Asymptotic solution of thermocapillary convection in a thin annular pool of silicon melt

Abstract

The free surface of the silicon melt in a thin annular pool is subjected to a radial temperature gradient. Since the surface tension depends on the temperature, it will create a thermocapillary force on the free surface and, in turn, yield to thermocapillary convection in the bulk of the liquid by the viscous traction. This paper presents an investigation on the steady two-dimensional thermocapillary convection in a differentially heated annular pool of the silicon melt using the asymptotical way. The pool is heated from the outer cylindrical wall and cooled at the inner wall. Bottom and top surfaces are adiabatic. The asymptotic solution is obtained in the core region in the limit as the aspect ratio, which is defined as the ratio of the pool height to the gap width, goes to zero. The numerical experiments are also carried out to compare to the asymptotic solution of the steady two-dimensional thermocapillary convection. The results indicate that the expressions of velocity and temperature fields in the core region from the asymptotic solution are found to be valid in the limit of small aspect ratio.