Interfacial Wave Theory of Pattern Formation [electronic resource] : Selection of Dendritic Growth and Viscous Fingering in Hele-Shaw Flow / by Jian-Jun Xu.
- Xu, Jian-Jun, 1940-
- Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 1998.
- 1st ed. 1998.
- Physical Description:
- XII, 296 pages : online resource
- Additional Creators:
- SpringerLink (Online service)
- Springer Series in Synergetics, 0172-7389 ; 68
- 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 -- 1.3 Macroscopic Continuum Model -- 2. Unidirectional Solidification and the Mullins-Sekerka Instability -- 2.1 Solidification with Planar Interface from a Pure Melt -- 2.2 Unidirectional Solidification from a Binary Mixture -- 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 -- 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 -- 4.4 Summary and Discussion -- 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 -- 5.3 The Nonclassic Steady State of Dendritic Growth with Nonzero Surface Tension -- 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 -- 6.3 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 -- 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 -- 7.4 Matching Conditions -- 7.5 The Spectra of Eigenvalues and Instability Mechanisms -- 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 -- 8.5 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 -- 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 -- 9.7 Eigenvalues Spectra and Instability Mechanisms -- 9.8 Fingering Flow with a Nose Bubble -- 9.9 The Selection Criteria of Finger Solutions.
- The stabiltiy mechanisms of a curved front and the pattern formation in dendrite growth and viscous fingering have been fundamental subjects in the areas of condensed-matter physics, materials science, crystal growth, and fluid mechanics for about half a century. This book studies interfacial instability 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 the key problems step by step and systematically derives their mathematical solutions, on the basis of the newly established interfacial wave theory. Finally, it carefully examines these results by comparisons with the available experimental results. The unified, asymptotic approach described in this book will be useful for investigation of pattern formation phenomena occurring in a much broader class of inhomogeneous dynamic systems. The results on global stability and selection mechanisms of pattern formation given in this book will be of particular interest to researchers. For the readers who are not familiar with this rapidly growing interdisciplinary field, this book will be a stimulating and valuable introduction.
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