On Thermal Convection in a Large Box

Daniels [4] has shown that the presence of imperfectly insulating sidewalls transforms the onset of Bénard convection in a layer of fluid confined by one pair of lateral boundaries from a bifurcation to a continuous transition as the Rayleigh number increases through the critical value. In this paper these ideas are extended to accommodate rectangular containers with large, horizontal dimensions. The appropriate boundary conditions for the amplitude equations are derived, and a limited discussion of the properties of the solutions is undertaken. In particular it is shown that, in agreement with earlier theories and experiments, there is a preference for rolls parallel to the shorter side.