Numerical Simulation of Steady Thermocapillary Flow in a Flat Channel Using Linear Approximation

Abstract

we performed a numerical study of steady thermocapillary flow of a viscous incompressible fluid in a flat channel with a stationary solid bottom and a free surface. We considered the creeping-flow regime, in which the only driving force was temperature-dependent surface tension. We used a quasi-one-dimensional expansion of the velocity, temperature, and pressure fields in the transverse coordinate, retaining terms up to second order in the longitudinal coordinate. We solved the resulting boundary-value problem for a system of ordinary differential equations using a fourth-order collocation method implemented in MATLAB with the bvp4c solver. We applied the computational algorithm for five values of the Marangoni number. We obtained profiles of the vertical and horizontal velocity components and the modal temperature and pressure, and constructed two-dimensional distributions of the stream function, pressure, and temperature. We found that two counter-flowing streams formed over the range of parameters considered, separated by a level of zero horizontal velocity, and determined the positions of the corresponding zeros. Analysis of the residuals of the governing equations confirmed the high accuracy of the numerical solution. We investigated the dependence of the flow structure on the governing parameter. The results provide a benchmark for testing numerical schemes for thermocapillary convection.