Thermocapillary motion of two viscous liquids in a cylindrical pipe

A study is made of an invariant solution of the equations motion of a viscous heat-conducting fluids, which is treated as unidirectional motion in a circular pipe with a common interface under the action of the thermocapillary force. A priori estimates of the velocity and temperature are obtained. The steady state is determined and it is shown that, at larger times, this state is the limiting one. It was established that the thermocapillary effect with surface curvature can induce the return flow. Using Laplace transformation properties the exact analytical solution was constructed. Some examples of numerical reconstruction of the velocities fields depending on geometric and physical parameters were considered.