TABLE OF CONTENTS
Cover Page Fly Leaf i Tittle Page ii
Declaration ……………………………………………………………………………………………………………….. i
Certification ……………………………………………………………………………………………………………… ii
Dedication ……………………………………………………………………………………………………………….. iii
Acknowledgement ……………………………………………………………………………………………………. iv
Abstract ……………………………………………………………………………. Error! Bookmark not defined.
Table of Contents ……………………………………………………………………………………………………… v
List of Figures ………………………………………………………………………………………………………… viii
List of Tables …………………………………………………………………………………………………………… xi
List of Appendices …………………………………………………………………………………………………… xii
Nomenclature ………………………………………………………………………………………………………… xiii
Dimensionless Quantities …………………………………………………………………………………………. xv
CHAPTER ONE ……………………………………………………………………………………………………….. 1
GENERAL INTRODUCTION …………………………………………………………………………………… 1
1.1 Introduction ……………………………………………………………………………………………. 1
1.2 Statement of the Problem …………………………………………………………………………. 2
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 Governing Equations ………………………………………………………………………. 6
CHAPTER TWO………………………………………………………………………………………………………. 7
LITERATURE REVIEW ………………………………………………………………………………………….. 7
2.1 Introduction ……………………………………………………………………………………………. 7
2.2 Natural convection flow …………………………………………………………………………… 7
2.3 Suction and Injection ……………………………………………………………………………….. 9
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2.4 Variable viscosity ………………………………………………………………………………….. 10
CHAPTER THREE ………………………………………………………………………………………………… 12
MATHEMATICAL ANALYSES ……………………………………………………………………………… 12
3.1 Introduction ………………………………………………………………………………………….. 12
3.2 Mathematical formulation and Geometrical Description ………………………………. 12
3.2.1 Natural convection flow with viscous dissipation and variable viscosity effects ……………………………………………………………………………………………. 13
3.3 Natural convection heat transfer flow in a vertical porous channel with viscous dissipation and variable viscosity effects. …………………………………………………….. 14
3.3.1 Mathematical formulation (Problem II) ………………………………………………. 15
3.4 Non-dimensionalization …………………………………………………………………………. 16
3.4.1 Non-dimensionalization of problem I (3.2.1) ……………………………………….. 16
3.4.2 Non-dimensionalization of problem II (3.3.1) ………………………………………. 17
CHAPTER FOUR …………………………………………………………………………………………………… 18
SOLUTION OF THE PROBLEMS ………………………………………………………………………….. 18
4.1 Introduction ………………………………………………………………………………………….. 18
4.2 Uncoupling and Linearization of resulting differential equations …………………… 18
4.2.1 Outline of Homotopy perturbation Method ………………………………………….. 18
4.2.2 Convergence of the Homotopy Perturbation Method …………………………….. 20
4.3 Solution to Problem 3.2 ………………………………………………………………………….. 21
4.4 Phase and Amplitude of Periodic temperature and velocity of problem 3.2 ……… 23
4.5 Nusselt number (𝑵𝒖) and Skin friction (𝝉) for problem 3.2 ………………………….. 24
4.6 Solution of Problem 3.3 ………………………………………………………………………….. 24
4.7 Phase and Amplitude of Periodic temperature and velocity of problem 3.3 …….. 27
4.8 Nusselt number (𝑵𝒖) and Skin friction (𝝉) of problem 3.3……………………………. 27
4.9 Convergence of the Series Solutions …………………………………………………………. 28
CHAPTER FIVE …………………………………………………………………………………………………….. 30
RESULTS AND DISCUSSION ………………………………………………………………………………… 30
5.1 Introduction ………………………………………………………………………………………….. 30
5.2 Flow between two vertical parallel plates (3.2) …………………………………………… 30
5.3 Flow in vertical porous channel (3.3) ………………………………………………………… 43
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5.4 Validation of the Results ………………………………………………………………………… 56
5.4.1 Validation of Result of Problem I ……………………………………………………………. 56
5.4.2Validation of results of Problem II …………………………………………………………… 57
CHAPTER SIX……………………………………………………………………………………………………….. 58
SUMMARY, CONCLUSION AND RECOMMENDATION ……………………………………….. 58
6.1 Summary ……………………………………………………………………………………………… 58
6.2 Conclusion …………………………………………………………………………………………… 58
6.3 Recommendation ………………………………………………………………………………….. 59
REFERENCES ……………………………………………………………………………………………………….. 61
APPENDICES ………………………………………………………………………………………………………… 65
Appendix I …………………………………………………………………………………………………….. 65
Appendix II …………………………………………………………………………………………………… 67
Appendix III ………………………………………………………………………………………………….. 71
Appendix IV ………………………………………………………………………………………………….. 74
CHAPTER ONE
GENERAL INTRODUCTION
1.1 Introduction
As humans our entire living from the very beginning through to the end is surrounded and facilitated by fluids; the very beginning being in the amniotic fluid (a mixture of liquids and gases). Coupled with this is the fact that these fluids are responsive to variations in temperature (heating and cooling) which in turn affects the normal functioning of our intelligently designed body systems causing it to seek redress in various reactive ways to maintain normalcy. In the same manner, the unintelligent inventions of science and engineering such as machine parts, electricaland electronic components, nuclear reactors, etc, that either generate heat or are subjected to heating must be harnessed to seek redress if they must last long and serve their purpose efficiently without going berserk. This can be effectively achieved if we understand the behaviour of fluids and fluid properties to these variations. For this, natural convection has continued to be and is still of extreme applicability as a means of cooling these systems. Again, fluids influence our movement by all kinds of means whether on land, water or in the air as well as the movement of mass and heat around (circulation), into (injection) and out (suction) of living things and inventions. This has made the study of fluid dynamics a worthwhile endeavor that cannot be over-emphasized.
The study of natural convection under different physical phenomena has received the attention of many authors. Gebhart (1962) showed that the heating of fluids by viscous
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dissipation cannot be neglected for fluids with high Prandtl number when steady free convective flow is considered because it can greatly affect the flow characteristics. When fluids are sheared, viscous dissipation is induced by the friction between the different layers of particles of the fluid as they move against each other, generating heat in the process. It therefore follows that the rate of shear-induced heat generation within a fluid is responsive to variations in viscosity. Inspired by the separate works of Sahin (1999) and Tasnim and Mahmud (2009), Jha and Ajibade (2011) reported that viscosity is the most sensitive property of fluid to rising temperatures. For instance, 10% rise in the temperature of water results to as much as 140% reduction in viscosity, at least, Schlichting and Gersten(2000). Bar-Cohen and Roshenow (1984) studied the fully developed fluid flow between two periodically heated parallel plates to capture miniaturizations of electrical and electronic components. Nanda and Sharma (1963) separated the temperature and velocity into steady and periodic parts while studying the effects of sinusoidal variations in surface temperatures. Wang (1988) investigated the effect of Strouhal number on the development of boundary layers between two periodically heated parallel plates by separating the steady and unsteady parts and concluded that Strouhal number exerts an inverse effect on unsteady flow, but the effect tends to zero as Strouhal number tends to infinity.
1.2 Statement of the Problem
Jha and Ajibade (2011) investigated the effects of viscous dissipation on natural convection flow in a vertical channel formed by two infinite parallel plates and concluded that the viscous dissipation heating becomes stronger than those of the plates when the thermal
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diffusivity of the fluid is relatively small causing transfer of heat from fluid to plates instead, but their results are limited to very narrow range of values of Eckert Number(E). Again the viscous dissipation parameter, E, is limited in its role in the work of Jha and Ajibade (2011) as it was used as a perturbation parameter only, being an off-shoot of the bottle necks of the general perturbation method. Above all, as important a fluid property as viscosity is due to its sensitivity to variations in temperature as demonstrated in the independent works of Sahin (1999) and Tasnim and Mahmud (2009), its interaction with viscous dissipation was not taken into account in any of the works mentioned above. And as viscosity is a physical property of all fluids, the combined effects of variable viscosity and viscous dissipation will be of general relevance, hence the motivation. In this dissertation therefore, we seek to find the solution to both the energy and momentum fields of the working fluid to capture viscous dissipation and variable viscosity effects.
1.3 Aim and Objectives of the study
The aim of this work is to investigate the effects of variable viscosity andviscous dissipation on steady, fully developed natural convection flow of incompressible fluids between infinitevertical parallel plates. This aim will be achieved through the following objectives; which are to:
(i) study the effect of varying viscosity on both the velocity and temperature profiles for the fluids in a vertical channel.
(ii) examine the contribution of suction/injection to both the energy and momentum of the fluids.
(iii) investigate the general effects of viscous dissipation on the fluid flow pattern and temperature.
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1.4 Research Methodology
To attain the set objectives, once the new problem was identified, the governing equations were obtained with their boundary conditions and necessary modifications were made to capture the new situations and objectives and scale the limitations of the previous works. A comprehensive study of earlier works was carried out by means of literature review. The dimensional equations of energy and momentum for the new problem was first separated into steady and periodic parts. These dimensional steady and periodic equations alongside their boundary conditions were reduced to dimensionless forms by the introduction of non-dimensional quantities. The resulting non-dimensional partial differential equations, being strongly non-linear and coupledwere subjected to the „Homotopy Perturbation‟ method which reduced them to a system of linear ordinary differential equations which were solved by the undetermined coefficients method. The expressions for both temperature and velocity fields were obtained as well as the skin friction and rate of transfer of heat on the bounding plates for the two cases. The results obtained are simulated for different values of the governing parameters and presented in graphical forms. The determination of the effects of the governing parameters on the flow elements as well as in the validation of obtained results are done and conclusions drawn.
1.5 Organization of the Dissertation
This dissertation is divided into six chapters with references and appendixes. Chapter one has the general introduction while chapter two is composed of the literature review. Chapter three carries the method that was employed in solving the problems of the
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research. Chapter four contains the solutions of the two problems, chapter five has the graphical analysis of the solutions and discussions, while chapter six is made up of the summary, conclusion and recommendation which is followed by references and appendices.
1.6 Basic definitions
(1) Compressible and incompressible flow: A compressible flow is a flow in which the fluid density varies significantly within the flow field. While in an incompressible flow the density does not change.
(2) Free orNatural Convection: Free or Natural convection is the process when a temperature difference produces a density difference which results in mass movement.
(3) Symmetric heating: this is a type of heating of the boundary layers in which equal amount of constant heating is applied on boundary surfaces of the system.
(4) Dimensionless quantity: is a quantity without an associated physical dimension.
(5) Skin friction: Is a component of drag, the force resisting the motion of a solid body through a fluid or the motion of fluid by a solid surface. It arises from the friction of the fluid against the skin of the object that is moving through it. Skin friction arises from the interaction between the fluid and the skin of the body, and is directly related to the area of the surface of the body that is in contact with the fluid. Skin friction follows the drag equation and rises with the square of the velocity. Skin friction can be reduced by shaping the moving body so that smooth flow is possible.
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(6) Suction: Is a force that causes fluids to be drawn into an interior space or to adhere
to a surface because of the difference between internal and external pressure.
(7) Injection: Is a force that causes a fluid to be drawn out of an inner space or to
adhere to a surface because of the difference between the external and internal
pressures.
1.7 Basic Governing Equations
Continuity Equation:
DivV 0
1.7.1
for incompressible fluids, the continuity equation becomes
.V 0.V 0
1.7.2
Navier-Stoke‟s Equation:
2 1
.
q
V q F P V
t
1.7.3
Energy Equation:
2
Cp
T
V T k T q
t
1.7.4
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