Poiseuille-type flow in a channel with permeable walls

In the framework of the NavierStokes equations, the flow of a viscous incompressible fluid between immovable parallel permeable walls is considered, on which only the longitudinal velocity component is equal to zero. Solutions are sought in which the velocity component transverse to the plane of the plates is constant. Both stationary and non-stationary solutions are obtained, among which there is a non-trivial solution with a constant pressure and a longitudinal velocity exponentially decaying with time. These solutions show the influence on the profile of the horizontal velocity component of the removal of the boundary layer into the depth of the flow from one plate with simultaneous suction of the boundary layer on the other plate. It is established that for stationary flows the removal of the boundary layer into the depth of the flow from one plate and, with simultaneous suction of the boundary layer on the other plate, leads to an increase in the drag compared to the classical Poiseuille flow. In the case of impermeable walls, an exact non-stationary solution is obtained, the velocity profile of which at fixed times differs from the profile in the classical Poiseuille flow and, in the limit (as time tends to infinity), corresponds to rest.