Injection of Ideal Fluid from a Slot into a Stream: Two Free Boundaries

A fluid is injected from a slot into a stream of another fluid.In a simple model this leads to a two-phase two-free-boundaryproblem with the jump relation |u^–|² – |u⁺|² = on the free boundary {u=0}, and |u| = ¹ on the free boundary{u > – Q}, where u is the stream function and Q isthe flux of the injected fluid. Using the variational theoryof Alt, Caffarelli & Friedman, we prove existence of (,₁, u) such that there is a smooth fit for both free boundaries.