Onset of the Thermal Convection in a Finite Two-Dimensional Box

Abstract

Thermal instability of a fluid in a finite rectangular box is investigated, assuming the flow field to be two-dimensional. The horizontal plates are assumed to be rigid and perfectly thermal conducting. Three cases of boundary conditions for the vertical side walls are considered, which are rigid and perfectly conducting, rigid and perfectly insulating, and stress free and perfectly insulating boundary conditions. The critical condition of the linear stability of the motionless state is obtained for various aspect ratios of the box. It is concluded that the stability characteristics are very different between the cases of rigid and stress free side walls. Asymptotic expressions of the critical Rayleigh number are obtained for very wide and very narrow boxes respectively. The Moffatt vortices are confirmed up to the second.