Symmetric Fluid Outflow from a Vessel Bounded by Two Parallel Walls through a Filter (Modification of Joukowsky’s Problem)

In the paper, we investigate the problem of a symmetric two-dimensional fluid outflow from a vessel bounded by two parallel walls. The outflow is through a filter modeled by a permeable surface. It is assumed that when a liquid passes through such a surface, the local seepage velocity is proportional to the local pressure drops between the input and output sides of the filter. It is also assumed that the filter has a directing effect: the outflowing liquid moves in the direction perpendicular to the plane of the filter. The liquid moves under the action of a given pressure difference at infinity in the vessel and in the outflowing jet. Inside the vessel, the liquid is assumed to be ideal and incompressible, and the flow is assumed to be potential. After passing through the filter, the liquid remains ideal and incompressible, but the unidirectional flow in the outflowing jet is no longer potential and will have a vorticity distribution. We have found an exact analytical solution of the problem using the method of conformal mapping. The proposed formulation modifies that by Joukowsky (Mat. Sbornik 15 , 1890), who studied a similar outflow through an open orifice with the formation of a jet bounded by free streamlines.