The stability mechanisms of a curved front and the selection of pattern formation in dendrite growth and viscous fingering have been long-standing fundamental subjects in the areas of condensed matter physics, material science, crystal growth and fluid mechanics. This book studies the interfacial stability and pattern formation in dynamic systems away from the equilibrium state. In particular, it deals with the two prominent prototype systems: dendrite growth in solidification and viscous fingering in Hele-Shaw flow. It elucidates problems step by step and systematically derives their mathematical solutions on the basis of the interfacial wave theory. Finally, it carefully examines these results by comparisons with the available experimental results.

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.