On the stability of a fluid heated from below

Abstract

In 1916 Lord Rayleigh showed that in certain circumstances a layer of fluid contained between two infinite plane horizontal surfaces could remain at rest with the density increasing upwards, and he enunciated a criterion which allows the critical temperature difference for the onset of convection to be found in terms of the depth of the fluid, the viscosity and conductivity of the gas and a reference temperature, such as the absolute temperature of the lower (heated) surface. His result was extended to a fluid bounded by two rigid plane surfaces by Jeffreys, and in this form has been examined experimentally by K. Chandra (using air) and D. T. E. Dassanayake (using carbon dioxide). Their results show (i) that the Rayleigh-Jeffreys criterion is confirmed for relatively deep layers, (ii) that in relatively shallow layers an entirely distinct mode (‘columnar’), which does not satisfy the Rayleigh-Jeffreys criterion, is initiated and (iii) that this mode passes into another mode when the temperature of the lower surface is greatly increased, at temperature differences which agree approximately with Rayleigh-Jeffreys criterion. The problem has been re-examined in the present paper, on the assumption that the new mode found by Chandra and Dassanayake arises from instability in a shallow boundary layer, whose depth is unrelated to the distance between the upper and lower rigid boundaries. It is shown that the criterion for the ‘columnar’ mode involves only the ratio of the absolute temperatures of the upper and lower surfaces, and an expression is derived and verified for the critical temperature difference at which the transition takes place from the ‘columnar’ to the ‘cellular’ mode. Finally, it is shown that the Rayleigh mode will occur if the depth of the fluid exceeds a certain value, but that for the more shallow layers, the ‘columnar’ mode will be generated initially, ultimately passing to the ‘cellular’ mode for increased temperature difference.