Subsonic Irrotational Flows in a Finitely Long Nozzle with Variable end Pressure

In this paper, we characterize a set of physically acceptable boundary conditions that ensure the existence and uniqueness of a subsonic irrotational flow in a finitely long flat nozzle. Our results show that if the incoming flow is horizontal at the inlet and an appropriate pressure is prescribed at the exit, then there exist two positive constants m 0 and m 1 with m 0 < m 1, such that a global smooth irrotational subsonic flow exists uniquely in the nozzle, provided that the incoming mass flux m ∈ [m 0, m 1). The horizontal velocity of the flow is always positive in the whole nozzle and the maximum speed of the flow will approach the sonic speed as the mass flux m tends to m 1. The flow is governed by the steady compressible irrotational Euler equations with nonlocal and nonlinear mixed boundary conditions. A new key issue is that the Bernoulli's constant of the irrotational flow is not given a priori, which can be determined uniquely by the end pressure and the incoming mass flux. To handle the nonlocal boundary condition raised from the mass flux, we introduce an auxiliary problem with Bernoulli's constant as a parameter, instead of the mass flux. Furthermore, the monotonicity between the mass flux and the Bernoulli's constant is established for given end pressure.