On the solution properties of boundary problem simulating thermocapillary flow

An inverse initial boundary value problem that arises as a result of mathematical modelling of specific thermocapillary 2D motion near an extreme point on solid wall is investigated. One of the velocity field components considered motion linearly depends on the longitudinal coordinate. This is a good agrement with the quadratic dependence of temperature field on the same coordinate. For stationary flow in the case of small Marangoni numbers the solution can be found by exact formulae. Nonstationary solution is found in quadratures in Laplace transformation space. The calculation results of zero and first solution approximations of this inverse stationary problem are given. If temperature on the solid wall is stabilized with time, then the nonstationary solution will converge to steady regime. The calculations are performed for different values of the Prandtl number and Bio number. Numerical results well support the theoretical conclusions on the example of modelling process arising the thermocapillary motion from a state of rest in the transformer oil layer. It is shown that choosing a specific thermal regime on a solid wall it is possible to control the fluid motion inside a layer.