The failure of a boundary layer model to describe certain cases of cellular convection

Abstract

A model has been developed to describe the two-dimensional cellular motion of a viscous fluid with Prandtl number of order unity heated from below at high Rayleigh number. This model is easily adapted to describe such flow for infinite Prandtl number provided all boundaries are free. It has been suggested that a good approximation to the physical problem of flow between rigid horizontal boundaries will be obtained by setting the Prandtl number equal to infinity, thus ignoring the momentum convection. Similar cellular motion occurs in flow in a porous medium at high Rayleigh numbers and a similar model has been proposed to describe this motion. The basic features of the model are outlined and it is shown that the model fails to describe either the motion of an infinite Prandtl number fluid between parallel rigid boundaries or fluid in a porous medium. An analysis of the model for large finite Prandtl number and large Rayleigh number shows that as the Prandtl number is increased the velocity boundary layers on the rigid horizontal boundaries thicken, and eventually fill the cell, thus losing their boundary layer identity and causing the breakdown of the model.