The evolution of the thermocapillary motion of three fluids in a plane layer

The unidirectional motion of three immiscible incompressible viscous heat-conducting liquids in a plane layer is considered. It is assumed that the motion occurs only under the action of thermocapillary forces from a state of rest. The analysis of the motion is reduced to solving linear conjugate initial boundary value problems for a system of parabolic equations. A non-stationary solution is sought by the Laplace transformation method and is obtained in the form of finite analytical expressions in transforms. It is proved that, as the time increases, the solution always reaches the steady state obtained earlier and an exponential estimate of the rate of convergence is given with an indicator which depends on the physical properties of the media and the layer thicknesses. The evolution of the velocity and temperature perturbation fields to a steady state for specific liquid media is obtained by numerical inversion of the Laplace transformation.