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Asymptotic study and weakly nonlinear analysis at the onset of Rayleigh-Bénard convection in Hele-Shaw cell. (English) Zbl 1195.76177

Summary: The aim of this paper is the derivation of the Ginzburg-Landau equations (as introduced by A. C. Newell and J. A. Whitehead [J. Fluid Mech. 38, 279–303 (1969; Zbl 0187.25102)]) from the hydrodynamic equations for an infinite Hele-Shaw cell. The dimensional analysis and the asymptotic study allow one to distinguish two nonlinear formulations; each one depends on the order of magnitude of the Prandtl number. The first formulation corresponds to the case \(\mathrm{Pr}=O(1)\) or \(\mathrm{Pr}\gg 1\), whereas the second corresponds to the case \(\text{Pr}=O(\epsilon^{\ast 2})\), where \(\epsilon^\ast \ll 1\) denotes the aspect ratio of the cell. Here a weakly nonlinear analysis is performed for the two formulations.

MSC:

76E06 Convection in hydrodynamic stability
76E30 Nonlinear effects in hydrodynamic stability
76M45 Asymptotic methods, singular perturbations applied to problems in fluid mechanics
76D27 Other free boundary flows; Hele-Shaw flows

Citations:

Zbl 0187.25102
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References:

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