On Maintained Convective Motion in a Fluid Heated from below

Anne Pellew, R. V. Southwell

Abstract

This paper examines the stability in viscous liquid of a steady regime in which the temperature decreases with uniform gradient between a lower horizontal surface which is heated and an upper horizontal surface which is cooled. The problem has been treated both experimentally and theoretically by Benard, Brunt, Jeffreys, Low and Rayleigh, and it is known that instability will occur at some critical value of gh$^{3}\Delta \rho /\rho $k$\nu $, h denoting the thickness of the fluid layer, $\Delta \rho /\rho $ the fractional excess of density in the fluid at the top as compared with the fluid at the bottom surface, k the diffusivity and $\nu $ the kinematic viscosity. The critical value depends upon the conditions at the top and bottom surfaces, which may be either 'free' or constrained by rigid conducting surfaces. The theoretical problem is solved here under three distinct boundary conditions, and greater generality than before is maintained in regard to the 'cell pattern' which occurs in plan. In addition an approximate method is described and illustrated, depending on a stationary property akin to that of which Lord Rayleigh made wide application in vibration theory. Within the assumptions of the approximate theory (i.e. with neglect of terms of the second order in respect of the velocities) a particular size is associated with every shape of cell (such that 'a$^{2}$' takes a preferred value), but no particular shape is more likely than another to occur in a layer of indefinite extent (section 31). The explanation of the apparent preference for a hexagonal cell pattern (section 5) must presumably be sought in a theory which takes account of second-order terms. This conjecture if correct goes some way towards explaining the rather indefinite nature of observed cell-formations (cf. Low 1930, figure 10).