  We study the inviscid limit problem of the incompressible flows in the
presence of both impermeable regular boundaries and a hypersurface transversal
to the boundary across which the inviscid flow has a discontinuity jump. In the
former case, boundary layers have been introduced by Prandtl as correctors near
the boundary between the inviscid and viscous flows. In the latter case, the
viscosity smoothes out the discontinuity jump by creating a transition layer
which has the same amplitude and thickness as the Prandtl layer. In the
neighborhood of the intersection of the impermeable boundary and of the
hypersurface, interactions between the boundary and the transition layers must
then be considered. In this paper, we initiate a mathematical study of this
interaction and carry out a strong convergence in the inviscid limit for the
case of the plane parallel flows introduced by Di Perna and Majda in \cite{DM}.
