Flow driven by Marangoni convection and rotating magnetic field in a floating-zone configuration

Abstract

This paper treats the axisymmetric flo w driven by Marangoni convection and a Rotating Magnetic Field (RMF) in a Floating-Zone configuration. For a low Prandtl number liquid, and different Marangoni values, we investigate the evolution of the flo w as the intensity of the RMF is increased. It evolves from a Marangoni dominated flo w to a RMF dominated flo w . We particularly emphasise and explain the transition re gion where both sources of motion surprisingly tend to cancel each other. Introduction Float-Zone is the method of choice to produce high-purity silicon crystals. A polycrystalline rod is moved through a heater. As the melt solidifies on a crystal seed a single crystal grows. This containerless technique yields dislocation-free crystal. The presence of thermally induced surface tension gradients along the free sur- face leads to a strong conv ection (called Marangoni convection) in the melt that produces undesirable macro and microsegregation. A drastic reduction of microse gregation due to a rotating magnetic field (RMF) in a crystal grown with floating-zone technique has been shown e xperimentally (1). The effect of the RMF is to produce a prescribed azimuthal body force in the melt. In the e xperiments (1), the flo w is in a regime where realistic numerical simulation is still difficult to achiev e. We study a simplified situation to un- derstand some of the mechanisms that occur when the flo w is dri ven by both sources of motion. The classical model for the study of Marangoni convection in a liquid bridge is to consider the liquid suspended between two planar circular isothermal disks at different temperature (2), (3). The b uoyant convection is usually neglected when compared to the Marangoni convection. This situation is often referred to as a half-zone. While the free- surface position is not known a-priori, a further simplification is to consider its location to be at a constant radius. In our study, we keep most of half-zone simplifications with the notable e xception that we take both disks to be at melting temperature and a heat flux is prescribed along the free surf ace. Such models have already been proposed (4), (5), (6). We use the same parabolic e xpression for the heat flux as the latter. 1. Pr oblem formulation We use cylindrical coordinates with the axis along the centerline of the cylinder with the origin at the middle of the liquid region and unit vectors . The free surface is located at a radius and we fix the distance between the feed rod and the crystal at . We make the assumption that the flo w remains axisymmetric and steady. We nor- malize length by and velocity by , where is the kinematic viscosity of the fluid. We use the vorticity , stream-function formulation where and the velocity . We introduce the