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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

    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