On the Onset of Rayleigh‐Bénard Convection in a Fluid Layer of Slowly Increasing Depth

Abstract

A two‐scaling approach is used to investigate the onset of convection in a fluid layer whose depth is a slowly increasing function of horizontal distance. It is shown that whatever the value of the imposed temperature difference between the boundaries (provided, of course, that the lower one is hotter) there are regions which are stable and regions which are unstable to small perturbations. As the depth increases the amplitude of steady solutions increases from exponentially small values to take on the familiar square‐root behavior of weakly nonlinear solutions. The solution in this narrow transition region is described in terms of the second Painlevé transcendent. In the exceptional case when the perturbation takes the form of longitudinal rolls, this equation needs some modification in that the second derivative is replaced by the fourth. The flow in a horizontal layer when the temperature difference between the boundaries increases slowly may be treated in exactly the same way. The necessary modifications to theory and results are given in an Appendix.