Existence and Stability of Multidimensional Transonic Flows through an Infinite Nozzle of Arbitrary Cross-Sections

Authors: Gui-Qiang Chen and Mikhail Feldman

Title: Existence and Stability of Multidimensional Transonic Flows through an Infinite Nozzle of Arbitrary Cross-Sections

Abstract
We establish the existence and stability of multidimensional steady transonic flows through an infinite nozzle with arbitrary cross-sections, including the slowly varying de Lavel nozzles. The transonic flow is governed by the inviscid potential flow equation with supersonic upstream at the entrance, uniform subsonic downstream at infinity, and the slip boundary condition on the nozzle boundary. The multidimensional transonic nozzle problem is reformulated into a free boundary problem, for which the free boundary is a transonic shock dividing two regions of $C^{1,\alpha}$ flow in the infinite nozzle, and the equation is hyperbolic in the upstream supersonic region and elliptic in the downstream subsonic region. We further develop a nonlinear iteration approach and employ its advantages to deal with such a free boundary problem in the unbounded domain and to solve the multidimensional transonic nozzle problem in a direct and simple fashion. Our results indicate that, for the transonic nozzle problem, there exists a unique transonic flow such that the flow is divided into a $C^{1,\alpha}$ subsonic flow up to the nozzle boundary in the unbounded downstream region from the supersonic upstream flow by a $C^{1,\alpha}$ multidimensional transonic shock that is orthogonal to the nozzle boundary at every point of their intersection, and the uniform velocity state at infinity in the downstream direction is uniquely determined by the supersonic upstream flow at the entrance which is sufficiently close to a uniform flow. The uniform velocity state at the infinity can not be apriori prescribed from the corresponding pressure for such a flow to exist. We further prove that the transonic flow with a transonic shock is stable with respect to the supersonic upstream flow at the entrance.
This article has appeared in:
Arch. Rational Mech. Anal. vol. 20, pages (2006)
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Author Address
    Gui-Qiang Chen
    Department of Mathematics
    Northwestern University
    Evanston, IL 60208-2730
    gqchen@math.nwu.edu