Boundary layer solution for the Rayleigh-Benard convection problem

A solution of a Rayleigh-Benard convection problem as a boundary layer problem using the method of matched asymptotic expansions (MAE) is described. Temperature and flow boundary layers are considered. It is shown that in the case with free boundary conditions no flow boundary layers occur. With rigid boundaries buoyancy effects are felt in the horizontal boundary layers even in the lowest order creating a flow boundary layer there. Temperature profiles for the boundary layers are obtained as series expansions in terms of temperature modes. Mode amplitudes may be determined, given the temperature profile at a nonsingular cross section of the boundary layer. The MAE, however, prescribes the initial temperature profiles for the layers in the corner, where the boundary layers become singular (infinite layer thickness). This difficulty can not be resolved within the framework of MAE, but a rational approximation to the solution can be devised.