On the two-component Benard problem a numerical solution

A numerical solution for the two-component Bénard problem is presented, taking into account the contribution of thermal diffusion to the total density gradient. The results are compared with the approximate solution obtained by the variational technique of the local potential introduced some years ago by Glansdorff and Prigogine. The results calculated by the two methods are in agreement when the Soret coefficient is not too large. But when the gradients become important, the exact numerical solution presented here shows a small divergence from the variational method.The critical Rayleigh number is also compared with the one extrapolated from the analytical solution obtained for free boundary conditions.