On the tendency towards hexagonal cells in steady convection

Abstract

This paper attempts to explain theoretically the observed results that (1) the cells in steady convection approach a hexagonal form, and (2) the occurrence of ascent or descent in the middle of the cell depends on how the kinematical viscosity varies with temperature. The theory is based on non-linear equations and, of course, a variable coefficient of viscosity. It is found that, due to the variation of viscosity with temperature, the non-linear terms contain a second-order term which is destabilizing. This second-order term regulates the development and leads to a final motion composed of regular hexagons with ascent or descent in the middle of the cell according as the viscosity decreases or increases with temperature. The influence of a variable viscosity on Rayleigh's result concerning the initiation of convection is obtained as a by-product.