Global subsonic Euler flows in an infinitely long axisymmetric nozzle

Abstract

In this paper, we consider global subsonic compressible flows through an infinitely long axisymmetric nozzle. The flow is governed by the steady Euler equations and has boundary conditions on the nozzle walls. Existence and uniqueness of global subsonic solution are established for an infinitely long axisymmetric nozzle, when the variation of Bernoulli's function in the upstream is sufficiently small and the mass flux of the incoming flow is less than some critical value. The results give a strictly mathematical proof to the assertion in Bers (1958) [2]: there exists a critical value of the incoming mass flux such that a global subsonic flow exists uniquely in a nozzle, provided that the incoming mass flux is less than the critical value. The existence of subsonic flow is obtained by the precisely a priori estimates for the elliptic equation of two variables. With the assumptions on the nozzle in the far fields, the asymptotic behavior can be derived by a blowup argument for the infinitely long nozzle. Finally, we obtain the uniqueness of uniformly subsonic flow by energy estimate and derive the existence of the critical value of incoming mass flux.

Subsonic Flows in a Multi-Dimensional Nozzle

“…Similar approach has been applied in 3D axis-asymmetric nozzles by Xie and Xin in [28]. Furthermore, these ideas are also useful to study the physically more important case, subsonic Euler flows, by Xie and Xin in [29] (see also the generalization in [11]). However, it seems difficult to apply the method in [27] and [28] in general multi-dimensional (n ≥ 3) nozzles, since the stream function formulation can not work in this case.…”

Subsonic non-isentropic Euler flows with large vorticity in axisymmetric nozzles

“…Applying such formulation is the key issue in the current paper to derive well-posedness with large vorticity for non-isentropic flows. One may also refer to the results in periodic nozzles [5], in bounded nozzles [6], in three dimensional axially symmetric nozzles [9,10], and [2,13] the results for the non-isentropic Euler system. Note that the results for the flows with non-vanishing vorticity mentioned above hold if the Bernoulli's function has small variation, which indeed implies that the vorticity of the incoming flows is sufficiently small in the whole nozzle.…”

“…satisfying the non-isentropic Euler system (7), the solid wall condition (9), the symmetric condition (10), the condition (11), and…”

Three dimensional steady subsonic Euler flows in bounded nozzles

“…1 When the flows have non-zero vorticity, steady incompressible Euler flows which are the zero Mach number limits of compressible subsonic flows were investigated in [1,20,27] and references therein. The existence of subsonic Euler flows in two dimensional and three dimensional axially symmetric nozzles was established in [31,14] via a stream function formulation for the Euler equations, see [9,5] for recent study on subsonic Euler flows in half plane.…”