Rayleigh–Bénard convection in two-dimensional arbitrary finite domains

In the present work, we consider the linear and nonlinear hydrodynamic stability problems of two-dimensional Rayleigh–Bénard convection in arbitrary finite domains. The effects of the domain shapes on the critical Rayleigh number and convection pattern are investigated by means of a linear stability analysis employing a Chebyshev pseudospectral method. An extension of the present technique to nonlinear stability analysis allows derivation of the Landau equation for arbitrary finite domains. The results of nonlinear stability analysis are confirmed by comparison with numerical solution of the Boussinesq set. The results of the present investigation may be exploited to enhance or suppress thermal convection by varying system domain.