Mathematical Models of Convection (eBook)

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Victor K. Andreev, Institute of Computational Modelling, Siberian Branch of the Russian Academy of Sciences, Krasnoyarsk, Russia; Yuri A. Gaponenko, Institute of Computational Modelling, Siberian Branch of the Russian Academy of Sciences, Krasnoyarsk, Russia; Olga N. Goncharova, Altai State University, Barnaul, Russia; and Vladislav V. Pukhnachev, Lavrentyev Institute of Hydrodynamics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia.

Preface 5
List of contributing authors 11
1 Equations of fluid motion 17
1.1 Basic hypotheses of continuum 17
1.2 Two methods for the continuum description. Translation formula 20
1.3 Integral conservation laws. Equations of continuous motion 23
1.4 Thermodynamics aspects 29
1.5 Classical models of liquids and gases 32
2 Conditions on the interface between fluids and on solid walls 40
2.1 Notion of the interface 40
2.2 Kinematic condition 41
2.3 Dynamic condition 42
2.4 Elements of thermodynamics of the interface 47
2.5 Conditions of continuity 49
2.6 Energy transfer across the interface 50
2.7 Free surfaces 55
2.8 Additional conditions 57
3 Models of convection of an isothermally incompressible fluid 60
3.1 Isothermally incompressible fluid 60
3.2 Equations of thermal convection of an isothermally incompressible fluid 62
3.3 Model of linear thermal expansion 63
3.4 Some submodels 65
3.5 On boundary conditions 67
3.6 Two problems of convection 69
4 Hierarchy of convection models in closed volumes 76
4.1 Initial relations 76
4.2 Similarity criteria 78
4.3 Transition to dimensional variables 80
4.4 Expansion in the small parameter 83
4.5 Equations of microconvection of an isothermally incompressible fluid 87
4.6 Oberbeck-Boussinesq equations 90
4.7 Linear model of the transitional process 91
4.8 Some conclusions 94
4.9 Convection of nonisothermal liquids and gases under microgravity conditions 97
4.10 Convection of a thermally inhomogeneous weakly compressible fluid 104
4.11 Exact solutions in an infinite band 109
4.12 Analysis of well-posedness of the initial-boundary problem for equations of convection of a weakly compressible fluid 121
5 Invariant submodels of microconvection equations 131
5.1 Basic model and its group properties 131
5.2 Optimal subsystems of the subalgebras T1 and T2, factor-systems, and some solutions 134
5.3 On one steady solution of microconvection equations in a vertical layer 142
5.4 Solvability of a nonstandard boundary-value problem 153
5.5 Unsteady solution of microconvection equations in an infinite band 160
5.6 Invariant solutions of microconvection equations that describe the motion with an interface 166
6 Group properties of equations of thermodiffusion motion 173
6.1 Lie group of thermodiffusion equations 173
6.2 Group properties of two-dimensional equations 190
6.3 Invariant submodels and exact solutions of thermodiffusion equations 198
7 Stability of equilibrium states in the Oberbeck-Boussinesq model 214
7.1 Convective instability of a horizontal layer with oscillations of temperature on the free boundary 214
7.2 Instability of a liquid layers with an interface 224
7.3 Convection in a rotating fluid layer under microgravity conditions 233
8 Small perturbations and stability of plane layers in the microconvection model 243
8.1 Equations of small perturbations 243
8.2 Stability of the equilibrium state of a plane layer with solid walls 247
8.3 Emergence of microconvection in a plane layer with a free boundary 257
8.4 Stability of a steady flow in a vertical layer 268
9 Numerical simulation of convective flows under microgravity conditions 279
9.1 Numerical methods used for calculations 279
9.2 Numerical study of unsteady microconvection in canonical domains with solid boundaries 290
9.3 Numerical study of steady microconvection in domains with free boundaries 307
9.4 Study of convection induced by volume expansion 323
9.5 Convection in miscible fluids 343
10 Convective flows in tubes and layers 363
10.1 Group-theoretical nature of the Birikh solution and its generalizations 363
10.2 An axial convective flow in a rotating tube with a longitudinal temperature gradient 371
10.3 Unsteady analogs of the Birikh solutions 379
10.4 Model of viscous layer deformation by thermocapillary forces 393
Bibliography 417
Index 431

lt;P>"This book presents a careful and detailed introduction to the modern mathematical models of convection. […] 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." Zentralblatt für Mathematik