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# A Study on MHD Free-Convective Flow in Micro-Channels

**A Study on MHD Free-Convective Flow in Micro-Channels**

**ABSTRACT**

This work analyses the steady natural convection flow of viscous, incompressible, electrically conducting fluid in a vertical parallel plate micro-channels with combined effects of transverse magnetic field and suction/injection in the presence of velocity slip and temperature jump at the micro-channel surfaces. The fully developed solutions of the velocity, temperature, volume flow rate, skin-friction and rate of heat transfer which is expressed as a Nusselt number are derived analytically. The solution obtained for the velocity has been used to compute the skin friction, while the temperature has been used to compute the Nusselt number. The effect of the various flow parameters such as suction/injection parameter, Hartmann number, rarefaction parameter, and fluid-wall interaction parameter are discussed with the aid of line graphs. During the course of numerical computations, results show that as suction/injection, rarefaction and fluid-wall interaction increase, the volume flow rate increases while it decreases with increase in Hartmann number. The implication of this on the flow is that the gas velocity near the wall will decrease.

**TABLE OF CONTENTS**

Cover Page………………….…………………….……………………………………………i

Fly Leaf………………………………….……………………………………………………..ii

Title Page…………………………………………………………………………………………………………………iii

Declaration…………………………….…………………………………………….………..iv

Certification…………………………..……………………………………….………………..v

Dedication…………………………..………………………………….……………………..vi

Acknowledgement………………..…………………………………………………………..vii

Abstract…………………………..…………………………………………………………..viii

Table of Contents………………………………………………………………………………ix

List of Figures………………………………..…………………………………………………x

List of Appendices……….………………….…………………………………………………xi

Notations……………………………………..………………………………………………xiii

Basic Definitions……………………………..……………………………………………….xv

Dimensionless Quantities……………………………………………………………………,xvi

CHAPTER ONE

GENERAL INTRODUCTION

1.0 Introduction …………………………………………………………………………………………………… i

1.1 Aim and Objectives of Study …………………………………………………………………………… 2

1.2 Research Methodology …………………………………………………………………………………… 2

1.3 Organization of the Thesis ………………………………………………………………………………. 3

CHAPTER TWO

LITERATURE REVIEW

2.0 Introduction ………………………………………………………………………………………………….. 4

2.1 MHD Natural Convection Flow in Vertical Parallel Plate Micro-channels ………………. 6

2.2 Suction/Injection on MHD Natural Convection in Vertical Parallel Plate Micro-channels ……………………………………………………………………………………………………………. 8

CHAPTER THREE

MATHEMATICAL ANALYSIS AND METHOD SOLUTIONS

3.0 Introduction …………………………………………………………………………………………………. xi

3.1 MHD Natural Convection Flow in Vertical Parallel Plate Micro-channels …………….. 11

3.1.1 Mathematical Analysis ……………………………………………………………………………. 11

3.2 Suction/Injection on MHD Natural Convection in Vertical Parallel Plate Micro channels………………………13

3.2.1 Mathematical Analysis ……………………………………………………………………………. 11

3.3 Non Dimensionalisation ………………………………………………………………………………… 15

3.4 Solution to Problems …………………………………………………………………………………….. 17

3.4.1 Solution for MHD Natural Convection in Vertical Parallel Plate Micro-channels.17

3.4.2 Solution to Suction/Injection on MHD Natural Convection in Vertical Parallel Plate Micro-channels ……………………….. 18

CHAPTER FOUR

RESULTS AND DISCUSSIONS

4.0 Introduction ………………………………………………………………………………………………… 20

4.1 Results for MHD Natural Convection Flow in Vertical Parallel Plates Micro-channels …………………………………….. 20

4.2 Results for Suction/Injection on MHD Natural Convection in Vertical Parallel Plate Micro-channels ……………….. 27

CHAPTER FIVE

SUMMARY, CONCLUSIONS AND RECOMMENDATIONS

5.0 Introduction ………………………………………………………………………………………………… 37

5.1 Summary ……………………………………………………………………………………………………. 37

5.2 Conclusion ………………………………………………………………………………………………….. 38

5.3 Recommendations………………………………………………………………………………………… 39

REFERENCES………..……………………………………………………………………40

APPENDIX I ……………………………………………………………………………………………………. 44

APPENDIX II ………………………………………………………………………………………………….. 45

APPENDIX III …………………………………………………………………………………………………. 46

LIST OF FIGURES

Figure 3.1: Flow configuration and coordinate system of the first problem………..…………………12

Figure 3.2: Flow configuration and coordinate system of the second problem……………………….14

Figure 4.1.1: Velocity profile for various values of βvKn (ln = 1.667, M = 2.0)……………………..23

Figure 4.1.2: Velocity profile for various values of ln (βvKn = 0.05, M = 2……………………………..23

Figure 4.1.3: Velocity profile for various values of M (βvKn = 0.05, ln = 1.667)……………………24

Figure 4.1.4: Relationship of Volume flow rate to βvKn for various values of ln……………………….24

Figure 4.1.5: Relationship of Volume flow rate to βvKn for various values of M……………………….25

Figure 4.1.6: Relationship of skin friction to various values of ln (Y = 0)……………………………25

Figure 4.1.7: Relationship of skin friction to various values of ln (Y = 1)……………………………26

Figure 4.1.8: Relationship of skin friction to various values of M (Y = 0)……………………………26

Figure 4.1.9: Relationship of skin friction to various values of M (Y = 1)……………………………27

Figure 4.2.1: Velocity profile for various values of βvKn (ln = 1.667, S = 0.5, M = 2.0, Pr = 0.71)…30

Figure 4.2.2: Velocity profile for various values of ln (βvKn = 0.05, S = 0.5, M = 2.0, Pr = 0.71)….31

Figure 4.2.3: Velocity profile for various values of M (βvKn = 0.05, S = 0.5, ln = 1.667, Pr = 0.71).31

Figure 4.2.4: Velocity profile for various values of S (βvKn = 0.05, M = 2.0, ln = 1.667, Pr=0.71)…32

Figure 4.2.5: Relationship of Volume flow rate to βvKn for various values of ln……………………….32

Figure 4.2.6: Relationship of Volume flow rate to βvKn for various values of M……………………….33

Figure 4.2.7: Variation of skin friction to various values of ln (S = 0.5, M = 2.0 (Y = 0))…………..xii33

Figure 4.2.8: Variation of skin friction to various values of ln (S = 0.5, M = 2.0 (Y = 1))……………34

Figure 4.2.9: Variation of skin friction to various values of M (S = 0.5, ln = 1.667 (Y = 0))………34

Figure 4.2.10 Variation of skin friction to various values of M (S = 0.5, ln = 1.667 (Y = 1))………..35

Figure 4.2.11: Variation of skin friction to various values of S (M = 2.0, ln = 1.667 (Y = 0))………Error! Bookmark not defined.35

Figure 4.2.12: Variation of skin friction to various values of S (M = 2.0, ln = 1.667 (Y = 1))……….36

LIST OF APPENDICES

Appendix I: – List of constants used in Problem 3.1 Appendix II: – List of constants used in Problem 3.2 Appendix III: – Computer program for the role of magnetohydrodynamic (MHD) natural convection flow in vertical parallel plate micro-channels Appendix IV: – Computer program for the combined role of transverse magnetic field suction/injection on the magnetohydrodynamic (MHD) natural convection flow in vertical micro-channels

**CHAPTER ONE**

**GENERAL INTRODUCTION**

**1.0 Introduction**

Micro-flow has been given great importance in recent research activities due to its new application in micro-fluidic system devices, such as biomedical sample injection, biochemical cell reaction, micro-electric ship cooling, etc. A fundamental understanding of the flow and thermal fields as well as the corresponding characteristics at micro-scale, which may deviate from those at macro-scale, is required for the technological demands. Gaseous flow in micro-scale devices have been in the vanguard of research activities and have received great deal of attentions in recent years, due to the rapid growth of application in micro-total analysis systems and micro-electromechanical systems (MEMS). These applications have raised the interest in understanding the physical aspects of fluid flow and convective heat transfer in both forced and natural forms through micron sized channels, known as micro-channels. A magnetohydrodynamic (MHD) flow, which is the simplest plasma model, has been the subject of a great number of empirical and theoretical investigations in many industrial fields. MHD flows associating with heat transfer have received considerable attention due to the fact that their applications reside in many industrial fields such as electric propulsion for space exploration, crystal growth in liquids, cooling of nuclear reactors, electronic packages, microelectronic devices, etc.

The most common type of body force, which acts on fluid, is attributed to gravity so that the body force vector can be deduced from the gravitational acceleration. On the other hand, when an electrically conducting fluid is subjected to a magnetic field, the fluid motion induces an electric current such that the fluid velocity is reduced on account of interaction between the electric current and the fluid motion. Therefore, in case of free convection of an electrically conducting fluid in the presence of a magnetic field, there should be two body forces, i.e., a buoyancy force and a Lorentz force. They interact with each other, and in turn influence the transport phenomena of heat and mass. Among various studies of MHD free flows, few studies have been accomplished for the confined enclosures. Seki et al. (1979) studied the laminar natural convection of mercury subjected to a magnetic field parallel to gravity in a rectangular enclosure. Numerical results were obtained and compared to their experiment in the consideration of a partially heated vertical wall by uniform heat generator. Rudraiah et al. (1995) performed a numerical simulation about natural convection in a two-dimensional cavity filled with an electrically conducting fluid in the presence of a magnetic field aligned to gravity. They selected the Grashof and Hartmann numbers as controlling parameters to examine the effect of a magnetic field on free convection and associated heat transfer. Some research works were carried out on natural convection in a vertical micro-channel. Example of which are Yu and Ameel (2001), Cheng and Weng (2005) and Jha et al. (2013). Details are given in literature review.

**1.1 Aim and Objectives of Study**

The aim of this work is to study a fully developed natural convection flow of viscous, incompressible, electrically conducting fluid in a micro-channels formed by two vertical parallel plates under the effects of transverse magnetic field and suction/injection and this is achieved through the following objectives:

i. investigate the influence of external applied transverse magnetic field on steady natural convection flow of conducting fluid in vertical parallel plate micro-channels, and

ii. investigate the combined influence of external applied transverse magnetic field and suction/injection on steady natural convection flow of conducting fluid in vertical parallel plate micro-channels

**1.2 Research Methodology**

The methodology adapted in this thesis in order to achieve the set objectives is by dividing the work into five-stages. In the first stage, we reviewed some existing literature and extended them to include new parameters. Second stage is concerned with solving the mathematical model to obtain analytical solutions. In the third stage, the numerical values of the analytical solutions of the developed equations are obtained by computer package (MATLAB). The fourth stage is the graphical representation of the problems, and the last stage is the interpretation of the graphs so as to discuss the influence of each of the governing parameters and conclusions.

**1.3 Organization of the Thesis**

A study on MHD free-convective flow in vertical parallel plates micro-channels is a five-chapter thesis in order to achieve the objectives of the work. Chapter one introduces the objectives of the study and the method adopted to achieve the set objectives. Chapter two basically reviews existing literature. Chapter three gives the mathematical analysis and solutions of the governing equations with their non-dimensionalisation. While chapters four and five gives the discussion of the results/graphs, summary, concluding remarks and recommendations.