D. Schwabe

Coupling and separation of buoyant and thermocapillary convection

Abstract We describe the results of convection-experiments with a liquid of Prandtl number Pr = 17 with free surface in rectangular configuration under the action of thermal buoyant forces and thermocapillary forces. These forces are varied independently by changing the geometry and by applying a surface temperature gradient which could be different from the bulk temperature gradient in magnitude and direction. By these means the coupling and separation of convection rolls driven mainly by one of the forces has been studied. For the large temperature gradients normally encountered in Czochralski growth of refractories and other high Prandtl number melts thermocapillary forces are always significant in a convection roll near the free surface. The velocities in this convection roll are very high compared to the velocities measured in the bulk volume. A separation of the surface tension driven flow from the motion in the bulk fluid has been found. This separation can give rise to a complicated convective structure in the bulk fluid which is sensitive to the ratio of the driving forces. This points out the need for better understanding and control of the surface properties of melts, e.g. in doped systems.

Surface-Tension-Driven Flow in Crystal Growth Melts

Convection is an important parameter in crystal growth from the melt. A driving force for convection besides buoyancy, crystal rotation and crucible rotation, is the surface-tension gradient due to temperature or concentration gradients in the free melt surface. This article points out that in the various melt-growth techniques with free melt surface, e.g. the floating-zone technique, the Czochralski technique and the open-boat zoning, surface-tension-driven flow is likely to occur and to influence the growth. Surface-tension-driven flow is especially important in small melt volumes, near the free surface of larger volumes, or under conditions of weightlessness (e.g. in Spacelab). Under the large temperature gradients encountered in crystal growth, surface-tension-driven flow can become time dependent. Numerical studies, experiments on surface-tension-driven flow, and observations in real growth systems are reviewed.

Oscillatory thermocapillary convection in open cylindrical annuli. Part 2. Simulations

Oscillatory thermocapillary convection in open cylindrical annuli heated from the outer wall is investigated numerically. Results at fixed inner/outer radius ratio of 0.5, aspect ratios (Ar )o f 1, 2.5, 3.33, and 8 ,zero Biot number, and a Prandtl number of 6.84 are obtained and compared with experiments (Part 1 of this paper). Convection is steady and axisymmetric at sufficiently low values of the Reynolds number (Re). Transition to oscillatory states occurs at critical values of Re which depend on Ar. With Ar =1 ,2 .5 and 3.33, we observe 5, 9 and 12 azimuthal wavetrains, respectively, travelling clockwise at the free surface near the critical Re .W ith Ar =8 , there are 20 standing waves near the critical Re .E xperimental results in Part 1 support this finding. A multi-roll structure appears beyond the critical Re in shallow liquid layers with Ar =3 .33 and 8. The critical Re and frequency are in qualitative but not in quantitative agreement with the experimental ones. Either heat loss from the free surface or heating from the surroundings to the free surface stabilizes the flow, and the critical Re increases with increasing Biot number while the critical period goes down. The numerical results agree better with the experimental ones if the free surface is assumed to be heated as shown in Part 1. We have also computed supercritical time-dependent states and find that while the non-dimensional frequency increases with increasing Re near the critical region, it approaches an asymptote at super- critical Re.

Unstable flow and solidification speed due to the interaction of thermocapillary and solutocapillary forces in directional solidification

During floating zone growth from a melt with a surface active impurity with segregation coefficient k eff <1, solutocapillary forces arise which counteract thermocapillary ones. A hydrodynamic instability of the surface-tension-driven flow (Marangoni effect) due to these counteracting forces has been found experimentally and its mechanism is discussed in this paper. In contrast to the well-known passive growth oscillations due to hydrodynamic instabilities in the melt, growth and remelting of the crystal actively participate in this instability.

Thermocapillary Convection in Cylindrical Geometries

Thermocapillary convection in two types of cylindrical geometries is studied by three-dimensional numerical simulations: an open cylindrical annulus heated from the outside wall and a liquid bridge. The non-deformable free surfaces are either flat or curved as determined by the fluid volume, V, and the Young-Laplace equation. Convection is steady and axisymmetric at sufficiently low values of the Reynolds number, Re, with either flat or curved surfaces. For the parameter ranges considered, it is found that only steady convection is possible at any Re in strictly axisymmetric computations. Transition to oscillatory three-dimensional motions occurs as Re increases beyond a critical value dependent on the aspect ratio, the Prandtl number, and V. Good agreement with available experiments is achieved in all cases.