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Mathematical models of convection. (English) Zbl 1257.76001

de Gruyter Studies in Mathematical Physics 5. Berlin: De Gruyter (ISBN 978-3-11-025814-1/hbk; 978-3-11-025859-2/ebook). xv, 417 p. (2012).
The study of convection is of relevance in many practical fields such as space technologies, power engineering, metallurgy, environmental science, meteorology, geo- and astrophysics, chemistry, crystal physics, petroleum technology, etc. Owing to the increasing accuracy of measurements and detailed mathematical models, it is possible to pose and successfully solve new problems in this field, e.g. the production of superpure materials under microgravity conditions. Many new approaches have also been developed to classical problems such as transport properties of porous media.
This book presents a careful and detailed introduction to the modern mathematical models of convection. The attention also was paid to numerical simulations of convective flows under microgravity conditions (unsteady microconvection in canonical domains with solid boundaries, steady microconvection in domains with free boundaries, etc.). The book consists of 10 chapters, a list of references and an index. The description of these chapters is, in short, as follows:
Chapter 1 “Equation of fluid motion” describes basic hypotheses of continuum, two methods for continuum description (translation formula), integral conservation laws, thermodynamics and classical models of liquids and gases. Chapter 2 “Conditions on the interface between fluids and on solid walls” presents the notion of interface, kinematic conditions, dynamic conditions, elements of thermodynamics of the interface, conditions of continuity, energy transfer across the interface, free surfaces and additional conditions. Chapter 3 “Models of convection of an isothermally incompressible fluid” deals with isothermally incompressible fluids, equations of thermal convection of an isothermally incompressible fluid, models of linear thermal expansion, boundary conditions and two problems of convection. Chapter 4 “Hierarcy of convection models in closed volumes” describes initial relations, similarity criteria, transition to dimensional variables, expansion in a small parameter, equations of microconvection of an isothermally incompressible fluid, Oberbeck-Boussinesq equations, linear models of transitional processes, convection of nonisothermal liquids and gases under microgravity conditions, exact solutions in an infinite band and analysis of well-posedness of the initial-boundary value problems for equations of convection of a weakly compressible fluid. Chapter 5 “Invariant submodels of microconvection equations” deals with basic models and their group properties, optimal subsystems of subalgebras and factor-systems, steady solution of microconvection equations in a vertical layer, solvability of a non-standard boundary value problem, unsteady solution of microconvection equations in an infinite band and invariant solutions of microconvection equations that describe the motion with an interface. Chapter 6 “Group properties of equations of thermodiffusion motion” is devoted to Lie group of thermodiffusion equations, group properties of two-dimensional equations, and invariant submodels and exact solutions of thermodiffusion equations. Chapter 7 “Stability of equilibrium states in the Oberbeck-Boussinesq model” presents the convective instability of a horizontal layer with oscillations of temperature on the free boundary, instability of a liquid layers with an interface, and convection in a rotating fluid layer under microgravity conditions. Chapter 8 “Small perturbations and stability of plane layers in the microconvection model” describes equations of small perturbations, stability of the equilibrium state of a plane layer with solid walls, emergence of microconvection in a plane layer with a free boundary, and stability of a steady flow in a vertical layer. Chapter 9 “Numerical simulation of convective flows under microgravity conditions” deals with numerical methods used for calculations, numerical study of unsteady microconvection in canonical domains with solid boundaries, numerical study of steady microconvection in domains with free boundaries, study of convection induced by volume expansion, and convection in miscible fluids. The final Chapter 10 “Convective flows in tubes and layers” examines the group-theoretical nature of Birikh solution and its generalizations, an axial convective flow in a rotating tube with a longitudinal temperature gradient, unsteady analogs of the Birkh solutions and model of viscous layer deformation by thermocapillary forces. A large list of 236 important books and papers concludes this book.
In the reviewer’s opinion, this book provides a fundamental and comprehensive presentation of the mathematical and physical theory of fluid flows in non-classical models of convection, pointing out the most important practical applications. The book is excellently written and readable. Results of numerical solutions are given graphically and in tabular form. The book will be of great interest to a wide range of specialists working in the area of convection. It can be also recommended as a text for seminars and courses, as well as for independent study. Some chapters of the book present state-of-the-art reviews, and they can stimulate new research in the area of convection flows of Newtonian and non-Newtonian fluids, both from theoretical and application point of views.

MSC:

76-02 Research exposition (monographs, survey articles) pertaining to fluid mechanics
76E06 Convection in hydrodynamic stability
76M60 Symmetry analysis, Lie group and Lie algebra methods applied to problems in fluid mechanics
80A20 Heat and mass transfer, heat flow (MSC2010)
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