Steady thermocapillary-buoyant convection in a shallow annular pool. Part 1: Single layer fluid

Abstract

This paper examines the steady thermocapillarybuoyant convection in a shallow annular pool subjected to a radial temperature gradient. A matched asymptotic theory is used to obtain the asymptotic solutions of the flow and thermal fields in the case of small aspect ratios, which is defined as the ratio of the layer thickness to the gap width. The flow domain is divided into the core region away from the cylinder walls and two end regions near each cylinder wall. Asymptotic solutions are obtained in the core region by solving the core and end flows separately and then joining them through matched asymptotic expansions. For the system of silicon melt, the asymptotic solutions are compared with the results of numerical simulations. It is found that the two kinds of solutions have a good agreement in the core region for a small aspect ratio. With the increase of aspect ratio, the applicability of the present asymptotic solutions decreases gradually.