Planform selection in Rayleigh–Bénard convection between finite slabs

Abstract

Thermal convection in a thin horizontal fluid layer enclosed between two rigid slabs of arbitrary thicknesses and conductivities has been investigated. We have found a mathematical transformation between this problem and the problem of the upper and lower slabs being interchanged. A weakly nonlinear expansion has been applied to reduce the governing equations to a set of Landau equations. Their extremum principle combined with an analytical solution for the case of insulating slabs has been used to prove that rhombuses and rolls are the only stable solutions. Hexagons, quasi-patterns and any solution involving higher numbers of modes, are proved to be unstable. Stability regions of rolls and rhombuses have been found numerically for a wide range of slab conductivities and thicknesses. The wavenumber selection has been investigated by studying two coupled Ginzburg–Landau equations. Earlier stability analyses of Proctor's equation valid for the limit of poorly conducting slabs has revealed that the wavenumbers of squares, i.e. rhombuses with orthogonal wave vectors, are restricted by a zigzag instability and by a truly three-dimensional instability. We show here that the wavenumber selection for more general cases with finite conductivities and thicknesses of the slabs are always restricted by the same types of instability. In addition, we show how the stability and wavenumber selection of another solution of the Ginzburg–Landau equations, the undulated rolls, is restricted by a cross-roll instability.