stability and simulation of a standing wave in porous media · a rayleigh-benard convection is...

STABILITY AND SIMULATION OF A STANDING WAVE

IN POROUS MEDIA

LAU SIEW CHING

UNIVERSTI TEKNOLOGI MALAYSIA

“I hereby certify that I have read this thesis and in my view, this thesis fulfills the

requirement for the award of the Degree of Bachelor in Science and Education

(Mathematics).”

STABILITY AND SIMULATION OF A STANDING WAVE

IN POROUS MEDIA

LAU SIEW CHING

This Thesis is submitted in Partial Fulfillment of the Requirement for the

Award of the Degree of Bachelor in Science and Education (Mathematics).”

FACULTY OF EDUCATION

UNIVERSITI TEKNOLOGI MALAYSIA

2006

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“I hereby declare that all the materials presented in this report are the results of my

own search except for the work I have been cited clearly in the reference.”

Signature

Author : LAU SIEW CHING

Date …31-3-2006……………………

iii

Especially for my parents and my brother, sister.

Thanks for all your love and support

I love you all

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ACKNOWLEDGEMENT

I wish to take this opportunity to express my thanks to my project supervisor,

Encik Ibrahim Mohd. Jais. He has guided me a lot in process doing this thesis. His

sharing let me learn a lot about the topic. Really thanks to him for spending time in

guiding me complete the thesis.

Then, I would like to thanks my encouraging supporters my family, course

mates, and my friends. They always give me support and encouragement.

Besides that, I am grateful to my dearest family. They always give me

encouragement even though they are at Sarawak. Their love and advice always

support me going on my thesis.

Finally, I also wish to say thank you to those help me directly and indirectly

in this thesis. Thank for their kindness and tolerance in all the way.

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Abstract

A Rayleigh-Benard Convection is being considered for a system in a porous media.

A system of convection in a fluid is observed when the fluid layer is heated from

below which eventually drives the flow. The temperature difference between the top

and the bottom layer causes density to differ thus induces motion. Here we take into

account a form of standing wave. A linear heating is being considered which has the

solution of a pitchfork bifurcation where both branches are valid solutions for the

convection. Simulation with full nonlinear equations shows stable solutions for both

cases. Numerical solutions for the full nonlinear equations show the flow

characteristics of various solutions depending on the varying Ra values.

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Abstrak

Perolakan Rayleigh-Bernard boleh dipertimbangkan sebagai suatu sistem dalam

media berporos. Sistem perolakan dalam suatu bendalir diperhatikan apabila lapisan

bendalir dipanaskan dari bahagian bawah yang mempengaruhi aliran. Perbezaan

suhu antara lapisan atas dan lapisan bawah menyebabkan ketumpatan berbeza. Di

sini, kita hanya mengambil kira satu bentuk gelombang lazim. Pemanasan linear

yang mempunyai penyelesaian bagi suatu graf super serampang di mana dua cabang

tersebut adalah penyelesaian stabil bagi perolakan. Simulasi dengan persamaan tak

linear penuh menunjukkan ciri aliran bagi pelbagai penyelesaian yang bergantung

kepada nilai Ra yang berbeza.

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CONTENTS

CHAPTER SUBJECTS PAGE

PENGESAHAN STATUS TESIS

SUPERVISOR ENDORSEMENT

TITLE i

DECLARATION ii

DEDICATION iii

ACKNOWLEDGEMENT iv

ABSTRACT v

ABSTRAK vi

CONTENTS vii

LIST OF FIGURES ix

LIST OF SYMBOLS xi

LIST OF APPENDIXES xii

I INTRODUCTION

1.1 Background of The Study 1

1.2 Objectives of Report 2

1.3 Rayleigh-Bernard Convection 2

1.4 Convection Cells 4

1.5 Stability of the Convection 5

1.6 Summary and the Outline of the Report 7

viii

II EQUATIONS OF CONVECTION

2.1 Introduction 8

2.2 Equation of Conservation of Mass 8

2.3 Energy Equation 10

2.4 Darcy Equation 13

III DERIVATION OF AMPLITUDE EQUATION

3.1 Introduction 15

3.2 Governing Equation 15

3.3 Weakly Nonlinear Analysis 20

3.4 Analysis of the Amplitude Equation 25

3.5 Numerical Solution 27

3.6 Stabilization of the Convection with Different

Rayleigh Number 30

IV CONCLUSION AND RECOMMENDATIONS

4.1 Conclusion 42

4.2 Recommendations 43

BIBLIOGRAPHY 44

APPENDIXES 46

ix

LIST OF FIGURES

No. Title Page

1.1 The pattern of the convection 2

1.2 Rayleigh-Bénard Convection 3

1.3 Cross-sectional view of cell illustrating convection cells

and the direction of the Rayleigh-Bénard cell 4

1.4 Cells in a two-dimensional view 5

1.5 Bifurcation diagram for the point Rac of convection 6

3.1 The bifurcation of Bt=RB-B3. 26

3.2 Contour of the Rayleigh number= 38 30











x



xi

LIST OF SYMBOLS

α - Isobaric thermal expansion coefficient

g - Vertical acceleration

∆T - Temperature difference

d - Cell thickness

Ra - Rayleigh number

Rac - Critical Rayleigh number

ρ - Fluid density

V - Volume

p - Pressure

v - Darcy velocity

t - Time

S - Surface

K - Permeability

κ - Hydraulic conductivity

µ - Fluid dynamic viscosity

A - Amplitude

xii

LIST OF APPENDIXES

Appendix Title Page

A Standing Wave Equation 46

B Table 3.1: The table of RB-B3=0. 47

C Command Matlab to get bifurcation of the amplitude

Equation 48

D Example to get the contour graph 49

CHAPTER I

INTRODUCTION

1.1 Background of the study

Porous media is a material that consist a solid matrix with an interconnected

void. We suppose that the solid matrix is either rigid or it undergoes small

deformation. The interconnectedness of the void allows the flow of one or more

fluid through the material. Porous media are irregular in shape in natural and

unnatural setting. Examples of natural porous media are beach sand, sandstone,

wood and the others.

Convection is the phenomena of fluid motion induced by buoyancy when a

fluid is heated from below. The temperature between the top layer and bottom

layer causes the density to differ. Convection is the study of heat transport

processes affected by the flow of fluids. The study of any convective heat

transfer problem must rest on a solid understanding of basic heat transfer and

fluid mechanics principles of convection. The examples of convection through

porous media maybe found in manmade systems such as fiber and granular

2

insulations, winding structures for high density electric machines, and the cores

of nuclear reactors.

Figure 1.1: The Pattern of The Convection

1.2 Objectives of Report

The report has the following objectives:

i. To identify the basic governing equation for standing wave, AARAA 2t −= .

ii. To solve these equations using linear perturbations.

iii. To analyze the amplitude equation derived from the governing equation and

its stability.

iv. To use numerical simulation of the full nonlinear equation.

1.3 Rayleigh-Bérnard Convection

Rayleigh-Bérnard convection is a simple system of convection which a fluid

layer heated from below drives the convective flow. It is resulting from the

buoyancy of the heated layer the magnitude of such forces depending on the

3

temperature difference prevailing between the top and the bottom portion of the fluid

layer.

Figure 1.2: Rayleigh-Bénard Convection

In 1900, Bénard noticed that there has a rather regular cellular pattern of

hexagonal convection cells during investigation of a fluid, with a free surface, heated

from below in a dish. Convection in a thin horizontal layer of fluid heated from

below is known as Rayleigh-Bénard Convection (RBC). It will present variety of

phenomena when the control parameter is modulated. The usual control parameter is

the Rayleigh number,

Ra = (αg 3d ∆T)/κv, (1.1)

where α is the isobaric thermal expansion coefficient, g the vertical acceleration, ∆T

the imposed temperature difference and d the cell thickness.

Rayleigh-Bénard convection is the instability of a fluid layer which is

confined between two thermally conducting plates and extended infinitely heated

from below. This produces a linear temperature difference in its simple form. Since

liquid has positive thermal expansion coefficient, the hot liquid at the bottom of the

cell expands and produces an unstable density gradient in the fluid layer.

4

Figure 1.3: Cross-sectional view of cell illustrating convection cells

Convection occurs if the amplifying effect exceeds the dissipative effect of

thermal diffusion and buoyancy. If the Rayleigh number is greater than 1708, then

convection occurs. This means that there is no convective flow if the Rayleigh

number is below 1708. So we will use the reduced Rayleigh number throughout the

pave, which is normalized to the onset value of 1708. If the temperature differences

is very large (Rayleigh number »1), then the fluid rises very quickly.

1.4 Convection Cells

When the critical Rayleigh number is exceeded, the instability set in. The hot

layer will go up simultaneously when the cold layer comes down. The process

creates convective cells, where adjacent cell has the opposite vorticity. Both cells

with the same vorticity will not happen side by side rather they are intermingled by a

convective cell with different vorticity, and the fluid will separate into a pattern of

convective cells. In each cell the fluid rotates in a closed orbit and the direction of

rotation alternates with successive cells.

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Heated from below

Figure 1.4: Cells in a two-dimensional view

1.5 Stability of The Convection

The instability occurs at the minimum temperature gradient at which a

balance can be steadily maintained between the entropy generated through heat

diffusion by the temperature fluctuation and the corresponding entropy carried away

by the velocity fluctuations. Pearson (“On Convection Cells Induced By Surface

Tension”, 1958) showed that the variation of surface tension with temperature could

lead to convection in the heated fluid layer, even in the absence of buoyancy. The

effect is due to the shearing forces produced in the surface layer of a fluid by

gradients of surface tension, and has been known as the Marangoni effect. Rayleigh

number is a dimensionless number which could be used in studying the convection in

the forced and free form. So, Rayleigh number can be defined as,

,...3,2,1,)(2

2222

=+

= jjRaα

απ (1.2)

where j is the number of convention cells vertically. Ra is minimum when j=1

and πα = . Thus, the critical Rayleigh number written as Rac can be written as,

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Rac= 24π = 39.48

For the higher-order modes (j= 2, 3,…),

224 jRaj π= and jπαj =

Conductive state remains stable for Ra< 24π . Instability appears as

convection when Ra is raised to 24π , and it appears in the form of a cellular motion

with horizontal wave numberπ based on linear studies. A layer of a single

component fluid is stable if the density decreases upward, but a similar layer of a

fluid consisting of two components may be dynamically unstable. In general, the

amplitude bifurcation of convection is shown in Figure (1.5).

Figure 1.5: Bifurcation Diagram For the Point Rac of Convection

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1.6 Summary and Outline of the Report

The report is compiled as follows,

Chapter 1 illustrates some basic concepts of convection and the Rayleigh-

Bénard convection. The stability of the Rayleigh-Bénard will be discussed also in

this chapter.

Chapter 2 is discussing the differential form of the governing equations of

fluid. The equations use the concept of conservation of mass, momentum and energy.

The fluid flow follows Darcy’s law and Boussinesq approximation is assume true.

In Chapter 3, the analysis is based on weakly nonlinear analysis which

derives amplitude equation. Study of stability of the amplitude equation will be

carried out. Analysis carried out will be analytical as well as numerical.

Chapter 4 is the conclusion of the study and the extension of this study for

future research.

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