J. Fluid Mech. zyxwvutsrq (1995), uol. 291, pp. 351-312 zyxwvutsr Copyright zyxwvutsrqp 0 1995 Cambridge University Press 357 Hydrodynamical instabilities of thermocapillary flow in a half-zone By MARTEN LEVENSTAMt AND GUSTAV AMBERG Department zyxwvutsr of Mechanics, Royal Institute of Technology, S-100 44 Stockholm, Sweden (Received 21 February 1994 and in revised form 14 February 1995) The stability of the flow in a half-zone configuration is analysed with the aid of direct numerical simulation. The work is concentrated on the small Prandtl numbers relevant for typical semiconductor melts. The axisymmetric thermocapillary flow is found to be unstable to a steady non-axisymmetric state with azimuthal wavenumber 2, for a zone with aspect ratio 1. The critical Reynolds number for this bifurcation is 1960. This three dimensional steady solution loses stability to an oscillatory state at a Reynolds number of 6250. For small Prandtl numbers, both bifurcations are seen to be quite insensitive to changes in the Prandtl number, and are thus hydrodynamic in nature. An analogy to the instability of thin vortex rings is made. This analogy suggests a physical mechanism behind the instability and also gives an explanation of how the azimuthal wavenumber of the bifurcated solution is selected. The implications of this for the floating-zone crystal growth process are discussed. 1. Introduction One method for the production of single crystals of semiconductors is the so-called float-zone method (FZ). In this method a drop of semiconductor melt is held by surface tension forces between two solid rods of the same material. The drop is kept molten by an intense heat source focused on it. The two suspending rods are cooled, and the heat source is passed slowly along the rod so that the material melts and re-solidifies as the heat source passes. After one or several such passes the purity and the crystal structure of the rod is greatly improved. The main advantage of this method is that it is containerless, and thus makes it possible to produce extremely pure crystals. It has long been known that the fluid motion in this system is caused both by gravitational and thermocapillary convection. The convection has undesired effects and as a means of suppressing the gravitational convection, the FZ-method has been proposed for space processing of materials. However, owing to the intense temperature gradients that are present over the drop surface, the thermocapillary convection may be significant, even on Earth. A lot of attention has been focused on the thermocapillary convection in this system. One of the first works to point out the importance of thermocapillary effects was Chang zyxw & Wilcox (1976). Experiments on the FZ method, both on Earth and in microgravity conditions in space, have shown a banded structure, striations, in the chemical composition of the finished crystals. From this it has been suggested that the flow in the FZ has been Present adress: Department of Mathematics, Chalmers University of Technology, S-411 22 Goteborg, Sweden.