Asymptotic solution of thermocapillary convection in two immiscible liquid layers in a shallow annular cavity

The steady laminar two-dimensional thermocapillary convection of two superposed horizontal liquid layers in a shallow annular cavity was investigated using asymptotical analysis. The liquids were supposed to be immiscible with a nondeformable interface. The cavity was heated from the outer cylindrical wall and cooled at the inner wall. Bottom and top surfaces were rigid and adiabatic. Asymptotic solutions were obtained in the core region in the limit as the aspect ratio, which was defined as the ratio of the lower layer thickness to the gap width, and trended to zero. The numerical experiments were also carried out to compare with the asymptotic solution of the steady two-dimensional thermocapillary convection. It is found that the expressions of velocity and temperature fields in the core region are valid in the limit of the small aspect ratio.