Interfacial wave theory of pattern formation : selection of dendritic growth and viscous fingering in Hele-Shaw flow

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Interfacial wave theory of pattern formation : selection of dendritic growth and viscous fingering in Hele-Shaw flow

Jian-Jun Xu

（Springer series in synergetics）

Springer, 1998

  • : pbk

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Includes bibliographical references and index

Description and Table of Contents

Volume

ISBN 9783540631453

Description

For the last several years, the study of interfacial instability and pattern formation phenomena has preoccupied many researchers in the broad area of nonlinear science. These phenomena occur in a variety of dynamical sys- tems far from equilibrium. In many practically very important physical sys- tems some fascinating patterns are always displayed at the interface between solid and liquid or between two liquids. Two prototypes of these phenomena are dendrite growth in solidification and viscous fingering in a Hele-Shaw cell. These two phenomena occur in completely different scientific fields, but both are described by similar nonlinear free boundary problems of partial- differential-equation systems; the boundary conditions on the interface for both cases contain a curvature operator involving the surface tension, which is nonlinear. Moreover, both cases raise the same challenging theoretical is- sues, interfacial instability mechanisms and pattern selection, and it is now found that these issues can be solved by the same analytical approach. Thus, these two phenomena are regarded as special examples of a class of nonlinear pattern formation phenomena in nature, and they are the prominent topics of the new interdisciplinary field of nonlinear science. This research monograph is based on a series of lectures I have given at McGill University, Canada (1993-1994), Northwestern Poly technical In- stitute, China (1994), Aachen University, Germany (1994), and the CRM summer school at Banff, Alberta, Canada (1995).

Table of Contents

  • 1. Introduction.- 1.1 Interfacial Pattern Formations in Dendrite Growth and Hele-Shaw Flow.- 1.2 A Brief Review of the Theories of Free Dendrite Growth.- 1.2.1 Maximum Velocity Principle (1976).- 1.2.2 Marginal Stability Hypothesis (1978).- 1.2.3 Microscopic Solvability Condition (MSC) Theory (1986-1990s).- 1.2.4 Interfacial Wave (IFW) Theory (1990).- 1.3 Macroscopic Continuum Model.- 1.3.1 Macroscopic Transport Equations.- 1.3.2 The Interface Conditions.- 1.3.3 The Scaling and the Dimensionless System.- References.- 2. Unidirectional Solidification and the Mullins-Sekerka Instability.- 2.1 Solidification with Planar Interface from a Pure Melt.- 2.1.1 Basic Steady State Solution.- 2.1.2 Unsteady Perturbed Solutions and Mullins-Sekerka Instability.- 2.1.3 Asymptotic Solutions in the Long-Wavelength Regime, k = O(?).- 2.1.4 Asymptotic Solutions in the Extremely Short-Wavelength Regime, k = O (1/ ?).- 2.2 Unidirectional Solidification from a Binary Mixture.- 2.2.1 Mathematical Formulation of the Problem.- 2.2.2 Basic Steady State.- 2.2.3 Unsteady Perturbed Solutions.- 2.2.4 Asymptotic Solutions in the Long-Wavelength Regime, k = O(?).- 2.2.5 Asymptotic Solutions in the Extremely Short-Wavelength Regime, k = O(1/ ?)
  • g = O1/ ?).- 2.2.6 Some Remarks on Unidirectional Solidification.- References.- 3. Mathematical Formulation of Free Dendrite Growth from a Pure Melt.- 3.1 Three-Dimensional Axially Symmetric Free Dendrite Growth.- 3.2 Two-Dimensional Free Dendrite Growth.- Reference.- 4. Steady State of Dendrite Growth with Zero Surface Tension and Its Regular Perturbation Expansion.- 4.1 The Ivantsov Solution and Unsolved Fundamental Problems..- 4.2 Three-Dimensional Axially Symmetric Steady Needle Growth.- 4.2.1 Mathematical Formulation.- 4.2.2 The Regular Perturbation Expansion Solutions (RPE) as ?? 0.- 4.2.3 The Asymptotic Behavior of the Regular Perturbation Expansion Solution as ? -> ?.- 4.3 Two-Dimensional, Steady Needle Crystal Growth.- 4.3.1 Mathematical Formulation of Two-Dimensional Needle Growth.- 4.3.2 The Regular Perturbation Expansion Solution as ? -> 0.- 4.3.3 Asymptotic Behavior of the Regular Perturbation Expansion Solution as ? -> ?.- 4.4 Summary and Discussion.- References.- 5. The Steady State for Dendrite Growth with Nonzero Surface Tension.- 5.1 The Nash-Glicksman Problem and the Classic Needle Crystal Solution.- 5.2 The Geometric Model and Solutions of the Needle Crystal Formation Problem.- 5.2.1 Geometric Model of Dendrite Growth.- 5.2.2 The Segur-Kruskal Problem.- 5.2.3 Nonclassic Steady Needle Growth Problem.- 5.2.4 Needle Crystal Formation Problem.- 5.3 The Nonclassic Steady State of Dendritic Growth with Nonzero Surface Tension.- 5.3.1 The Complete Mathematical Formulation for Free Dendrite Growth.- References.- 6. Global Interfacial Wave Instability of Dendrite Growth from a Pure Melt.- 6.1 Linear Perturbed System Around the Basic State of Three-Dimensional Dendrite Growth.- 6.2 Outer Solution in the Outer Region away from the Tip.- 6.2.1 Zeroth-Order Approximation.- 6.2.2 First-Order Approximation.- 6.2.3 Singular Point ?cof the Outer Solution.- 6.3 The Inner Solutions near the Singular Point ?c.- 6.4 Tip Inner Solution in the Tip Region.- 6.5 Global Trapped-Wave Modes and the Quantization Condition.- 6.6 Global Interfacial Wave Instability of Two-Dimensional Dendrite Growth.- 6.7 The Comparison of Theoretical Predictions with Experimental Data.- References.- 7. The Effect of Surface Tension Anisotropy and Low-Frequency Instability on Dendrite Growth.- 7.1 Linear Perturbed System Around the Basic State.- 7.2 Multiple Variable Expansion Solution in the Outer Region.- 7.3 The Inner Equation near the Singular Point ?c.- 7.3.1 Case I: ?0= O(l).- 7.3.2 Case II: |?0| ? 1.- 7.3.3 A Brief Summary.- 7.4 Matching Conditions.- 7.5 The Spectra of Eigenvalues and Instability Mechanisms.- 7.5.1 The Global Trapped-Wave Instability.- 7.5.2 The Low-Frequency Instability.- 7.6 Low-Frequency Instability for Axially Symmetric Dendrite Growth.- 7.7 The Selection Conditions for Dendrite Growth.- References.- 8. Three-Dimensional Dendrite Growth from Binary Mixtures.- 8.1 Mathematical Formulation of the Problem.- 8.2 Basic State Solution for the Case of Zero Surface Tension.- 8.3 Linear Perturbed System for the Case of Nonzero Surface Tension.- 8.4 The MVE Solutions in the Outer Region.- 8.4.1 The Zeroth-Order Approximation.- 8.4.2 The First-Order Approximation.- 8.5 The Inner Solutions near the Singular Point ?c.- 8.6 Global Modes and the Quantization Condition.- 8.7 Comparisons of Theoretical Results with Experimental Data.- References.- 9. Viscous Fingering in a Hele-Shaw Cell.- 9.1 Introduction.- 9.2 Mathematical Formulation of the Problem.- 9.3 The Smooth Finger Solution with Zero Surface Tension.- 9.4 Formulation of the General Problem in Curvilinear Coordinates (?,?)and the Basic State Solutions.- 9.5 The Linear Perturbed System and the Outer Solutions.- 9.6 The Inner Equation near the Singular Point ?c.- 9.6.1 Case I: |?0| = O(1).- 9.6.2 Case II: |?0| ?1.- 9.7 Eigenvalues Spectra and Instability Mechanisms.- 9.7.1 The Spectrum of Complex Eigenvalues and GTW Instability.- 9.7.2 The Spectrum of Real Eigenvalues and the LF Instability.- 9.8 Fingering Flow with a Nose Bubble.- 9.8.1 The Basic State of Finger Formation with a Nose Bubble and Its Linear Perturbation.- 9.8.2 The Quantization Conditions for the System with a Nose Bubble.- 9.9 The Selection Criteria of Finger Solutions.- References.
Volume

: pbk ISBN 9783642804373

Description

For the last several years, the study of interfacial instability and pattern formation phenomena has preoccupied many researchers in the broad area of nonlinear science. These phenomena occur in a variety of dynamical sys- tems far from equilibrium. In many practically very important physical sys- tems some fascinating patterns are always displayed at the interface between solid and liquid or between two liquids. Two prototypes of these phenomena are dendrite growth in solidification and viscous fingering in a Hele-Shaw cell. These two phenomena occur in completely different scientific fields, but both are described by similar nonlinear free boundary problems of partial- differential-equation systems; the boundary conditions on the interface for both cases contain a curvature operator involving the surface tension, which is nonlinear. Moreover, both cases raise the same challenging theoretical is- sues, interfacial instability mechanisms and pattern selection, and it is now found that these issues can be solved by the same analytical approach. Thus, these two phenomena are regarded as special examples of a class of nonlinear pattern formation phenomena in nature, and they are the prominent topics of the new interdisciplinary field of nonlinear science. This research monograph is based on a series of lectures I have given at McGill University, Canada (1993-1994), Northwestern Poly technical In- stitute, China (1994), Aachen University, Germany (1994), and the CRM summer school at Banff, Alberta, Canada (1995).

Table of Contents

  • 1. Introduction.- 1.1 Interfacial Pattern Formations in Dendrite Growth and Hele-Shaw Flow.- 1.2 A Brief Review of the Theories of Free Dendrite Growth.- 1.2.1 Maximum Velocity Principle (1976).- 1.2.2 Marginal Stability Hypothesis (1978).- 1.2.3 Microscopic Solvability Condition (MSC) Theory (1986-1990s).- 1.2.4 Interfacial Wave (IFW) Theory (1990).- 1.3 Macroscopic Continuum Model.- 1.3.1 Macroscopic Transport Equations.- 1.3.2 The Interface Conditions.- 1.3.3 The Scaling and the Dimensionless System.- References.- 2. Unidirectional Solidification and the Mullins-Sekerka Instability.- 2.1 Solidification with Planar Interface from a Pure Melt.- 2.1.1 Basic Steady State Solution.- 2.1.2 Unsteady Perturbed Solutions and Mullins-Sekerka Instability.- 2.1.3 Asymptotic Solutions in the Long-Wavelength Regime, k = O(?).- 2.1.4 Asymptotic Solutions in the Extremely Short-Wavelength Regime, k = O (1/ ?).- 2.2 Unidirectional Solidification from a Binary Mixture.- 2.2.1 Mathematical Formulation of the Problem.- 2.2.2 Basic Steady State.- 2.2.3 Unsteady Perturbed Solutions.- 2.2.4 Asymptotic Solutions in the Long-Wavelength Regime, k = O(?).- 2.2.5 Asymptotic Solutions in the Extremely Short-Wavelength Regime, k = O(1/ ?)
  • g = O1/ ?).- 2.2.6 Some Remarks on Unidirectional Solidification.- References.- 3. Mathematical Formulation of Free Dendrite Growth from a Pure Melt.- 3.1 Three-Dimensional Axially Symmetric Free Dendrite Growth.- 3.2 Two-Dimensional Free Dendrite Growth.- Reference.- 4. Steady State of Dendrite Growth with Zero Surface Tension and Its Regular Perturbation Expansion.- 4.1 The Ivantsov Solution and Unsolved Fundamental Problems..- 4.2 Three-Dimensional Axially Symmetric Steady Needle Growth.- 4.2.1 Mathematical Formulation.- 4.2.2 The Regular Perturbation Expansion Solutions (RPE) as ?? 0.- 4.2.3 The Asymptotic Behavior of the Regular Perturbation Expansion Solution as ? -> ?.- 4.3 Two-Dimensional, Steady Needle Crystal Growth.- 4.3.1 Mathematical Formulation of Two-Dimensional Needle Growth.- 4.3.2 The Regular Perturbation Expansion Solution as ? -> 0.- 4.3.3 Asymptotic Behavior of the Regular Perturbation Expansion Solution as ? -> ?.- 4.4 Summary and Discussion.- References.- 5. The Steady State for Dendrite Growth with Nonzero Surface Tension.- 5.1 The Nash-Glicksman Problem and the Classic Needle Crystal Solution.- 5.2 The Geometric Model and Solutions of the Needle Crystal Formation Problem.- 5.2.1 Geometric Model of Dendrite Growth.- 5.2.2 The Segur-Kruskal Problem.- 5.2.3 Nonclassic Steady Needle Growth Problem.- 5.2.4 Needle Crystal Formation Problem.- 5.3 The Nonclassic Steady State of Dendritic Growth with Nonzero Surface Tension.- 5.3.1 The Complete Mathematical Formulation for Free Dendrite Growth.- References.- 6. Global Interfacial Wave Instability of Dendrite Growth from a Pure Melt.- 6.1 Linear Perturbed System Around the Basic State of Three-Dimensional Dendrite Growth.- 6.2 Outer Solution in the Outer Region away from the Tip.- 6.2.1 Zeroth-Order Approximation.- 6.2.2 First-Order Approximation.- 6.2.3 Singular Point ?cof the Outer Solution.- 6.3 The Inner Solutions near the Singular Point ?c.- 6.4 Tip Inner Solution in the Tip Region.- 6.5 Global Trapped-Wave Modes and the Quantization Condition.- 6.6 Global Interfacial Wave Instability of Two-Dimensional Dendrite Growth.- 6.7 The Comparison of Theoretical Predictions with Experimental Data.- References.- 7. The Effect of Surface Tension Anisotropy and Low-Frequency Instability on Dendrite Growth.- 7.1 Linear Perturbed System Around the Basic State.- 7.2 Multiple Variable Expansion Solution in the Outer Region.- 7.3 The Inner Equation near the Singular Point ?c.- 7.3.1 Case I: ?0= O(l).- 7.3.2 Case II: |?0| ? 1.- 7.3.3 A Brief Summary.- 7.4 Matching Conditions.- 7.5 The Spectra of Eigenvalues and Instability Mechanisms.- 7.5.1 The Global Trapped-Wave Instability.- 7.5.2 The Low-Frequency Instability.- 7.6 Low-Frequency Instability for Axially Symmetric Dendrite Growth.- 7.7 The Selection Conditions for Dendrite Growth.- References.- 8. Three-Dimensional Dendrite Growth from Binary Mixtures.- 8.1 Mathematical Formulation of the Problem.- 8.2 Basic State Solution for the Case of Zero Surface Tension.- 8.3 Linear Perturbed System for the Case of Nonzero Surface Tension.- 8.4 The MVE Solutions in the Outer Region.- 8.4.1 The Zeroth-Order Approximation.- 8.4.2 The First-Order Approximation.- 8.5 The Inner Solutions near the Singular Point ?c.- 8.6 Global Modes and the Quantization Condition.- 8.7 Comparisons of Theoretical Results with Experimental Data.- References.- 9. Viscous Fingering in a Hele-Shaw Cell.- 9.1 Introduction.- 9.2 Mathematical Formulation of the Problem.- 9.3 The Smooth Finger Solution with Zero Surface Tension.- 9.4 Formulation of the General Problem in Curvilinear Coordinates (?,?)and the Basic State Solutions.- 9.5 The Linear Perturbed System and the Outer Solutions.- 9.6 The Inner Equation near the Singular Point ?c.- 9.6.1 Case I: |?0| = O(1).- 9.6.2 Case II: |?0| ?1.- 9.7 Eigenvalues Spectra and Instability Mechanisms.- 9.7.1 The Spectrum of Complex Eigenvalues and GTW Instability.- 9.7.2 The Spectrum of Real Eigenvalues and the LF Instability.- 9.8 Fingering Flow with a Nose Bubble.- 9.8.1 The Basic State of Finger Formation with a Nose Bubble and Its Linear Perturbation.- 9.8.2 The Quantization Conditions for the System with a Nose Bubble.- 9.9 The Selection Criteria of Finger Solutions.- References.

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