## ABSTRACT

In this dissertation, the problems of Magnetohydrodynamic unsteady free convection flow

past vertical porous plates with suction and oscillating boundaries are studied. The linear

and nonlinear partial differential equations governing the flow problems and boundary

conditions were transformed into dimensionless form, and the perturbation techniques

applied in getting analytical solutions for the velocity, temperature, the skin friction

coefficient and Nusselt number. It was observed that an increase in the values of thermal

Grashof number, Eckert number and heat source increases velocity profile, while an

increase in Darcy term retards the velocity profile. An increase in heat source and Grashof

number, also increases the Heat transfer coefficient. The effects of various parameters on

the flow fields have been presented with the help of graphs and tables.

xii

## TABLE OF CONTENTS

CONTENT PAGE

DEDICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

CERTIFICATION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

TABLE OF CONTENTS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

LIST OF TABLES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

DIMENSIONLESS NUMBERS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

GREEK SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

NOMENCLATURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

ABSTRACT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

CHAPTER ONE

INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Background of the study . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Statement of the Problems . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Aim and Objectives of the Study . . . . . . . . . . . . . . . . . . . . 3

1.4 Significance of the Study . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.5 Definitions of Basic Concepts . . . . . . . . . . . . . . . . . . . . . . 4

1.6 Structure of the Dissertation . . . . . . . . . . . . . . . . . . . . . . . 5

CHAPTER TWO

LITERATURE REVIEW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Some Related Literature Review . . . . . . . . . . . . . . . . . . . . 6

CHAPTER THREE

METHODOLOGY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.2 Regular perturbation expansions . . . . . . . . . . . . . . . . . . . . 10

3.3 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.3.1 Magnetohydrodynamic Unsteady Free Convection Flow Past

Vertical Porous Plates with Heat Deposition . . . . . . . . . . 12

3.3.2 Magnetohydrodynamic Unsteady Free Convection Flow Past

an Infinite Vertical Porous Plates with Heat Deposition . . . . 17

3.3.3 Darcy Forchcheimer Magnetohydrodynamic Unsteady Free

Convection Flow Past Vertical Porous Plates with Heat Deposition

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

v

CHAPTER FOUR

RESULTS AND DISCUSSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.2 Magnetohydrodynamic Unsteady Free Convection Flow Past Vertical

Porous Plates with Heat Deposition . . . . . . . . . . . . . . . . . 50

4.2.1 Velocity field . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.2.2 Temperature field . . . . . . . . . . . . . . . . . . . . . . . . 54

4.2.3 Skin friction coefficient and Nusselt number . . . . . . . . . . 56

4.3 Magnetohydrodynamic Unsteady Free Convection Flow Past an Infinite

Vertical Porous Plates with Heat Deposition . . . . . . . . . . . 57

4.3.1 Velocity field . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.3.2 Temperature field . . . . . . . . . . . . . . . . . . . . . . . . 59

4.3.3 Skin friction coefficient and Heat transfer coefficient . . . . . 61

4.4 Darcy Forchcheimer Magnetohydrodynamic Unsteady Free Convection

Flow Past Vertical Porous Plates with Heat Deposition . . . . . . 61

4.4.1 Velocity field . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.4.2 Temperature field . . . . . . . . . . . . . . . . . . . . . . . . 65

4.4.3 Skin friction coefficient and Heat transfer coefficient . . . . . 67

CHAPTER FIVE

SUMMARY, CONCLUSION AND RECOMMENDATIONS . . . . . . . . . . . . . . . . . . 69

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.3.1 Magnetohydrodynamic Unsteady Free Convection Flow Past

Vertical Porous Plates with Heat Deposition . . . . . . . . . . 70

5.3.2 Magnetohydrodynamic Unsteady Free Convection Flow Past

an Infinite Vertical Porous Plates with Heat Deposition . . . . 70

5.3.3 Darcy Forchcheimer Magnetohydrodynamic Unsteady Free

Convection Flow Past Vertical Porous Plates with Heat Deposition

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.4 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.5 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.6 Limitations of the Study . . . . . . . . . . . . . . . . . . . . . . . . . 72

REFERENCES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

vi

## CHAPTER ONE

INTRODUCTION

1.1 Background of the study

As understanding of the natural world has grown, human civilization and communities

have consistently been established at locations that feature a viable source of fluid flowing.

Throughout history, people have continuously attempted to manipulate the natural fluid

flow, in order to effect an improvement in such areas as agricultural stability, living

environment, and transportation.

The Magnetohydrodynamic (MHD) channel flow, was first described theoretically by

Hartmann (1937), who considered plane Poiseuille flow with a transverse magnetic field.

Since then, the study of MHD has been an active area of research because of its geophysical

and astrophysical applications. Ahmed and Batin (2013), investigated the effects of

conduction-radiation and porosity of the porous medium on laminar convective heat transfer

flow of an incompressible, viscous, electrically conducting fluid over an impulsively started

vertical plate embedded in a porous medium in presence of transverse magnetic field.

Modern technologies have emerged, and we have become increasingly reliant on the

fundamental principles of fluid flow. Humanity has come to depend upon the development

and design of modern transport, such as cars, ships and air-crafts, which are rooted in an

essential understanding and knowledge of fluid flows and this knowledge area, is an integral

area for solving aerodynamic problems. The area also provides a plethora of engineering

problems concerning energy conservation and transmission. Time past methodological

engineering, and even biomedical studies, have proven the universally accepted tenant that

understanding fluid flow is critical to the development of applied knowledge.

The effect of radiation, chemical reaction and variable viscosity on hydromagnetic heat and

mass transfer in the presence of magnetic field are studied by Seddeek and Almushigeh

1

(2010). Ahmed et al. (2012), considered MHD mixed convection and mass transfer from an

infinite vertical porous plate with chemical reaction in presence of a heat source. Uwanta

and Isah (2012) studied the boundary layer fluid flow in a channel with heat source, soret

effects and slip condition.

1.2 Statement of the Problems

Fluid flow is still a growing area, due to its wide application in technology, engineering,

science and medicine. This wide interest makes it necessary to undertake research, where

the governing equations are a set of unsteady, coupled and invariably the equations are also

nonlinear, at the same time, many of the phenomena that fluid flow shows for oscillating

boundary conditions.

Balasubramanyam et al. (2010) analysed the combined effects of magnetic field and viscous

dissipation on convective heat and mass transfer flow through a porous medium in a vertical

channel in the presence of heat generating sources. However, they had neither considered

the effect of unsteady state nor coupled within the governing equation or presence of Darcy

term.

Therefore, there is need for further research to address the above mention problems. The

governing equations of an adapted model are modified. Three problems of heat transfer

were formulated, both are unsteady and coupled. The first problem is linear with fixed

plates, while the second problem is non linear with infinity boundary and the third problem

is non linear with fixed boundary. The problems considered are:

(i) Magnetohydrodynamic unsteady free convection flow past vertical porous plates

with heat deposition.

(ii) Magnetohydrodynamic unsteady free convection flow past an infinite vertical porous

plates with heat deposition.

2

(iii) Darcy Forcheimer Magnetohydrodynamic unsteady free convection flow past vertical

porous plates with heat deposition.

The governing equations are continuity equations, momentum equations and energy equations

which were solved using perturbation technique with the help of fitting boundaries.

The process of nondimensionalising the governing equations prior to the start of developing

the perturbation approximation will also be addressed.

1.3 Aim and Objectives of the Study

The main aim of this dissertation is to modify an existing work of Balasubramanyam

et al. (2010), by introducing conditions that do change with time, coupled with and in the

presence of modified Darcy term. With the following specific objectives of the study, to

solve:

(i) Magnetohydrodynamic unsteady free convection flow past vertical porous plates

with heat deposition.

(ii) Magnetohydrodynamic unsteady free convection flow past an infinite vertical porous

plates with heat deposition.

(iii) Darcy Forcheimer Magnetohydrodynamic unsteady free convection flow past vertical

porous plates with heat deposition.

1.4 Significance of the Study

This study, augments to the body of knowledge, based on the following rationales, which

were not taken care of in the work of Balasubramanyam et al. (2010):

(i) Unsteady state of the modified adapted work is to be considered.

(ii) The presence of coupled between momentum and energy equations of problems

(3.3.2) and (3.3.3) surface.

3

(iii) Inclusion of Darcy Forcheimer term in the problem formation.

1.5 Definitions of Basic Concepts

Eckert number (Ec): Is the ratio of kinetic energy to enthalpy change, Raisinghania

(2003).

Grashof number (Gr): It is the ratio of the product of the inertial force and the buoyant

force to the square of viscous force in the convection, Raisinghania (2003).

Hall effect: It is the production of a voltage difference across the electric conductor,

transverse to the electric current in the conductor and a magnetic field perpendicular to the

current, Raisinghania (2003).

Heat dissipation: It si that energy which is dissipated in a viscous liquid in motion of

account of the internal friction, Douglas and Gibilisco (2003).

Heat transfer: Is the transfer of heat energy from one body to another as a result of

temperature difference, Douglas and Gibilisco (2003).

Incompressible fluid: Is one that requires a large variation in pressure to produce some

appreciable variation in density, John et al. (2011).

Magnetohydrodynamic (MHD): Is an important branch in fluid dynamics, which is

concerned with the interaction of electrically conducting fluids and electromagnetic fluids,

Raisinghania (2003).

Nusselt number or Heat transfer coefficient (Nu): It is defined as ratio of convective

heat transfer to conductive heat transfer across the boundary, Raisinghania (2003).

Prandtl number (Pr): It is defined as the ratio of kinematic viscosity (v) to thermal

diffusivity (k) of a fluid, Raisinghania (2003).

Porous medium: Is a solid matrix containing holes either connected or non connected,

dispersed with in the medium in a regular or random manner provided such holes occur

4

frequently in the medium, Raisinghania (2003).

Porous parameter: Is defined as the ratio of Darcy resistance to viscous force, Borowski

and Borwein (2005).

Reynold’s number (Re): The number ensures dynamic similarity at corresponding points

near the boundaries where viscous effects are most important. Its reciprocal is called

Reynold’s number and is denoted by Re, John et al. (2011).

Skin friction: The dimensionless shear stress at the surface, John et al. (2011).

Steady flow: Is one in which the velocity, pressure and concentration may vary from point

to point but do not change with time, John et al. (2011).

Unsteady flow: Is one in which properties and conditions associated with the motion of

the fluid depend on the time so that the flow pattern varies with time, John et al. (2011).

Viscous dissipation: The heat generated by internal friction within the fluid element of

the fluid per unit time, John et al. (2011).

Viscosity: is the internal friction of a fluid which makes it resist flowing past a solid surface

or other layers of the fluid, John et al. (2011).

1.6 Structure of the Dissertation

The dissertation has been presented in five (5) chapters. Chapter one serves as the general

introduction to the research report. It provides background of the study, statement of the

problem, its significance, aim and objectives of the study as well as definitions of the basic

concept. Chapter two examines the related literature review, while Chapter three discusses

the methodology adapted for the research report, which includes model formation and

solutions to the mathematical models under study, Chapter four presents the results and

discussion to illustrate the flow characteristics for the velocity, temperature, skin friction

coefficient and Nusselt number. Finally Chapter five draws together the research project.

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