Convection in a low Prandtl number fluid

Abstract

As a simple two-dimensional model of convection in the liquid phase during crystal growth using the Bridgman technique, we consider the fluid flow in a shallow rectangular cavity heated from one side. For an aspect ratio of 4 (aspect ratio = length height ), previous numerical studies have shown the existence of two types of oscillatory solution. The first of these has been well-studied for a range of Prandtl numbers from 0 to 0.015. However, the form of the second has been observed only in a time-dependent study for a Prandtl of zero. We locate the new Hopf bifurcation which gives rise to this latter solution and study its dependence on Prandtl number. Before the onset of oscillations, various steady corotating multi-cell solutions are found depending on the aspect ratio of the cavity. We compute how the transition between three corotating cells and two corotating cells takes place for changing aspect ratio. Understanding can be facilitated by comparison with analogous problems. We consider tilting the cavity to the Bénard configuration, where the fluid is heated from below rather than from the side. In this case a no-flow solution exists where heat is transferred by conduction alone. This solution becomes unstable to counter-rotating cells at a symmetry-breaking bifurcation. We study how this symmetry-breaking bifurcation disconnects as the cell is tilted and the solutions evolve into the side-wall cavity solutions. In addition we trace the saddle-node and Hopf bifurcations found in the side-wall heated problem for changing tilt and reveal that they also exist in the Bénard convection limit; disconnected solutions in the Bénard problem have not been studied previously.