Hydrodynamical instabilities of thermocapillary flow in a half-zone

Abstract

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.

Linear stability of thermocapillary convection in cylindrical liquid bridges under axial magnetic fields

“…Figure 11 shows the dependence of critical Reynolds numbers and wavenumbers on P r for Bi = Gr = 0, Γ = 1 and various Hartmann numbers. Our value of Re c = 1899 for P r = 0.01 without magnetic field is in excellent agreement with the result of Levenstam & Amberg (1995) who found Re c = 1960.…”

“…As in previous studies without magnetic field (Rupp et al 1989;Kuhlmann & Rath 1993;Levenstam & Amberg 1995) no critical axisymmetric (i.e. m = 0) mode was found despite extensive parameter variations.…”

“…the instability of the non-axisymmetric steady state) and was not considered in the present study. Levenstam & Amberg (1995) found the transition to the unsteady state to take place at Re = 6250 for P r = 0.01 and unit aspect ratio. Values given by Rupp et al (1989) for aspect ratio Γ = 1.2 are higher.…”

“…Hence, it follows that the low Prandtl number (P r 1) instability is due to the large strain rate present in the basic axial shear flow. Up to now, the stationary instability in small Prandtl number fluids was only observed in numerical studies (Rupp et al 1989;Kuhlmann & Rath 1993;Levenstam & Amberg 1995;Wanschura et al 1995). Experimental evidence is still missing since model experiments are usually carried out with large Prandtl number liquids (e.g.…”

Effect of Oxygen on Thermocapillary Convection in a Molten Silicon Column under Microgravity

“…There have also been many reports on numerical simulations of thermocapillary convection in a half-zone bridge in low Prandtl number fluids such as molten silicon. [8][9][10][11][12][13][14][15] These simulations indicated that transition processes occur first, from an axisymmetric flow to a three-dimensional steady flow and then from a threedimensional steady flow to a three-dimensional oscillatory flow. Most studies, however, have not taken into consideration the influence of oxygen partial pressure in the ambient atmosphere on thermocapillary convection of molten silicon.…”