TRANSIENT MHD FREE CONVECTION FLOW IN A VERTICAL MICROTUBE

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ABSTRACT

This project analyses the transient magnetohydrodynamic free convective flow in vertical microtube in the presence of velocity slip and temperature jump at the inner surface of the microtube. The Laplace transform technique has been used to find the solutions for the velocity and temperature fields by solving the governing partial differential equations in Laplace domain. However, the Riemann-sum approximation method is used to invert the Laplace domain to the time domain. The solution derived is validated by assenting comparison with exact solutions derived for the steady state which has been derived separately. An agreement was found for transient and steady state at large value of time. The solution obtained for the velocity has been used to compute the skin friction. The effect of various flow parameters entering into the problem such as time, Prandtl number , Hartmannnumber , rarefaction parameter , and fluid-wall interaction parameter are discussed with the aid of line graphs.

 

TABLE OF CONTENTS

DECLARATION.. ii

CERTIFICATION.. iii

DEDICATION.. iv

ACKNOWLEDGEMENTS. v

ABSTRACT. vi

TABLE OF CONTENTS. vii

CHAPTER ONE. 1

INTRODUCTION.. 1

1.1 Background of the Study. 1

1.2 Statement of the Problem.. 2

1.3 Aim and Objectives of the Study. 2

1.5 Scope and Limitation. 3

DIMENSIONLESS QUANTITIES. 3

1.Prandtl number . 3

2. Hartmann Number 4

1.7 Definition of the Basic Terms. 4

1.7.1 Magnetohydrodynamic (MHD) 4

1.7.2 Transient 4

1.7.3 Free Convection. 4

1.7.4 Vertical Microtube. 4

1.7.5 Skin Friction. 4

1.7.6 Steady State. 5

1.7.7 Mass Flux. 5

1.6.8 Laplace transform.. 5

1.7.9 Riemann-Sum Approximation. 6

1.7.10Viscous/ Non viscous fluids. 6

1.7.11Compressible/Incompressible fluids. 6

1.7.12 Steady flow.. 6

1.7.13 Unsteady flow.. 7

1.7.14 Newtonian fluid. 7

1.7.15 Non-Newtonian fluid. 7

CHAPTER TWO.. 8

LITERATURE REVIEW… 8

2.1 Introduction. 8

CHAPTER THREE. 11

CHAPTER FOUR.. 15

4.1 RESULTS AND DISCUSSION.. 15

CHAPTER FIVE. 26

SUMMARY, RECOMMENDATION CONCLUSION.. 26

5.1 Summary. 26

5.2 Recommendation. 26

5.3 Conclusion. 27

REFERENCES. 28

 

 

                                                       

 

CHAPTER ONE

INTRODUCTION

1.1 Background of the Study

A wide range of applications of micro-electromechanical systems and -technology have given a fillip to research area where a non-continuum behavior is present. In this work, we are interested in studying the surface-fluid interaction where slip flow regime occurs. In this regard, Knudsen number  is a deciding factor, which is a measure of molecular mean free path to characteristic length. When the Knudsen number  is very small, no slip is observed between the surface and the fluid and is in tune with the essence of continuum mechanics. Furthermore, when Knudsen number  lies in the range 0.001-0.1, slip occurs at the surface–fluid interaction and is generally studied under the light of model Maxwell–Smoluchowskn first-order slip boundary conditions.Extensive investigations have been conducted recently in the field of micro geometry flow, but the literature lacks studies that take into account the role of wall surface curvature on transient magneto hydro dynamic (MHD) free convective flow in vertical microtube. However, to cite a few work in this direction, Khadrawi et al.( 2005) investigated analytically the transient thermal behaviour of a stagnant gas confined in a horizontal micro-channel under the effect of the dual-phase-lag heat conduction model. Khadrawi(2011) studied the transient hydrodynamic and thermal behaviors’ of fluid flow in a vertical porous micro-channel under the effect of hyperbolic heat conduction model.

The unsteady hydrodynamics and thermal behaviour of fluid flow in an open-ended vertical parallel-plate micro-channel are investigated semi analytically under the effect of the dual-phase-lag heat conduction model by Khadrawi and Al-Nimr. (2007) Also, Weng and Chen (2009) studied the impact of wall surface curvature on steady fully developed natural convection flow in an open-ended vertical micro tube with an a symmetric heating of annulus surface. Recently,further extended the work of Weng and Chen (2009)by taking into account suction/injection on vertical annular micro-channel. It is observed that skin friction decreases at the outer surface of the inner porous cylinder with an increase in fluid–wall interaction parameter reverse at the inner surface of the outer porous cylinder. Avci and Aydin (2009) studied the fully developed mixed convective heat transfer of a Newtonian fluid in a vertical micro tube. On the other hand, the MHD phenomenon has received considerable attention during the last two decades due to its importance from the energy generation point of view, and one may envisage MHD generators for power generation.

MHD pumps are already in use in chemical energy technology for pumping electrically conducting fluids at some of the atomic energy center. Besides these applications, when the fluid is electrically conducting, the free convection flow is appreciably influenced by an imposed magnetic field.Therefore, to refer to few works in this direction,Sheikholeslami et al.(2014) investigated the magnetic field effect on nano-fluid flow and heat transfer in a semi annulus enclosure via control volume-based finite element method. Sheikholeslami and Gorji-Bandpy(2014) presented the numerical solution for free convection offer rofluid in a cavity heated from below in the presence of external magnetic field, while the MHD natural convection of nanofluid in a vertical microtube.

1.2 Statement of the Problem

This work analyses the transient magneto hydrodynamic free convective flow in vertical microtube in the presence of velocity slip and temperature jump at the inner surface of the cylinder. The effect of various flow parameters entering into the problem such as time, Prandtl number, Hartmann number, rarefaction parameter, and fluid–wall interaction parameter are discussed with the aid of line graphs.

1.3 Aim and Objectives of the Study

The aim of this project is to study transient MHD Fluid free convection in a vertical microtube. With the following objectives:

1. to analyze the behavior of MHD fluid flow in the presence of a transverse magnetic field and velocity slip in a microtube

• to discuss the effects of the time Prandtl number Hartmann number rarefaction parameter ) and fluid–wall interaction parameter on the fluid temperature, velocity, volume flow rate, rate of heat transfer, and skin friction
• to present and discuss the effect of the parameters above graphically using MATLAB (2016b).

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