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

ISBN-13: 9783319526621 / Angielski / Twarda / 2017 / 591 str.

Jian-Jun Xu

Opis

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).

Recenzje

"I was particularly pleased with Xu's 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)

Spis treści

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.

O autorze

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.

 

Blurb

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 phenomena 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.