Distant side-walls cause slow amplitude modulation of cellular convection

Abstract

The effect of vertical boundaries on convection in a shallow layer of fluid heated from below is considered. By means of a multiple-scale perturbation analysis, results for horizontally unbounded layers are modified so that they ‘fit in’ to a rectangular region. The critical Rayleigh number and critical wave-number are determined. Motion is predicted to have the form of finite ‘rolls’ whose axes are parallel to the shorter sides of the dish. Aspects of the non-linear development and stability of this motion are studied. The general question of convective pattern selection in a bounded layer is discussed in the light of available theoretical and experimental results.