Two-dimensional thermocapillary fluid motion in an open channel

The problem of two-dimensional thermocapillary fluid motion in a flat channel is studied. The temperature in the liquid is distributed according to the quadratic law, which is consistent with the velocity field of the Himentz type. At the bottom of the channel, the temperature depends on the time, which allows you to control the movement inside the layer. The Oberbeck-Boussinesq equations are taken as a mathematical model. The resulting initial - boundary value problem is highly nonlinear and inverse with respect to the pressure gradient along the channel. To solve it, a modified Galerkin method was used, where Legendre polynomials were chosen as the basis functions. The expansion coefficients are functions of time for which a system of nonlinear ODES was obtained. As a result of the application of the Runge-Kutta-Felberg method, a solution was found that, with increasing time, tends to solve a stationary problem if the temperature at the bottom of the channel stabilizes.