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Laplacian Growth: Interfacial Evolution in a Hele-Shaw Cell

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Laplacian Growth: Interfacial Evolution in a Hele-Shaw Cell Laplacian Growth: Interface Evolution in a Hele-Shaw Cell A dissertation presented to the faculty of the College of Arts and Sciences of Ohio University In partial fulfillment of the requirements for the degree Doctor of Philosophy Khalid R. Malaikah August 2013 © 2013 Khalid R. Malaikah. All Rights Reserved. 2 This dissertation titled Laplacian Growth: Interface Evolution in a Hele-Shaw Cell by KHALID R. MALAIKAH has been approved for the Department of Mathematics and the College of Arts and Sciences by Tatiana Savin Associate Professor of Mathematics Robert Frank Dean, College of Arts and Sciences 3 ABSTRACT MALAIKAH, KHALID R., Ph.D., August 2013, Mathematics Laplacian Growth: Interface Evolution in a Hele-Shaw Cell (101 pp.) Director of Dissertation: Tatiana Savin Laplacian growth is the interface dynamics where the normal component of velocity of a free boundary is proportional to the normal derivative of a harmonic function defined in a moving domain. The interface evolution in a Hele-Shaw cell is described by the Laplacian growth model. In this study we derive governing equations in terms of the Schwarz function of the interface for some specific Hele-Shaw flows in which the interface is not equipotential. This is a generalization of the well-known equation ( ) derived for the free boundary one-phase Hele-Shaw problem. Here, is the complex potential and is the time derivative of the Schwarz function. The structure of the thesis is as follows: o In Chapter 1 we give an introduction to the history of the problem, and discuss the methods and the state of the art. o Chapter 2 is devoted to the Schwarz function equation for the two-phase Hele- Shaw flows. Here we re-derive the equations earlier obtained by D. Crowdy using a slightly different method. Our derivation is based on an introduction of a single- valued complex velocity potential. o In Chapter 3 we derive the Schwarz function equation for a class of generalized Hele-Shaw flows and apply it to the case of an interior problem in a cell with the 4 time-dependent gap. This generalizes the governing equation of the interfacial motion in a Hele-Shaw cell in the presence of an arbitrary external potential. 5 DEDICATION To my parents: Reda and Zakia. To my family: my wife Marwa and my daughter Puneen. To all my brothers and sisters. 6 ACKNOWLEDGMENTS I would like to express my special gratitude to my advisor Dr. Tatiana Savin, for her excellent guidance, support, and patience throughout the course of this research. I would like to thank my committee members, Dr. Archil Gulisashvili, Dr. David Tees, and Dr. Xiaoping A. Shen, for their encouragement. I would like to acknowledge the Government of Saudi Arabia for providing me financial support and Taibah University for granting me academic leave for higher studies at Ohio University. 7 TABLE OF CONTENTS Page Abstract ............................................................................................................................... 3 Dedication ........................................................................................................................... 5 Acknowledgments............................................................................................................... 6 List of Figures ..................................................................................................................... 9 1 Preliminaries ................................................................................................................ 10 1.1 Introduction ........................................................................................................... 10 1.2 Mathematical background .................................................................................... 15 1.2.1 Conformal invariance ..................................................................................... 15 1.2.2 Schwarz function for an analytic curve ......................................................... 18 1.2.3 Cauchy transform ........................................................................................... 25 1.2.4 Subharmonic and superharmonic functions ................................................... 27 1.3 Newtonian fluid .................................................................................................... 28 1.3.1 Equations of motion ....................................................................................... 29 1.3.2 Potential flow ................................................................................................ 36 1.3.3 Reynolds number ........................................................................................... 42 1.4 Summary of results and future plan ...................................................................... 43 1.5 Organization of thesis ........................................................................................... 49 2 Two-phase displacement in a Hele-Shaw cell ............................................................. 53 2.1 One-phase Hele-Shaw flow .................................................................................. 53 2.1.1 Thin film approximation ................................................................................ 56 2.1.2 Explicit Solutions ........................................................................................... 59 2.2 Governing equation for the two-phase Hele-Shaw flow ...................................... 68 2.3 Mother Body ......................................................................................................... 73 8 2.3.1 Mother body and continuation of the gravitational field ............................... 73 2.3.2 A dynamic mother body in a Hele-Shaw flow ............................................... 74 3 Generalized Hele-Shaw flows ...................................................................................... 79 3.1 Hele-Shaw flow with external potential field ....................................................... 79 3.2 Governing equation for a class of the generalized Hele-Shaw flow ..................... 84 3.3 Governing equation for the time-dependent gap Hele-Shaw flow ....................... 90 References ......................................................................................................................... 95 9 LIST OF FIGURES Page Figure 1: Laplacian growth process ................................................................................11 Figure 2: Hele-Shaw cell ................................................................................................12 Figure 3: Rotation of a tangent vector ............................................................................22 Figure 4: Newton’s parallel plate experiment .................................................................29 Figure 5: The system and the fixed control volume .......................................................30 Figure 6: Surface force of two dimensional fluid’s medium ..........................................34 Figure 7: Geometric interpretation of streamlines ..........................................................39 Figure 8: Elementary plane flows ...................................................................................41 Figure 9: Typical length scale in Hele-Shaw cell ...........................................................56 Figure 10: Polubarinova-Galin Equation ........................................................................60 Figure 11: Saffman-Taylor finger ..................................................................................87 10 1 PRELIMINARIES 1.1 Introduction A wide range of physical and mathematical applications are observed in nature as a contour dynamic where an interface moves between two immiscible phases. The interfacial motion is generated due to a driven mechanism such as mass distribution and/or heat fluxes. The present thesis investigates some examples of interface dynamic in which the free boundary moves with a velocity proportional to the gradient of a harmonic field. This pattern is called the Laplacian growth process (see Figure 1), which governs a variety of natural growth phenomena such as viscous fingering and the growth of bacterial colonies [31]. Laplacian growth is a non-linear complex dynamics which has been attracting the attention of mathematicians and physicists for more than a century. Specifically, we have a simply-connected domain Ω( ) with an interface ( ); inside the region Ω( ) the potential satisfies the Laplace’s equation . (1.1.1) The moving boundary is supposed to be equipotential or constant-field, i.e., an interface satisfying the boundary condition constant. (1.1.2) Furthermore, the velocity of the point of the moving boundary ( ) is defined as the gradient of a scalar field ( ) , (1.1.3) 11 where is called the velocity potential. Assume the continuity of the velocity potential within the simply-connected domain up to the boundary. Given the initial domain, one wants to study the evolution of the simply-connected domain which occupied by a viscous fluid whose motion is generated by a driven mechanism such as several sources or sinks. In this study we are interested in the interfacial motion where the interface is no longer equipotential so that the velocity potential equals a specific function of the boundary point. ( ) ( ) Figure 1: Laplacian growth process The central theme of the present dissertation is a characterization of an interfacial motion between two immiscible phases in a Hele-Shaw cell which is an
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