Thermocapillary Motion. Nonlinear Problems

Statement of the problem. Let us consider the problem of a steady-state thermocapillary motion in a liquid layer of thickness h. The motion is assumed to be two-dimensional. The dependence of the surface tension on temperature is assumed to be quadratic according to (5.9.19). The thermogravitational effect is not taken into account. It is assumed that the linear temperature distribution is maintained on the hard lower surface, and the plane surface of the layer is thermally insulated. The origin of the Cartesian coordinates X, Y is placed on the solid surface at the point with temperature To. The velocity and temperature fields are described by Eqs. (5.9.1)-(5.9.4) with jg = 0. 

The boundary conditions taking into account the quadratic dependence of the surface tension (5.9.19) on temperature have the form 

the no-slip and no-flow conditions hold on the hard surface, and a linear temperature distribution is maintained. Condition (5.9.21) says that the no-flow condition on the free surface and the condition of zero heat flux through the free surface must hold, and the balance of tangential thermocapillary and viscous stresses must be provided. Taking into account the quadratic dependence (5.9.19) of the surface tension on temperature, we rewrite the right-hand side of the last condition in (5.9.21) using the relation 

Mass and Heat Transfer Under Complicating Factors 

For the unknown functions ip(y), Q(y), and f(y) and the constant A, treated as an eigenvalue, we obtain the following problem  

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