through a porous channel of variable permeability, and provided a review

of the popular choices of variable permeability. These include the work of Rees and Pop,

[66] who employed the following model in their analysis of vertical free convection in

porous media:

Ly

w eKKKk /)(

(1.6)

wherein L is the length scale over which permeability varies; K is permeability of

ambient medium; and wK is permeability at the wall.

Alloui et al [9] employed an exponential model of the following form in their analysis of

convection in binary mixtures:

cyek (1.7)

where, when c is small ( c approaches zero) then the permeability behaves like the linear

function cyk 1 .

In their study of vortex instability of mixed convection flow, Hassanien et al [34] employed

an exponential model of the form:

]1[ / ydeKk

(1.8)

where K is permeability at the edge of boundary layer, d is a constant and is a

parameter.

In modeling flow through a free-space channel bounded by a porous layer, Nield and

Kuznetsov [58] avoided the permeability discontinuity at the interface between the regions

by introducing a porous layer of variable permeability sandwiched between the channel

9

and the porous layer. They modeled the permeability in such a way that the reciprocal of

the permeability varies linearly across the layer, namely

yakLK /1 (1.9)

where L is the overall porous medium thickness, a is related to defining the thickness of

the transition layer, 1K is the transition layer permeability, and k is the constant

permeability of the underlying porous layer.

Abu Zaytoon [1] extended the work of Nield and Kuznetsov and introduced specialized

forms of variable permeability for Brinkman’s equation in his study of flow through the

transition layer. His analysis reduced Brinkman’s equation to Airy’s, generalized Airy’s

and Weber differential equations. Abu Zaytoon [1] also reviewed some of the available

variable permeability models and reported that for inhomogeneous soil layers the following

form of variable permeability has been used (cf. Schiffman and Gibson [71] Mahmoud and

Deresiewicz [47]).

ncykk )1(0 (1.10)

where ck ,0 and n are treated as curve fitting parameters. Cheng [21] used a value of 2n

in equation (1.10), and discussed other forms of the permeability distribution.

1.4 Flow of Fluids with Pressure-Dependent Viscosity

Newtonian fluid flow with variable viscosity is governed by the Navier-Stokes equations,

written here in the following form suitable for incompressible fluids:

10

.)()( gvvpvvv T

t

(1.11)

It has long been realized that viscosity of a Newtonian fluid is a function of both

temperature and pressure, and this realization goes back to the 19th century’s works of

Stokes, [78] and Barus [13]. Stokes’ investigations [78] emphasized the dependence of

viscosity on pressure, and the need to consider viscosity as a non-constant in flow situations

where pressure varies significantly. The experimental work of Barus [13] provided the

following relationship between viscosity, , and pressure, p:

ape0 (1.12)

where 0 and a are positive constants. Equation (1.12) suggests an increase in viscosity

with increasing pressure.

The early investigations of Stokes and Barus still permeate today due to the many

applications associated with the flow of fluids with pressure-dependent viscosities. These

applications span industries such as thin film lubrication, fuel oil pumping, ground water

recovery systems design, the pharmaceutical industry, food processing industry, polymer

melts, Tanner et al [36] in addition to natural geothermal and geophysical applications and

mantle flow, Shen and Forsyth [73]. More recently, Afsar Khan et al [5] discussed

applications of variable viscosity in peristaltic transport.

Different forms of relations between viscosity and pressure have been investigated and

proposed by various authors. For instance, the following relationship has been used by

11

many authors including Housiadas and Georgiou [35] in their study of weakly

compressible Newtonian flows of pressure-dependent viscosity,

}]{1[0 refpp (1.13)

is the shear viscosity, 0 is the shear viscosity at the reference pressure refp , and is

a positive parameter. Szeri [79] and Fusi et al [27] used linear relationships of the form

p (1.14)

]1[0 p . (1.15)

Other relationships can be found in the works of Andrade [11] and Malek et al [48].

Investigations of the effect of pressure on the properties of liquids in the early part of the

20th century are given by Bridgman [18] while more recent analysis can be found in the

works of Cutler [22] Griest et al [29] Johnson and Cameron [39] Schmelzer et al [72]

among others. For a more detailed discussion on the available literature one is referred to

the work of Srinivasan and Rajagopal [75] and the references therein.

Interest in the flow of fluids with pressure dependent viscosity in porous media stems from

the many applications this type of flow enjoys. In particular, pumping of fuel oil from

reservoirs, ground water recovery, lubrication mechanisms in domains with porous lining,

fluidics, and biomedical research in human and animal tissues at high pressures all

represent flow through porous media where variations in viscosity are critical. These and

12

many other applications have been reviewed and discussed in great details in Kanan and

Rajagopal [40] and the references therein.

Two aspects in this field are important: modelling the flow of a fluid with pressure-

dependent viscosity through porous structures, and solutions to initial and boundary value

problems. Both of these aspects have been pioneered by Rajagopal and coworkers, among

many others. A number of models have been developed by Malek et al [48] Kanan and

Rajagopal [40] using homogenization methods and mixture theory approach. Mathematical

models of filtration with applications to oil flow have been developed and discussed by

Fusi et al [27]. Existence of solutions, the nature of solutions, and obtaining various

solutions of flow through porous media of fluids with pressure-dependent viscosities in

various flow domains have been championed by Rajagopal et al [64].

Solutions to the Navier-Stokes equations with pressure-dependent viscosity are abundant

and have been considered by various authors. Of relevance to the current work are the large

number of solutions obtained by Naeem and co-workers [53].

1.5 Scope of the Current Work

In spite of the large number of articles discussing the flow of fluids with pressure-

dependent viscosities through porous media, and the many advancements in this field of

study, evidence is abundant that the subject matter is still at its infancy. The roles of Darcy

resistance, Forchheimer effects, and effects of variations in porosity and permeability

remain unclear. A typical view is seen in the model developed by Kanan and Rajagopal

[40] where Darcy resistance is replaced by a product of a function of pressure and filtration

13

velocity. Furthermore, models development based on volume averaging are lacking in the

existing literature.

In order to provide partial answers to the above situation, this work is intended to fill some

gaps that exist in the literature by developing models of flow through porous media of

fluids with pressure-dependent viscosities based on intrinsic volume averaging while

taking porosity as a variable function of position. The role of Darcy resistance and its

reduction to Kanan and Rajagopal’s pressure function is discussed. The developed model

is then analytically solved under a Riabouchinski flow assumptions. The problem of

coupled parallel flow under variable viscosity assumptions is considered, for the first time,

in this work, and fills in a gap in the existing literature.

Effects of permeability variations in two-dimensional flow are considered in this work by

studying the flow behind a two dimensional grid. Selection of permeability function is

made in such a way that the governing equations are satisfied.

This work is organized as follows. In Chapter 2, a set of equations that describes the flow

in a variable viscosity for constant and variable porosity is developed using intrinsic

volume averaging of continuity equation and the Navier-Stokes equations. The model

obtained is compared to existing models in the literature. In Chapter 3, the problem of

Riabouchinsky flow with pressure-dependent viscosity through variable permeability

porous media is considered. Exact solution to the governing equations is obtained for a

selection of permeability functions. In Chapter 4, formulation and solution of the problem

of coupled parallel flow through two porous layers with variable viscosity are presented.

Conditions at the interface between layers are analyzed and suitable exponential variations

of viscosity are selected. In Chapter 5, viscous, incompressible fluid flow through a porous

14

medium with variable permeability in two dimensions is considered. Solution is obtained

for a given vorticity distribution and a suitable permeability function. In Chapter 6,

analytical solutions to the Darcy-Lapwood-Brinkman equation with Variable permeability

are obtained and classified. Comparison is made with solutions to the Darcy-Lapwood-

Brinkman equation when permeability is constant in order to illustrate the effects of

variable permeability. In Chapter 7, effects of permeability variations on laminar flow

through a porous medium behind a two-dimensional grid are studied to illustrate the

interdependence between Reynolds number and variable permeability.This work is

concluded with Chapter 8 which discusses conclusions and recommendations for future

work.

15

Chapter 2

Flow of a Fluid with Pressure-Dependent Viscosity through Porous

Media

2.1 Chapter Introduction

It has long been recognized that viscosity of a fluid can vary with changes in temperature

and pressure, Barus [13,14] Bridgman [17] although in the absence of temperature

variations it has been customary to assume constant shear viscosity when Newtonian liquid

flow is considered. However, many industrial applications involve processes with high

pressures that warrant consideration of pressure-dependent viscosity variation of

incompressible fluid flow both in free space and in porous media, Fusi et al [27] Rajagopal

et al [63] Savatorova and Rajagopal [70] Srinivasan et al [77] Srinivasan and Rajagopal

[76] Szeri [79]. These applications involve chemical and process technologies, such as

medical tablet production, crude and fuel oil pumping, food processing, and fluid-film

lubrication theory, and microfluidics, Martinez-Boza et al [49] Szeri [79]. These and other

applications of pressure-dependent viscosity variations of fluid flow in porous media

motivate the current work in which we model the unsteady flow of a variable viscosity,

incompressible fluid through a porous medium. The goal is to develop a set of governing

equations that describe the flow in variable porosity porous media. This is accomplished

in the current work by intrinsic volume averaging of the continuity equation and the

Navier-Stokes equations over a control volume composed of solid and pore space.

16

2.2 Governing Equations

In free space, the flow of an incompressible fluid with pressure-dependent viscosity is

governed by the equation of continuity and the equations of linear momentum, namely

0 v

(2.1)

gTpvvvt

)( (2.2)

where

TvvT )(

(2.3)

v

is the velocity vector field, p is the pressure, is the fluid density, )( p is the

viscosity of the fluid that is assumed to be a function of pressure.

In order to develop a set of field equations governing the flow of an incompressible fluid

with pressure-dependent viscosity through an isotropic porous medium, the equations

governing the flow in free space will be averaged over a Representative Elementary

Volume (REV), introduced in Bachmat and Bear [12]. The effects of the porous

microstructure on the flowing fluid will be accounted for through the concept of a

Representative Unit Cell (RUC), introduced in Du Plessis and Masliyah [25].

Typical condition on the velocity vector is the no-slip assumption on the solid matrix. This

is implemented in this work and translates to 0

v on the stationary solid matrix. At the

outset, equation (2.2) is written in the following dyadic form that is suitable for volume

averaging:

gTpvvvt

, (2.4)

and the equations to be averaged are thus Equations (2.1) and (2.4).

17

Following Bachmat and Bear [12] a Representative Elementary Volume, REV, is a control

volume, V , composed of a fluid-phase and a (stationary) solid-phase. Fluid is contained in

the pore space, V , and the solid-phase is contained in the porous matrix solid of volume

sV , in the same proportion as the whole porous medium. An REV is therefore a control

volume whose porosity is the same as that of the whole porous medium. The porosity, ,

is defined in terms of a fluid-phase function, , as follows. Define at a point x in V as:

sVx

Vxx

;0

;1)(

(2.5)

then porosity is then defined as

V

VdV

VdV

VVV

111

. (2.6)

Length scales associated with the REV are microscopic and macroscopic length scales, l

and L respectively. An REV is chosen such that 33 LVl , wherein the microscopic

length scale, l , could be the average pore diameter, and the macroscopic length scale, L ,

could be a depth of a channel. In order to develop the equations of flow through a porous

structure we define the volume average (volumetric phase average) of a fluid quantity F

per unit volume, as:

< F > =

VVV

FdVV

FdVV

FdVV

111 (2.7)

18

and the intrinsic phase average (that is, the volumetric average of F over the effective pore

space, V ) as:

< F > =

VVV

FdVV

FdVV

FdVV

111. (2.8)

Relationship between the volumetric phase average and the intrinsic phase average can be

seen from equations (2.7), and (2.8), and the definition of porosity, (2.6), as

FF , and the deviation of an averaged quantity from its true (microscopic)

value is given by the quantity FFF .

The following averaging theorems are then applied to equations (2.1) and (2.4). Letting F

and H be volumetrically additive scalar quantities, F

a vector quantity, and c a constant

(whose average is itself), then, Du Plessis and Masliyah [25]:

(i) FccF =c F

(ii) S

dSnFV

FF 1

where S is the surface of the solid matrix in the REV that is in contact with the fluid, and

n is the unit normal vector pointing into the solid, and a surface integral of the form S

dSn

has been abbreviated as S

dSn .

(iii) HFHF HFHF

(iv) FHFH HFHF

19

(v) F

S

dSnFV

F .1

(vi) tF tt FF )(

Rule (vi) is valid under the assumption that the fluid-solid interface in the REV has a zero

microscopic velocity.

(vii) F S

dSnFV

F .1

(vii) Due to the no-slip condition, a surface integral is zero if it contains the fluid velocity

vector explicitly.

Porosity, , may or may not be independent of time. It is time dependent in cases of flow

of a corrosive material through a porous sediment.

2.3 Averaging the Governing Equations

Taking the average of both sides of equation (2.1), and using rule (v), we obtain:

S

dSnvV

vv 01

. (2.9)

By Gauss’ divergence theorem, namely VS

dVFdSnF , equation (2.9) takes the

form when equation (2.1) is invoked:

.0 v

(2.10)

For the incompressible flow at hand, continuity of mass flow translates into vanishing

normal component of velocity.

20

Taking the averages of both sides of equation (2.4) and applying rule (iii), we get:

gTpvvvt

. (2.11)

The averages in equation (2.11) are evaluated term by term, as follows.

Using rules (i) and (vi), we obtain:

tttt vvvv )(

. (2.12)

Using rules (i), (v) and (iv), we obtain:

vvvv

+ vv + .

S

dSnvvV

(2.13)

Using rules (i) and (ii), we have have:

pp S

dSnpV

p 1 . (2.14)

Using rule (v), we get:

T

S

dSnTV

T .1

(2.15)

Using rule (i), we get

gg

. (2.16)

Now, substituting (2.12)-(2.16) in (2.11), we obtain:

vvv t

)(

vv + S

dSnvvV

S

dSnpV

p 1

S

dSnTV

T 1

g

. (2.17)

21

Equations (2.10) and (2.17) represent the intrinsic volume averaged continuity and

momentum equations. The deviation terms and surface integrals appearing in equation

(2.17) contain information on the interactions between the fluid-phase and solid-phase in

the control volume. These are analyzed in what follows.

2.4 Analysis of the Deviation Terms and Surface Integrals

Using Rule (vii), the term S

dSnvvV

in (2.17) contains the velocity vector explicitly,

hence vanishes. The term vv is related to the hydrodynamic dispersion of the

average velocity. Hydrodynamic dispersion through porous media is the sum of mechanical

dispersion and molecular diffusion. Mechanical dispersion is due to tortuosity of the flow

path within the porous microstructure, and molecular diffusion arises due to diffusion of

the fluid vorticity, Bachmat and Bear [12] Du Plessis and Masliyah [25]. Now, the above

deviation term is an inertial representative of mechanical dispersion. In the absence of high

velocity and high porosity gradients, the deviations in the velocity vector are small, and the

product of deviations is negligible. Hence, the term vv is negligible.

The term S

dSnTV

1 is the interfacial viscous stress exchange, which corresponds to the

microscopic momentum exchange of the Newtonian fluid with the solid matrix. It can be

argued that this term depends on the morphology of the porous matrix, viscosity of the

fluid and, if present, the relative velocity of the fluid-phase and the solid-phase. As a first

approximation, the following expression can be used for this surface integral

22

k

vvfdSnT

VS

2

1, wherein

kf

and k is the permeability. The

term S

dSnpV

1 represents pressure fluctuations on the fluid-solid interface, and is arguably

small, hence neglected. Equation (2.17) thus takes the form

vvv t

)( p

T

k

v

2

g

. (2.18)

2.5 Final Form of Governing Equations

Equations (2.10) and (2.18) represent the intrinsic volume-averaged continuity and

momentum equations, respectively, written in terms of the specific discharge (superficial

average of the velocity vector), v

. They are valid for variable porosity media.

Letting vq

, gG

, pP and TI

we can write (2.10) and

(2.18) in the following forms:

Continuity equation (2.10) is written as:

.0 q

(2.19)

Momentum equations (2.18) are written as:

/qqqt

P I

k

q

G

. (2.20)

Expanding the dyadic form on LHS of (2.20) and using (2.19), equation (2.20) takes the

form:

23

qqqt

P I

k

q

G

. (2.21)

Using (2.3), we obtain:

TI

Tvv

(2.22)

Defining and using

qv

in (2.22), we obtain:

I

T

qq

(2.23)

Using (2.23) in (2.21) we obtain:

qqqt

P

T

qq

q

k

G

. (2.24)

In equation (2.24), the product is that of true fluid viscosity and porosity, which is an

average viscosity in the porous medium. We shall take this product to be the average

viscosity and thus write (2.24) as:

qqqt

P

T

qq

q

k

G

. (2.25)

Equation (2.25) is valid for variable porosity media. If porosity is constant, we obtain

the governing equations as follows. Defining vu

, equations (2.10) and (2.18) yield,

respectively, (after dividing throughout by ):

.0 u

(2.26)

24

uuut

P Tuu )(

u

k

G

. (2.27)

or

uuut

P Tuu )(

u

k

G

. (2.28)

The term Tuu )(

in equation (2.28) can also be written as:

TTT uuuuuu )()()()(

(2.29)

whence, equation (2.28) takes the form:

uuut

P TT uuuu )()()(

u

k

G

(2.30)

2.6 Chapter Conclusion

A set of equations governing the flow of a Newtonian fluid with variable viscosity through

an isotropic porous medium has been derived. Equations have been derived for both

variable-porosity and constant porosity material. In addition, we have taken viscosity to be

variable without specifying the form of the functional dependence on pressure.

25

Chapter 3

Riabouchinsky Flow of a Pressure-Dependent Viscosity Fluid in Porous

Media


It has long been recognized that viscosity of a flowing fluid changes with flow conditions.

For example, changes in temperature result in changes in viscosity, and excessive increase

in pressure could affect the viscosity (although it has been customary to assume constant

shear viscosity when Newtonian liquids flow in the absence of temperature variations), (cf.

Housiadas et al [36] and the references therein). An assumption of constant viscosity

clearly ignores effects of pressure on viscosity in fluid flow conditions when the fluid

pressure is of the order of 1000 atm. (Denn [23] Rajagopal and Saccomandi [62] Renardy

[67]. A further factor that has an influence on viscosity is the flow domain (such as flow in

constrictions and flow through porous media) where it has been suggested by a large

number of researchers that the effective viscosity of a fluid in a porous structure is different

than the viscosity of the base fluid (cf. Almalki and Hamdan [10] and the references

therein).

High pressures arise in industrial applications that involve chemical and process

technologies, such as medical tablet production, crude and fuel oil pumping, food

processing, fluid-film lubrication theory, and in microfluidics, Denn [23] Hron et al [37]

Kannan and Rajagopal [40] Fusi et al [27] Martin-Alfonso et al [50,51]. In these and many

other applications, the form of momentum equations used must include a shear viscosity

26

that is a function of pressure. It has been argued that in some applications the effects of

high pressure on increasing viscosity is much more significant than the effect of pressure

on increasing density, Housiadas et al [36]. Accordingly, one can ignore compressibility

effects but must take into account the dependence of viscosity on pressure, Denn [23]

Goubert et al [28]. This realization is even more important when high pressures are

encountered in the flow of fluids through porous media due to the important applications

of groundwater recovery and oil production. Fusi et al [27] Rajagopal [60,61] Rajagopal

and Saccomandi [62]. In this regard, governing equations must be derived to accommodate

the different types of fluid flow, flow conditions, and the anticipated porous structure, Du

Plessis and Masliyah [25] Rajagopal [60]. While many models governing fluid flow

through porous media have been developed over the past sixteen decades, equations

governing flow through porous media with variable viscosity have been developed more

recently, Alharbi et al [7] Hron et al [37] Kannan and Rajagopal [40] Rajagopal [59]

Rajagopal and Saccomandi [62].

Solutions to the available models are at least as challenging as solutions to Navier-Stokes

equations with variable viscosity, due in part to the additional drag term that involves

permeability. When permeability is a variable function of position (and time, when

clogging is taken into account), a further complication is added to the equations. Solutions,

and modelling of flow, thus necessitate approximations and simplifying assumptions,

Rajagopal [60]. Considerations of special cases of flow, linearization, and simplifying

assumptions are not at all new in the study of Navier-Stokes flow in general, and pressure-

dependent viscosity analysis in particular, as has been amply demonstrated in the work of

Naeem [53] and the work of Naeem and co-workers (cf. [53] and the references therein).

27

Solutions to flow of a fluid with pressure-dependent viscosity have been obtained under

various assumptions and using a plethora of techniques have been reviewed, reported and

implemented in the work of Naeem [53] and the vast literature reported in his work. One

such technique is based on the concept of Riabouchinsky flow in which the streamfunction

of a two-dimensional flow is assumed to be a function of one space variable, or

combinations of functions of single variables. This approach has received considerable

success in the understanding of flow phenomena and the introduction of methodologies

based on this approach. Naeem [53] successfully implemented this approach in his study

of Navier-Stokes flow of a fluid with pressure-dependent viscosity. This approach is also

conveniently valid, and suitable in what follows, where we study the same flow in a porous

structure that is of either constant or variable permeability. We hasten to point out that

solutions of motions of fluids with pressure-dependent viscosities through variable

permeability porous media is at its infancy.


The equations governing variable-viscosity fluid flow through porous media, in the

absence of heat transfer effects, are given by the following continuity and linear momentum

equations, Alharbi et al [7]:

Continuity Equation:

0 u

(3.1)

Momentum Equations:

uu

p TT uuuu )()()(

uk

. (3.2)

28

wherein u

is the velocity field, is the fluid density, is the viscosity, p is the pressure

and k is the permeability.

For two-dimensional flow, we take ),( vuu

, and write equation (3.1) as:

0 yx vu . (3.3)

Equations (3.2) are written as:

uk

vuuPuvvq yxyxxxyxx

])}({)2[(

1)()(

2

1 2 (3.4)

vk

vuvPuvuq xxyyyyyxy

])}({)2[(

1)()(

2

1 2 (3.5)

where

pP (3.6)

and the square of the speed is defined by

222 vuq . (3.7)

Assuming that viscosity depends on the pressure according to the relationship:

Pa (3.8)

where a is a constant, then using (3.8) in (3.5) and (3.6) yields, respectively:

uk

aPvuaPuvaPPauuvv

qxyyyyxxxyxx )(][)12()()

2(

2

(3.9)

vk

aPvuaPuvaPPauuvu

qxyxxyxyxyxy )(][)12()()

2(

2

(3.10)

29

We note that equations (3.9) and (3.10) are valid for both variable and constant

permeability. Now, equation (3.3) implies the existence of the streamfunction such that:

yu (3.11)

xv (3.12)

and the vorticity, , is defined as:

yx uvu

(3.13)

and expressed in terms of the streamfunction as

.2 yyxx (3.14)

Using equations (3.11) and (3.12) in equations (3.9) and (3.10), we obtain, respectively

yxxyyyyxyxxx

xy

k

aPaPaPPa

)()12()

2( 22

22

(3.15)

xxxyyxxyxyyy

xy

k

aPaPaPPa

)()12()

2( 22

22

. (3.16)

Equations (3.15) and (3.16) must be satisfied by the streamfunction ),( yx and the

pressure ),( yxP , if the permeability is constant. However, if ),( yxkk is variable, then

(3.15) and (3.16) must be satisfied by the three functions. This implies that at least one of

the functions must be specified.

30

3.3 Method of Solution

Solutions to (3.15) and (3.16) are obtained by assuming a functional form of ),( yx and

then solving for the pressure ),( yxP . Viscosity is then determined using equation (3.8).

Assuming that the streamfunction is a function of y only, namely

)(),( yfyx (3.17)

we obtain

0 xyxxx (3.18)

)(' yfy (3.19)

)(" yfyy . (3.20)

Velocity components and vorticity, equations (3.11), (3.12), and (3.13), take the following

forms, respectively

)(' yfu (3.21)

0v (3.22)

)(" yf . (3.23)

Using (3.17)-(3.20) in equations (3.15) and (3.16), we obtain, respectively

)(')(")('" yfk

aPyfaPyaPfP yx (3.24)

)('' yfaPP xy . (3.25)

Upon substituting (3.25) in (3.24), and simplifying, we obtain

)](''1[

)('

)](''1[

)('"2222 yfak

yaf

yfa

yaf

P

Px

. (3.26)

31

The terms on the RHS of (3.26) can be expressed as follows:

)](''1[

)](''1[ln

2

1

)](''1[2

)('''

)](''1[2

)('''

])(''1[

)('''22 yaf

yaf

dy

d

yaf

yaf

yaf

yaf

yfa

yaf

(3.27)

)](''1[2

)('

)](''1[2

)('

])(''1[

)('22 yafk

yaf

yafk

yaf

yfak

yaf

. (3.28)

Equation (3.26) is then expressed as:

)](''1[2

)('

)](''1[2

)('

)](''1[

)](''1[ln

2

1)(ln

yafk

yaf

yafk

yaf

yaf

yaf

dy

d

x

P

. (3.29)

Integrating (3.29) with respect to x, we obtain:

)(ln)](''1[2

)('

)](''1[2

)('

)](''1[

)](''1[ln

2

1ln yFx

yafk

yaf

yafk

yaf

yaf

yaf

dy

dP

(3.30)

or

)](2

exp[)( yGx

yFP (3.31)

where )(yF is a function to be determined, and

)](''1[

)('

)](''1[

)('

)](''1[

)](''1[ln)(

yafk

yaf

yafk

yaf

yaf

yaf

dy

dyG

. (3.32)

From equation (3.31) we obtain

)](2

exp[2

)()(yG

xyGyFPx (3.33)

)](2

exp[)]()(2

)([ yGx

yGyFx

yFPy . (3.34)

Using (3.33) and (3.34) in equation (3.25), we obtain

)()()(''2

)]()(2

)([ yGyFyfa

yGyFx

yF (3.35)

and, upon equating coefficients of similar powers of x, we obtain

0)()( yGyF (3.36)

32

and

)()(''2)(

)(yGyf

a

yF

yF

. (3.37)

Equations (3.36) and (3.37) yield:

dyyGyf

ayF )()(''

2exp)( (3.38)

and

2)( CyG (3.39)

where 2C is a constant.

Using (3.39), and integrating the RHS of equation (3.38) once, equation (3.38) is replaced

by

)](2

exp[)( 24 yf

aCCyF (3.40)

where 4C is a constant. In addition, using (3.39), equation (3.32) becomes

2)](''1[

)('

)](''1[

)('

)](''1[

)](''1[ln C

yafk

yaf

yafk

yaf

yaf

yaf

dy

d

(3.41)

and the pressure equation (3.31) is replaced by:

]2

exp[)( 2xCyFP . (3.42)

Using (3.40) in (3.42) we obtain

))]((2

exp[ 24 yfax

CCP . (3.43)

The pressure distribution and flow quantities thus hinge on the function )(yf that must be

chosen such that (3.41) is satisfied. Equation (3.41) can be expressed in the following form:

33

a

C

k

yff

aCf

2

)('

2

222 . (3.44)

In the analysis to follow, we will select and substitute a suitable function )(yf in equation

(3.44), one that satisfies (3.44) and helps determine the constant 2C . With the knowledge

of 2C , the pressure function in (3.43) can then be determined and other flow quantities and

viscosity can be determined from equations (3.8), (3.17), and (3.21)-( 3.23). The next

section will provide some examples.

3.4 Results and Analysis

3.4.1 Example 1

Let BAyyf )( , where A and B are known constants, and assume that permeability k is

constant. Equation (3.44) yields k

aAC

22

and equation (3.43) gives the following

pressure distribution:

)](exp[4 aAxk

aACP

. (3.45)

Equation (3.8) renders the viscosity function as:

)](exp[4 aAxk

aACa

. (3.46)

The constant 4C in the pressure distribution can be determined with the imposition of a

condition on the pressure. For instance, if 0)0,0( PP then (3.45) yield ],exp[22

04k

AaPC

and the pressure distribution and viscosity become, respectively,

34

]exp[0 xk

aAPP . (3.47)

]exp[0 xk

aAPa . (3.48)

The flow quantities of streamfunction, vorticity and velocity components are determined

from equations (3.17), (3.21), (3.22) and (3.23) as:

BAy (3.49)

Au (3.50)

0v (3.51)

0 . (3.52)

Although equation (3.47) is a decaying exponential function in x , and does not warrant a

graph, we nevertheless illustrate the effects of the combinations of the parameters involved

graphically.

In Figures 3.1 and 3.2, we provide a sketch of 0P

P vs. x, as given by equation (3.47). Figure

3.1 illustrates the effect of parameter a that appears in equation (3.8). We therefore take

10 x and 10 y and choose the following values of permeability k and constants A

to be fixed and vary a. The following three cases are considered:

Case 1: k=1, A=1, a=1

Case 2: k=1, A=1, a=5

Case 3: k=1, A=1, a=10.

35

Figure 3.1 shows that while increasing a, when other parameters are fixed, the pressure

decreases. This can also be seen in equation (3.47) where the pressure decreases

exponentially with increasing a.

Figure 3.2 illustrates the effect of increasing permeability while holding other parameters

fixed.

Case 1: k=0.1, A=1, a=1

Case 2: k=1, A=5, a=5

Case 3: k=10, A=1, a=1

Figure 3.1: Effect of parameter a on0P

P, eq. (3.47), fixed permeability and parameter A

36

Figure 3.2: Effect of parameter combinations on0P

P, eq.(3.47)

37

3.4.2 Example 2

Let CByAyyf 2)( , where A, B, C are known coefficients, and BAyk 2 a

variable permeability. Equation (3.44) yields 14

2222

Aa

aC , hence equation (3.43) gives

the following pressure distribution:

)}]2({14

exp[224 BAyax

Aa

aCP

(3.53)

and equation (3.8) renders the viscosity as:

)}]2({14

exp[224 BAyax

Aa

aCa

. (3.54)

If we impose the condition 0)0,0( PP then equation (3.53) gives ].41

exp[22

2

04Aa

BaPC

Pressure and viscosity distributions thus become:

]14

2exp[

22

2

0

Aa

AyaaxPP (3.55)

]14

2exp[

22

2

0

Aa

AyaaxPa . (3.56)



CByAy 2 (3.57)

BAyu 2 (3.58)

0v (3.59)

A2 . (3.60)

38

Three-dimensional pressure distribution described by (3.55) is illustrated in Figures 3.3

and 3.4, in order to illustrate the effects of varying parameter a. Increasing the value of a

results in increasing the pressure with decreasing x. We point out in this case that the

permeability value is tied in with the value of parameter A and the constant pressure 0P .

Figure 3.3: 0P

P, eq.(3.55), for A=1, and a=1

39

Figure 3.4: 0P

P, eq.(3.55), for A=1, and a=10, eq.(3.55)

3.4.3 Example 3

Let )cos(sin)( yyAyf where A is a known coefficient, and choose a variable

permeability of the form

]cos[sin]cos[sin

]sin[cos2

2 yyyyaAC

yyk

. (3.61)

Provided that 122 Aa , 2C can be any constant that is tied to the permeability through

equation (3.61). Under these conditions, the pressure and viscosity distributions are given,

for selected values of 2C , a, and A, respectively by

)}]sin(cos{2

exp[ 24 yyaAx

CCP (3.62)

)}]sin(cos{2

exp[ 24 yyaAx

CCa . (3.63)

40

If we impose the condition 0)0,0( PP then equation (3.62) yields ]2

exp[ 204

aACPC

,

and the pressure and viscosity distributions take the form:

)]1sin(cos22

exp[ 220 yy

aACxCPP (3.64)

)].1sin(cos22

exp[ 22 yyaACxC

a (3.65)



)cos(sin yyA (3.66)

)sin(cos yyAu (3.67)

0v (3.68)

)cos(sin yyA . (3.69)

For the sake of illustration, the permeability distribution, equation (3.61), is graphed in

Figures 3.5, 3.6, 3.7 and 3.8 for 5,2,12 C and 10, respectively. With increasing 2C , the

location of the jump in permeability shifts closer to the plane 0y .

The effect of 2C on 0P

P is illustrated in Figures 3.9 to 3.10, which demonstrate the relative

shift of the location of maximum and minimum pressure towards the line y = 0 as 2C

increases from 1 to 10.

41

Figure 3.5: Permeability function, eq.(3.61), for 1,12 aAC


42



43

Figure 3.9: 0P

P, eq.(3.64), for 1,12 aAC

Figure 3.10: 0P

P, eq.(3.64),for 1,102 aAC

44


In this work, we considered the two-dimensional flow of a fluid with pressure-dependent

viscosity through a porous medium with variable permeability, in general. The

permeability function was selected so that the governing equations are satisfied and the

arbitrary constants appearing in the pressure distribution can be determined. The variable

fluid viscosity was taken to be proportional to the pressure, and the effect of the constant

of proportionality on the pressure distribution was illustrated using pressure distribution

graphs. We have illustrated in this Chapter how to handle and solve the governing

equations of flow through variable permeability. The same methodology can be applied

when other forms of viscosity-pressure relations are employed.

45

Chapter 4

Coupled Parallel Flow of Fluids with Viscosity Stratification through

Composite Porous Layers


Coupled parallel flow through a channel bounded by a porous layer has received

considerable attention in the literature due to the importance of this flow in physical

applications. These range from ground water recovery and oil and gas production, to

lubrication mechanism design and the design of heating and cooling systems. Conditions

at the interface between the flow domains were first formalized by Beavers and Joseph,

Hron et al [37] whose experimental and theoretical work was based on Navier-Stokes flow

in the channel and Darcy’s flow in the porous layer. Much has been accomplished in this

field since the introduction of Beavers and Joseph condition. In fact, various models of

flow through porous media have been implemented in the analysis of interfacial conditions,

and many excellent advancements in the field have contributed significantly to our state of

knowledge. A number of excellent reviews of the problem, the models, and the knowledge

base, are available (cf. Vafai and Thiyagaraja [83] Rudraiah [68] Allan and Hamdan [8]

Ford and Hamdan [26] Williams [86] Vafai [81] Kaviany [41] Abu Zaytoon, Alderson and

Hamdan [2] Nield and Kuznetsov [58]) and the references therein).

Associated with the above coupled parallel flow is the flow through two or more porous

layers. This situation was identified by Thiagaraja and Vafai [83] as the second problem of

interest in this field. This situation arises in physical flows such as the flow of ground water

46

in the (essentially) layered soil and bedrock, in agriculture and irrigation problems, and in

biomedical applications involving flow of blood in animal tissues. The literature reports on

a fairly large number of contributions in this field (cf. Almalki and Hamdan [10] and the

references therein).

While the study of fluid flow with variable viscosity in general represents a rather

established field of work, Alharbi, Alderson, and Hamdan [7] and variations of viscosity

are attributed to temperature and pressure, the flow of fluids with variable viscosity through

porous structures is, by comparison, less established. Applications of this type of flow are

found in the oil and pharmaceutical industries, among others, Hron et al [37] Kannan and

Rajagopal [40] Fusi et al [27] Martin-Alfonso et al [50,51] Rajagopal [60,61] Rajagopal

and Saccomandi [62] Renardy [67] and various flow models and problems have been

discussed in the pioneering work of Rajagopal and coworkers (cf. Hron et al [37] Kannan

and Rajagopal [40] Rajagopal [60,61] Rajagopal and Saccomandi [62] and the references

therein). Less studied, however, is the coupled parallel flow where permeability of the

porous medium and viscosity of the fluid vary with position and flow conditions. In the

current work, coupled parallel flows of fluids with variable, stratified viscosities through a

composite of porous layers is considered. Matching conditions at the interface between

layers are discussed and applied to a flow situation in which viscosities are stratified and

vary with position. There is no assumption that viscosity variations are due to pressure or

temperature variations in the current work. However, we set the stage for future

considerations of the flow of fluids with pressure-dependent viscosities.

47

4.2 Problem Formulation

Consider the flow in the configuration shown in Figure 4.1.

Figure 4.1 : Representative Sketch

Porous layer 1 is described by 0,0),( 1 yLxyx , and is bounded by an impermeable

wall at 1Ly , while porous layer 2 is described by 20,0),( Lyxyx , and is bounded

by an impermeable wall at 2Ly . An assumed sharp, permeable interface, exists between

the two layers along the line 0y . The fluids saturating the layers possess stratified,

variable viscosities, and their motions are described by the following equations, for layers

1 and 2, respectively, and are based on model equations derived by Alharbi et al [7]:

x

pukdy

du

dy

d

dy

ud11

1

111

2

1

2

1

(4.1)

x

pukdy

du

dy

d

dy

ud22

2

222

2

2

2

2

(4.2)

48

where, for layer 2,1j , )( jjj y is the variable viscosity of the fluid, )(yuu jj is

the velocity of the fluid, 0Cp jx is the constant, common, driving pressure gradient,

and jk is the permeability.

Equations (4.1) and (4.2) are to be solved subject to the following boundary and interfacial

conditions:

No-slip condition at the impermeable walls:

0)()( 2211 LuLu . (4.3)

Conditions at the interface 0y :

At the interface, the velocity, shear stress, and viscosity of the fluid are assumed

continuous. Therefore, we have:

)0()0( 21

uu (4.4)

00

12

11

ydy

du

ydy

du (4.5)

00

21yy

. (4.6)

It is assumed that the viscosity in each layer is stratified in the y-direction, hence a function

of y only, and is of the exponential forms:

)exp()( 1101 fy (4.7)

)exp()( 2202 fy (4.8)

49

where 0 is a constant reference viscosity (which could be taken here as the fluid viscosity

at atmospheric pressure and room temperature), the forms of functions )(1 yf and )(2 yf

are to be assumed, and 1 , 2 are constants that must satisfy conditions derived from the

forms of )(1 yf and )(2 yf , as discussed below in equations (4.19) and (4.22).

Now, using the chain rule, we obtain

y

f

df

d

dy

d

1

1

11 (4.9)

y

f

df

d

dy

d

2

2

22 (4.10)

and upon substituting (4.9) and (4.10) in (4.1) and (4.2), we obtain the following governing

equations:

)](exp[])(

[ 11

01

11112

1

2

yfC

k

u

dy

du

dy

ydf

dy

ud

(4.11)

)](exp[])(

[ 22

02

22222

2

2

yfC

k

u

dy

du

dy

ydf

dy

ud

. (4.12)

Once the functions )(1 yf and )(2 yf are specified, we can then solve (4.11) and (4.12) for

1u and 2u .

4.3 Solution Methodology

When the permeabilities 1k and 2k are constant, and )(1 yf and )(2 yf are given by

111 )( byayf (4.13)

222 )( byayf (4.14)

50

then the viscosity expressions (4.7) and (4.8) take the forms:

])[exp()( 11101 byay (4.15)

])[exp()( 22202 byay (4.16)

where 2121 ,,, bbaa are constants to be specified or to be determined. Viscosity at the

interface is given by the following expressions:

)exp()0( 1101 b (4.17)

)exp()0( 2202 b (4.18)

which, upon invoking condition (4.6) and simplifying, yield the following relationship

between the constants 121 ,, b and 2b :

2211 bb . (4.19)

In order to determine the effects of the porous layer thicknesses, we first find a relationship

between the thicknesses and the constants that determine the forms of the viscosity

expressions. This is accomplished by letting L and u be the viscosities of the fluids at

the lower and upper walls, respectively, namely:

])[exp()( 1111011 bLaL L (4.20)

])[exp()( 2222022 bLaL u . (4.21)

Now, taking Lu at the walls, and using (4.19), equations (4.20) and (4.21) yield:

222111 LaLa . (4.22)

Equations (4.19) and (4.22) must be satisfied by the choice of coefficients in the viscosity

functions (4.15) and (4.16). The governing equations (4.11) and (4.12) thus take the forms,

which must be solved for 1u and 2u :

51

)](exp[ 111

01

11112

1

2

byaC

k

u

dy

dua

dy

ud

. (4.23)

)](exp[ 222

02

22222

2

2

byaC

k

u

dy

dua

dy

ud

. (4.24)

General solutions to (4.23) and (4.24) take the following forms, respectively

))exp(()exp()exp( 11211

0

22111 bymmkC

ymcymcu

(4.25)

))exp(()exp()exp( 22212

0

24132 bynnkC

yncyncu

(4.26)

where 4321 ,,, cccc are arbitrary constants, and

1

211111

1)

2(

2 k

aam

(4.27)

1

211112

1)

2(

2 k

aam

(4.28)

2

222221

1)

2(

2 k

aan

(4.29)

2

222222

1)

2(

2 k

aan

. (4.30)

The arbitrary constants in solutions (4.25) and (4.26) can be determined using conditions

of velocity and shear stress continuity at the interface, and the no-slip conditions on the

walls, namely conditions (4.3), (4.4) and (4.5). The following linear equations for the

arbitrary constants are thus obtained:

]exp[]exp[ 222111

0

4321 bkbkC

cccc

(4.31)

52

)](exp[)exp( 1111111

0

11221 bLaLmkC

Lmmcc

(4.32)

)](exp[)exp( 2222212

0

21243 bLaLnkC

Lnncc

(4.33)

and

1]exp[]exp[]exp[1]exp[]exp[

*]exp[])exp(1[])exp(1[

11111111

0

22221

222

0

21241122

LaLmbkC

LaLn

bkC

LnncLmmc

. (4.34)

The solution to the linear system of equations, (4.31)-(4.34), takes the following forms (in

the order they are evaluated):

][

][

][

][

21

651

21

4324

AA

AAB

AA

AABc

(4.35)

5142412 ABABcAc (4.36)

41321 BBBcc (4.37)

124213 BBcccc (4.38)

where

11212

212121

)exp(

)exp(

Lmmmm

LnnnnA

(4.39)

])exp(1[

])exp(1[

112

2122

Lmm

LnnA

(4.40)

])exp(1[

1]exp[

112

222213

Lmm

LaLnA

(4.41)

11212

222222114

)exp(

]exp[

Lmmmm

aLaLnnA

(4.42)

53

11212

111111115

)exp(

]exp[

Lmmmm

aLaLmmA

(4.43)

])exp(1[

1]exp[

112

111116

Lmm

LaLmA

(4.44)

0

1111

]exp[

bCkB

(4.45)

0

2222

]exp[

bCkB

(4.46)

1123 )exp( LmmB (4.47)

and

]exp[ 111114 LaLmB . (4.48)

4.4 Results and Discussion

Results have been obtained for the following choices of parameters.

1) For the sake of illustration, select 121 bb and 121 . These choices satisfy

condition (4.19).

2) Select 10

C and 10

0

C.

3) For each value of 0

C the following combinations of layer thicknesses and viscosity

coefficients are chosen:

11 L , 12 L , 1,1 21 aa

11 L , 22 L , 5.0,1 21 aa

21 L , 12 L , 2,1 21 aa

54

4) For each of the selections in 3), above, a subset of the following permeability values

are selected:

1,1 21 kk

1.0,1 21 kk

1,1.0 21 kk

1.0,01.0 21 kk

For the above values of parameters, the arbitrary constants 321 ,, ccc and 4c have been

calculated using (4.35)-(4.38). The computed values are used in the computations of

velocity and shear stress at the interface, and the determination of velocity profiles across

the layers. Velocity at the interface, 0y , is obtained using either of the following

expressions, obtained from (4.25) and (4.26), respectively:

)exp()0( 111

0

211 bkC

ccu

(4.49)

)exp()0( 222

0

432 bkC

ccu

. (4.50)

Shear stress at the interface is obtained using either of the following expressions, obtained

from (4.25) and (4.26), respectively:

)exp()()0( 11211

0

22111 bmmk

Cmcmc

dy

du

(4.51)

)exp()()0( 22212

0

24132 bnnk

Cncnc

dy

du

. (4.52)

55

Values of the velocity and shear stress at the interface, for the selected combination of

parameters, are listed in Tables 4.1 and 4.2. Both the velocity and shear stress at the

interface are influenced by the permeability, driving pressure gradients, layer thicknesses

and coefficients of y in the viscosity function.

Effects of permeability and pressure gradients on the interfacial velocity and shear stress

are illustrated in Table 4.1, which shows the increase in these quantities with pressure

gradient, for a given permeability. For a fixed pressure gradient, interfacial velocity and

shear stress increase with permeability. In addition, they increase with increasing

permeability in either of the layers when the permeability of the other layer remains

unchanged. With increasing permeability there is an increase of momentum transfer from

one layer to another, thus influencing the velocity in the region near the interface.

Since layer thicknesses and coefficients of y must satisfy 222111 LaLa , then for

,21 the effects of these parameters is illustrated for the combinations shown in Table

4.2, which shows an increase in the velocity and shear stress at the interface with decreasing

lower layer thickness. This may be attributed to the lesser influence the upper layer has on

the lower layer when the lower layer is of greater thickness.

56

Velocity at

Interface

Shear stress at

Interface

10

C, 121 kk

0.2642485855 0.3551029315

100

C, 121 kk

2.642485855 3.551029315

10

C, 1.0,01.0 21 kk

0.01692744072 0.1429570892

100

C, 1.0,01.0 21 kk

0.1692744072 1.429570892

10

C, 1.0,1 21 kk

1.060520029 1.856567696

10

C, 1,1.0 21 kk

0.0914581620 0.2455583205

Table 4.1: Velocity and shear stress at interface for different 0

Cand permeabilities

121 bb , 121 , 11 L 12 L , 11 a , 12 a .

Velocity at

Interface

Shear stress at

Interface

11 L 12 L , 11 a 12 a 0.2642485855 0.3551029315

11 L 22 L , 11 a , 5.02 a 1.003176454 1.748439545

21 L 12 L , 11 a 22 a

0.1956139595 0.1174141298

Table 4.2: Velocity and shear stress at interface for different 1L , 2L , 1a and 2a , 121 bb ,

121 , 121 kk , 10

C.

57

Viscosity expressions are given by equations (4.15) and (4.16). It is clear that the most

important parameter in each profile is the coefficient of y. The effects of 1b and 2b are

manifested in magnifying the viscosity distribution. Therefore, we set 12121 bb

and illustrate the effects of 1a and 2a on the viscosity profiles, through graphing 0

1 )(

yand

0

2 )(

y versus y, for different parameters. The values of 11 L , 12 L , 5,2,11 a and

2,1,5.02 a are selected for illustrations in Figures 4.2 and 4.3. These figures are intended

to provide a visual illustration of the viscosity profiles of equations (4.15) and (4.16). They

show the relative exponential decrease in viscosity as we move away from the solid

boundary towards the interface, for all the parameters tested. The choice of viscosity

functions used in this work produce a continuous viscosity stratification that could

potentially be used as a base for comparison with other future choices of viscosity profiles,

and sets the stage for future analysis. It is emphasized here that 1a must be negative and

2a must be positive in order to produce viscosity variations such that the wall viscosities

are highest.

Velocity solutions (4.25) and (4.26) are represented in Figures 4.4 to 4.7 for various values

of parameters tested. In Figures. 4.4 and 4.5, the velocity profiles are shown for ,10

C

while Figures. 4.6 and 4.7 show the profiles for 100

C.

Comparing Figures 4.5 and 4.7 illustrates the effect of changing the pressure gradient ratio

0

Cfrom 1 to 10 , while keeping other parameters fixed. Figures 4.5 and 4.7 show an

58

approximate ten-fold increase in the magnitude of velocities in both layers corresponding

to the ten-fold increase in the magnitude of pressure gradient. The velocity profiles retain

their basic shape in both layers, however.

Figures 4.4-4.7 also illustrate the effects of increasing permeability on the velocity profile

in each layer. For all other parameters considered, increasing permeability in each layer

results in an increase in the velocity throughout the layer. This behavior is expected in light

of the increase in flow rate with increasing permeability.

Condition (4.22), namely 222111 LaLa , reduces to 2211 LaLa when 21 . While it

is not easy to isolate the influence of 1L and 2L , or 1a and 2a individually on the velocity

profiles in Figures 4.4-4.7, it is possible however to rely on the effects of 1a and 2a on the

viscosity, shown in Figures 4.2 and 4.3, as follows.

Figure 4.2 illustrates that decreasing 1a from -1 to -5 results in increasing the viscosity in

the lower layer. Since increasing viscosity of a fluid results in slower flow, we conclude

that decreasing 1a from -1 to -5 should result in a slower flow in the lower layer. Likewise,

increasing 2a from 0.5 to 2 in the upper layer results in increasing viscosity of the fluid, as

shown in Figure 4.3, thus implying slower flow with increasing 2a .

Now, in light of the condition 2211 LaLa , an increase in 2a must be accompanied by a

decrease in 2L for a fixed value of 11La , and a decrease in 1a must be accompanied with a

decrease in 1L for a fixed value of 22La . We can thus arrive at the following conclusions,

which are exhibited in Figures 4.4-4.7:

59

(1) Increasing 2a and decreasing 2L results in slowing down the flow in the upper

layer when all other parameters are fixed.

(2) Decreasing 1a and decreasing 1L results in slowing down the flow in the lower

layer when all other parameters are fixed.

Figure 4.2: Viscosity Profile in the Lower Layer, eq.(4.15)

60

Figure 4.3: Viscosity Profile in the Upper Layer, eq.(4.16)

Figure 4.4: Velocity profiles, eq.(4.25) and eq.(4.26), when 11 L , ,22 L

5.0,1 21 aa and 10

C

61

Figure 4.5: Velocity profiles, eq.(4.25) and eq.(4.26), when 21 L , 12 L , 2,1 21 aa

and 10

C

Figure 4.6: Velocity profiles, eq.(4.25) and eq.(4.26), when 11 L , 12 L , 1,1 21 aa

and 100

C

62

Figure 4.7: Velocity profiles when, eq.(4.25) and eq.(4.26), 21 L , 12 L , 2,1 21 aa

and ,100

C


In this Chapter, we formulated and solved the problem of coupled parallel flow through

two porous layers when the fluid viscosities are stratified. Appropriate interfacial

conditions have been stated, and exponential variations in viscosity have been considered

to produce the necessary high wall viscosities. Effects of permeability, pressure gradient

and other flow parameters have been discussed. This work initiates and sets the stage for

future analysis of coupled parallel flow of fluids with pressure-dependent and temperature-

dependent variable viscosities.

63

Chapter 5

Exact Solution of Fluid Flow through Porous Media with Variable

Permeability for a Given Vorticity Distribution


Variable permeability considerations in the study of flow through porous media offer a

more realistic approach in the flow through natural porous settings and in the simulation

of flow through porous layers, Cheng [21] Hamdan [30] Hamdan and Kamel [32]. In fact,

considerations of flow in the transition zone mandates taking a non-constant permeability

in order to avoid permeability discontinuity at the interface between layers, and to

circumvent invalidity arguments of Brinkman’s equation, (cf. Abu Zaytoon [1] Abu

Zaytoon, Alderson and Hamdan [2,3,4] Nield and Kuznetsov [58] and the references

therein). However, when the permeability is a variable function of position then an

additional variable is introduced into the governing equations. This results in an under-

determined system of more unknowns than equations unless, of course, a condition on the

permeability is introduced or the permeability function is defined externally or specified,

Alharbi, Alderson and Hamdan [6].

When a fluid flow model such as the Darcy-Lapwood-Brinkman model is used, where its

structure is similar to that of the Navier-Stokes equations, one must deal with the inherent

nonlinearity that arises due to the convective inertial terms. Methods available for

approximating the Navier-Stokes equations are also applicable to the Darcy-Lapwood-

64

Brinkman model. A number of methods have been available to linearize the Navier-Stokes

equations, or to solve the Navier-Stokes equations under simplifying assumptions or under

the assumptions of special types of flow (cf. Naeem [53] Jamil et al [38] Naeem and Jamil

[54] Naeem and Younus [55] Siddiqui [74] Chandna and Labropulu [19] Chandna and

Oku-Ukpong [20] Wang [84] Taylor [80] Kovasznay [43] Lin and Tobak [46] Ranger [65]

Hamdan [30,31] Merabet et al [52] and the references therein). An approach that has

received considerable attention is the assumption of vorticity being proportional to the

stream function of the two- dimensional flow, Taylor [80] Kovasznay [43] Lin and Tobak

[46]. This approach and other methods have been used successfully to study various fluid

flows, Naeem [53] Jamil et al [38] Naeem and Jamil [54] Naeem and Younus [55] Siddiqui

[74] Chandna and Labropulu [19] Chandna and Oku-Ukpong [20]. We will employ the

assumption of vorticity proportional to the streamfunction in the current work. We thus

consider the two-dimensional flow of an incompressible fluid through a porous medium

with variable permeability. We obtain an exact solution to the flow equations for a given

vorticity distribution. We will assume that the vorticity distribution is proportional to the

stream function of the flow.


Equations governing the steady, two-dimensional flow of a viscous fluid through a porous

medium with variable permeability are given by the continuity equation and the Darcy-

Lapwood-Brinkman equation, given respectively by:

Continuity Equation:

0 v

(5.1)

65

Momentum Equations:

vk

vpvv

2*)( (5.2)

where v

is the velocity vector, is the fluid density, is the base viscosity of the fluid,

* is the effective viscosity of the fluid in the porous medium, p is the pressure and k is

the permeability. Without loss of generality of the method of solution introduced in this

work, we will take * .

For the two-dimensional flow at hand we take ),( vuv

, and equations (5.1) and (5.2) can

be written as:

0 yx vu (5.3)

uk

uPvuuu xyx

2 (5.4)

vk

vPvvuv yyx

2 (5.5)

where

pP .

Equations (5.3), (5.4) and (5.5) represent a system of three scalar equations in the

unknowns Pvu ,, as functions of x and y. The variable permeability, k, is also an unknown

function of x and y. This results in a system of equations that is underdetermined. We must

therefore devise a method of solution where the permeability is determined by satisfaction

of a permeability condition. This condition is derived based on the integrability condition.

However, we first introduce the vorticity and streamfunction of the flow.

66

Continuity equation (5.3) implies the existence of the streamfunction such that:

yu (5.6)

and

xv (5.7)

and vorticity, , is defined as:

.yx uvu

(5.8)

Using (5.6) and (5.7) in (5.8), we obtain the streamfunction equation:

.2 yyxx (5.9)

Now, using (5.6) and (5.7), equations (5.4) and (5.5) take the following forms, respectively

yyyyxxyyxyxyxk

P

])[( . (5.10)

xxyyxxxyxxxyyk

P

])[( . (5.11)

Multiplying (5.9) by x and subtracting from (5.10) and rearranging, we obtain:

yyyyxxxyxyxxxxk

P

])[( . (5.12)

Multiplying (5.9) by y and subtracting from (5.11) and rearranging, we obtain:

xxyyxxyyyyxyxyk

P

])[( . (5.13)

Now, defining the following generalized pressure function

)(2

1)(

2

1 2222

yxPvuPL (5.14)

67

and differentiating (5.14) once with respect to x and once with respect to y , we obtain

)( yxyxxxxx pL (5.15)

)( yyyxyxyy pL (5.16)

Equation (5.15) is the LHS of (5.12) and equation (5.16) is the LHS of (5.13). Equations

(5.12) and (5.13) are thus written respectively as

yyyyxxxxk

L

])[( (5.17)

xxyyxxyyk

L

])[( (5.18)

The governing equations are thus (5.17) and (5.18), in the unknowns and , with the

vorticity defined by (5.9). Continuity equation (5.3) is automatically satisfied by virtue of

introducing the streamfunction in (5.6) and (5.7). Once and are determined, velocity

components can be calculated from (5.6) and (5.7). The pressure, ),( yxP , can then be

determined from (5.14) once the generalized pressure function, ),( yxL , is determined from

(5.17) and (5.18).

5.3 Method of Solution

5.3.1 Integrability Condition and Permeability Equation

In this work we assume that the vorticity is proportional to the streamfunction of the flow.

We thus assume that:

(5.19)

68

where is a nonzero constant.

Using (5.19) in (5.9), (5.17) and (5.18), we obtain, respectively:

yyxx (5.20)

yxyxx Ak

L

][ (5.21)

xyxyy Ak

L

][ (5.22)

where

][k

A

. (5.23)

From (5.21) and (5.22) we obtain:

yyyyyxxyxy AAL (5.24)

xxxxxyyxyx AAL . (5.25)

Setting yxxy LL yields the following integrability condition:

0 AAA xxyyxyyx . (5.26)

Integrability condition (5.26) must be met if (5.19) is to hold and (5.17) and (5.18) are

satisfied.

Now, from (5.19) we get

xx (5.27)

yy . (5.28)

Upon using (5.27) and (5.28) in (5.26), we obtain

69

0 AAA xxyy . (5.29)

Equation (5.29) is to be solved for A (hence the permeability function) once the form of

is determined.

5.3.2 Determination of Streamfunction, Vorticity and Velocity Components

In order to determine , we rely on equation (5.20), which is a Helmholtz equation that

admits plane wave solution of the form

)(),( yx (5.30)

where

sincos yx ; . (5.31)

From (5.31) we obtain the following derivatives:

.0;0;sin;cos yyxxyx (5.32)

Using the chain rule, the following derivatives of are established:

cos)(x (5.33)

sin)(y (5.34)

2cos)(xx (5.35)

2sin)(yy (5.36)

Using (5.30), (5.35) and (5.36) in (5.20), we obtain

)()( . (5.37)

70

Equation (5.37) is a homogeneous, second order ordinary differential equation with

auxiliary equation given by

02 r . (5.38)

Solution to (5.38) gives rise to the following cases:

1) 0;2 nn , and

2) 0;2 mm

Case 1: When 0;2 nn

Solution to (5.37) takes the form

))(cos()()( 21 CnC (5.39)

where )(1 C and )(2 C are real constants that depend on ),[ .

Using (5.30), (5.31) and (5.39), we obtain

))()sincos(cos()(),( 21 CyxnCyx . (5.40)

Velocity components are then obtained from (5.6), (5.7) and (5.40), respectively as

))()sincos(sin(sin)(),(),( 21 CyxnnCyxyxu y (5.41)

))()sincos(sin(cos)(),(),( 21 CyxnnCyxyxv x (5.42)

and the vorticity is obtained using (5.19) and (5.40) as

))()sincos(cos()( 21 CyxnC . (5.43)

Case 2: When 0;2 mm


71

]exp[)(]exp[)()( 21 mBmB (5.44)

where )(1 B and )(2 B are real constants that depend on ),[ .

Using (5.30), (5.31) and (5.44), we obtain

)]sincos(exp[)()]sincos(exp[)(),( 21 yxmByxmByx . (5.45)

Velocity components are then obtained from (5.6), (5.7) and (5.45), respectively as

)]sincos(exp[sin)()]sincos(exp[sin)(),( 21 yxmmByxmmByxu

(5.46)

)]sincos(exp[cos)()]sincos(exp[cos)(),( 21 yxmmByxmmByxv

(5.47)

and the vorticity is obtained using (5.19) and (5.45) as

]]sincos(exp[)()]sincos(exp[)([ 21 yxmByxmB . (5.48)

5.3.3 Determination of the Permeability Function

Equation (5.29) must be satisfied by the streamfunction and the permeability function.

With the knowledge of the streamfunction, for the two cases discussed above and given by

equations (5.40) and (5.45), we substitute the streamfunction expressions in equation (5.29)

and determine the permeability function.

Using (5.33) and (5.34), equation (5.29) takes the form

0)]([]cos)([]sin)([ AAA xy . (5.49)

72

Case 1:

Using (5.31) and (5.39), we write (5.54) as

0))](cos()([

]cos))(sin()([]sin))(sin()([

21

2121

CnCA

CnnCACnnCA xy. (5.50)

Dividing (5.50) by ))(sin()( 21 CnC , we obtain

0))]([cot(]cos[]sin[ 2 CnAnAnA xy . (5.51)

Now, using the chain rule, we obtain

cosAAx (5.52)

sinAAy . (5.53)

Using (5.52) and (5.53) in (5.51), and simplifying, we obtain

0))]([cot( 2

Cnn

AA . (5.54)

Using separation of variables, we write (5.54) as

dCnnA

dA))]([cot( 2 . (5.55)

Solution to (5.55) is given by

)(ln))](ln[sin(ln 322

CCn

nA (5.56)

which we write as

22

))]())sincos(()[sin(())]()[sin(( 2323nn CyxnCCnCA

(5.57)

where )(3 C is an arbitrary function of ),[ .

Using (5.23) and (5.57), we obtain the permeability function as:

73

2

))]())sincos(()[sin((

),(

23nCyxnC

Ayxk

(5.58)

Case 2:

Using (5.31) and (5.44), we write (5.49) as

0)]exp()()exp()([)]exp()(

)exp()([cos)]exp()()exp()([sin

212

121

mBmBAmmB

mmBAmmBmmBA xy. (5.59)

Using (5.52) and (5.53), we write (5.59) as:

0)]exp()()exp()([)]exp()()exp()([ 2121 mBmBAmBmBmA

(5.60)

Equation (5.60) is variable separable and can be written as

d

mBmB

mBmB

mA

dA

)]exp()()exp()([

)]exp()()exp()([

21

21

(5.61)

whose solution is given by

)(ln)]exp()()exp()(ln[ln 3212

BmBmB

mA (5.62)

Or

2

)]exp()()exp()()[( 213mmBmBBA

. (5.63)

Permeability function is then obtained by substituting (5.63) in (5.23) to obtain

74

2

))]sincos(exp()())sincos(exp()()[(

),(

213myxmByxmBB

Ayxk

(5.64)

5.3.4 Determination of Pressure

Equations (5.21) and (5.22) take the following forms in terms of the variable :

]sin)([)]()][([coscos AL (5.65)

]cos)([)]()][([sinsin AL . (5.66)

Multiply (5.65) by cos and (5.66) by sin , and adding, we get

)]()][([ L . (5.67)

Integrating (5.73) we get

)()]([2

)( 4

2

CL (5.68)

where )(4 C is an arbitrary function of ),[ .

The pressure is then determined from equation (5.14) as

)(2

1 22 vuLP (5.69)

where L is given in (5.68) and u and v are given in terms of as:

cos)(v (5.70)

sin)(u . (5.71)

Using (5.68), (5.70) and (5.71) in (5.69), we obtain the following expression for pressure:

75

)()]([2

1)]([

24

22

CP . (5.72)

Corresponding to solutions (5.39) and (5.44), the pressure function takes the following

forms. In Case 1, using (5.39) in (5.72) gives:

)())](sin()([2

))](cos()([2

)( 4

2

21

22

21

CCnCn

CnCP (5.73)

and, upon using (5.31), we obtain

)())]()sincos(sin()([2

))]()sincos(cos()([2

),(

4

2

21

2

2

21

CCyxnCn

CyxnCyxP

. (5.74)

In Case 2, using (5.44) in (5.72) gives:

)()]()()exp()([2

)]exp()()exp()([2

)( 4

2

21

22

21

CmeBmBm

mBmBP

(5.75)

and, upon using (5.31), we obtain

)()]}sincos(exp[)()]sincos(exp[)({2

)]]sincos(exp[)()]sincos(exp[)([2

),(

4

2

21

2

2

2

1

CyxmByxmBm

yxmByxmByxP

.

(5.76)

5.3.5 Summary of Solutions

Flow variables based on the two cases of solution are summarized as follows, where we

keep their equation numbers as they appeared in the text.

76

Case 1:

))()sincos(cos()(),( 21 CyxnCyx . (5.40)

))()sincos(sin(sin)(),(),( 21 CyxnnCyxyxu y (5.41)

))()sincos(sin(cos)(),( 21 CyxnnCyxv x (5.42)

))()sincos(cos()( 21 CyxnC . (5.43)

2

))]())sincos(()[sin((

),(

23nCyxnC

Ayxk

. (5.58)

)())]()sincos(sin()([2

))]()sincos(cos()([2

),(

4

2

21

2

2

21

CCyxnCn

CyxnCyxP

. (5.74)

Case 2:

)]sincos(exp[)()]sincos(exp[)(),( 21 yxmByxmByx . (5.45)

)]sincos(exp[sin)()]sincos(exp[sin)(),( 21 yxmmByxmmByxu

(5.46)

)]sincos(exp[cos)()]sincos(exp[cos)(),( 21 yxmmByxmmByxv

(5.47)

]]sincos(exp[)()]sincos(exp[)([ 21 yxmByxmB . (5.48)

2

))]sincos(exp()())sincos(exp()()[(

),(

213myxmByxmBB

Ayxk

.

77

(5.64)

)()]}sincos(exp[)()]sincos(exp[)({2

)]]sincos(exp[)()]sincos(exp[)([2

),(

4

2

21

2

2

2

1

CyxmByxmBm

yxmByxmByxP

.

(5.76)

The above solutions do not involve the permeability functions explicitly. This is due to the

fact that the permeability function was derived based on an integrability condition that in

terms of a pre-determined streamfunction that was determined independent of the

permeability. The above solutions for the streamfunction, vorticity, velocity and pressure

are valid for both cases of constant permeability and variable permeability. In fact, they are

valid for all permeability range (in particular, as permeability approached infinity and the

flow reduces to the Navier-Stokes flow). In other words, these are the same solutions to

the Navier-Stokes flow when vorticity is proportional to the streamfunction. We therefore

interpret the permeability functions as the needed permeability distributions to generate the

flow variables given in the above equations.

Typical three-dimensional plots for the above solutions are given below for selected values

of the parameters appearing in Case 1 and Case 2. Figures 5.1-5.24 are for the Case 1

results and Figures 5.25-5.48 are for Case 2. The intention here is to illustrate qualitatively

the shape distributions of the computed flow quantities over the two-dimensional flow

domain. Quantitative comparisons of the effects of parameters involved are not easily

detected from these three dimensional plots.

78

Figure 5.1: Case 1 Permeability Distribution,

eq.(5.58), 1 , 1)(1 D , 1n , 3/ , 1)(2 C


eq.(5.58), 1 , 1)(1 D , 1n , 6/ , 1)(2 C


eq.(5.58), 1 , 1)(1 D , 1n , 3/ , 1)(2 C

79


eq.(5.58), 1 , 1)(1 D , 1n , 6/ , 1)(2 C


eq.(5.58), 1 , 4)(1 D , 5.0n , 6/ , 1)(2 C


eq.(5.58), 10 , 1)(1 D , 1n , 4/ , 1)(2 C

80

Figure 5.7: Case 1 Pressure Distribution,

eq.(5.74), 1 , 1)(1 C , 2n , 3/ , 1)(2 C , 2)(4 C


eq.(5.74), 1 , 1)(1 C , 2n , 6/ , 1)(2 C , 2)(4 C


eq.(5.74), 2 , 7)(1 C , 1n , 3/ , 1)(2 C , 2)(4 C

81


eq.(5.74), 4 , 3)(1 C , 1n , 6/ , 1)(2 C , 2)(4 C


eq.(5.74), 1 , 4)(1 C , 5.0n , 6/ , 1)(2 C , 2)(4 C


eq.(5.74), 10 , 1)(1 C , 1n , 4/ , 1)(2 C , 2)(4 C

82

Figure 5.13: Case 1 Streamsurfaces,

eq.(5.40), 1 , 1)(1 C , 1n , 3/ , 1)(2 C


eq.(5.40), 1 , 1)(1 C , 1n , 6/ , 1)(2 C


eq.(5.40), 1 , 7)(1 C , 1n , 3/ , 1)(2 C

83


eq.(5.40), 1 , 3)(1 C , 1n , 6/ , 1)(2 C


eq.(5.40), 1 , 4)(1 C , 5.0n , 6/ , 1)(2 C


eq.(5.40), 10 , 1)(1 C , 1n , 4/ , 1)(2 C

84

Figure 5.19: Case 1 Vorticity Distribution,

eq.(5.43), 1 , 1)(1 C , 1n , 3/ , 1)(2 C


eq.(5.43), 1 , 1)(1 C , 1n , 6/ , 1)(2 C


eq.(5.43), 1 , 7)(1 C , 1n , 3/ , 1)(2 C

85


eq.(5.43), 1 , 3)(1 C , 1n , 6/ , 1)(2 C


eq.(5.43), 1 , 4)(1 C , 5.0n , 6/ , 1)(2 C


eq.(5.43), 10 , 1)(1 C , 1n , 4/ , 1)(2 C

86


eq.(5.64), 3 , 5.0)(1 B , 5.0)(2 B , 1m , 36/7 , 1)(2 D


eq.(5.64), 5 , 1)(1 B , 1)(2 B , 9.0m , 3/ , 1)(2 D


eq.(5.4), 4 , 5.0)(1 B , 5.0)(2 B , 1m , 6/ , 1)(2 D

87


eq.(5.64), 1 , 5.0)(1 B , 5.0)(2 B , 6.0m , 4/ , 1)(2 D


eq.(5.64), 1 , 5.0)(1 B , 5.0)(2 B , 1m , 3/ , 1)(2 D


eq.(5.64), 5.0 , 5.0)(1 B , 5.0)(2 B , 1m , 3/ , 1)(2 D

88

Figure 5.31: Case 2 Pressure,

eq.(5.76), 3 , 5.0)(1 B , 5.0)(2 B , 1m , 36/7 , 1)(4 C


eq.(5.76), 5 , 1)(1 B , 1)(2 B , 9.0m , 3/ , 1)(4 C


eq.(5.76), 4 , 5.0)(1 B , 5.0)(2 B , 1m , 6/ , 1)(4 C

89


eq.(5.76), 1 , 5.0)(1 B , 5.0)(2 B , 6.0m , 4/ , 1)(4 C


eq.(5.76), 1 , 5.0)(1 B , 5.0)(2 B , 1m , 3/ , 1)(4 C

Figure 5.36: Case 2 Pressure

eq.(5.76), 5.0 , 5.0)(1 B , 5.0)(2 B , 1m , 3/ , 1)(4 C

90


eq.(5.45), 3 , 5.0)(1 B , 5.0)(2 B , 1m , 36/7


eq.(5.45), 5 , 1)(1 B , 1)(2 B , 9.0m , 3/


eq.(5.45),, 4 , 5.0)(1 B , 5.0)(2 B , 1m , 6/

91


eq.(5.45), 1 , 5.0)(1 B , 5.0)(2 B , 6.0m , 4/


eq.(5.45), 1 , 5.0)(1 B , 5.0)(2 B , 1m , 3/


eq.(5.45), 5.0 , 5.0)(1 B , 5.0)(2 B , 1m , 3/

92


eq.(5.48), 3 , 5.0)(1 B , 5.0)(2 B , 1m , 36/7


eq.(5.48), 5 , 1)(1 B , 1)(2 B , 9.0m , 3/


eq.(5.48), 5 , 1)(1 B , 1)(2 B , 9.0m , 3/

93


eq.(5.48), 1 , 5.0)(1 B , 5.0)(2 B , 6.0m , 4/


eq.(5.48), 1 , 5.0)(1 B , 5.0)(2 B , 1m , 3/


eq.(5.48), 5.0 , 5.0)(1 B , 5.0)(2 B , 1m , 3/

94


The main theme of this work has been the devising of a method to obtain the permeability

distribution in a variable permeability porous medium. To accomplish this, exact solutions

were obtained under the assumption of vorticity being a function of the streamfunction of

the flow. Expressions for the permeability, pressure, velocity components, streamfunction

and vorticity were successfully obtained. Three-dimensional figures are provided to

illustrate the distributions obtained.

95

Chapter 6

Analytic Solutions to the Darcy-Lapwood-Brinkman Equation with

Variable Permeability


Flow through variable permeability porous media has applications in oil and gas recovery,

industrial and biomechanical processes and in natural environmental settings and

agriculture, Hamdan [31]. Naturally occurring media are of variable porosity and

permeability, and the flow through which is governed by flow models with permeability

tensor, Hamdan [31] Hamdan and Kamel [32]. In some idealization of heterogeneous and

inhomogeneous media, and in two-dimensional flow simulations, the permeability can be

taken as a variable function of one or two independent variables, Hamdan and Kamel [32].

A number of studies have implemented this approach (cf. Hamdan and Kamel [32] Abu

Zaytoon et al [2] Nield and Kuznetsov [58] and the references therein). Variable

permeability simulation has also proved to be indispensable in the study of the transition

layer, Nield and Kuznetsov [58] (defined here as a thin layer that is sandwiched between a

constant permeability porous layer and a free-space channel, and the flow through which

is governed by Brinkman’s equation).

Models of flow through porous media come in a variety of forms depending on whether

viscous shear effects and inertial effects are important. In the presence of solid boundaries,

shear effects are important and it has been customary to use Brinkman’s equation to model

the flow, Hamdan [31]. When, in addition, inertial effects are important one resorts to the

96

Darcy-Lapwood-Brinkman (DLB) equation (discussed in the current Chapter), which takes

into account macroscopic inertial effects and viscous shear effects. If micro-inertial effects

are important, one resorts to a Forchheimer-type inertial model, Hamdan [31].

The DLB equation, discussed in Chapter 2 of this work, resembles the Navier-Stokes

equations, and involves a viscous damping (Darcy-like) term. Not unlike the Navier-Stokes

equations, exact solutions are rare due to the nonlinearity of the equations and the

inapplicability of the Superposition Principle to nonlinear partial differential equations, (cf.

Chandna and Labropulu [19] Chandna and Oku-Ukpong [20] Dorrepaal [24] Labropulu et

al [45] Wang [84,85] Taylor [80]) identified the source of nonlinearity as the convective

inertial terms, which vanish in two-dimensional flows when the vorticity of the flow is a

function of the streamfunction of the flow.

By taking the vorticity to be proportional to the streamfunction of the flow, Taylor’s

solution [80] represents a double infinite array of vortices decaying exponentially with

time. Kovasznay [41] extended Taylor’s approach and linearized the Navier–Stokes

equations by taking the vorticity to be proportional to the streamfunction perturbed by a

uniform stream. Kovasznay’s two-dimensional solution represents the flow behind

(downstream of) a two-dimensional grid. Two solutions representing the reverse flow over

a flat plate with suction and blowing were obtained by Lin and Tobak [46] who extended

Kovasznay’s approach. Various other authors have obtained exact solutions to the Navier–

Stokes and other equations for special types of flow (cf. Ranger [65] Hamdan [30] Hamdan

and Ford [26] and the reviews in Wang [84] and [85]).

Most methods used in linearizing the Navier-Stokes equations have been used in the

analysis of the DLB equation with constant permeability. The case of variable permeability

97

is treated in this Chapter, where we consider two-dimensional flow through a porous

medium governed by a variable-permeability DLB equation and find three analytic

solutions for a prescribed permeability function of one space variable, when the vorticity

of the flow is a function of the streamfunction of the flow. We take the variable

permeability to be a function of position and a function of Reynolds number, since in the

case of flow with constant permeability, Reynolds number and permeability are

interdependent. The current study may prove to be of importance in stability studies of

flow through variable permeability periodic porous structures and in the study of flows that

deviate from base flows in porous structures.


The steady flow of an incompressible fluid through a porous medium composed of a mush

zone is governed by the equations of continuity and momentum, written respectively as,

Hamdan [31]

0 v

(6.1)

vk

vpvv

2*)( (6.2)

where v

is the velocity vector field, p is the pressure, is the fluid density, is the

viscosity of the base fluid, * is the effective viscosity of the fluid as it occupies the porous

medium, k is the permeability (considered here a scalar function of position), is the

98

gradient operator and 2 is the Laplacian. In the absence of definite information about the

relationship between * and , we will take * in this work.

Considering the flow in two space dimensions, x and y, we take ),( vuv

, ),( yxkk and

),( yxpp . Using the dimensionless quantities defined by

2

*

2

***** ;;),(

),(;),(

),(U

pp

L

kk

U

vuvu

L

yxyx

(6.3)

where L is a characteristic length and U a characteristic velocity, equation (6.1) then takes

the following dimensionless form after dropping the asterisk (*):

0 yx vu (6.4)

and momentum equations (6.2) are expressed in the following dimensionless form:

k

uuPvuuu xyx 2)Re( (6.5)

k

vvPvvuv yyx 2)Re( (6.6)

where

ULRe is the Reynolds number.

System (6.4), (6.5), and (6.6) can be conveniently written in streamfunction-vorticity form

as follows. Equation (6.4) implies the existence of a dimensionless streamfunction ),( yx

such that

y

u

(6.7)

x

v

. (6.8)

99

Dimensionless vorticity, , in two dimensions is defined as:

.yx uvv

(6.9)

Using (6.7) and (6.8) in (6.9), we obtain the streamfunction equation

.2 yyxx (6.10)

Vorticity equation is obtained from equations (6.5) and (6.6) by eliminating the pressure

term through differentiation, and can be written in the following equivalent forms

22

2 1]Re[

k

k

k

k

k

yyxxyxxy

(6.11)

22

2 1]Re[

k

ku

k

kv

kvu

yxyx . (6.12)


In order to solve equations (6.10) and (6.11 (or (6.12)) for and , we assume vorticity

to be a function of the streamfunction defined by

Re

y (6.13)

where is a parameter to be determined. Since equations (6.10) and (6.11) or (6.12)

represent two equations in the two unknowns and , we must assume the form of the

permeability function, ).,( yxk In the current Chapter, we assume k to be a function of x

only or a function of y only. Since the vorticity in (6.13) involves y explicitly, we will

assume that

100

x

xkkRe

)(

. (6.14)

Equation (6.14) is based on the assumption that the model equations (Darcy-Lapwood-

Brinkman equation (6.2)) is valid when inertial effects are significant, hence 0Re . It is

clear that when permeability increases, the flow is faster, and Re increases, and conversely.

This choice of permeability function is characteristic of flow in a porous domain where

permeability decreases downstream as x increases. We also assume here that is a

permeability-adjustment parameter in the sense that it is a parameter that depends on the

local value of permeability along a given line x constant.

If 0Re , then the Darcy-Lapwood-Brinkman model, equation (6.2), reduces to the

inertia-free Brinkman’s equation, which warrants different choice of permeability function.

Now, substituting (6.13) and (6.14) in (6.11), and simplifying, we obtain

yxx

x ]Re

Re[

1Re1Re

Re)( 2

22

2

(6.15)

Equation (6.15) has the integration factor given by

]1Re

)Re2/(exp[..

2

22

xxFI (6.16)

and solution given by

]1Re

)Re2/(exp[)(

Re 2

22

xxyf

y (6.17)

where )(yf is an arbitrary function of y .

101

Using (6.10), (6.13) and (6.17), we obtain the following equation that must be satisfied by

)(yf

0)(]}ReRe2[)1Re(1Re

{)( 2422

22

2

2

yfxxyf

(6.18)

Equating coefficients of x to power, we obtain:

0)()1Re( 22

2

yf

(6.19)

0)()1Re(

Re222

4

yf

(6.20)

0)(})1Re(

Re

1Re{)(

22

26

2

yfyf

. (6.21)

Equations (6.19) and (6.20) yield 0)( yf when 0 and 0Re . Using 0)( yf in (6.17)

gives Re

y , with horizontal streamlines constant, and velocity components 0v ,

and Re

1u which is a decreasing horizontal velocity with increasing Reynolds number.

If 0)( yf , then by letting

22

26

2 )1Re(

Re

1Re

(6.22)

equation (6.21) takes the form

0)()( yfyf . (6.23)

Auxiliary equation of (6.23) is:

02 m (6.24)

102

with characteristic roots given by

m . (6.25)

Three cases arise depending on the value of :

Case 1: 0


yy

ececyf

21)( (6.26)

where 1c and 2c are arbitrary constants. The streamfunction, solution (6.17) thus becomes

]1Re

)Re2/(exp[][

Re 2

22

21

xx

ececy yy

(6.27)

with velocity components given by

]1Re

)Re2/(exp[][

Re

12

22

21

xx

ececuyy

y (6.28)

]1Re

)Re2/(exp[]

1Re

Re)(][[

2

22

2

2

21

xxx

ececvyy

x (6.29)

and vorticity takes the form

]1Re

)Re2/(exp[][

Re 2

22

21

xx

ececy yy

. (6.30)

Case 2: 0

If 0 , then equation (6.26) is replaced by

yccyf 21)( (6.31)

103

and the streamfunction, solution (6.27), is thus replaced by

]1Re

)Re2/(exp[][

Re 2

22

21

xxycc

y (6.32)


]1Re

)Re2/(exp[

Re

12

22

2

xxcu y (6.33)

]1Re

)Re2/(exp[]

1Re

Re)(][[

2

22

2

2

21

xxxyccv x (6.34)


]1Re

)Re2/(exp[][

Re 2

22

21

xxycc

y. (6.35)

Case 3: 0

In this case, the characteristic roots in equation (6.25) are of the form

im (6.36)

and equation (6.26) is replaced by

ycycyf sincos)( 21 . (6.37)

The streamfunction, solution (6.27) is thus replaced by

]1Re

)Re2/(exp[]sincos[

Re 2

22

21

xxycyc

y (6.38)


104

]1Re

)Re2/(exp[]cossin[

Re

12

22

21

xxycycu y (6.39)

]1Re

)Re2/(exp[]

1Re

Re)(][sincos[

2

22

2

2

21

xxxycycv x

(6.40)


]1Re

)Re2/(exp[]sincos[

Re 2

22

21

xxycyc

y. (6.41)

6.4 Sub-Classification of Flow

6.4.1 Determining the values of

Equation (6.14) gives the variable permeability as a function of x and shows its

dependence on parameter and on Reynolds number. For choices of and Re , the value

of dimensionless variability must be such that 1)(0 xk . This implies that 0 . The

value of also affects the value of , as given in its definition in equation (6.22) which

also shows the influence of Re on . Reynolds number must be greater than zero so that

the permeability takes a positive value.

Equation (6.14) ties in together the values of permeability function at different values of ,x

and the values of parameter for a given permeability value. We emphasize here that in

obtaining the solutions discussed in this work, we assumed that 0Re and 0x so that

the permeability has a positive numerical value, hence 0 . A minimum value of is

105

chosen so that the dimensionless permeability does not exceed 1. In what follows we will

show that must be greater than unity.

Limiting cases on the flow:

When 0Re , equation (6.22) shows that 2 , and equation (6.14) shows that 0k

. This represents the limiting case for an impermeable solid.

Another limiting case is obtained when Re is large ( 1Re ), equation (6.22) shows that

2 .

If 0 then 02 , or 0 or 1 . If ,0 then 0 and the fluid is inviscid

(which is not the case in the current flow problem). If 1 , we run into problems

determining values of Reynolds number, as discussed in what follows.

The cases of 0 , 0 , and 0 :

When 0 , equation (6.22) reduces to:

02Re3Re)( 2254 . (6.42)

If 1 then Re is negative. A positive Re is therefore obtained when 0 and 0

and 1 . Solution to equation (6.42) in terms of is given by:

)(2

813Re

23

(6.43)

where we choose the sign of root that renders a positive value for Re , for a given 0

1 . This value of Re , for the choice of represents the critical value that makes 0

When 0 equation (6.22) reduces to:

106

02Re3Re)( 22232 (6.44)

and we must have 1 in order to have a positive Reynolds number, given in terms of

as:

)(2

813Re

23

. (6.45)

A choice of positive Re that is less than or equal to the value calculated using equation

(6.45) guarantees that 0 .

When 0 equation (6.22) reduces to:

02Re3Re)( 22232 (6.46)

and we must have 10 in order to have a positive Reynolds number, given in terms

of as:

)(2

813Re

23 aa

a

. (6.47)

A choice of Re greater or equal to the value calculated using equation (6.47) guarantees

that 0 .

6.4.2 Stagnation Points

The flow described by equations (6.27)-( 6.30), (6.32)-(6.35) and (6.38)-( 6.41) can be sub-

classified according to the values of the arbitrary constants 1c and 2c . In particular when

0 vu stagnation points in the flow occur.

Case 1: 0

Setting 0 vu in (6.28) and (6.29) gives the following values of ),( yx at stagnation:

107

Re2x (6.48)

2/1

2

2

25

11

2

2

25

121

2

25

1

1Re

2/Reexp[Re4

1Re

2/Reexp[Re4

1Re

2/Reexp[Re2

1ln

1

cc

ccc

c

y (6.49)

In order to have a positive argument of the natural logarithmic function, we must choose

01 c . The value of 2c must be chosen such that 01Re

2/Reexp[Re4

2

2

25

121

ccc or

2

2

25

1

12

1Re

2/Reexp[Re4

c

cc . (6.50)

We note that permeability to the fluid along the vertical lines (6.48) is given by

1Re

xk . (6.51)

When 12 cc , solutions represent a reversing flow over a flat plate (situated to the right of

the y-axis) with suction ( 0Re

y

) or blowing ( 0Re

y

). For non-negative 1c ,

suction occurs, and when 01 c blowing occurs. When 21 cc , the flow is non-reversing

with suction if 02 c or blowing if 02 c .

Case 2: 0

108

Equations (6.33) and (6.34) give the following stagnation points:

2

1

c

cy (6.52)

)Re

1ln(

)1Re(2ReRe

2

222

cx

. (6.53)

For )Re

1ln(

2c to be defined, we much have 02 c . In addition, we much have

0)Re

1ln(

)1Re(2Re

2

222

c

(6.54)

which is guaranteed by choosing 02 c such that

])1Re(2

Reexp[Re

1

2

232

c . (6.55)

In order to have a positive value for permeability, we must have 0x . Values of the

parameters in (6.53) must be chosen to guarantee 0x . These solutions represent flow

over a porous flat plate with suction or blowing.

Case 3: 0

Setting 0u and 0v in equations (6.39) and (6.40) results in

]Re))[ln(/1Re(ReRe 2242 x (6.56)

1tan1 y (6.57)

where

109

2

1

c

c (6.58)

2

21

1)(

cc . (6.59)

It is clear that 21,cc ; however the parameters in (6.56) must be chosen such that 0x

so that permeability is positive. The obtained solution represents a flow field consisting of

alternating vortices that are superposed on the main flow, perpendicular to their planes.

6.4.3 Comparison with Constant Permeability Solutions

When the permeability is constant in the DLB equation, the three exact solutions, below,

have been obtained by Merabet et al [52]:

][ Ry (6.60)

2

2

111

kR

(6.61)

Re

1R ; 1 (6.62)

Case 1: 10

Ry

2

2

2

2

2

1

11exp

11exp

k

kR

R

y

k

k

R

xc

k

kR

R

y

k

k

R

xc

(6.63)

110

k

k

R

1

1Re . (6.64)

Case 2: 0

k

k

R

xyddRy

1exp)( 21 (6.65)

k

k

R

1

1Re . (6.66)

Case 3: 0

2

2

1

1cos

1exp R

k

k

R

ye

k

k

R

xRy

22

2

1sin

1exp R

k

k

R

ye

k

k

R

x

(6.67)

k

k

R

1

1Re . (6.68)

In both cases of constant or variable permeability, Reynolds number is connected to

permeability. However, is defined differently and its range is different in both flow

types, while 1 for constant permeability flow.

111

Figure 6.1: Permeability, eq.(6.14), when

1Re

Figure 6.2: Permeability, eq.(6.14), when

3Re

112

Figure 6.3: Case 1 Velocity )(yu , eq.(6.28), when,

1Re , 1,1,1 21 cc

Figure 6.4: Case 1 Velocity )(yu , eq.(6.28), when

3Re , 1,1,1 21 cc

113

Figure 6.5: Case 1 Velocity )(xv , eq.(6.29), when

1Re , 5.021 cc


3Re , 5.021 cc

114


1Re , 12 c


3Re , 12 c

115


1Re , 5.021 cc


3Re , 5.021 cc

116


1Re , 1,1 21 cc , 1


3Re , 1,1 21 cc , 1

117


1Re , 1,1 21 cc


3Re , 1,1 21 cc

118


In this Chapter we have obtained three exact solutions to the Darcy-Lapwood-Brinkman

equation with variable permeability. Permeability has been defined as a function of one

space dimension, and vorticity is prescribed as a function of the streamfunction of the flow.

Ranges of parameters have been determined, and comparison is made with the solutions

obtained for the constant permeability case. Solutions obtained represent fields of flow

over a flat plate with blowing or suction.

119

Chapter 7

Permeability Variations in Laminar Flow through a Porous Medium

Behind a Two-dimensional Grid


The nonlinearity of the Navier-Stokes equations and the rare existence of their exact

solution are also inherent in the Darcy-Lapwood-Brinkman (DLB) equation that governs

the flow through porous media, or flow through mushy zones undergoing rapid freezing,

Hamdan [31]. The DLB equation contains the same convective inertial terms, viscous shear

terms and pressure gradient as the Navier-Stokes equation, in addition to a viscous damping

term that contains the permeability (cf. equation (7.2) in section 2, below). More

complications are added to the process of finding exact solutions of the DLB equation when

the permeability is non-constant. A variable permeability function adds one more unknown

to the governing equations without adding an additional equation to render the system of

governing equations determinate. This necessitates selecting a permeability function that,

together with the flow variables, satisfies the governing equations, Hamdan and Kamel

[32]. While in general finding a permeability function that satisfies the DLB equation might

be formidable, the problem is simplified by using linearization techniques that are popular

in finding exact solutions to the Navier-Stokes equations. An excellent review of the

popular methods is provided by Wang [84] and other methods have been introduced by

various authors (cf. Dorrepaal [24] Ranger [65] Hamdan [30] and the references therein).

120

An early method of solution was introduced close to a century ago by Taylor [80] who

observed that in two-dimensional flow, the nonlinear convective acceleration vanishes in

the Navier-Stokes equations when the vorticity is a function of the Stokes streamfunction.

Kovasznay [43] obtained a linearization of the Navier-Stokes equation by using Taylor’s

approach and taking vorticity proportional to the streamfunction, and Lin and Tobak [46]

extended Kovasznay’s linearization to obtain reverse flow over a flat plate with blowing

and suction.

In the study of flow through porous media, Hamdan and Ford [26] used Kovasznay’s

approach to study laminar flow behind a two-dimensional grid, and used the DLB equation

with constant permeability. Their work, [26] illustrated the effects of constant permeability

on the vorticity of the two-dimensional flow.

While flow through porous media with constant permeability has received considerable

attention in the literature, Hamdan [31] Abu Zaytoon et al [2] flow problems in natural and

industrial settings involve porous media with variable permeability. In addition, validity of

some models of flow through porous media hinges on considerations of variable

permeability in the models. In a recent article, Nield and Kuznetsov [58] demonstrated the

need for, and the application of a variable permeability porous layer (used in their work as

a transition layer) in analyzing flow over porous layers. Flow through the variable

permeability transition layer used by Nield and Kuznetsov [58] was governed by

Brinkman’s equation. The DLB equation is the same in structure as the Brinkman equation

except for the inertial terms, Alharbi et al [6]. These convective terms will be eliminated

by linearization in this work, where the problem of laminar flow through a variable-

121

permeability porous medium behind a two-dimensional grid is considered. This is the same

problem considered by Hamdan and Ford, [26] except that the current problem involves a

permeability function that must be determined so that the vorticity equation is satisfied. An

extension to the approach followed in Ford and Hamdan [26] will be used in the current

work, and derivations of equations that the permeability functions must satisfy are

obtained, then solved. Analysis carried out in this work might be of utility in stability

analysis of flow through porous media with variable permeability.


Consider the flow through a porous medium of the type where the Darcy-Lapwood-

Brinkman (DLB) equation is valid; that is, one in which viscous shear and macroscopic

inertia are important. The flow of an incompressible fluid through the medium is governed

by the equations of continuity and momentum, given by Alharbi, Alderson and Hamdan

[6]:

0 v

(7.1)

vk

vpvv

2*)( (7.2)

where v

is the velocity vector field, p is the pressure, is the fluid density, is the

viscosity of the base fluid, * is the effective viscosity of the fluid as it occupies the porous

medium, k is the permeability (considered here a scalar function of position), is the

122

gradient operator and 2 is the Laplacian. In the absence of definite information about the

relationship between * and , we will take * in this work.

Considering the flow in two space dimensions, x and y, we take ),( vuv

, ),( yxkk and

),( yxpp . The equation of continuity can be written as:

0 yx vu (7.3)

and momentum equations can be expressed in the x- and y-directions, respectively, as:

uk

uPvuuuu xyx

2

0 )]([ (7.4)

vk

vPvvvuu yyx

2

0 )]([ (7.5)

where 0u is the average velocity in the x-direction.

The governing equations (7.3), (7.4), and (7.5) are conveniently expressed in vorticity-

velocity form as follows. Letting be the vorticity of the flow, defined as:

yx uvv

(7.6)

and eliminating the pressure term from equations (7.4) and (7.5) by differentiating (7.4)

with respect to y and differentiating (7.5) with respect to x, then subtracting, gives:

22

2

0][k

ku

k

kv

kvuu

yxyx

(7.7)

where

is the kinematic viscosity.

Equations (7.3), (7.6) and (7.7) represent three equations to be solved for the unknowns

vu, and . The permeability distribution is yet to be determined. These equations can be

123

rendered dimensionless with respect to a characteristic length L that represents grid

spacing, and a characteristic velocity (the average velocity 0u ) by defining:

02

000 Re,/,/,/,/,/,/Lu

LkKLyYLxXuLuvVuuU (7.8)

where Re is Reynolds number. Equations (7.3), (7.6) and (7.7) thus take the following

dimensionless forms, respectively:

0 YX VU (7.9)

YX UV (7.10)

22

2 11ReRe]1[

KY

KU

KX

KV

KYV

XU

(7.11)

It is thus required to solve equations (7.9), (7.10) and (7.11) for the unknowns VU , and

, for a given dimensionless permeability distribution.


Hamdan and Ford [26] and the references therein discussed the following ways to linearize

the vorticity equation when permeability is constant:

(1) If inertial terms are small compared with viscous effects, then Reynolds number is small

and the vorticity equation is reduced to a linear equation.

(2) If changes in velocity are small compared with the average velocity, the quadratic terms

may be neglected.

124

Due to variations of permeability, as can be seen from equation (7.11), the second

linearization approach is followed in this Chapter. Assuming that the changes in velocity

are small compared with the average velocity, we neglect the quadratic terms and set:

0

YV

XU (7.12)

Equation (7.11) is then reduced to:

22

2 11Re

KY

KU

KX

KV

KX

(7.13)

Now, if the changes in velocity are not small, then we attempt to find solutions for which

the quadratic terms vanish, and (7.12) holds.

In order to facilitate the solution to equations (7.9), (7.10) and (7.13), we introduce the

dimensionless streamfunction of the flow, ),( YX , the existence of which is implied by

the dimensionless equation of continuity (7.9), and defined in terms of the dimensionless

velocity components by:

YU (7.14)

and

2 (7.15)

Using (7.14) and (7.15) in (7.10), the following streamfunction equation is obtained:

2 (7.16)

The equations to be solved are therefore the streamfunction equation (7.16) and the

vorticity equation (7.13). Velocity components can then be evaluated from (7.15) and

(7.16).

125

Assuming that the streamfunction is of the periodic form:

)2sin()( YXF (7.17)

where )(XF is to be determined, then the velocity components and vorticity take the

following forms, respectively:

)2cos()(2 YXFU Y (7.18)

)2sin()( YXFV X (7.19)

)2sin()]()(4[ 2 YXFXF (7.20)

Substituting (7.18), (7.19) and (7.20) in (7.12) yields:

0)4sin()]()()()([ YXFXFXFXF (7.21)

Equation (7.21) is satisfied when ,...3,2,1,0;4

nn

Y , or when:

0)()()()( XFXFXFXF (7.22)

Equation (7.22) can be written in the form:

)(

)(

)(

)(

XF

XF

XF

XF

(7.23)

which, upon integrating once and simplifying, gives:

)()( XFXFc (7.24)

where c is an arbitrary constant.

Equation (7.24) is satisfied by )(XF of the form:

XAeXF )( (7.25)

126

where A and are constants.

Using (7.25) in (7.17) gives:

)2sin( YAe X (7.26)

and using (7.26) in (7.18), (7.19) and (7.20) yields, respectively:

)2cos(2 YAeU X (7.27)

)2sin( YAeV X (7.28)

]4[)2sin(]4[ 2222 YAe X (7.29)

For the case of flow through porous media with constant permeability, Hamdan and Ford

[26] it was argued that equation (7.29) gives the vorticity in terms of the streamfunction,

by virtue of its definition, and is not based on solving the vorticity equation (equation (7.13)

with constant permeability). The choice of )(XF was based on the satisfaction of (7.12),

while the solution for vorticity must satisfy both (7.12) and (7.13). Hence, the form of

vorticity is determined by (7.29), namely:

)2sin()( YXG (7.30)

where )(XG is a function to be determined by substituting (7.30) in (7.13) when

permeability is constant.

Since in the current Chapter the permeability is a variable function of position, the above

process of solving (7.13) while implementing (7.30) is cumbersome. We modify the above

approach by taking (7.29) to represent the solution for vorticity and find the permeability

distribution that satisfies (7.13) when vorticity is given by (7.29).

127

In order to determine the permeability distribution that guarantees that the vorticity given

by (7.29) satisfies the vorticity equation (7.13), equations (7.27), (7.28) and (7.29) are

substituted in (7.13) to get:

222222

22 1

]4[

)2cot(21

]4[

1]4[Re

KY

KY

KX

K

K

(7.31)

Equation (7.31) is solved for the cases of )(XKK only, and )(YKK only.

Case 1: )(XKK

Equation (7.31) in this case reduces to

)1

(]4[

)1

(]4[Re22

22

KdX

d

K

(7.32)

which is a linear, first order ordinary differential equation in the unknownK

1. Integration

factor, I.F., is given by:

}4

exp{..22

XFI

(7.33)


}4

exp{4Re

122

1

22 XC

K

(7.34)

where 1C is an arbitrary constant.

Case 2: )(YKK

Equation (7.31) in this case reduces to

)1

(]4[

)2cot(2)

1(]4[Re

22

22

KdY

dY

K

(7.35)

128

which is a linear, first order ordinary differential equation in the unknown K

1. Integration

factor, I.F., is given by:

])2cos(ln4

4exp[..

2

22

YFI

(7.36)


])2cos(ln4

4exp[]4[Re

1

2

22

2

22 YC

K

(7.37)

where 2C is an arbitrary constant and ,...3,2,1;4

nn

Y .

7.4 Results and Discussion

7.4.1 Total Flow

Equations (7.26)-(7.28) represent the solutions for ,U , V and . The corresponding

quantities for the total flow are given by:

)2cos(211 YAeU X (7.38)

)2sin( YAeV X (7.39)

)2sin(* YAeY X (7.40)

)2sin(]4[ 22 YAe X (7.41)

129

where )( 0

* is the total streamfunction such that 1*

U

Y , and 0 is the

perturbing stream.

7.4.2 Stagnation Points

Stagnation points of the total flow occur where 0V and 01U . Locations of the

stagnation points are at ,...3,2,1,0);2

,0(),( nn

YX , as explained as follows. Taking

0V in (7.39), and noting that 0 and 0A , then 0)2sin( Y , 2

nY ; ...3,2,1,0n

. Taking 01U in (7.38) gives .)2cos(2

1

YeA

X

Assuming that stagnation occurs

at 0X when 2

nY , gives .

)cos(2

1

nA

Now, if the zero streamline passes through

the stagnation point, then 0* when 0Y (that is, when 0n ), then .2

1

A

Taking2

1A in (7.38)-(7.41), gives:

)2cos(11 YeU X (7.42)

)2sin(2

YeV X

(7.43)

)2sin(2

1* YeY X

(7.44)

)2sin(]2

2[2

Ye X

(7.45)

130

7.4.3 Determination of Constants

The value2

1A , which appears in the total flow quantities, was determined in the

previous subsection through consideration of stagnation points. It is the same value

obtained in Hamdan and Ford [26] for the case flow through a porous medium of constant

permeability since the locations of stagnation points are the same for the problem at hand

as for the cases of constant and variable permeability by virtue of the fact that the

streamfunction and velocity components in the total flow are the same.

In the assumption of the form of vorticity function, and in determination in the case of

constant permeability, the value of the constant was tied to the roots of the auxiliary

equation, Hamdan and Ford [26] of the governing ordinary differential equation for )(XG

of equation (7.30). For the current problem where permeability is non-constant, finding a

value for may not be as easy, however, there are some restrictions that can be stated as

follows:

(a) In order to have a non-zero vorticity in the flow field, equation (7.45) suggests that

2 .

(b) The permeability functions, equations (7.34) and (7.37), depend on Reynolds

number, and 1C (or 2C ). Their values must be chosen such that the permeability

at any point in the flow field is positive, hence the following must hold:

0}4

exp{4Re22

1

22

XC

(7.46)

131

0])2cos(ln4

4exp[]4[Re

2

22

2

22

YC

(7.47)

For instance, if 0Re then at the stagnation points ,...3,2,1,0);2

,0(),( nn

YX , we must

have:

22

1 4 C (7.48)

and

22

2 4 C . (7.49)

7.4.4 Comparison with the Case of Constant Permeability and Kovasnay’s

Solution

Kovasnay’s approach [43] for Navier-Stokes flow was followed both in this work and in

the work of Hamdan and Ford [26] for flow through constant permeability media.

Subsequently, the form of the streamfunction (hence the velocity components) is the same

in all three studies. Differences occur in the vorticity and permeability.

Kovasnay [43] obtained the following vorticity expression for the Navier-Stokes flow:

)2sin(2

ReYe

m mX

(7.50)

where

2

2

42

(Re)

2

Re m (7.51)

132

Hamdan and Ford [26] obtained the following vorticity expression for the DLB case with

constant permeability:

)2sin(]2

Re

2

1[ Ye

m

K

mX

(7.52)

where m is as given in (7.51). Equation (7.52) demonstrates the effect of the constant

permeability on the flow, and how it enters the vorticity equation.

This Chapter calculates the vorticity through its definition in terms of the streamfunction,

as given in (7.45), and takes into account the effects of the permeability function and

Reynolds number through equations (7.34) and (7.37), wherein Reynolds number enters

the definition of the permeability and influences the values of permeability without the

need to guess a suitable range of Re.

0}4

exp{4Re22

1

22

XC

(7.53)

0])2cos(ln4

4exp[]4[Re

2

22

2

22

YC

(7.54)

133

1C 2C Re Condition on

0 0 0 ),2()2,(

0 0 1

),2

1611()

2

1611,(

22

0 0 Re

),2

16ReRe()

2

16ReRe,(

2222

r r Re

),2

]4[4ReRe()

2

]4[4ReRe,(

222222

rr

1r 2r Re When K=K(X):

),2

]4[4ReRe()

2

]4[4ReRe,(

2

1

222

1

22

rr

When K=K(Y):

),2

]4[4ReRe()

2

]4[4ReRe,(

2

2

222

2

22

rr

Table 7.1: Ranges of Values of at the Stagnation Points

134

7.4.5 Graphical Representation of Solutions

In what follows we present the solutions for cases 1 and 2 graphically, using two- and

three-dimensional plots for permeability, streamfunction and vorticity over a sub-region of

the flow domain.

For Case 1:

Equation (7.34) can be shown in the following two-dimensional figures for the values of

2C = 1, 2, 5, 10 and 20, and Re = 0, 1, and 10. The values of used are in accordance with

Table 7. 2.

Re

2

]14[4ReRe10

222

0 114 2

1 22 )14(412

1

2

21

10 22 )14(42515

Table 7.2: Choice of used in Case 1.

For the range of Re tested, permeability changes are very small between stagnation points

(at Y=0.5 and Y=1) and a noticeable increase takes place as we get closer to the stagnation

point. The amount of increase in permeability is greater for greater values of 2C , as shown

in Figures 7.1-7.5. A typical three-dimensional graph of )(YK is given in Figure 7.6 which

shows the permeability distribution as we approach the stagnation point at Y=1.

135

Figure 7.1: )(YK , eq.(7.34), when 12 C


136



137


Figure 7.6: )(YK , eq.(7.34), when 12 C , 10Re

138

For Case 2:

Equation (7.34) is shown graphically in the following Figures 7.7 and 7.8. While small

changes in the permeability distribution )(XK take place as we increase Re, a noticeable

increase takes place as we get closer to the stagnation point at 0X . The choice of in

this case is determined using Table 7.3.

Re

2

]44[4ReRe10

222

0 144 2

1 22 )44(41

2

1

2

21

10 22 )44(42515

Table 7.3: Choice of used in Case 2.

139

Figure 7.7: )(XK , eq.(7.37), when 11 C

Figure 7.8: )(XK , eq.(7.37), when 11 C , 10Re

140

The streamfunction distribution is given by equation (7.44). Streamsurfaces are shown in

Figures 7.10-7.12 for different values of . The values of are selected here are for

illustration purposes and show regions of changes in streamfunction. Changes in result

in changes in the location of the maximum and minimum locations as we approach

stagnation points.

Figure 7.9: Streamsurface, eq.(7.44), when 1


141



142

Vorticity distribution, (7.45), is shown in Figures 7.13-7.16 for different values of . The

values of are selected here for illustration purposes and show the significant changes in

vorticity. Changes in result in changes in the location of the maximum and minimum

locations as we approach stagnation points.

Figure 7.13: Vorticity Distribution, eq.(7.45), when 1


143



144


The problem of flow through a variable-permeability porous structure behind a two-

dimensional grid was considered in this work in order to determine the permeability

functions under which the streamfunction is periodic in Y. A method of linearization bases

on Kovasznay’s approach has been followed in this work, and solutions have been found

with the assumptions that ,...3,2,1;4

nn

Y . and 2 . Conditions on the arbitrary

constants that are involved in the permeability functions have been stated in (7.46) and

(7.47). Stagnation points of the flow occur at ,...3,2,1,0);2

,0(),( nn

YX .

145

Chapter 8

Conclusions and Recommendations

In this dissertation, we provided investigations in the effects of variable viscosity and

variable permeability on the flow characteristics through porous structures. This was

accomplished in Chapter 2 by deriving a set of equations governing the flow of a

Newtonian fluid through a porous structure where macroscopic inertial effects, Darcy

effects and Brinkman’s viscous shear effects have been taken into account. The developed

model is a Darcy-Lapwood-Brinkman model with pressure-dependent viscosity.

Since the model was developed using intrinsic volume averaging, it naturally involved

Darcian effects with a viscous damping term that involves permeability to the fluid. As

compared to existing models in the literature, which replace the Darcy coefficient in the

Darcy term by a pressure function, the current model provides flexibility in studying the

effects of variable permeability. Microscopic inertial terms (Forchheimer’s effects) have

not been taking into account in the current model. It is therefore recommended that:

Recommendation 1: Future work should take into account other models that implicitly

account for permeability by replacing the Darcy coefficient by a pressure function and

compare and contrast their solutions with the model developed in this work.

Recommendation 2: Future work should model the Forchheimer effects when the flow

through the porous medium involves fluids with pressure-dependent viscosity.

In Chapter 3, we considered the two-dimensional flow of a fluid with pressure-dependent

viscosity through a porous medium with variable permeability, in general. The

permeability function was selected so that the governing equations are satisfied and the

146

arbitrary constants appearing in the pressure distribution can be determined. The variable

fluid viscosity was taken to be proportional to the pressure, and the effect of the constant

of proportionality on the pressure distribution was illustrated using pressure distribution

graphs. This work illustrated how to handle and solve the governing equations of flow

through variable permeability. The same methodology can be applied when other forms of

viscosity-pressure relations are employed. It is therefore recommended that:

Recommendation 3: Provide further analysis to this problem to facilitate the determination

of a large number of arbitrary constants and the generation of new flow patterns.

In Chapter 4, we formulated and solved the problem of coupled parallel flow through

two porous layers when the fluid viscosities are stratified. Appropriate interfacial

conditions have been stated, and exponential variations in viscosity have been

considered to produce the necessary high wall viscosities. Effects of permeability,

pressure gradient and other flow parameters have been discussed. This work initia tes

and sets the stage for future analysis of coupled parallel flow of fluids with pressure -

dependent variable viscosities. It is therefore recommended that:

Recommendation 4: Further analysis should be carried out on viscosity stratification in

coupled-parallel flow in order to generalize the forms of viscosity variations.

Recommendation 5: Further analysis should be carried out on viscosity stratification in

coupled-parallel flow with permeability variations in composite porous layers.

In Chapter 5, devising of a method to obtain the permeability distribution in a

variable permeability porous medium was proposed. To accomplish this, exact

solutions were obtained under the assumption of vorticity being a function of the

streamfunction of the flow. Expressions for the permeability, pressure, velocity

147

components, streamfunction and vorticity were successfully obtained. Three-

dimensional figures are provided to illustrate the distributions obtained. Further

detailed work is needed to consider a full range of permeability distributions, and to

relax the assumption of vorticity being a function of the streamfunction of the flow.

It is therefore recommended that:

Recommendation 6: Further analysis of two-dimensional flow through variable

permeability porous media be carried out in general in order to find conditions that

permeability distributions need to satisfy.

In Chapter 6, we have obtained three exact solutions to the Darcy-Lapwood-Brinkman

equation with variable permeability. Permeability has been defined as a function of one space

dimension, and vorticity is prescribed as a function of the streamfunction of the flow. Ranges

of parameters have been determined, and comparison is made with the solutions obtained for

the constant permeability case. Solutions obtained represent fields of flow over a flat plate

with blowing or suction. This work did not provide a general form for the permeability

function. It is therefore recommended that:

Recommendation 7: The problem discussed in Chapter 6 should be reworked with a two-

dimensional permeability function.

In Chapter 7, the problem of flow through a variable-permeability porous structure behind

a two-dimensional grid was considered in this work in order to determine the permeability

functions under which the streamfunction is periodic in one of the space variables. A method

of linearization has been followed in this work, and solutions have been found subject to

certain assumptions. Conditions on the arbitrary constants that are involved in the

permeability functions have been stated and locations of the stagnation points have been

identified. While the method of solution is sound, and results are useful, it is important to

148

expand this work to provide more robust permeability functions and to relax some of the

restrictive assumptions. It is therefore recommended that:

Recommendation 8: Solutions found in Chapter 7 should be expanded to include general

forms of the permeability function and to relax some of the assumptions imposed.

149

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Appendix

Appendix A. Streamsurface Figures

Streamsurfaces of Case1 are shown in Figures A.1-A.10 when and in Figures

A.11-A.20 when . Streamsurfaces of Case 2 are shown in Figures A.21-A.25

when and in Figures A.26-A.30 when . Streamsurfaces of Case 3 are

shown in Figures A.31-A.40 when and in Figures A.41-A.50 wen .

Figure A.1: Case 1 Streamsurface,

eq.(6.27), , ,

,


eq.(6.27), , ,

1Re

3Re

1Re

3Re

1Re

3Re

121 cc 1.1

1/1.1Re 1

121 cc 1.1 1/1.1Re

10

160


eq.(6.27), , , ,


eq.(6.27), , , ,


eq.(6.27), , , ,


eq.(6.27), , , ,


eq.(6.27), , , ,


eq.(6.27), , , ,

121 cc 2 1/2Re

1

121 cc 2 1/2Re

10

121 cc 5 1/5Re

1

121 cc 5 1/5Re

10

121 cc 10 1/10Re

1

121 cc 10 1/10Re

10

161


eq.(6.27), , , ,


eq.(6.27), , , ,


eq.(6.27), , ,

,


eq.(6.27), , ,

,


eq.(6.27), , , ,


eq.(6.27), , , ,

121 cc 20 1/20Re

1

121 cc 20 1/20Re

10

121 cc 1.1

3/1.1Re 1

121 cc 1.1

3/1.1Re 10

121 cc 2 3/2Re

1

121 cc 2 3/2Re

10

162


eq.(6.27), , , ,


eq.(6.27), , , ,


eq.(6.27), , 10 , 3/10Re ,

1


eq.(6.27), , ,

,


eq.(6.27), , ,

,


eq.(6.27), , ,

,

121 cc 5 3/5Re

1

121 cc 5 3/5Re

10

121 cc

121 cc 10

3/10Re 10

121 cc 20

3/20Re 1

121 cc 20

3/20Re 10

163


eq.(6.32), , ,


eq.(6.32), , ,


eq.(6.32), , ,


eq.(6.35), , ,


eq.(6.32), , ,


eq.(6.32), , ,

121 cc 1.1

1/1.1Re

121 cc 2 1/2Re

121 cc 5 1/5Re

121 cc 10 1/10Re

121 cc 20 1/20Re

121 cc 1.1

3/1.1Re

164


eq.(6.35), , ,


eq.(6.35), , ,


eq.(6.35), , ,


eq.(6.35), , 20 , 3/20Re


eq.(6.38), , ,

,


eq.(6.38), , ,

,

121 cc 2 3/2Re

121 cc 5 3/5Re

121 cc 10 3/10Re

121 cc

121 cc 1.1

1/1.1Re 1

121 cc 1.1

1/1.1Re 10

165


eq.(6.35), , , ,


eq.(6.35), , , ,


eq.(6.35), , , ,


eq.(6.35), , , ,


eq.(6.35), , , ,


eq.(6.35), , , ,

121 cc 2 1/2Re

1

121 cc 2 1/2Re

10

121 cc 5 1/5Re

1

121 cc 5 1/5Re

10

121 cc 10 1/10Re

1

121 cc 10 1/10Re

10

166


eq.(6.35), , , ,


eq.(6.35), , , ,


eq.(6.35), , ,

,


eq.(6.35), , ,

,


eq.(6.35), , , ,


eq.(6.35), , , ,

121 cc 20 1/20Re

1

121 cc 20 1/20Re

10

121 cc 1.1

3/1.1Re 1

121 cc 1.1

3/1.1Re 10

121 cc 2 3/2Re

1

121 cc 2 3/2Re

10

167


eq.(6.35), , , ,


eq.(6.35), , , ,


eq.(6.35), , ,

,


eq.(6.35), , ,

,


eq.(6.35), , ,

,


eq.(6.35), , ,

,

121 cc 5 3/5Re

1

121 cc 5 3/5Re

10

121 cc 10

3/10Re 1

121 cc 10

3/10Re 10

121 cc 20

3/20Re 1

121 cc 20

3/20Re 10

168

Appendix B. Vorticity Figures

Vorticity of Case 1 are shown in Figures A.51-A.60 when and in Figures A.61-

A.70 when . Vorticity of Case 2 are shown in Figures A.71-A.75 when

and in Figures A.76-A.80 when . Vorticity of Case 3 are shown in Figures A.81-

A.90 when and in Figures A.91-A.100 when .

Figure A.51: Case 1 Vorticity, eq.(6.30),

, , ,


, , ,


, , ,


, , ,

1Re

3Re

1Re

3Re

1Re

3Re

121 cc 1.1 1/1.1Re 1

121 cc 1.1 1/1.1Re 10

121 cc 2 1/2Re 1

121 cc 2 1/2Re 10

169


, , ,


, , ,


, , ,


, , ,


, , ,


, , ,

121 cc 5 1/5Re 1

121 cc 5 1/5Re 10

121 cc 10 1/10Re 1

121 cc 10 1/10Re 10

121 cc 20 1/20Re 1

121 cc 20 1/20Re 10

170


, , ,


, , ,


, , ,


, , ,


, , ,


, , ,

121 cc 1.1 3/1.1Re 1

121 cc 1.1 3/1.1Re 10

121 cc 2 3/2Re 1

121 cc 2 3/2Re 10

121 cc 5 3/5Re 1

121 cc 5 3/5Re 10

171


, , ,


, , ,


, , ,


, , ,


, ,


, ,

121 cc 10 3/10Re 1

121 cc 10 3/10Re 10

121 cc 20 3/20Re 1

121 cc 20 3/20Re 10

121 cc 1.1 1/1.1Re

121 cc 2 1/2Re

172


, ,


, ,


, ,


, ,


, ,


, ,

121 cc 5 1/5Re

121 cc 10 1/10Re

121 cc 20 1/20Re

121 cc 1.1 3/1.1Re

121 cc 2 3/2Re

121 cc 5 3/5Re

173


, ,


, ,


, , ,


, , ,


, , ,


, , ,

121 cc 10 3/10Re

121 cc 20 3/20Re

121 cc 1.1 1/1.1Re 1

121 cc 1.1 1/1.1Re 10

121 cc 2 1/2Re 1

121 cc 2 1/2Re 10

174


, , ,

Figure A.86: Case 3 Vorticity,eq.(6.41),

, , ,


, , ,


, , ,


, , ,


, , ,

121 cc 5 1/5Re 1

121 cc 5 1/5Re 10

121 cc 10 1/10Re 1

121 cc 10 1/10Re 10

121 cc 20 1/20Re 1

121 cc 20 1/20Re 10

175


, , ,


, , ,


, , ,


, , ,


, , ,


, , ,

121 cc 1.1 3/1.1Re 1

121 cc 1.1 3/1.1Re 10

121 cc 2 3/2Re 1

121 cc 2 3/2Re 10

121 cc 5 3/5Re 1

121 cc 5 3/5Re 10

176


, , ,


, , ,


, , ,

Figure A.100:Case 3 Vorticity,eq.(6.41),

, , ,

121 cc 10 3/10Re 1

121 cc 10 3/10Re 10

121 cc 20 3/20Re 1

121 cc 20 3/20Re 10

Curriculum Vitae

Sayer Obaid Bathi Alharbi is a Saudi citizen who was born in Saudi Arabia in 1985. After

completing his General Certificate of Education (GCE Grade 12) in 2004, he attend the

Qassim University where he completed a Bachelor of Science in Mathematics in 2007. He

enrolled at UNBs graduate studies programs and his completion of a Master of Science in

Mathematics in May 2012. This was followed by Ph.D Mathematics program at UNB in

Fredericton where he completed most of his course work and comprehensive examinations.

He transferred to the Saint John campus in summer of 2014, to complete his comprehensive

, research seminars and work on his dissertation in the area of Fluid Mechanics and flow

through porous media.

Universities attended:

1. University of New Brunswick (2010-2012), Master of Mathematics.

2. Qassim University (2004-2007), Bachelor of Science in Mathematics.

Publications:

[1] S. O. Alharbi and M. H. Hamdan, High-Order finite difference schemes for the first

derivative in von Mises coordinates, Journal of Applied Mathematics and Physics, Vol.

4 No.3, 2016, pp.524-545.

[2] S.O. Alharbi, T.L. Alderson and M.H. Hamdan, Analytic solutions to the Darcy-

Lapwood-Brinkman equation with variable, permeability, Int. Journal of Engineering

Research and Applications, Vol. 6 No. 4, (Part - 5), April 2016, pp.42-48.

[3] S.O. Alharbi, T.L. Alderson and M.H. Hamdan, Coupled parallel flow of fluids with

viscosity stratification through composite porous layers, IOSR Journal of Engineering,

Vol. 6 No. 5, May 2016, pp.32-41.

[4] S.O. Alharbi, T.L. Alderson and M.H. Hamdan, Flow of a fluid with pressure-

dependent viscosity through porous media, Advances in Theoretical and Applied

Mechanics, Vol. 9 No. 1, 2016, pp.1 – 9.

[5] S.O. Alharbi, T.L. Alderson and M.H. Hamdan, Exact solution of fluid flow through

porous media with variable permeability for a given vorticity distribution, Int. J. of

Enhanced Research in Science, Technology & Engineering, Vol. 5 No. 5, 2016, pp.262-

276.

[6] S.O. Alharbi, T.L. Alderson and M.H. Hamdan, Permeability variations in laminar

flow through a porous medium behind a two-dimensional grid, IOSR Journal of

Engineering, Vol. 6 No. 5, May 2016, pp.42-55.

[7] S.O. Alharbi, T.L. Alderson and M.H. Hamdan, Riabouchinsky flow of a pressure-

dependent viscosity fluid in porous media, Asian Journal of Applied Sciences, Vol. 4

No. 3, 2016, pp.637-651.

Conference Presentations:

1. Effects of Variable Viscosity and Variable Permeability on Fluid Flow Through

Porous Media, Inter-Campus Seminar Day, November 2016, Saint John, NB

2. Fourth-Order-Accurate Finite Difference Scheme with Non-uniform Grid in von

Mises Coordinates. CMS Summer Meeting, June 2015, Charlottetown, PEI

3. Non-Uniform Grid in Computational Fluid Dynamics, Inter-Campus Seminar Day,

April 2012, Saint John, NB

4. Riabouchinsky Flow of a Pressure-Dependent Viscosity Fluid in Porous Media,

Science Atlantic MSCS Conference, October 2016, Sydney, NS

effects of variable viscosity and variable permeability on ... · effects of variable viscosity and...

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