STABILITY AND SIMULATION OF A STANDING WAVE IN POROUS MEDIA LAU SIEW CHING UNIVERSTI TEKNOLOGI MALAYSIA

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1 STABILITY AND SIMULATION OF A STANDING WAVE IN POROUS MEDIA LAU SIEW CHING UNIVERSTI TEKNOLOGI MALAYSIA

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3 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).

4 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

5 ii 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

6 iii Especially for my parents and my brother, sister. Thanks for all your love and support I love you all

7 iv 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.

8 v 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.

9 vi 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.

10 vii CONTENTS CHAPTER SUBJECTS PAGE PENGESAHAN STATUS TESIS SUPERVISOR ENDORSEMENT TITLE DECLARATION DEDICATION ACKNOWLEDGEMENT ABSTRACT ABSTRAK CONTENTS LIST OF FIGURES LIST OF SYMBOLS LIST OF APPENDIXES i ii iii iv v vi vii ix xi xii I INTRODUCTION 1.1 Background of The Study Objectives of Report Rayleigh-Bernard Convection Convection Cells Stability of the Convection Summary and the Outline of the Report 7

11 viii II EQUATIONS OF CONVECTION 2.1 Introduction Equation of Conservation of Mass Energy Equation Darcy Equation 13 III DERIVATION OF AMPLITUDE EQUATION 3.1 Introduction Governing Equation Weakly Nonlinear Analysis Analysis of the Amplitude Equation Numerical Solution Stabilization of the Convection with Different Rayleigh Number 30 IV CONCLUSION AND RECOMMENDATIONS 4.1 Conclusion Recommendations 43 BIBLIOGRAPHY 44 APPENDIXES 46

12 ix LIST OF FIGURES No. Title Page 1.1 The pattern of the convection Rayleigh-Bénard Convection Cross-sectional view of cell illustrating convection cells and the direction of the Rayleigh-Bénard cell Cells in a two-dimensional view Bifurcation diagram for the point Ra c of convection The bifurcation of B t =RB-B Contour of the Rayleigh number= Contour of the Rayleigh number= Contour of the Rayleigh number= Contour of the Rayleigh number= Contour of the Rayleigh number= Contour of the Rayleigh number= Contour of the Rayleigh number= Contour of the Rayleigh number= Contour of the Rayleigh number= Contour of the Rayleigh number= Contour of the Rayleigh number= 48 38

13 x 3.13 Contour of the Rayleigh number= Contour of the Rayleigh number= 50 41

14 xi LIST OF SYMBOLS α - Isobaric thermal expansion coefficient g - Vertical acceleration T - Temperature difference d - Cell thickness Ra - Rayleigh number Ra c - 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

15 xii LIST OF APPENDIXES Appendix Title Page A Standing Wave Equation 46 B Table 3.1: The table of RB-B 3 =0. 47 C Command Matlab to get bifurcation of the amplitude Equation 48 D Example to get the contour graph 49

16 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

17 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: 2 i. To identify the basic governing equation for standing wave, A = RA A A. t ii. iii. To solve these equations using linear perturbations. 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

18 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 d 3 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.

19 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 So we will use the reduced Rayleigh number throughout the pave, which is normalized to the onset value of 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.

20 5 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, ( j π + α ) Ra =, j = 1,2,3,... 2 α (1.2) where j is the number of convention cells vertically. Ra is minimum when j=1 andα = π. Thus, the critical Rayleigh number written as Ra c can be written as,

21 6 Ra c = 2 4π = For the higher-order modes (j= 2, 3, ), Raj 4π j 2 2 = and αj = jπ 2 Conductive state remains stable for Ra< 4π. Instability appears as 2 convection when Ra is raised to 4π, 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 Ra c of Convection

22 7 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.