Andreev V.K.  

On the direct and inverse problems of a creeping motions in a thin layers

In this report, a plane analogue of Hiemenz solution of Navier-Stokes equations is studied in the framework of the Oberbeck-Boussinesq model. The first part of the report is devoted to review the results of the direct mathematical problems. As a rule, these problems allow to describe liquid flow in microchannel with a free boundary (Andreev V.K., 2012). The second part of the report demonstrates some new results concerning inverse problem. This problem describes common motions of two immiscible liquids in a plane channel under the action of thermocapillary forces. As Marandoni number is small (so-called creeping flow) the problem becomes a linear one. Some a priori estimates are obtained and restructions on initial data for tending the solution to stationary one are found. In Laplace images the exact solution is obtained as quadratures and some numerical results of velocities and temperature behavior in microchannels are presented.