A Solution Method of Nonlinear Convective Stability Problems in Finite Domains

Abstract

A Chebyshev pseudospectral method is generalized to solve the linear and nonlinear hydrodynamic stability problems of thermal convection in a two-dimensional rectangular box with rigid sidewalls, where there may exist a heat source or a magnetic field to enhance or suppress the convection. The incompressibility condition is imposed rigorously on all boundaries. The effects of box aspect ratio, heat source, and magnetic field on the critical Rayleigh number and convection cell size are examined and compared with the results of other investigators. We have extended the present technique to nonlinear stability analysis and derived the Landau equation that describes the temporal evolution of the strength of convection in the rectangular box with rigid sidewalls. The results of nonlinear stability analysis are compared with the exact results obtained by the numerical solution of the Boussinesq equation. The present technique solves linear and nonlinear convective stability problems accurately and can be employed to solve other hydrodynamic stability problems in finite domains.