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
5.10: Pressure Distribution, 4 , 3)(1 C , 1n , 6/ , 1)(2 C , 2)(4 C
........................................................................................................................................... 81
5.11: Pressure Distribution, 1 , 4)(1 C , 5.0n , 6/ , ,1)(2 C 2)(4 C
........................................................................................................................................... 81
5.12: Pressure Distribution, 10 , 1)(1 C , 1n , 4/ , 1)(2 C , 2)(4 C
........................................................................................................................................... 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
6.4: Velocity )(yu when
3Re , 1,1,1 21 cc .................................................. 112
6.5: Velocity )(xv when
1Re , 5.021 cc ............................................................. 113
6.6: Velocity )(xv when
3Re , 5.021 cc ............................................................. 113
6.7: Velocity )(yu when
1Re , 12 c ....................................................................... 114
6.8: Velocity )(yu when
3Re , 12 c ...................................................................... 114
6.9: Velocity )(xv when
1Re , 5.021 cc ............................................................ 115
6.10: Velocity )(xv when
3Re , 5.021 cc .......................................................... 115
6.11: Velocity )(yu when
1Re , 1,1 21 cc , 1 .............................................. 116
6.12: Velocity )(yu when
3Re , 1,1 21 cc , 1 ............................................... 116
xiii
6.13: Velocity )(xv when
1Re , 1,1 21 cc ......................................................... 117
6.14: Velocity )(xv when
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
(2.23)
Using (2.23) in (2.21) we obtain:
qqqt
P
T
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
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
3.1 Chapter Introduction
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.
3.2 Governing Equations
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)
The flow quantities of streamfunction, vorticity and velocity components are determined
from equations (3.17), (3.21), (3.22) and (3.23) as:
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)
The flow quantities of streamfunction, vorticity and velocity components are determined
from equations (3.17), (3.21), (3.22) and (3.23) as:
)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
Figure 3.6: Permeability function, eq.(3.61), for 1,22 aAC
42
Figure 3.7: Permeability function, eq.(3.61), for 1,52 aAC
Figure 3.8: Permeability function, eq.(3.61), for 1,102 aAC
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
3.5 Chapter Conclusion
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
4.1 Chapter Introduction
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
4.5 Chapter Conclusion
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
5.1 Chapter Introduction
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.
5.2 Governing Equations
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
Solution to (5.37) takes the form
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
Figure 5.2: Case 1 Permeability Distribution,
eq.(5.58), 1 , 1)(1 D , 1n , 6/ , 1)(2 C
Figure 5.3: Case 1 Permeability Distribution,
eq.(5.58), 1 , 1)(1 D , 1n , 3/ , 1)(2 C
79
Figure 5.4: Case 1 Permeability Distribution,
eq.(5.58), 1 , 1)(1 D , 1n , 6/ , 1)(2 C
Figure 5.5: Case 1 Permeability Distribution,
eq.(5.58), 1 , 4)(1 D , 5.0n , 6/ , 1)(2 C
Figure 5.6: Case 1 Permeability Distribution,
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
Figure 5.8: Case 1 Pressure Distribution,
eq.(5.74), 1 , 1)(1 C , 2n , 6/ , 1)(2 C , 2)(4 C
Figure 5.9: Case 1 Pressure Distribution,
eq.(5.74), 2 , 7)(1 C , 1n , 3/ , 1)(2 C , 2)(4 C
81
Figure 5.10: Case 1 Pressure Distribution,
eq.(5.74), 4 , 3)(1 C , 1n , 6/ , 1)(2 C , 2)(4 C
Figure 5.11: Case 1 Pressure Distribution,
eq.(5.74), 1 , 4)(1 C , 5.0n , 6/ , 1)(2 C , 2)(4 C
Figure 5.12: Case 1 Pressure Distribution,
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
Figure 5.14: Case 1 Streamsurfaces,
eq.(5.40), 1 , 1)(1 C , 1n , 6/ , 1)(2 C
Figure 5.15: Case 1 Streamsurfaces,
eq.(5.40), 1 , 7)(1 C , 1n , 3/ , 1)(2 C
83
Figure 5.16: Case 1 Streamsurfaces,
eq.(5.40), 1 , 3)(1 C , 1n , 6/ , 1)(2 C
Figure 5.17: Case 1 Streamsurfaces,
eq.(5.40), 1 , 4)(1 C , 5.0n , 6/ , 1)(2 C
Figure 5.18: Case 1 Streamsurfaces,
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
Figure 5.20: Case 1 Vorticity Distribution,
eq.(5.43), 1 , 1)(1 C , 1n , 6/ , 1)(2 C
Figure 5.21: Case 1 Vorticity Distribution,
eq.(5.43), 1 , 7)(1 C , 1n , 3/ , 1)(2 C
85
Figure 5.22: Case 1 Vorticity Distribution,
eq.(5.43), 1 , 3)(1 C , 1n , 6/ , 1)(2 C
Figure 5.23: Case 1 Vorticity Distribution,
eq.(5.43), 1 , 4)(1 C , 5.0n , 6/ , 1)(2 C
Figure 5.24: Case 1 Vorticity Distribution,
eq.(5.43), 10 , 1)(1 C , 1n , 4/ , 1)(2 C
86
Figure 5.25: Case 2 Permeability Distribution,
eq.(5.64), 3 , 5.0)(1 B , 5.0)(2 B , 1m , 36/7 , 1)(2 D
Figure 5.26: Case 2 Permeability Distribution,
eq.(5.64), 5 , 1)(1 B , 1)(2 B , 9.0m , 3/ , 1)(2 D
Figure 5.27: Case 2 Permeability Distribution,
eq.(5.4), 4 , 5.0)(1 B , 5.0)(2 B , 1m , 6/ , 1)(2 D
87
Figure 5.28: Case 2 Permeability Distribution,
eq.(5.64), 1 , 5.0)(1 B , 5.0)(2 B , 6.0m , 4/ , 1)(2 D
Figure 5.29: Case 2 Permeability Distribution,
eq.(5.64), 1 , 5.0)(1 B , 5.0)(2 B , 1m , 3/ , 1)(2 D
Figure 5.30: Case 2 Permeability Distribution,
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
Figure 5.32: Case 2 Pressure,
eq.(5.76), 5 , 1)(1 B , 1)(2 B , 9.0m , 3/ , 1)(4 C
Figure 5.33: Case 2 Pressure,
eq.(5.76), 4 , 5.0)(1 B , 5.0)(2 B , 1m , 6/ , 1)(4 C
89
Figure 5.34: Case 2 Pressure,
eq.(5.76), 1 , 5.0)(1 B , 5.0)(2 B , 6.0m , 4/ , 1)(4 C
Figure 5.35: Case 2 Pressure,
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
Figure 5.37: Case 2 Streamsurfaces,
eq.(5.45), 3 , 5.0)(1 B , 5.0)(2 B , 1m , 36/7
Figure 5.38: Case 2 Streamsurfaces,
eq.(5.45), 5 , 1)(1 B , 1)(2 B , 9.0m , 3/
Figure 5.39: Case 2 Streamsurfaces,
eq.(5.45),, 4 , 5.0)(1 B , 5.0)(2 B , 1m , 6/
91
Figure 5.40: Case 2 Streamsurfaces,
eq.(5.45), 1 , 5.0)(1 B , 5.0)(2 B , 6.0m , 4/
Figure 5.41: Case 2 Streamsurfaces,
eq.(5.45), 1 , 5.0)(1 B , 5.0)(2 B , 1m , 3/
Figure 5.42: Case 2 Streamsurfaces,
eq.(5.45), 5.0 , 5.0)(1 B , 5.0)(2 B , 1m , 3/
92
Figure 5.43: Case 2 Vorticity Distribution,
eq.(5.48), 3 , 5.0)(1 B , 5.0)(2 B , 1m , 36/7
Figure 5.44: Case 2 Vorticity Distribution,
eq.(5.48), 5 , 1)(1 B , 1)(2 B , 9.0m , 3/
Figure 5.45: Case 2 Vorticity Distribution,
eq.(5.48), 5 , 1)(1 B , 1)(2 B , 9.0m , 3/
93
Figure 5.46: Case 2 Vorticity Distribution,
eq.(5.48), 1 , 5.0)(1 B , 5.0)(2 B , 6.0m , 4/
Figure 5.47: Case 2 Vorticity Distribution,
eq.(5.48), 1 , 5.0)(1 B , 5.0)(2 B , 1m , 3/
Figure 5.48: Case 2 Vorticity Distribution,
eq.(5.48), 5.0 , 5.0)(1 B , 5.0)(2 B , 1m , 3/
94
5.4 Chapter Conclusion
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
6.1 Chapter Introduction
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.
6.2 Governing Equations
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)
6.3 Solution Methodology
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
Solution to (6.23) takes the form
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)
with velocity components given by
]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)
and vorticity takes the form
]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)
with velocity components given by
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)
and vorticity takes the form
]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
Figure 6.6: Case 1 Velocity )(xv , eq.(6.29), when
3Re , 5.021 cc
114
Figure 6.7: Case 2 Velocity )(yu , eq.(6.33), when
1Re , 12 c
Figure 6.8: Case 2 Velocity )(yu , eq.(6.33), when
3Re , 12 c
115
Figure 6.9: Case 2 Velocity )(xv , eq.(6.34), when
1Re , 5.021 cc
Figure 6.10: Case 2 Velocity )(xv , eq.(6.34), when
3Re , 5.021 cc
116
Figure 6.11: Case 3 Velocity )(yu , eq.(6.39), when
1Re , 1,1 21 cc , 1
Figure 6.12: Case 3 Velocity )(yu , eq.(6.39), when
3Re , 1,1 21 cc , 1
117
Figure 6.13: Case 3 Velocity )(xv , eq.(6.40), when
1Re , 1,1 21 cc
Figure 6.14: Case 3 Velocity )(xv , eq.(6.40), when
3Re , 1,1 21 cc
118
6.5 Chapter Conclusion
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
7.1 Chapter Introduction
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.
7.2 Governing Equations
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.
7.3 Solution Methodology
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)
and solution given by
}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)
and solution given by
])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
Figure 7.2: )(YK , eq.(7.34), when 22 C
136
Figure 7.3: )(YK , eq.(7.34), when 52 C
Figure 7.4: )(YK , eq.(7.34), when 102 C
137
Figure 7.5: )(YK , eq.(7.34), when 202 C
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
Figure 7.10: Streamsurface, eq.(7.44), when 2
141
Figure 7.11: Streamsurface, eq.(7.44), when 5
Figure 7.12: Streamsurface, eq.(7.44), when 10
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
Figure 7.14: Vorticity Distribution, eq.(7.45), when 2
143
Figure 7.15: Vorticity Distribution, eq.(7.45), when 5
Figure 7.16: Vorticity Distribution, eq.(7.45), when 10
144
7.5 Chapter Conclusion
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), , ,
,
Figure A.2: Case 1 Streamsurface,
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
Figure A.3: Case 1 Streamsurface,
eq.(6.27), , , ,
Figure A.4: Case 1 Streamsurface,
eq.(6.27), , , ,
Figure A.5: Case 1 Streamsurface,
eq.(6.27), , , ,
Figure A.6: Case 1 Streamsurface,
eq.(6.27), , , ,
Figure A.7: Case 1 Streamsurface,
eq.(6.27), , , ,
Figure A.8: Case 1 Streamsurface,
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
Figure A.9: Case 1 Streamsurface,
eq.(6.27), , , ,
Figure A.10: Case 1 Streamsurface,
eq.(6.27), , , ,
Figure A.11: Case 1 Streamsurface,
eq.(6.27), , ,
,
Figure A.12: Case 1 Streamsurface,
eq.(6.27), , ,
,
Figure A.13: Case 1 Streamsurface,
eq.(6.27), , , ,
Figure A.14: Case 1 Streamsurface,
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
Figure A.15: Case 1 Streamsurface,
eq.(6.27), , , ,
Figure A.16: Case 1 Streamsurface,
eq.(6.27), , , ,
Figure A.17: Case 1 Streamsurface,
eq.(6.27), , 10 , 3/10Re ,
1
Figure A.18: Case 1 Streamsurface,
eq.(6.27), , ,
,
Figure A.19: Case 1 Streamsurface,
eq.(6.27), , ,
,
Figure A.20: Case 1 Streamsurface,
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
Figure A.21: Case 2 Streamsurface,
eq.(6.32), , ,
Figure A.22: Case 2 Streamsurface,
eq.(6.32), , ,
Figure A.23: Case 2 Streamsurface,
eq.(6.32), , ,
Figure A.24: Case 2 Streamsurface,
eq.(6.35), , ,
Figure A.25: Case 2 Streamsurface,
eq.(6.32), , ,
Figure A.26: Case 2 Streamsurface,
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
Figure A.27: Case 2 Streamsurface,
eq.(6.35), , ,
Figure A.28: Case 2 Streamsurface,
eq.(6.35), , ,
Figure A.29: Case 2 Streamsurface,
eq.(6.35), , ,
Figure A.30: Case 2 Streamsurface,
eq.(6.35), , 20 , 3/20Re
Figure A.31: Case 3 Streamsurface,
eq.(6.38), , ,
,
Figure A.32: Case 3 Streamsurface,
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
Figure A.33: Case 3 Streamsurface,
eq.(6.35), , , ,
Figure A.34: Case 3 Streamsurface,
eq.(6.35), , , ,
Figure A.35: Case 3 Streamsurface,
eq.(6.35), , , ,
Figure A.36: Case 3 Streamsurface,
eq.(6.35), , , ,
Figure A.37: Case 3 Streamsurface,
eq.(6.35), , , ,
Figure A.38: Case 3 Streamsurface,
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
Figure A.39: Case 3 Streamsurface,
eq.(6.35), , , ,
Figure A.40: Case 3 Streamsurface,
eq.(6.35), , , ,
Figure A.41: Case 3 Streamsurface,
eq.(6.35), , ,
,
Figure A.42: Case 3 Streamsurface,
eq.(6.35), , ,
,
Figure A.43: Case 3 Streamsurface,
eq.(6.35), , , ,
Figure A.44: Case 3 Streamsurface,
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
Figure A.45: Case 3 Streamsurface,
eq.(6.35), , , ,
Figure A.46: Case 3 Streamsurface,
eq.(6.35), , , ,
Figure A.47: Case 3 Streamsurface,
eq.(6.35), , ,
,
Figure A.48: Case 3 Streamsurface,
eq.(6.35), , ,
,
Figure A.49: Case 3 Streamsurface,
eq.(6.35), , ,
,
Figure A.50: Case 3 Streamsurface,
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),
, , ,
Figure A.52: Case 1 Vorticity, eq.(6.30),
, , ,
Figure A.53: Case 1 Vorticity, eq.(6.30),
, , ,
Figure A.54: 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
Figure A.55: Case 1 Vorticity, eq.(6.30),
, , ,
Figure A.56: Case 1 Vorticity, eq.(6.30),
, , ,
Figure A.57: Case 1 Vorticity, eq.(6.30),
, , ,
Figure A.58: Case 1 Vorticity, eq.(6.30),
, , ,
Figure A.59: Case 1 Vorticity, eq.(6.30),
, , ,
Figure A.60: Case 1 Vorticity, eq.(6.30),
, , ,
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
Figure A.61: Case 1 Vorticity, eq.(6.30),
, , ,
Figure A.62: Case 1 Vorticity, eq.(6.30),
, , ,
Figure A.63: Case 1 Vorticity, eq.(6.30),
, , ,
Figure A.64: Case 1 Vorticity, eq.(6.30),
, , ,
Figure A.65: Case 1 Vorticity, eq.(6.30),
, , ,
Figure A.66: Case 1 Vorticity, eq.(6.30),
, , ,
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
Figure A.67: Case 1 Vorticity, eq.(6.30),
, , ,
Figure A.68: Case 1 Vorticity, eq.(6.30),
, , ,
Figure A.69: Case 1 Vorticity, eq.(6.30),
, , ,
Figure A.70: Case 1 Vorticity, eq.(6.30),
, , ,
Figure A.71: Case 2 Vorticity, eq.(6.35),
, ,
Figure A.72: Case 2 Vorticity, eq.(6.35),
, ,
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
Figure A.73: Case 2 Vorticity, eq.(6.35),
, ,
Figure A.74: Case 2 Vorticity, eq.(6.35),
, ,
Figure A.75: Case 2 Vorticity, eq.(6.35),
, ,
Figure A.76: Case 2 Vorticity, eq.(6.35),
, ,
Figure A.77: Case 2 Vorticity, eq.(6.35),
, ,
Figure A.78: Case 2 Vorticity, eq.(6.35),
, ,
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
Figure A.79: Case 2 Vorticity, eq.(6.35),
, ,
Figure A.80: Case 2 Vorticity, eq.(6.35),
, ,
Figure A.81: Case 3 Vorticity, eq.(6.41),
, , ,
Figure A.82: Case 3 Vorticity, eq.(6.41),
, , ,
Figure A.83: Case 3 Vorticity, eq.(6.41),
, , ,
Figure A.84: Case 3 Vorticity, eq.(6.41),
, , ,
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.85: Case 3 Vorticity, eq.(6.41),
, , ,
Figure A.86: Case 3 Vorticity,eq.(6.41),
, , ,
Figure A.87: Case 3 Vorticity,eq.(6.41),
, , ,
Figure A.88: Case 3 Vorticity,eq.(6.41),
, , ,
Figure A.89: Case 3 Vorticity,eq.(6.41),
, , ,
Figure A.90: 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
Figure A.91: Case 3 Vorticity,eq.(6.41),
, , ,
Figure A.92: Case 3 Vorticity,eq.(6.41),
, , ,
Figure A.93: Case 3 Vorticity,eq.(6.41),
, , ,
Figure A.94: Case 3 Vorticity,eq.(6.41),
, , ,
Figure A.95: Case 3 Vorticity,eq.(6.41),
, , ,
Figure A.96: Case 3 Vorticity,eq.(6.41),
, , ,
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.97: Case 3 Vorticity,eq.(6.41),
, , ,
Figure A.98: Case 3 Vorticity,eq.(6.41),
, , ,
Figure A.99: Case 3 Vorticity,eq.(6.41),
, , ,
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