CONVECTION IN A LOW PRANDTL NUMBER FLUID

As a simple two-dimensional model of convection in the liquid phase du ring crystal growth using the Bridgman technique, we consider the flui d flow in a shallow rectangular cavity heated from one side. For an as pect ratio of 4 (aspect ratio = length/height), previous numerical stu dies have shown the existence of two types of oscillatory solution. Th e 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 onl y in a time-dependent study for a Prandtl of zero. We locate the new H opf bifurcation which gives rise to this latter solution and study its dependence on Prandtl number. Before the onset of oscillations, vario us steady corotating multi-cell solutions are found depending on the a spect ratio of the cavity. We compute how the transition between three corotating cells and two corotating cells takes place for changing as pect ratio. Understanding can be facilitated by comparison with analog ous problems. We consider tilting the cavity to the Benard configurati on, 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 cond uction alone. This solution becomes unstable to counter-rotating cells at a symmetry-breaking bifurcation. We study how this symmetry-breaki ng bifurcation disconnects as the cell is tilted and the solutions evo lve into the side-wall cavity solutions. In addition we trace the sadd le-node and Hopf bifurcations found in the side-wall heated problem fo r changing tilt and reveal that they also exist in the Benard convecti on limit; disconnected solutions in the Benard problem have not been s tudied previously.