Classical cellular convection with a spatial heat source

Abstract

This paper considers the problem of the stability of an infinite horizontal layer of a viscous fluid which loses heat throughout its volume at a constant rate. The variation of the critical Rayleigh number, Rt, and the cell aspect ratio, a, with the rate of heat loss, is calculated with two sets of boundary conditions corresponding to two free and two rigid boundaries. In both cases we find that, as the rate of heat loss increases, Rt decreases, showing that the layer becomes more unstable, and a increases, showing that the cells become narrower. We also consider the possibility that a double layer of cells is formed for large values of the rate of heat loss, by the stable layer at the top, and find that this does not occur while the temperature of the upper surface of the layer is less than that of the lower.