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Interfacial Wave Theory of Pattern Formation

Selection of Dendritic Growth and Viscous Fingering in Hele-Shaw Flow

  • Book
  • © 1998

Overview

Authors:
  1. Jian-Jun Xu

    1. Department of Mathematics and Statistics, McGill University, Montreal, Canada

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  • The methods described in this book use analytic methods.
  • This as well as the many applications given in considerable detail make this monograph a unique welcomed addition to the existing literature.

Part of the book series: Springer Series in Synergetics (SSSYN, volume 68)

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Table of contents (9 chapters)

  1. Front Matter

    Pages I-XII
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  2. Introduction

    • Jian-Jun Xu
    Pages 1-26

  3. Unidirectional Solidification and the Mullins-Sekerka Instability

    • Jian-Jun Xu
    Pages 27-67

  4. Mathematical Formulation of Free Dendrite Growth from a Pure Melt

    • Jian-Jun Xu
    Pages 69-78

  5. Steady State of Dendrite Growth with Zero Surface Tension and Its Regular Perturbation Expansion

    • Jian-Jun Xu
    Pages 79-103

  6. The Steady State for Dendrite Growth with Nonzero Surface Tension

    • Jian-Jun Xu
    Pages 105-123

  7. Global Interfacial Wave Instability of Dendrite Growth from a Pure Melt

    • Jian-Jun Xu
    Pages 125-180

  8. The Effect of Surface Tension Anisotropy and Low-Frequency Instability on Dendrite Growth

    • Jian-Jun Xu
    Pages 181-212

  9. Three-Dimensional Dendrite Growth from Binary Mixtures

    • Jian-Jun Xu
    Pages 213-241

  10. Viscous Fingering in a Hele-Shaw Cell

    • Jian-Jun Xu
    Pages 243-290

  11. Back Matter

    Pages 291-296
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Keywords

About this book

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).

Authors and Affiliations

  • Department of Mathematics and Statistics, McGill University, Montreal, Canada

    Jian-Jun Xu

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