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• 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

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

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

This dissertation is accepted by the

THE UNIVERSITY OF NEW BRUNSWICK

October, 2016

• 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

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

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

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

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

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

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