ABSTRACT

This dissertationconsiders steady fully developed free and mixed convective flows of a viscous, incompressible, electrically conducting fluid in vertical concentric annuli, formed by two infinite and vertical concentric cylinders, filled with porous material having variable porosity in the presence of radial magnetic field. Unified exact solutions are derived by taking into account an isothermal or isoflux thermal boundary condition at the outer surface of inner cylinder. The solutions obtained are graphically represented and the effects of various controlling parameters such as Hartmann number, Darcy number, ratio of viscosity and the gap between the cylinders on the flow formation are discussed. The major results reveal that velocity of the fluid in the first problem is higher in case of isothermal heating of outer surface of inner cylinder compared to constant heat flux heating when the gap between cylinders is less or equal to radius of inner cylinders while reverse phenomena occur when the gap between cylinders is greater than radius of inner cylinder. In the second problem, velocity is approximately the same for both isothermal and isoflux thermal boundary conditions.
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TABLE OF CONTENTS

Title Page……………………………………………………………………….….….i
Declaration………………………………………………………………………..….ii
Certification…………………………………………………………………….…….iii
Dedication……………………………………………………………………………..iv
Acknowledgement…………………………………………………………….………v
Abstract………………………………………………………………….………….viii
Tables of Contents …………………………………………….…………………….ix
List of Figures……………………………………………….……………………….xii
List of Tables…………………………………………………….…………………xiv
List of Appendices……………………………………………………….…………..xv
Nomenclature and Greek Letters ………………………………………………….xvi
CHAPTER ONE………………………………………….…………………………1
GENERAL INTRODUCTION……………………………………………………….1
1.1Introduction……………………………………………………………….………1
1.2 Statement of the Problem……………………………………………….…………3
1.3 Aim and Objectives of the Study……………………………………….…………3
1.4 Research Methodology……………………………………………………………4
1.5 Organization of the Dissertation………………………………………………….4
1.6 Basic Definitions…………………………………………………………………5
1.7 Basic Hydrodynamic Equations………………………………………………..…6
CHAPTER TWO……………………………………………………………………8
LITERATURE REVIEW…………………………………………………………….8
2.1 Introduction……………………………………………………………………….8
2.2 Natural Convection……………………………………………………………….8
2.3 Magnetohydrodynamic (MHD)………………………………………………….10
2.4 Mixed Convection……………………………………………………………….12
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CHAPTER THREE…………………………………………………………..……14
MATHEMATICAL ANALYSIS AND SOLUTIONS………………………………14
3.1 Introduction………………………………………………………………….…14
3.2 Natural convection flow in vertical concentric annuli filled with porous material of variable porosity in presence of radial magnetic field………….….14
3.2.1 Mathematical Description……………………………………………………14
3.3 MHD Mixed Convection Flow in a Vertical Concentric Annuli Filled with Porous Material having Variable Porosity in Presence of Radial Magnetic Field…………………………………………………………………………….17
3.3.1 Mathematical Description……………………………………………….…..17
3.4 Non-Dimensionalization……………………………………………………….19
3.5 Transformation for Linear Second Order Differential Equations………………21
3.6 Solution to Problem 3.2 ………………………………..……………….……..22
3.6.1 Skin Friction of Problem 3.2………………………………………………..23
3.7 Solution to Problem 3.3…………………………………………………………24
3.7.1 Skin Friction of Problem 3.3………………………………………………..25
3.8 Validation of the Method……………………………………………….………26
CHAPTER FOUR…………………………………………………………..………27
RESULTS AND DISCUSSIONS……………………………………………………27
5.1 Introduction …………………………………………………………………..…27
5.2 Results and Discussion of problems 3.2 …………………………………………27
5.2 Results and Discussions of Problem 3.3…………………………………………35
CHAPTER FIVE……………………………………………………………………56
SUMMARY, CONCLUSION AND RECOMMENDATIONS……………………..56
5.1 Summary ………………….…………………………………………………..…56
5.2 Conclusion………………………………………………………………………56
5.3 Recommendation…………………………………………………………………58
REFERENCES ……………..………….………………………………………….59
APPENDICES………………………………………………………………………64

CHAPTER ONE

GENERAL INTRODUCTION
1.1 Introduction
To appreciate the importance of fluid dynamics in life demands little more than just a glance around us. In general, life as we know would not exist if there are no fluids and the behavior they exhibit. The water and air we respectively drink and breathe are fluids. In addition, our body fluids are mostly water based. Essential to our healthy living is the proper movement of these fluids within our bodies. In a more practical setting, like in our transportation systems, recreation, entertainment (sound from radio speakers for example) and our sleep (water beds), fluids greatly influence our comfort. It is clear to see from this that engineers need a clear knowledge of fluid behavior to handle many systems of their encounter. The study of electrically conducting fluid (e.g. liquid metals) called magnetohydrodynamics (MHD) flow is found in numerous pieces of literature. This is due to its many areas of industrial applications in geophysics and engineering to determine the desired convective flow in cases like the design of MHD power generators, cooling of nuclear reactors and changing of metals solidification processes. Natural convection flow in a vertical channel is vital in many transport processes both in nature and engineering applications. Examples of flow and heat transfer include natural circulation in geothermal reservoirs, porous insulators, solar power collectors, spreading of pollutants and so on. Similarly, combined natural and forced convection flow known as mixed convection flow in a vertical channel has many applications both in nature and engineering in many transport processes such as in cooling of electronic devices
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by fans, solar collectors exposed to wind currents, cooling of nuclear reactors during
emergency shutdown and heat exchangers placed in a low-velocity environment.
Mixed convection flow occurs either when the impact of external forces in natural
convection or buoyancy forces in forced convection is significant. The combination
of natural and forced convection is noticeable, especially when the forced flow
velocity is low and/or the temperature gradient is large. Convection flow through
porous media has also received much attention. The process is vital to many
engineering and scientific applications. These include; separation of oil from sand by
steam, porous heat exchangers, geological flows and movement of nutrient in
mammalian tissues. When these kinds of flows are slowed down, are modeled by the
use of Darcy‟s law
K
u L
p
 ‘
  (where these quantities are as defined in the
nomenclatures). This expression relates pressure gradient p spanning the length L of
the medium to fluid velocity u’ , averaged over a large scale. The study of transport
processes in annular geometry has also become an interesting area of research due to
its wide range of applications, such as in cooling of turbine rotors and high speed gas
bearings, electrical devices cooling system, heat exchangers and in drilling operation
of oil and gas wells. Several researchers such as Singh et al (1997), Singh and Singh
(2012) have investigated natural convection flows in vertical concentric annuli under
different phenomena.It is in this background that our choice of this study is
articulated.
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1.2 Statement of the problem
Singh et al (1997) studied natural convection in vertical concentric annuli in the
presence of a radial magnetic field a physical situation of heating or cooling the inner
cylinder either isothermally or at constant heat flux. This problem however, is
limited in application particularly in situations where the gap between the cylinders
is filled with a material of varying porosity and the fluid transport process is actuated
not only by the buoyancy forces but also by pressure forces. It is significant
therefore, to stretch the work of Singh et al (1997) to capture these situations. Hence,
this research endeavors to investigate the impact of variable porosity of the porous
material trapped between the cylinders. Also, the influences of channel gap,
magnetic field and pressure gradient as well as ratio of viscous forces on flow
formation in natural convection in a vertical concentric annuli filled with porous
material of variable porosity in the presence of a radial magnetic field are
investigated.
1.3 Aim and objectives of the study
The overall aim of this research is to study natural and mixed convection flows in a
vertical concentric annuli filled with porous material of varying porosity in presence
of a radial magnetic field.
The objectives to attain the set aim are to:
(i) examine the flow behaviour of the fluid with variable porosity (
ao D ) of the
porous material trapped between the cylinders
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(ii) examine the effect of the gap between the cylinders (  ) on the flow formation
(iii) investigate the effect of Hartmann number M on flow formation
(iv) investigate fluid flow behavior with the ratio of viscosity  
(v) consider the impact of pressure gradient Pon the fluid flow in the annuli.
1.4 Research methodology
To attain the above set objectives, a survey of the existing works on natural and
mixed convection flow in annuli under different phenomena is conducted and
stretched to capture newly identified physical situations where the previous works
are limited. The mathematical problems of “natural and mixed convection flow in a
vertical concentric annuli filled with porous material of variable porosity in the
presence of a radial magnetic field, which is nonlinear,were solved analytically using
method of undetermined coefficients after transformation to linearity. The skin
friction and mass flux are obtained in analytical form. Using a computer package,
MATLAB 7.5 (2007b), the results obtained were presented graphically and in
numerical values. The graphs and numerical values are analyzed in order to ascertain
the impact of each of the controlling parameters on the flow.
1.5 Organization of the dissertation
This dissertation is grouped into five chapters with references and appendices.
Chapter one comprises basically the general introduction of the dissertation.
Literature review and the methodology employed in the research together with the
solutions of the problemsare contained in chapters two and three respectively. While
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chapter four deals with the results and discussions. Chapter five presents summary
and conclusion. It is then, followed by references and appendices.
1.6 Basic Definitions
Annulus: is the area bounded by two concentric cylinders.
Boussinesq approximation: is the assumption that the flow is taken under
sufficiently small difference in temperature and density.
Convection: is the transfer of heat from one place to another by the movement of a
heated fluid.
Darcy Number: relative effect of permeability of a medium verses cross-sectional
area. It is denoted by 2 d
K
D o
a 
Dimensionless quantity: is a quantity without an associated physical dimension.
Fluid: is any substance that deforms continuously when subjected to a shear stress,
no matter how small.
Forced convection: is a mechanism or form of heat transport in which fluid motion
is generated by an external source (like suction device fan and pump).
Hartmann Number: is the ratio of electromagnetic force to the viscous force. It is
denoted by


H BL a 
Magnetic Field: is the magnetic effect of electric currents and magnetic material
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Mass flux: is the rate of mass flow per unit area.
MHD: implies Magnetohydrodynamics which is the study of the magnetic properties
of moving conductive fluids.
Mixed convection: is the type of heat transport caused by both natural and force
convections.
Natural convection: is a mechanism or form of heat transport in which fluid motion
is generated only by density differences in fluid cause by temperature gradients.
Porous: admitting the passage of fluid through pores of a solid surface.
Pressure gradient: is the change in atmospheric pressure per unit of horizontal
distance in the direction which it changes most rapidly.
Skin friction: is the shear stress that occurs between the fluid and the solid surface
of boundaries.
Viscosity: is a fluid property through which a fluid offers resistance to shear stresses.
1.7 Basic Hydrodynamic Equations
Continuity equation:
.   0


q
t


where, q is velocity of the flow.
If  is constant then, the equation reduces to
.q  0
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(ii) Momentum equation( Navier-Stokes Equation)
q q F p q
t
q 2 .      



 


  
Where p is the specific thermodynamic work and F an eternal field
(iii) Energy equation
    k T 
Dt
DT
C 2
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