Thermal convection in an infinite channel with no-slip sidewalls

Abstract

We obtain exact solutions of the linearized perturbation equations for the convective motions in a Boussinesq fluid contained in an infinite rectangular channel and heated from below. The top and bottom are assumed to be perfect heat conductors. The sides can either be conducting or insulating. We assume that the sidewalls are rigid, but allow the top and bottom to be free so that we can separate variables.

We find that the preferred modes of convection closely resemble transverse ‘finite rolls’ [as predicted by Davis (1967) for convection in a box] for channels with height to width ratios outside the range 0·1 to 1. Inside this range they show noticeable departures from roll form. We prove, however, that, except when the sidewalls are relaxed to infinity, ‘finite rolls’ are never exact solutions of the linearized equations, even though in most cases they are good approximations.

We also find that bringing the sidewalls closer together inhibits convection and, generally, produces thinner cells at the onset of convection.