Comparative analysis of the analytical and numerical solution of the problem of thermocapillary convection in a rectangular channel

The paper compares the exact solution of one-dimensional and two-dimensional stationary convective flow equations with a free boundary for a flat liquid channel. Constant temperature gradient is set on the bottom solid wall. On the upper free boundary the surface tension coefficient is linearly dependent on temperature. Zero heat flux and velocities are set on the side walls of the two-dimensional problem. The deviation of the one-dimensional exact solution is determined for different aspect ratio and Marangoni number.