The non-linear interaction of a finite number of disturbances to a layer of fluid heated from below

Abstract

It is predicted that, at a temperature difference a little less than that at which motion starts according to linear stability theory, a steady hexagonal convective pattern will develop from finite-amplitude instabilities in a horizontal layer of fluid heated from below. This is because the first disturbances to start growing must be the triplet of two-dimensional ‘rolls’ which form angles of 60° with each other and whose amplitudes and phases first fall in certain critical ranges. The growth of these disturbances stabilizes all other disturbances and is such that ultimately the right phases and amplitudes occur to give hexagonal cells. If the temperature difference is increased somewhat beyond its critical value, the hexagonal pattern becomes unstable and a two-dimensional roll pattern is predicted. In an intermediate temperature range, rolls are unstable but transport more heat than hexagons. ‘Free–free’ boundary conditions, a viscosity which varies with temperature, and a fixed disturbance wave-number are assumed in this extension of the work of Palm (1960) and Segel & Stuart (1962). Other theoretical results and some experimental results are compared with the present predictions.