Asymptotic Solution Of Thermal Convection In Two Immiscible Liquid Layers In A Shallow Annular Pool

The Czochralski method is one of the main methods of producing crystals from the melt. In this method, the coupled thermocapillary and buoyancy-driven convection in the melt and the evaporation of some volatile components may have bad effects on the quality of the grown crystal. The liquid-encapsulated crystal (LEC) growth technique is used for solving these problems. However, the additional liquid-liquid interface between the melt and the encapsulant in this technique can be a source of new and complex dynamics that may significantly influence crystal quality. In order to obtain crystal of high quality, the thermal convection in two annular immiscible liquid layers must be better understood and controlled. The steady pure thermocapillary convection and thermocapillary-buoyancy convection in a system of two horizontal superimposed immiscible liquid layers filling a lateral heated thin annular pool were investigated using matched asymptotic theory. The flow domain was divided into a core region away from the cylinder walls and two end regions near each cylinder wall. Asymptotic solutions of the temperature and velocity fields were obtained in the core region under different top boundary conditions (the pool is open or closed) and heating conditions (heated at the inner wall or at the outer wall). For evaluating the validity of the obtained asymptotic solutions, numerical simulations were also carried out and the results were compared with the asymptotic solutions. The influences of the thermocapillary Reynolds number, the curvature, the aspect ratio and the thickness ratio of the two layers on the asymptotic solutions were estimated. The valid ranges of the asymptotic solutions were determined in all cases.It is found that when the inner radius of the annular pool tends to infinity, the present asymptotic solutions of pure thermocapillary convection are the same as the results obtained in a rectangular cavity. In all cases, the asymptotic solutions of the temperature and the radial velocity in the core region show good agreements with the simulation results. The present asymptotic solutions are valid in most region of the pool except in the region closed to the cylinder walls. The accuracy and valid range of the obtained asymptotic solutions decrease with the increase of the thermocapillary Reynolds number, the aspect ratio and the thickness ratio of the two layers. The influence of the curvature on the accuracy and valid range of the asymptotic solutions is very small.