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E-raamat: Interfacial Wave Theory of Pattern Formation in Solidification: Dendrites, Fingers, Cells and Free Boundaries

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  • Sari: Springer Series in Synergetics
  • Ilmumisaeg: 23-May-2017
  • Kirjastus: Springer International Publishing AG
  • Keel: eng
  • ISBN-13: 9783319526638
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Interfacial Wave Theory of Pattern Formation in Solidification: Dendrites, Fingers, Cells and Free BoundariesSuurem pilt
  • Formaat: PDF+DRM
  • Sari: Springer Series in Synergetics
  • Ilmumisaeg: 23-May-2017
  • Kirjastus: Springer International Publishing AG
  • Keel: eng
  • ISBN-13: 9783319526638
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This comprehensive work explores interfacial instability and pattern formation in dynamic systems away from the equilibrium state in solidification and crystal growth. Further, this significantly expanded 2nd edition introduces and reviews the progress made during the last two decades. In particular, it describes the most prominent pattern formation phenomena commonly observed in material processing and crystal growth in the framework of the previously established interfacial wave theory, including free dendritic growth from undercooled melt, cellular growth and eutectic growth in directional solidification, as well as viscous fingering in Hele-Shaw flow. It elucidates the key problems, systematically derives their mathematical solutions by pursuing a unified, asymptotic approach, and finally carefully examines these results by comparing them with the available experimental results.The asymptotic approach described here will be useful for the investigation of pattern formation p

henomena occurring in a much broader class of inhomogeneous dynamical systems. In addition, the results on global stability and selection mechanisms of pattern formation will be of particular interest to researchers working on material processing and crystal growth.The stability mechanisms of a curved front and the pattern formation have been fundamental subjects in the areas of condensed-matter physics, materials science, crystal growth, and fluid mechanics for some time now. This book offers a stimulating and insightful introduction for all physicists, engineers and applied mathematicians working in the fields of soft condensed-matter physics, materials science, mechanical and chemical engineering, fluid dynamics, and nonlinear sciences.

Introduction.- Unidirectional Solidification and the Mullins-sekkerka instability.- Mathematical formulation of free dendrite growth from a pure melt.- Basic steady state of axi-symmetric free dendritic growth.- The steady state for dendritic growth with nonzero surface tension.- Global interfacial wave instability of dendrite growth from a pure melt.- Two dimensional dendritic growth.- Three dimensional dendritic growth from undercooled binary mixture.- Viscous fingering in a hele-shaw cell.- Spatially-periodic deep-cellular growth in hele-shaw cell.- Steady lamellar eutectic growth.

Arvustused

I was particularly pleased with Xus coverage of material that is often missing from conventional texts on pattern growth and nonequilibrium phenomena in condensed matter. such an ambitious, advanced text will surely be of great interest to mathematical physicists and researchers in chemical physics, engineering, and materials science. (Domenico Truzzolillo, Physics Today, March, 2018)

1 Introduction
1(28)
1.1 Interfacial Pattern Formation in Dendritic Growth and Hele--Shaw Flow
1(7)
1.2 A Brief Review of the Theories of Free Dendritic Growth
8(5)
1.2.1 Maximum Velocity Principle (1976)
9(1)
1.2.2 Marginal Stability Hypothesis (1978)
10(1)
1.2.3 Microscopic Solvability Condition (MSC) Theory (1986--1990s)
11(1)
1.2.4 Interfacial Wave (IFW) Theory (1990)
12(1)
1.3 Macroscopic Continuum Model
13(16)
1.3.1 Macroscopic Transport Equations
14(2)
1.3.2 The Interface Conditions
16(6)
1.3.3 The Scaling and the Dimensionless System
22(3)
References
25(4)
2 Unidirectional Solidification and Mullins--Sekerka Instability
29(46)
2.1 Solidification with Planar Interface from a Pure Melt
29(23)
2.1.1 Basic Steady-State Solution
31(1)
2.1.2 Unsteady Perturbed Solutions and Mullins-Sekerka Instability
31(10)
2.1.3 Asymptotic Solutions in the Long-Wave Regime, k = O(ε)
41(5)
2.1.4 Asymptotic Solutions in the Extremely Short-Wave Regime, k = O(1/ε)
46(6)
2.2 Unidirectional Solidification from a Binary Mixture
52(23)
2.2.1 Mathematical Formulation of the Problem
52(2)
2.2.2 Basic Steady State
54(3)
2.2.3 Unsteady Perturbed Solutions
57(8)
2.2.4 Asymptotic Solutions in the Long-Wave Regime, k = O(ε)
65(3)
2.2.5 Asymptotic Solutions in the Extremely Short-Wave Regime, {k = O(3/ε); g = O(1/ε)}
68(5)
2.2.6 Some Remarks on Unidirectional Solidification
73(1)
References
73(2)
3 Mathematical Formulation of Free Dendritic Growth from a Pure Melt
75(14)
3.1 Three-Dimensional Free Dendritic Growth
76(6)
3.2 Axisymmetric Free Dendrite Growth
82(2)
3.3 Two-Dimensional Free Dendritic Growth
84(5)
Reference
88(1)
4 Basic Steady State of Axisymmetric Dendritic Growth and Its Regular Perturbation Expansion
89(20)
4.1 The Ivantsov Solution and Unsolved Fundamental Problems
89(2)
4.2 Axially Symmetric Steady Needle Growth with Nonzero Surface Tension
91(15)
4.2.1 Mathematical Formulation
91(1)
4.2.2 Regular Perturbation Expansion Solutions (RPE) as ε → 0
92(8)
4.2.3 The Asymptotic Behavior of the Regular Perturbation Expansion Solution as ξ → ∞
100(2)
4.2.4 Some Numerical Results of the Interface Shape
102(4)
4.3 Summary and Discussion
106(3)
References
107(2)
5 The Steady State for Dendritic Growth with Nonzero Surface Tension
109(20)
5.1 The Nash--Glicksman Problem and the Classical Needle Crystal Solution
109(3)
5.2 The Geometric Model and Solutions of Needle Crystal Growth
112(14)
5.2.1 Geometric Model of Dendritic Growth
112(1)
5.2.2 The Segur--Kruskal Problem
113(2)
5.2.3 Steady Nonclassical Needle Growth Problem
115(8)
5.2.4 Needle Crystal Formation Problem
123(3)
5.3 The Nonclassical Steady State of Dendritic Growth with Nonzero Surface Tension
126(3)
5.3.1 The Complete Mathematical Formulation for Free Dendrite Growth
126(2)
References
128(1)
6 Global Interfacial Wave Instability of Dendritic Growth from a Pure Melt
129(82)
6.1 Linear Perturbed System Around the Basic State of Axisymmetric Dendritic Growth
130(2)
6.2 Outer Solution in the Outer Region Away from the Tip
132(20)
6.2.1 Zeroth-Order Approximation
138(6)
6.2.2 First-Order Approximation
144(6)
6.2.3 Singular Point ζc of the Outer Solution
150(2)
6.3 The Inner Solutions near the Singular Point ζc
152(11)
6.4 Tip Inner Solution in the Tip Region
163(3)
6.5 Global Trapped-Wave Modes and the Quantization Condition
166(8)
6.6 Comparison of Theoretical Predictions with Experimental Data
174(9)
6.6.1 The Dendrite Tip Velocity and Tip Radius
175(1)
6.6.2 The Critical Number ε*
176(3)
6.6.3 The Universal Scaling Parameter, ε* σ*?
179(2)
6.6.4 The Nature of the Dendrite-Tip: Steady or Oscillatory?
181(2)
6.7 Three-Dimensional Nonaxisymmetric Spiral Dendritic Modes of Perturbed States
183(25)
6.7.1 Mathematical Formulation of General Three-Dimensional Unsteady Dendritic Growth
184(1)
6.7.2 The Basic State for Dendritic Growth with Nonzero Surface Tension
185(1)
6.7.3 3D Linear Perturbed System
186(2)
6.7.4 Multiple Variables Expansion Solution in the Outer Region
188(2)
6.7.5 Zeroth-Order Approximation of Outer Solution
190(3)
6.7.6 First-Order Approximation of the Outer Solution
193(2)
6.7.7 The Inner Solution near the Singular Point ζc of the Outer Solution
195(2)
6.7.8 Tip Inner Solution in the Tip Region
197(3)
6.7.9 Global Trapped-Wave (GTW) Modes and Quantization Condition
200(8)
6.8 A Brief Summary
208(3)
References
209(2)
7 Free Dendritic Growth with Anisotropy
211(80)
7.1 Mathematical Formulation for 2D Dendritic Growth with Anisotropy of Surface Tension
212(3)
7.2 RPE for Basic Steady-State Solutions
215(8)
7.2.1 The Zeroth-Order Approximation (ε = 0)
215(1)
7.2.2 The First-Order Approximation, (ε2)
216(5)
7.2.3 Asymptotic Behavior of the Regular Perturbation Expansion Solution as ξ → ∞
221(2)
7.3 Global Interfacial Wave Instabilities of Two-Dimensional Dendritic Growth
223(22)
7.3.1 Linear Perturbed System Around the Basic State
223(1)
7.3.2 Multivariate Expansion Solution in the Outer Region
224(11)
7.3.3 The Inner Equation near the Singular Point ζc
235(9)
7.3.4 Matching Procedure and Connection Conditions
244(1)
7.4 The Quantization Condition of Global Trapped-Wave Modes
245(8)
7.5 The Quantization Condition of Global Low-Frequency Modes
253(12)
7.6 The Selection Conditions for 2D Dendritic Growth
265(5)
7.7 The Effect of Kinetic Attachment at the Interface on Dendritic Growth
270(19)
7.7.1 Linear Perturbed System Around the Basic State
272(3)
7.7.2 The Complex Spectrum of Eigenvalues with |σ0| = O(1) and GTW Instability
275(3)
7.7.3 The Real Spectrum of Eigenvalues with |σ0| << 1 and LF Instability
278(11)
7.8 Axially Symmetric Dendritic Growth with Anisotropy
289(2)
References
290(1)
8 Three Dimensional Dendritic Growth from an Undercooled Binary Mixture
291(48)
8.1 Mathematical Formulation of the Problem
292(3)
8.2 Basic Steady-State Solution of the System
295(1)
8.2.1 The Zeroth-Order Approximation Solution
295(1)
8.3 Three-Dimensional Linear Perturbed States Around the Basic State
296(2)
8.4 Multiple Variables Expansion Solution in the Outer Region
298(3)
8.5 The MVE Solutions in the Outer Region
301(12)
8.5.1 The Zeroth-Order Approximation
301(4)
8.5.2 First-Order Approximation
305(8)
8.6 The Inner Solutions near the Singular Point ζc
313(7)
8.6.1 Leading-Order Approximation
315(5)
8.7 Tip Inner Solution in the Tip Region
320(4)
8.8 Global Trapped-Wave (GTW) Modes and Quantization Condition
324(3)
8.9 Axisymmetric Global Modes (m = 0)
327(4)
8.10 Comparisons of Theoretical Results with Experimental Data
331(8)
References
337(2)
9 Viscous Fingering in a Hele--Shaw Cell
339(58)
9.1 Introduction
339(6)
9.2 Mathematical Formulation of the Problem
345(2)
9.3 The Smooth Finger Solution with Zero Surface Tension
347(6)
9.4 Mathematical Formulation of the Problem with Zero Surface Tension
353(5)
9.4.1 The System of Curvilinear Coordinates (ξ, η))
353(1)
9.4.2 Mathematical Formulation of the Problem in the (ε, η) Coordinate System
354(2)
9.4.3 The Regular Perturbation Expansion Solution for the Basic State as ε → 0
356(2)
9.5 The Linear Perturbed System and the Outer Solutions
358(9)
9.5.1 The Linear Perturbed System and the Multiple Variables Expansions
358(4)
9.5.2 The Zeroth-Order Approximation Solutions
362(5)
9.6 The Inner Equation near the Singular Point ζc
367(7)
9.6.1 Case I: |σ0| = O(1)
370(1)
9.6.2 Case II: |σ0| << 1
371(3)
9.7 Eigenvalue Spectra and Instability Mechanisms
374(9)
9.7.1 The Spectrum of Complex Eigenvalues and GTW Instability
374(4)
9.7.2 The Spectrum of Real Eigenvalues and LF Instability
378(5)
9.8 Fingering Flow with a Nose Bubble
383(6)
9.8.1 The Basic State of Finger Formation with a Nose Bubble and Its Linear Perturbation
383(3)
9.8.2 The Quantization Conditions for the System with a Nose Bubble
386(3)
9.9 The Selection Criteria of Finger Solutions
389(8)
References
394(3)
10 Spatially Periodic Deep-Cellular Growth
397(106)
10.1 Introduction
397(2)
10.2 Steady State of the System of Deep-Cellular Growth from a Binary Mixture
399(27)
10.2.1 Mathematical Formulation of the Problem
399(2)
10.2.2 Mathematical Formulation of the Problem in a Curvilinear Coordinate System (ξ, η)
401(1)
10.2.3 The Basic Steady-State Solutions in the Far Field
402(1)
10.2.4 The Mathematical Formulation of the Problem in the Near Field
403(1)
10.2.5 Generalized Asymptotic Solution in the Outer Region
404(2)
10.2.6 Regular Perturbation Expansion of the Solution in the Outer Region
406(12)
10.2.7 Singular Perturbation Expansion Part of the Solution in the Outer Region
418(8)
10.3 The Inner Steady-State Solution in the Root Region and Interface Closure of Deep Cellular Growth
426(24)
10.3.1 Mathematical Formulation of the Problem in the Root Region
428(3)
10.3.2 The SPE Part of the Root Solution in the Root Region
431(2)
10.3.3 The Zeroth-Order Approximation O(1)
433(3)
10.3.4 The Root Solution in Subregion (II)
436(2)
10.3.5 Root Solution in Subregion (I)
438(2)
10.3.6 The Root Inner Equation in the Vicinity of (ξc, 0)
440(3)
10.3.7 Matching
443(1)
10.3.8 The Root Inner Solutions in Subregion (T) Matching the Root Solution in the Sector (S2)
443(1)
10.3.9 The Root Inner Solutions in Subregion (T) Matching the Root Solution in the Sector (S1)
444(1)
10.3.10 The Quantization Condition for the Eigenvalue ξ*
445(5)
10.4 Global Instabilities, Origin and Essence of Side Branches
450(37)
10.4.1 Mathematical Formulation of General Unsteady Cellular Growth
451(1)
10.4.2 The Basic Steady-State Solutions
452(1)
10.4.3 Linear Perturbed System
453(1)
10.4.4 Multiple-Variable Expansion (MVE) Solutions of Perturbed States in the Outer Region
454(9)
10.4.5 The Inner Solution in the Vicinity of the Singular Point (ζc, 0)
463(1)
10.4.6 Leading-Order Inner Solution in the Vicinity of ζc
464(4)
10.4.7 Global Instability Mechanisms
468(11)
10.4.8 Global Stability Diagram and Selection Principle of Arrayed-Cellular Growth
479(1)
10.4.9 Arrayed-Cellular Pattern Formation and Comparisons with Experimental Data
480(2)
10.4.10 Transition from Cellular Array to Dendritic Array
482(2)
10.4.11 The Origin and Essence of Side-Branching in a Strong Oscillatory Dendritic Array
484(3)
10.5 A Brief Summary
487(16)
References
499(4)
11 Lamellar Eutectic Growth
503(82)
11.1 Introduction
503(2)
11.2 Mathematical Formulation of Eutectic Growth from a Binary Mixture
505(7)
11.2.1 Scales and Dimensionless Parameters
505(3)
11.2.2 Formulation of the Problem in the Liquid Phase
508(2)
11.2.3 The Formulation of the Problem in the Solid Phase
510(1)
11.2.4 The Formulation of the Problem in the Far Field
511(1)
11.3 Formulation of the Problem with Multiple Variables in the Near Field
512(2)
11.4 The Steady-State Solution in the Liquid Phase
514(6)
11.4.1 The Solutions for the Concentration Field in Approximations of Orders (0, 0) and (1, 0)
514(1)
11.4.2 The Solutions for the Concentration Field in the Approximation of Order (0, 1)
514(1)
11.4.3 The Solutions for the Triple Point Location in the Approximation of Order (1, 0)
515(1)
11.4.4 Solutions for the Concentration Field in the Approximation of Order (1, 1)
516(2)
11.4.5 Approximations of Higher Order
518(2)
11.5 The Solution for the Interface Shape Between Liquid and Solid
520(11)
11.5.1 The Outer Solution for the Interface Shape in the Outer Region Away from the Triple Point
520(2)
11.5.2 The Inner Solution for the Interface Shape in the Inner Region near the Triple Point
522(3)
11.5.3 The Composite Solution for the Interface Shape
525(6)
11.6 Further Numerical Computations of Asymptotic Solutions
531(3)
11.7 Comparisons of Theoretical Solutions with Experimental Data
534(3)
11.8 Global Interfacial Instabilities of Eutectic Growth
537(19)
11.8.1 Dimensionless Form of the System for Unsteady Eutectic Growth
539(2)
11.8.2 Basic States
541(1)
11.8.3 Linear Perturbed States of Lamellar Eutectic Growth
542(2)
11.8.4 Multiple Variables Expansion Form of the Perturbed System
544(2)
11.8.5 Leading-Order Approximation
546(8)
11.8.6 Perturbed State Solutions in the Subinterval (0 ≤ x < w0)
554(1)
11.8.7 Perturbed State Solutions in the Subinterval (w0 < x ≤ 1)
555(1)
11.9 The Global Modes Solutions and Quantization Conditions
556(5)
11.10 The Global Steady (ST) Mode of Perturbed States
561(5)
11.10.1 The Untilted ST-Modes
561(5)
11.11 The Global Instability Mechanism
566(3)
11.12 Discussions on Selection of Eutectic Pattern Formation
569(2)
11.12.1 Comparison with Experimental Data
570(1)
11.13 A Brief Summary
571(1)
11.14 Appendix 1: The Derivation of the Basic Solution in the Far Field
572(2)
11.15 Appendix 2: The Fourier Series of Some Special Functions
574(11)
References
584(1)
Bibliography 585(2)
Index 587
Dr Jian-Jun Xu is a professor in the Department of Mathematics and Statistics at McGill University, Canada. He is an outstanding applied mathematician, working in the interdisciplinary area of applied mathematics, condensed matter physics, material science and fluid dynamics. He has published four monographs and about one hundred research papers. His expertise includes but is not limited to: asymptotics and numerical analysis, dynamical systems of non-Linear PDE with particular emphasis in the areas of solidification physics,  interfacial wave theory, pattern formation and crystal growth.











 
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