Steady collision of two jets issuing from two axially symmetric channels

Abstract: In the classical survey (Chapter 16.2, {\it Mathematics in industrial problem}, Vol. 24, Springer-Verlag, New York, 1989), A. Friedman proposed an open problem on the collision of two incompressible jets emerging from two axially symmetric nozzles. In this paper, we concerned with the mathematical theory on this collision problem, and establish the well-posedness theory on hydrodynamic impinging outgoing jets issuing from two coaxial axially symmetric nozzles. More precisely, we showed that for any given mass fluxes $M_1>0$ and $M_2<0$ in two nozzles respectively, that there exists an incompressible, inviscid impinging outgoing jet with contact discontinuity, which issues from two given semi-infinitely long axially symmetric nozzles and extends to infinity. Moreover, the constant pressure free stream surfaces of the impinging jet initiate smoothly from the mouths of the two nozzles and shrink to some asymptotic conical surface. There exists a smooth surface separating the two incompressible fluids and the contact discontinuity occurs on the surface. Furthermore, we showed that there is no stagnation point in the flow field and its closure, except one point on the symmetric axis. Some asymptotic behavior of the impinging jet in upstream and downstream, geometric properties of the free stream surfaces are also obtained. The main results in this paper solved the open problem on the collision of two incompressible axially symmetric jets in [24].