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**The energy method, stability, and nonlinear convection.
2nd rev. ed.**
*(English)*
Zbl 1032.76001

Applied Mathematical Sciences. 91. New York, NY: Springer. xii, 447 p. (2004).

The author presents energy method in various convection problems, and obtains nonlinear energy stability results by means of an integral inequality technique. In combination with Lyapunov methods, this approach is named generalized energy method. Separate attention is paid to nonlinear stability. The basic ideas of energy method and its advantages are illustrated in Chapter 2 on examples of diffusion equations in different situations, in particular on the system of interaction-diffusion equations interesting in many chemical and biological processes. This system is a prelude to fluid dynamical stability problems.

Chapter 3 contains energy stability results for Navier-Stokes equations of viscous incompressible fluid, for Bénard problem in a layer of fluid heated from below, and for Darcy equations of convection in porous media. Chapter 4 develops the method of coupling parameters in the theory of nonlinear energy stability for classical Bénard problem as well for porous convection (Darcy, Forchheimer and Brinkman equations). Symmetry and competing effects in convection problems are examined. Convection with internal heat generation and in a variable gravity fields is investigated. Chapter 5 is devoted to convection problems in half-space.

In Chapter 6 the author considers convection problems where different generalized energies (i.e. Lyapunov functionals) are employed to examine different effects. These are the effect of rotation in Bénard problem, nonlinear stability in rotating porous convection, bio-convection, and convection in a porous vertical slab. In Chapter 7 two geophysical applications are investigated: patterned (polygonal) ground formation, and convection in thawing subsea permafrost. Chapter 8 is devoted to surface tension driven convection in the presence of surface film and in a fluid overlying a porous material. Chapter 9 investigates convection in non-Newtonian fluids. In Chapter 10 the author considers the problems of time-varying gravity and surface temperature and patterned ground formation with time-dependent surface heating.

In Chapters 11, 12 convection problems of electrohydrodynamics, magnetohydrodynamics and ferrohydrodynamics are investigated by generalized energy method. Chapter 13 is devoted to convective instabilities for reacting viscous fluids far from equilibrium. Chapter 14 contains applications to convection diffusion in multicomponent media. Convection problems in compressible fluids are investigated in Chapter 15. Here Zeytounian model [R. Kh. Zeytounian, Int. J. Eng. Sci. 27, 1361-1366 (1989; Zbl 0693.76055)] governing thermal convection in a deep fluid layer, Hills and Roberts model [R. N. Hills and P. H. Roberts, Stab. Appl. Anal. Cont. Media, 1, 205-212 (1991)] and Berezin-Hutter model [Y. A. Berezin and K. Hutter, Math. Models Methods Appl. Sci. 7, 113-123 (1997; Zbl 0873.76028)] of convection in slightly compressible fluids are studied by generalized energy method. Convection problems for fluids with temperature-dependent properties are considered in Chapter 16. Chapter 17, “Penetrative convection”, contains unconditional stability results for the cases of quadratic buoyancy and in the presence of internal heat source, for convection with radiation heating and for penetrative convection in porous media. Chapter 18 is devoted to nonlinear stability in ocean circulation models.

In the concluding Chapter 19, the sections “Numerical solution of eigenvalue problem” and “Useful inequalities” (important for convection investigation by generalized energy method mathematical means) are separated. Comparatively with the previous edition [for review of the 1st ed. see (1992; Zbl 0743.76006)], the reviewed monograph contains much more physical applications, especially in Chapters 15-17.

Chapter 3 contains energy stability results for Navier-Stokes equations of viscous incompressible fluid, for Bénard problem in a layer of fluid heated from below, and for Darcy equations of convection in porous media. Chapter 4 develops the method of coupling parameters in the theory of nonlinear energy stability for classical Bénard problem as well for porous convection (Darcy, Forchheimer and Brinkman equations). Symmetry and competing effects in convection problems are examined. Convection with internal heat generation and in a variable gravity fields is investigated. Chapter 5 is devoted to convection problems in half-space.

In Chapter 6 the author considers convection problems where different generalized energies (i.e. Lyapunov functionals) are employed to examine different effects. These are the effect of rotation in Bénard problem, nonlinear stability in rotating porous convection, bio-convection, and convection in a porous vertical slab. In Chapter 7 two geophysical applications are investigated: patterned (polygonal) ground formation, and convection in thawing subsea permafrost. Chapter 8 is devoted to surface tension driven convection in the presence of surface film and in a fluid overlying a porous material. Chapter 9 investigates convection in non-Newtonian fluids. In Chapter 10 the author considers the problems of time-varying gravity and surface temperature and patterned ground formation with time-dependent surface heating.

In Chapters 11, 12 convection problems of electrohydrodynamics, magnetohydrodynamics and ferrohydrodynamics are investigated by generalized energy method. Chapter 13 is devoted to convective instabilities for reacting viscous fluids far from equilibrium. Chapter 14 contains applications to convection diffusion in multicomponent media. Convection problems in compressible fluids are investigated in Chapter 15. Here Zeytounian model [R. Kh. Zeytounian, Int. J. Eng. Sci. 27, 1361-1366 (1989; Zbl 0693.76055)] governing thermal convection in a deep fluid layer, Hills and Roberts model [R. N. Hills and P. H. Roberts, Stab. Appl. Anal. Cont. Media, 1, 205-212 (1991)] and Berezin-Hutter model [Y. A. Berezin and K. Hutter, Math. Models Methods Appl. Sci. 7, 113-123 (1997; Zbl 0873.76028)] of convection in slightly compressible fluids are studied by generalized energy method. Convection problems for fluids with temperature-dependent properties are considered in Chapter 16. Chapter 17, “Penetrative convection”, contains unconditional stability results for the cases of quadratic buoyancy and in the presence of internal heat source, for convection with radiation heating and for penetrative convection in porous media. Chapter 18 is devoted to nonlinear stability in ocean circulation models.

In the concluding Chapter 19, the sections “Numerical solution of eigenvalue problem” and “Useful inequalities” (important for convection investigation by generalized energy method mathematical means) are separated. Comparatively with the previous edition [for review of the 1st ed. see (1992; Zbl 0743.76006)], the reviewed monograph contains much more physical applications, especially in Chapters 15-17.

Reviewer: Boris V.Loginov (Ul’yanovsk)

### MSC:

76-02 | Research exposition (monographs, survey articles) pertaining to fluid mechanics |

76R05 | Forced convection |

76E06 | Convection in hydrodynamic stability |

76R10 | Free convection |

76E15 | Absolute and convective instability and stability in hydrodynamic stability |

76E30 | Nonlinear effects in hydrodynamic stability |