Nonlinear convection in a layer with nearly insulating boundaries

Abstract

A general class of solutions is studied describing three-dimensional steady convection flows in a fluid layer heated from below with boundaries of low thermal conductivity. Non-linear properties of the solutions are analysed and the physically realizable convection flow is determined by a stability analysis with respect to arbitrary three-dimensional disturbances. The most surprising result is that square-pattern convection is preferred in contrast to two-dimensional rolls that represent the only form of stable convection in a symmetric layer with highly conducting boundaries. The analysis is carried out in the limit of infinite Prandtl number and for a particular boundary configuration. But it is shown that the results hold for arbitrary Prandtl number to the order to which they have been derived and that other assumptions about the boundaries require only minor modifications as long as their thermal conductance remains low.