Interfacial wave theory of pattern formation. Selection of dendritic growth and viscous fingering in Hele-Shaw flow
Pattern formation has fascinated people well before science was born. However, our understanding of the simplest patterns is still very premature. Here the author distinguishes two major types of patterns, stationary and fractal, and focuses entirely on the former. Furthermore, the author limits himself to the two prominent examples -- dendrite growth in solidification and viscous fingering in a Hele-Shaw cell. These two pattern formation processes are described by similar partial differential equations and both involve a free interface which makes the problem very difficult. A handful of theoretical approaches have been suggested to treat dendrite growth and viscous fingering. In the first few chapters, the author reviews previous theories, presents governing equations, explains Mullins-Sekerka instability, and derives the classical Ivantsov steady state solution for dendrite growth with zero surface tension. This solution actually gives a family of dendrite whose tip velocity is undetermined. The selection of the tip velocity is an outstanding issue which was the focus of research activity over last 20 years. The main body of the book is an exposition of the interfacial wave theory developed by the author to handle the selection problem. This theory is presented in full detail for dendrite growth from a pure melt and from binary mixtures. The last chapter deals with formation of viscous fingers in a Hele-Shaw cell. The book does not assume preliminary knowledge of pattern formation phenomena. However, the literacy in asymptotic analysis (matched asymptotic expansions, Stokes lines, etc.) is required, or at least very helpful. The first part of the book will be a valuable introduction to pattern formation for a broad readership.
scientific article; zbMATH DE number 1113646