The motion of a binary mixture and viscous liquid in a circular pipe under the action of an unsteady pressure gradient

We study an invariant solution of the equations of thermodiffusion motion and of a viscous heat-conducting fluid, which is treated as an unidirectional motion in a circular pipe with a common interface under the action of an unsteady pressure gradient. A priori estimates of the velocity, temperature and concentration are obtained. The steady state is determined and it is shown (under some conditions on the pressure gradient) that, at larger times, this state is the limiting one. On the other hand, if the pressure gradient of binary mixture tends to zero sufficiently fast, then the motion in pipe is slowed down by the viscous friction.