ABSTRACT

The work reported in this thesis investigates analytically the hydrodynamic and thermal behaviour of fluid: for a fully developed natural convection flow in a vertical micro-porous-annulus (MPA), taking into account the velocity slip and temperature jump at the outer surface of the inner porous cylinder and inner surface of outer porous cylinder. A closed – form solution is presented for velocity, temperature, volume flow rate, skin-friction and rate of heat transfer expressed as a Nusselt number. The effect of various flow parameters on hydrodynamic and thermal behaviour such as suction/injection parameter, rarefaction parameter, fluid – wall interaction parameter and radius ratio are discussed with the aid of graphs. During the course of investigation, it is found that as suction/injection on the cylinder walls increases, the fluid velocity and temperature is enhanced. In addition, it is observed that wall surface curvature has a significant effect on flow and thermal characteristics, and also the effect of the velocity slip and temperature jump at the inner moving porous cylinder wall is much greater than its effects at the outer stationary one.

TABLE OF CONTENTS

Cover Page……………………………………………………………………………………………………………………….i
Fly Leaf…………………………………………………………………………………………………………………………..ii
Title Page……………………………………………………………………………………………………………………….iii
Declaration………………………………………………………………………………………..iv
Certification……………………………………………………………………………………….v
Acknowledgement………………………………………………………………………………..vi
Abstract…………………………………………………………………………………………..vii
Table of Contents………………………………………………………………………………..viii
List of Figures…………………………………………………………………………………….xi
List of Appendices………………………………………………………………………….……xv
Nomenclature…………………..…………………………………………………………………xvi
Basic definitions……………………………………………………………………………….xviii
Dimensionless Quantities…………………………………………………………………………xx
CHAPTER ONE
GENERAL INTRODUCTION
1.0 Introduction……………………………………………………………………………………1
1.1 The Objective of the Study……………………………………………………………………2
1.2 Research Methodology………………………………………………………………………..2
1.3 Organisation of the Research Work..………………………………………………………….2
CHAPTER TWO
LITERATURE REVIEW
Introduction………………………………………………………………………………………..4
viii
2.1 The Micro-Geometries of Micro-Porous Annulus………………………………………….5
2.2 The Effect of Suction/Injection on Micro-Porous –Annulus………………………………………..8
CHAPTER THREE
MATHEMATICAL ANALYSIS
3.0 Introduction………………………………………………………………………………11
3.1 The Micro-Geometries of Micro-Porous Annulus…….………………………………….11
3.2 The Effect of Suction/Injection on Micro-Porous –Annulus………………………………………15
CHAPTER FOUR
SOLUTIONS OF PROBLEMS
4.0 Introduction………………………………………………………………………………19
4.1 Solution of Problem 3.1……………….…………………………………………………19
4.2 Solution of Problem 3.2………………………………………………………………………..23
CHAPTER FIVE
RESULTS AND DISCUSSIONS
5.1 Discussing the Results of Problem 3.1..………………………………………………….26
5.2 Discussing the Results of Problem 3.2…………………………………………………………39
ix
CHAPTER SIX
SUMMARY AND CONCLUSIONS
6.1 SUMMARY………………………………………………………………………………62
6.2 CONCLUSIONS………………………………………………………………………………………………..64
REFERENCES……………………………………………………………………………………………………66
APPENDIX I………………………………………………………………………………………………………69
APPENDIX II……………………………………………………………………………………………………..70
APPENDIX III…………………………………………………………………………………………………….72
APPENDIX IV…………………………………………………………………………………………………….78
x

CHAPTER ONE

GENERAL INTRODUCTION
1.0 INTRODUCTION
The study of flow through microchannels has been an active area of research because of the potential applications of micro devices and other scientific areas. These devices would invariably involve fluid flow and therefore a clear understanding of microfluidic is imperative.
Fluid flow and heat transfer over a heated wall in a vertical microannulus have been extensively studied by many researchers. The research of fluid flow in microannulus has received great attention by several engineering work. The applications of these researches involve microelectromechanical systems and microfluidic systems such as in engineering, medical, biomedical use, computer chips, inkjet printer heads and gas chromatography.
Another area which needs to be studied is the combined effect of suction/injection and wall surface curvature in microchannels. So many investigators have studied the fully developed natural convection flow in vertical microchannels. Examples are: Chen and Weng (2005); Khadrawi et al (2005); Haddad et al (2005) and Avci and Aydin (2007).
The analytical hydrodynamic and thermal behaviour of fluid flow in micro-porous-annulus cylinders taking into account the velocity slip and temperature jump at the outer surface of inner porous cylinder and inner surface of outer porous cylinder and the behaviours of the Knudsen number are also studied in this work.
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1.1 Aim and Objectives of the Study
The aim of this work is to investigate the Flow Formation in Micro-Porous-Annulus. To achieve this aim, the following objectives were considered;
(i) Investigate the effect of the annular geometries of micro-porous annulus.
(ii) Investigate the effect of suction/injection on micro-porous –annulus.
1.2 Research Methodology
The methodology used in this research work is as follows: A review of some existing literatures by introducing or adding new parameters and boundary conditions to the existing equations to obtain new equations. The mathematical model or the equations obtained are solved to obtain the analytical solutions by the use of Laplace transformation method of solving differential equation. The numerical values of the analytical solutions of the developed equations are obtained by computer package (MATLAB).The numerical values obtained by the use of MATLAB are represented graphically. The last part is the representation and discussion of the graphs and the governing parameters.
1.3 Organization of the Research Work
In order to achieve the objective of the work, the work was divided into chapters:
Chapter one introduces the work, objectives of the study and methodology of the research work. While chapter two contains the general review of related literatures.
The third chapter contains the mathematical analysis of the governing equations of the problem with their non-dimensionalisation.
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While chapter four contains the solutions of the governing equations of the problems.
The results, discussion of the graphs obtained by the use of MATLAB are contained in chapter five.
Finally, the summary and conclusion of the thesis are presented in chapter six.
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