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Theory of solidification. (English) Zbl 0991.76002

Cambridge Monographs on Mechanics and Applied Mathematics. Cambridge: Cambridge University Press. xiv, 385 p. (2001).
This is a good book where, using the equations of fluid flow and heat and mass transfer, the author develops step by step the theory of solidification from the melt. The book is divided into the following chapters: (1) Introduction; (2) Pure substances; (3) Binary substances; (4) Nonlinear theory of directional solidification; (5) Anisotropy; (6) Nonequilibrium; (7) Dendrites; (8) Eutectics; (9) Microscale fluid flow; (10) Mesoscale fluid flow; (11) Phase-field models.
Chapter 2 deals with the solidification in pure liquids in the presence of thermal gradients. By assuming uniform temperature, by many examples the author explains how solid is created by growth from a saturated liquid. Chapter 3 describes the solidification in binary liquids. It is well-known that in binary liquid, during freezing process, solute is not soluble in crystalline form. Here the author discusses the mechanism of morphological instability in a dilute binary alloy. The nonlinear theory of directional solidification is discussed in chapter 4 in both two- and three-dimensional cases. Chapter 5 is devoted to crystal formation by assuming anisotropic medium, whereas in chapter 6 the author examines the effects of focussing high-power electron or laser beams on the body surface. These cause nonequilibrium, and the effect of nonequilibrium is also discussed in this chapter. The formation of dendrites is presented in chapter 7, and the theory of eutectics is discussed in chapter 8 where good photographs of this phenomenon are given.
Chapter 9 is quite interesting, being devoted to microscale fluid flows. There are three types of convection that can occur in the liquid: 1) convection due to different densities of liquid and solid; 2) buoyancy-driven convection due to density gradients; 3) when fluid-fluid interfaces are present, steep temperature and concentration gradients on these interfaces can cause steady convection. All these steady flows can become unstable. Here the author gives a general formulation of solidification combined with convection . A large variety of situations are analyzed where the fluid flows play the major role in the solidification process. Chapter 10 describes the solidification in mushy zone, where both dendrites and interstitial liquid are present. The flow in such a zone is treated as the flow in a porous region, and both linear and nonlinear convective instabilities are studied in this chapter.
In many previous studies, the surface formed by two fluid layers was assumed to be a mathematical surface of zero thickness, endowed with surface properties deemed to be appropriate to the physics of the problem. This is not always in reality, and it is practically important to develop an adequate model of solid-liquid system which explains the onset of soldification. In the last chapter 11, the author introduces such a model, the phase-field model, discusses some problems related to this model, and examines the domain of applicability of this model.
All the chapters are supplemented with detailed and quite recent references, and this can be considered as an additional advantage of this book. The reviewer strongly recommends the book to those who want to start their research in the area of viscous flows with heat and mass transfer.

MSC:

76-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to fluid mechanics
76Txx Multiphase and multicomponent flows
76Rxx Diffusion and convection
80A22 Stefan problems, phase changes, etc.
80A20 Heat and mass transfer, heat flow (MSC2010)
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