Mathematics & Physics 2016, 9(4), 404–415 УДК 517.9 2D Thermocapillary Motion of Three Fluids in a Flat Channel Victor K. Andreev∗ Elena N.Cheremnykh† Institute of computational modelling SB RAS Akademgorodok, 50/44, Krasnoyarsk, 660036 Institute of Mathematics and Computer Science Siberian Federal University Svobodny, 79, Krasnoyarsk, 660041 Russia Received 17.05.2016, received in revised form 12.08.2016, accepted 15.09.2016 Two-dimensional creeping motion of three immiscible, incompressible viscous fluids in a flat channel bounded by fixed solid walls, on which the temperature distribution is known, is investigated. <...> Unsteady motion is described by finite analytic formulas obtained by Laplace transform in images. <...> The evolution of the velocity fields to the stationary regime for specific liquids is obtained by the numerical inversion of Laplace transformation. <...> Nowadays, there are exact solutions of the Marangoni convection [5–7]. <...> This is the Poiseuille stationary flow of two immiscible liquids in an inclined channel. <...> As for non-stationary thermocapillary flows, studying of them began recently [11, 12]. <...> Thermocapillary convection problem for two incompressible liquids separated by a closed interface in a container was investigated in [13]. <...> Local (in time) unique solvability of the problem was obtained in Holder classes of functions. <...> Moreover, its unique solvability in Holder spaces with a power-like weight at infinity was established. <...> Velocity vector field decreases at infinity in the same way as the initial data and mass forces, the temperature diverges to the constant which is the limit of the initial temperature at infinity. ∗andr@icm.krasu.ru †elena cher@icm.krasn.ru ⃝ Siberian Federal University. <...> All rights reserved c – 404 – Victor K. Andreev, Elena N.Cheremnykh 2D Thermocapillary Motion of Three Fluids in a Flat Channel The present work is devoted to studying of solutions of a conjugate boundary value problem arising as a result of linearization of the Navier-Stokes system asupplemented with temperature equation. <...> The description of the 2D creeping joint motion of three viscous heat conducting fluids in flat layers is also provided here. <...> The motion arises due to thermocapillary forces imposed along two interfaces, after which the unsteady Marangoni convection begins. <...> Here u, v are the components of the velocity vector, p is the pressure <...>