The energy method, stability, and nonlinear convection.
type: Item
instance of: scholarly article
title: The energy method, stability, and nonlinear convection.
zbMATH Open document ID: 1032.76001
author: Brian Straughan
published in: Applied Mathematical Sciences
publication date: 2003-11-10T00:00:00Z
review text: 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 surfa…
Mathematics Subject Classification ID: 76-02
Mathematics Subject Classification ID: 76R05
Mathematics Subject Classification ID: 76E06
Mathematics Subject Classification ID: 76R10
Mathematics Subject Classification ID: 76E15
Mathematics Subject Classification ID: 76E30
zbMATH DE Number: 2002519
zbMATH Keywords: generalized energy method
zbMATH Keywords: nonlinear stability
zbMATH Keywords: diffusion equations
zbMATH Keywords: Navier-Stokes equations
zbMATH Keywords: Bénard problem
zbMATH Keywords: Darcy equations
zbMATH Keywords: porous convection
zbMATH Keywords: chemically reacting mixtures
zbMATH Keywords: geophysical applications
zbMATH Keywords: compressible fluids
zbMATH Keywords: internal heat generation
zbMATH Keywords: Lyapunov functionals
zbMATH Keywords: bio-convection
zbMATH Keywords: patterned ground formation
zbMATH Keywords: thawing subsea permafrost
zbMATH Keywords: surface tension
zbMATH Keywords: non-Newtonian fluids
zbMATH Keywords: time-varying gravity
zbMATH Keywords: electrohydrodynamics
zbMATH Keywords: magnetohydrodynamics
zbMATH Keywords: ferrorhydrodynamics
zbMATH Keywords: multicomponent media
zbMATH Keywords: Zeytounian model
zbMATH Keywords: Hills-Roberts model
zbMATH Keywords: Berezin-Hutter model
zbMATH Keywords: penetrative convection
zbMATH Keywords: radiation heating
zbMATH Keywords: ocean circualtion
reviewed by: Boris V. Loginov
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2002519