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Convection with temperature dependent viscosity in a porous medium: nonlinear stability and the Brinkman effect.

Lorna Richardson, Brian Straughan (1993)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We establish a nonlinear energy stability theory for the problem of convection in a porous medium when the viscosity depends on the temperature. This is, in fact, the situation which is true in real life and has many applications to geophysics. The nonlinear analysis presented here would appear to require the presence of a Brinkman term in the momentum equation, rather than just the normal form of Darcy's law.

Convective Instability of Reaction Fronts in Porous Media

K. Allali, A. Ducrot, A. Taik, V. Volpert (2010)

Mathematical Modelling of Natural Phenomena

We study the influence of natural convection on stability of reaction fronts in porous media. The model consists of the heat equation, of the equation for the depth of conversion and of the equations of motion under the Darcy law. Linear stability analysis of the problem is fulfilled, the stability boundary is found. Direct numerical simulations are performed and compared with the linear stability analysis.

Influence of Vibrations on Convective Instability of Reaction Fronts in Liquids

K. Allali, F. Bikany, A. Taik, V. Volpert (2010)

Mathematical Modelling of Natural Phenomena

Propagation of polymerization fronts with liquid monomer and liquid polymer is considered and the influence of vibrations on critical conditions of convective instability is studied. The model includes the heat equation, the equation for the concentration and the Navier-Stokes equations considered under the Boussinesq approximation. Linear stability analysis of the problem is fulfilled, and the convective instability boundary is found depending on...

Influence of Vibrations on Convective Instability of Reaction Fronts in Porous Media

H. Aatif, K. Allali, K. El Karouni (2010)

Mathematical Modelling of Natural Phenomena

The aim of this paper is to study the effect of vibrations on convective instability of reaction fronts in porous media. The model contains reaction-diffusion equations coupled with the Darcy equation. Linear stability analysis is carried out and the convective instability boundary is found. The results are compared with direct numerical simulations.

Non-parallel plane Rayleigh Benard convection in cylindrical geometry

A. Golbabai (1995)

Applicationes Mathematicae

This paper considers the effect of a perturbed wall in regard to the classical Benard convection problem in which the lower rigid surface is of the form z = ε 2 g ( s ) , s=ε r, in axisymmetric cylindrical polar coordinates (r,ϕ,z). The boundary conditions at s=0 for the linear amplitude equation are found and it is shown that these conditions are different from those which apply to the nonlinear problem investigated by Brown and Stewartson [1], representing the distribution of convection cells near the center....

Régularité du problème de Kelvin–Helmholtz pour l’équation d’Euler 2D

Gilles Lebeau (2002)

ESAIM: Control, Optimisation and Calculus of Variations

Nous prouvons que pour toute solution u du problème de Kelvin–Helmholtz des nappes de tourbillons pour l’équation d’Euler bi-dimensionnelle, définie localement en temps, la courbe de saut de u et la densité de tourbillon sont analytiques (sous une hypothèse de régularité Holderienne de la courbe de saut). Nous donnons également un résultat de régularité partielle de la trace de u sur t = 0 lorsque u est définie sur un demi-interval [ O , T [ .

Régularité du problème de Kelvin–Helmholtz pour l'équation d'Euler 2d

Gilles Lebeau (2010)

ESAIM: Control, Optimisation and Calculus of Variations

Nous prouvons que pour toute solution u du problème de Kelvin–Helmholtz des nappes de tourbillons pour l'équation d'Euler bi-dimensionnelle, définie localement en temps, la courbe de saut de u et la densité de tourbillon sont analytiques (sous une hypothèse de régularité Holderienne de la courbe de saut). Nous donnons également un résultat de régularité partielle de la trace de u sur t=0 lorsque u est définie sur un demi-interval [O,T[.

Unconditional nonlinear exponential stability in the Bénard problem for a mixture: necessary and sufficient conditions

Giuseppe Mulone, Salvatore Rionero (1998)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

The Lyapunov direct method is applied to study nonlinear exponential stability of a basic motionless state to imposed linear temperature and concentration fields of a binary fluid mixture heated and salted from below, in the Oberbeck-Boussinesq scheme. Stress-free and rigid surfaces are considered and absence of Hopf bifurcation is assumed. We prove the coincidence of the linear and (unconditional) nonlinear critical stability limits, when the ratio between the Schmidt and the Prandtl numbers is...

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