ASYMPTOTIC STUDY AND WEAKLY NONLINEAR-ANALYSIS AT THE ONSET OF RAYLEIGH-BENARD CONVECTION IN HELE-SHAW CELL

The aim of this paper is the derivation of the Ginzburg-Landau equation [as introduced by A. C. Newell and J. A. Whitehead, J. Fluid Mech. 38, 279 (1969)] 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 Pr=O(1) or Pr≫1, whereas the second corresponds to the case Pr=O(ε*2), where ε*≪1 denotes the aspect ratio of the cell. Here a weakly nonlinear analysis is performed for the two formulations. © 1995 American Institute of Physics.