Interfacial Wave Theory of Pattern Formation : Selection of Dendritic Growth and Viscous Fingering in Hele-Shaw Flow

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).

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.