An analytical solution for thermocapillary-driven convection of superimposed fluids at zero Reynolds and Marangoni numbers

This study aims to investigate thermocapillary-driven convection in two superimposed fluids in zero gravity. The fluids occupy the space between the walls of a horizontal microchannel which is heated from below by imposing the top wall to a uniform temperature and the bottom wall to a sinusoidal temperature that is higher (on the average) than the temperature of the top wall. The goal is to mimic thermocapillary convection as a result of the variation of the heights of the fluids along the microchannel and to explore the parameters that affect the fluid flow and interface deformation. This is achieved by solving the equations of conservation of mass and momentum and the balance of thermal energy and negligible analytically in both fluids, in the limit of creeping flow regime and negligible convection of heat. It is shown that the induced flow is characterized by periodic convection cells whose period is the same as the period of the imposed temperature field and extend from the interface to the walls in the vertical direction. The flow strength depends on the relative thicknesses of the fluid layers and the ratio of material properties. The maximum flow strength is achieved at a relative thickness that is set by the competition between the thermal and hydrodynamic effects. An estimate of the interface deformation is provided and it is shown that the sense of interface deformation is set by the relative thickness of the fluid layers and the viscosity ratio.