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Stability and wave motion in porous media. (English) Zbl 1149.76002
Applied Mathematical Scienes 165. New York, NY: Springer (ISBN 978-0-387-76541-9/hbk). xiv, 437 p. (2008).
This book contains 9 chapters. A brief description of Darcy, Forchheimer and Brinkman models is given in introduction, where the model of Nunziato-Cowin for materials with voids is also described. Structural stability is studied in chapter 2. Chapter 3 examines the spatial decay for Darcy, Forchheimer and Krishnamurti models. Convection in porous media is studied in chapter 4. The stability of some biological processes, micropolar flows, viscoelastic convection and second-grade fluids are topics of chapter 5. The last chapters are concerned with fluid-porous interface problems, elastic materials with voids, poroacoustic waves, and numerical solutions of eigenvalue problems. Some complex mathematical tools are used: functional Sobolev spaces and inequalities, Babushka-Aziz and Horgan-Payne inequalities, the maximum principle. A large list of references is given.
Unfortunately, the book does not contain studies concerning saturation and capillary phenomena in porous media. In such models, a wave-type solution exists (for example, in the absence of capillary pressure there exists Buckley-Leverett solution). The stability of these waves is studied in several papers – see [G. I. Barenblatt, V. M. Entov and V. M. Ryzhik, Theory of fluid flows through natural rocks. Dordrecht: Kluwer Academic Publishers (1990; Zbl 0769.76001)].

MSC:
76-02 Research exposition (monographs, survey articles) pertaining to fluid mechanics
76S05 Flows in porous media; filtration; seepage
76Exx Hydrodynamic stability
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