Unsteady flow of two binary mixtures in a cylindrical capillary with changes in the internal energy of the interface

The problem of two-dimensional unsteady flow of two immiscible incompressible binary mixtures in a cylindrical capillary in the absence of mass forces is studied. The mixtures are contacted through a common interface on which the energy condition is taken into account. The temperature and concentration of mixtures are distributed according to the quadratic law. It is in good agreement with the velocity field of the Hiemenz type. The resulting conjugate boundary value problem is a non-linear problem. It is also an inverse problem with respect to the pressure gradient along the axis of the cylindrical tube. To solve the problem the tau-method is used. It was shown that with increasing time the solution of the non-stationary problem tends to a steady state. It was established that the effect of increments of the internal energy of the inter-facial surface significantly affects the dynamics of the flow of mixtures in the layers.