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Model of a viscous layer deformation by thermocapillary forces
Vladislav V. Pukhnachev
Three-dimensional nonstationary flow of viscous incompressible liquid is investigated in a layer generated by nonuniform distribution of temperature on its free boundaries. If the temperature given on the layer boundaries is quadratically dependent on homogeneous coordinates, the external mass forces are absent and the motion arises from the rest state then the problem with free boundaries for the Navier-Stokes equations has the "exact" solution determined from system of equations with two independent variables. Here the free boundaries of the layer remain parallel planes and the distance between them must be also determined. Formulated in present paper are the conditions of both the unique solvability of the reduced problem on the whole in time and the collapse of the solution in finite time. Moreover studied in this paper are qualitative properties of the solution such as its behaviour at large time in the case of global solvability of the problem and the asymptotics of the solution near the collapse moment in the opposite case.