Stability of squares and rolls in Rayleigh–Bénard convection in an infinite-Prandtl-number fluid between slabs

Abstract

Nonlinear solutions in the form of squares and rolls are investigated for Rayleigh–Bénard convection in an infinite-Prandtl-number fluid enclosed between two symmetric slabs. It is found that the heat transfer depends strongly on the thickness and thermal conductivity of the slabs, but hardly on the planform of convection. Examples of stability regions of rolls are calculated, showing that for certain slab selections, rolls remain stable at even larger Rayleigh numbers than with fixed temperatures at the boundaries. The region of stable squares is restricted by a zigzag and a long-wavelength cross-roll instability in addition to a new three-dimensional instability. As the slab conductivity is increased, the stability region of the squares shrinks onto a point located well above the critical point for the onset of convection. For a small range of slab conductivities, stability regions for squares and rolls both exist for the same set-up. In the present calculations, the regions never overlap. An example, where both patterns are stable at the same Rayleigh number, provides an explanation for the co-existence of rolls and squares where transparent slabs with a low thermal conductivity were applied.