Two-phase fluids in collision of incompressible inviscid fluids effluxing from two nozzles

This paper is devoted to the mathematical theory of two-dimensional injection of incompressible, irrotational, and inviscid fluids issuing from two infinitely long nozzles into a free stream. In general, there is a free interface with constant jump of the Bernoulli constant on it, which is different and more difficult than the previous related works. Physically, it is called the collision fluid. The main result in this paper is that for given two co-axis symmetric infinitely long nozzles, imposing the incoming mass fluxes in the two nozzles, there exists a unique piecewise smooth collision fluid, such that the free interface of the collision fluid is a C¹ curve, and the pressure is continuous across the interface. As byproducts, the asymptotic behaviors, the positivity of the vertical velocity, monotone relationship between the location of the interface and the incoming mass fluxes are also established.