Incompressible Jet Flows in a de Laval Nozzle with Smooth Detachment

In this paper, we are concerned with the well-posedness theory of steady incompressible jet flow in a de Laval type nozzle with given end pressure at the outlet. The main results show that for any given incoming mass flux Q > 0 in the upstream and end pressure at the outlet, there exists an admissible interval to the Bernoulli’s constant, if the Bernoulli’s constant lies in the interval, there exists a unique smooth incompressible jet flow issuing from the nozzle. Moreover, the free boundary of the jet flow initiates smoothly from the surface of the divergent area of the de Laval nozzle. In particular, it is shown that the initial point of the free boundary lies behind the throat of the de Laval nozzle wall and varies continuously and monotonically with respect to the Bernoulli’s constant. As a direct corollary, imposing initial point of the free boundary on the divergent part of nozzle wall, there exists a unique incompressible jet for given incoming mass flux Q > 0 and pressure P_e at the outlet. This work is inspired by the significant works (Alt et al. in Commun Pure Appl Math 35:29–68, 1982; Arch Rational Mech Anal 81:97–149, 1983) for the incompressible jet flow imposing the initial point of the free boundary at the endpoint of the nozzle.