Note: Intro -- Preface of the Second Edition -- Preface of the First Edition -- Contents -- 1 Introduction -- 1.1 Interfacial Pattern Formation in Dendritic Growth and Hele-Shaw Flow -- 1.2 A Brief Review of the Theories of Free Dendritic 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 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 -- 2.1.2.1 Zeroth-Order Approximation Solutions -- 2.1.2.2 First-Order Approximation Solutions -- 2.1.3 Asymptotic Solutions in the Long-Wave Regime, k=O() -- 2.1.3.1 O(0) -- 2.1.3.2 O(2) -- 2.1.4 Asymptotic Solutions in the Extremely Short-Wave Regime, k=O(1) -- 2.1.4.1 O(0) -- 2.1.4.2 O(2) -- 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 -- 2.2.3.1 Zeroth-Order Approximation Solutions -- 2.2.3.2 First-Order Approximation Solutions -- 2.2.4 Asymptotic Solutions in the Long-Wave Regime, k=O() -- 2.2.5 Asymptotic Solutions in the Extremely Short-Wave Regime, k= O(1/E) -- g=O(1/E) -- 2.2.6 Some Remarks on Unidirectional Solidification -- References -- 3 Mathematical Formulation of Free Dendritic Growth from a Pure Melt -- 3.1 Three-Dimensional Free Dendritic Growth -- 3.2 Axisymmetric Free Dendrite Growth -- 3.3 Two-Dimensional Free Dendritic Growth -- Reference. , 4 Basic Steady State of Axisymmetric Dendritic Growth and Its Regular Perturbation Expansion -- 4.1 The Ivantsov Solution and Unsolved Fundamental Problems -- 4.2 Axially Symmetric Steady Needle Growth with Nonzero Surface Tension -- 4.2.1 Mathematical Formulation -- 4.2.2 Regular Perturbation Expansion Solutions (RPE) as E→0 -- 4.2.2.1 The Zeroth-Order Approximation O(0) -- 4.2.2.2 The First-Order Approximation O(2) -- 4.2.3 The Asymptotic Behavior of the Regular Perturbation Expansion Solution as ξ→∞ -- 4.2.4 Some Numerical Results of the Interface Shape -- 4.3 Summary and Discussion -- References -- 5 The Steady State for Dendritic Growth with Nonzero Surface Tension -- 5.1 The Nash-Glicksman Problem and the Classical Needle Crystal Solution -- 5.2 The Geometric Model and Solutions of NeedleCrystal Growth -- 5.2.1 Geometric Model of Dendritic Growth -- 5.2.2 The Segur-Kruskal Problem -- 5.2.3 Steady Nonclassical Needle Growth Problem -- 5.2.3.1 The Outer Expansion -- 5.2.3.2 The Composite Solution -- 5.2.3.3 The Properties of the Composite Solutions -- 5.2.4 Needle Crystal Formation Problem -- 5.3 The Nonclassical 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 Dendritic Growth from a Pure Melt -- 6.1 Linear Perturbed System Around the Basic State of Axisymmetric Dendritic 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 ζc of the Outer Solution -- 6.3 The Inner Solutions near the Singular Point ζc -- 6.4 Tip Inner Solution in the Tip Region -- 6.5 Global Trapped-Wave Modes and the Quantization Condition -- 6.6 Comparison of Theoretical Predictions with Experimental Data. , 6.6.1 The Dendrite Tip Velocity and Tip Radius -- 6.6.2 The Critical Number * -- 6.6.3 The Universal Scaling Parameter, * or σ*? -- 6.6.4 The Nature of the Dendrite-Tip: Steady or Oscillatory? -- 6.7 Three-Dimensional Nonaxisymmetric Spiral Dendritic Modes of Perturbed States -- 6.7.1 Mathematical Formulation of General Three-Dimensional Unsteady Dendritic Growth -- 6.7.2 The Basic State for Dendritic Growth with Nonzero Surface Tension -- 6.7.3 3D Linear Perturbed System -- 6.7.4 Multiple Variables Expansion Solution in the Outer Region -- 6.7.5 Zeroth-Order Approximation of Outer Solution -- 6.7.6 First-Order Approximation of the Outer Solution -- 6.7.7 The Inner Solution near the Singular Point ζc of the Outer Solution -- 6.7.8 Tip Inner Solution in the Tip Region -- 6.7.9 Global Trapped-Wave (GTW) Modes and Quantization Condition -- 6.8 A Brief Summary -- References -- 7 Free Dendritic Growth with Anisotropy -- 7.1 Mathematical Formulation for 2D Dendritic Growth with Anisotropy of Surface Tension -- 7.2 RPE for Basic Steady-State Solutions -- 7.2.1 The Zeroth-Order Approximation (=0) -- 7.2.2 The First-Order Approximation, O(2) -- 7.2.3 Asymptotic Behavior of the Regular Perturbation Expansion Solution as ξ→∞ -- 7.3 Global Interfacial Wave Instabilities of Two-Dimensional Dendritic Growth -- 7.3.1 Linear Perturbed System Around the Basic State -- 7.3.2 Multivariable Expansion Solution in the Outer Region -- 7.3.2.1 Zeroth-Order Approximation -- 7.3.2.2 First-Order Approximation -- 7.3.3 The Inner Equation near the Singular Point ζc -- 7.3.3.1 Case I: |σ0| = O(1) -- 7.3.3.2 Case II: |σ0| 1 -- 7.3.3.3 A Brief Summary -- 7.3.4 Matching Procedure and Connection Conditions -- 7.4 The Quantization Condition of Global Trapped-Wave Modes -- 7.5 The Quantization Condition of Global Low-FrequencyModes. , 7.6 The Selection Conditions for 2D Dendritic Growth -- 7.7 The Effect of Kinetic Attachment at the Interface on Dendritic Growth -- 7.7.1 Linear Perturbed System Around the Basic State -- 7.7.2 The Complex Spectrum of Eigenvalues with |σ0|=O(1) and GTW Instability -- 7.7.2.1 Zeroth-Order Approximation Solutions for GTW Instability -- 7.7.2.2 First-Order Approximation Solutions for GTW Instability -- 7.7.3 The Real Spectrum of Eigenvalues with |σ0|1 and LF Instability -- 7.8 Axially Symmetric Dendritic Growth with Anisotropy -- References -- 8 Three Dimensional Dendritic Growth from an Undercooled Binary Mixture -- 8.1 Mathematical Formulation of the Problem -- 8.2 Basic Steady-State Solution of the System -- 8.2.1 The Zeroth-Order Approximation Solution -- 8.3 Three-Dimensional Linear Perturbed States Around the Basic State -- 8.4 Multiple Variables Expansion Solution in the Outer Region -- 8.5 The MVE Solutions in the Outer Region -- 8.5.1 The Zeroth-Order Approximation -- 8.5.2 First-Order Approximation -- 8.6 The Inner Solutions near the Singular Point ζc -- 8.6.1 Leading-Order Approximation -- 8.7 Tip Inner Solution in the Tip Region -- 8.8 Global Trapped-Wave (GTW) Modes and QuantizationCondition -- 8.9 Axisymmetric Global Modes (m=0) -- 8.10 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 Mathematical Formulation of the Problem with Zero Surface Tension -- 9.4.1 The System of Curvilinear Coordinates (ξ, η) -- 9.4.2 Mathematical Formulation of the Problem in the (ξ,η) Coordinate System -- 9.4.3 The Regular Perturbation Expansion Solution for the Basic State as →0 -- 9.5 The Linear Perturbed System and the Outer Solutions. , 9.5.1 The Linear Perturbed System and the Multiple Variables Expansions -- 9.5.2 The Zeroth-Order Approximation Solutions -- 9.6 The Inner Equation near the Singular Point ζc -- 9.6.1 Case I: |σ0| = O(1) -- 9.6.2 Case II: |σ0| 1 -- 9.7 Eigenvalue Spectra and Instability Mechanisms -- 9.7.1 The Spectrum of Complex Eigenvalues and GTW Instability -- 9.7.1.1 Range 1: 12 ≤λ0 < -- 1 or 0 ≤a< -- 1 -- 9.7.1.2 Range 2: 0 < -- λ0 < -- 12 or a > -- 1 -- 9.7.2 The Spectrum of Real Eigenvalues and 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 -- 9.8.2.1 The GTW Mechanism -- 9.8.2.2 The LF Mechanism -- 9.9 The Selection Criteria of Finger Solutions -- Appendix: The Forms of Some Operators in the System of the Curvilinear Coordinate System (ξ,η) -- References -- 10 Spatially Periodic Deep-Cellular Growth -- 10.1 Introduction -- 10.2 Steady State of the System of Deep-Cellular Growth from a Binary Mixture -- 10.2.1 Mathematical Formulation of the Problem -- 10.2.2 Mathematical Formulation of the Problem in a Curvilinear Coordinate System (ξ, η) -- 10.2.3 The Basic Steady-State Solutions in the Far Field -- 10.2.4 The Mathematical Formulation of the Problem in the Near Field -- 10.2.5 Generalized Asymptotic Solution in the Outer Region -- 10.2.6 Regular Perturbation Expansion of the Solution in the Outer Region -- 10.2.6.1 Zeroth-Order Approximation O(ε0) -- 10.2.6.2 First-Order Approximation O(ε1) -- 10.2.6.3 The Second-Order Approximation O(ε2) -- 10.2.6.4 The Third-Order Approximation O(ε3) -- 10.2.7 Singular Perturbation Expansion Part of the Solution in the Outer Region -- 10.3 The Inner Steady-State Solution in the Root Region and Interface Closure of Deep Cellular Growth. , 10.3.1 Mathematical Formulation of the Problem in the Root Region.