Спектр краевой задачи, описывающей двумерное плоское стационарное термокапиллярное течение в канале

The problem of two-dimensional thermocapillary fluid flow in a channel with heated bottom is considered. Condition of thermal contact is set on the upper free boundary. The velocity field is linear with respect to the longitudinal coordinate, and the temperature and pressure fields are quadratic functions of the same coordinate. The analysis of the compatibility of the Navier-Stokes equations and the equation of heat transfer leads to a non-linear eigenvalue problem for finding the flow field in the layer. The spectrum of this problem is studied analytically for small Marangoni numbers (second approximation) and numerically for abitrary Marangoni numbers. The non-uniqueness of the solution is established. It is typical for problems of this kind