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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.
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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
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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
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(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.
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(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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