Side Wall Effect on the Pattern Formation of the Rayleigh-Benard Convection

Abstract

The effect of stress-free side walls on the pattern formation of the Rayleigh-Bénard convection is studied for the case in which the mode of the l-roll pattern and the mode of the l + 1-roll pattern bifurcate from the conduction state successively. In this case the mode of the l-roll pattern prevails when the Prandtl number is small and the aspect ratio (ratio of the side length to the height of the fluid layer) is small. The mode interaction is also studied for the case in which rolls of one mode have axes parallel to the x-direction and rolls of the other mode have axes parallel to the y-direction. In this case the mode interaction produces an unstable grid pattern. In this paper a system of equations is derived from the Boussinesq equations by reductive perturbation method to explain this mode interaction.