 Solution

The total velocity fields on the two sides areEach base flow is divergence-free. Incompressibility therefore makes both disturbance velocity potentials satisfy the Laplace equation:At the fixed horizontal walls, impermeability requires zero vertical velocity:The discontinuity in tangential base velocity at the interface is a vortex sheet. The assumption of irrotational flow is understood separately in each open fluid region; it cannot hold distributionally across that sheet when $U\ne 0$.