Contribution to the theory of cellular thermal convection

Abstract

In a previous paper on cellular thermal convection (Palm 1960) the importance of the effect caused by temperature variation of kinematic viscosity was pointed out. It was demonstrated that this effect would, owing to non-linear interactions, lead to a tendency towards hexagonal cells. For mathematical simplicity, only the interaction of two wave-components was taken into account.Segel & Stuart (1962), working with the same equations, have examined the stability of the various equilibrium solutions. They arrive at the important conclusion that a necessary condition for the solution corresponding to hexagons to be stable is that the variation of viscosity with temperature be sufficiently great.In the present paper the problem is discussed from a somewhat more general point of view. First it is shown that, when the variation of viscosity with temperature is sufficiently great, the solution corresponding to hexagons is the only stable one if only two wave-components are taken into account. To examine if this result is also true when the motion consists of an arbitrary number of wave-components, the case of three wave-components is studied. It turns out that in this case also the only possible mode is the pattern consisting of hexagons. The validity of this result is easily extended to a more general class of wave-components. It is shown that the solution corresponding to hexagons is stable for all small disturbances which can possibly occur. To prove this it is necessary to take into account non-linear disturbance theory.A reasonable conclusion from the paper by Segel & Stuart and the present paper is that a hexagonal pattern is observed only when a condition of the form (6.9) is fulfilled. Experiments concerning this problem are, however, lacking.

A Review of Mesoscale Cellular Convection

“…Segel and Stuart further suggest that more work should, if possible, be done in the interaction of many disturbances of varying cell shapes and wavenumbers in order to determine the convective mode of a particular wavenumber. Palm and Oiann (1964) have concluded that a hexagonal pattern is observed only when the following condition holds: where / depends only on boundary conditions. P = v/k is the Prandtl number; K the coefficient of thermal expansion and A(3 is the difference between the actual initial temperature gradient p and the critical temperature gradient for the onset of convection, p0.…”

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

“…where the Xi are positive constants given by Palm & Oiann (1964). In a subcritical state the square bracket in (6.1) is negative so the hexagonal pattern is in fact stable to all disturbances.…”

“…It will be assumed that the reader is familiar with the classical linear stability theory concerned with this problem-as in the book by Chandrasekhar (1961, ch. 2)-and also with the broad outlines of the non-linear investigations of Palm (1960), Segel & Stuart (1962), and Palm & Oiann (1964). We point out, however, that all one needs to know in advance is outlined in the paper preceding this one (Segel 1965) so that the two papers taken together are virtually selfcontained.…”