Approximate Analytical Solution of Natural Convection Flow with Viscous Dissipation and Variable Viscosity Effects

Approximate Analytical Solution of Natural Convection Flow with Viscous Dissipation and Variable Viscosity Effects.

 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.

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TABLE OF CONTENTS

Cover Page

Fly Leaf    i

Tittle Page    ii

Declaration……. i

Certification  ii

Dedication  iii

Acknowledgement…. iv

Abstract

Table of Contents..   v

List of Figures…     viii

List of Tables……. xi

List of Appendices.    xii

Nomenclature……..   xiii

Dimensionless Quantities..xv

CHAPTER ONE GENERAL INTRODUCTION

• Introduction….    1
• Statement of the Problem….     2
• Aim and Objectives of the study….  3
• Research Methodology…… 4
• Organization of the Dissertation…      4
• Basic definitions……….   5
• Basic Governing Equations 6

CHAPTER TWO LITERATURE REVIEW

• Introduction….. 7
• Natural convection flow……….    7
• Suction and Injection.9
• Variable viscosity….   10

CHAPTER THREE MATHEMATICAL ANALYSES

• Introduction………..     12
• Mathematical formulation and Geometrical Description……… 12
  • Natural convection flow with viscous dissipation and variable viscosity effects          13
• Natural convection heat transfer flow in a vertical porous channel with viscous dissipation and variable viscosity effects…. 14
  • Mathematical formulation (Problem II)….. 15
• Non-dimensionalization………..    16
  • Non-dimensionalization of problem I (3.2.1).  16
  • Non-dimensionalization of problem II (3.3.1)…       17

CHAPTER FOUR SOLUTION OF THE PROBLEMS

• Introduction………    18
• Uncoupling and Linearization of resulting differential equations……   18
  • Outline of Homotopy perturbation Method…………    18
  • Convergence of the Homotopy Perturbation Method….    20
• Solution to Problem 2…………………  21
• Phase and Amplitude of Periodic temperature and velocity of problem    23
• Nusselt number (??) and Skin friction (?) for problem 2.. 24
• Solution of Problem 3……………….      24
• Phase and Amplitude of Periodic temperature and velocity of problem 3.      27
• Nusselt number (??) and Skin friction (?) of problem 3.  27
• Convergence of the Series Solutions…  28

CHAPTER FIVE RESULTS AND DISCUSSION

• Introduction…….    30
• Flow between two vertical parallel plates (3.2)…    30
• Flow in vertical porous channel (3.3)…..      43
• Validation of the Results……..    56
  • Validation of Result of Problem I…….    56
  • Validation of results of Problem II……….   57

CHAPTER SIX SUMMARY, CONCLUSION AND RECOMMENDATION

• Summary.       58
• Conclusion……  58
• Recommendation..    59

REFERENCES..   61

APPENDICES….. 65

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REFERENCES

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