Abstract

This paper concerns the following transonic shock phenomena in a three-dimensional de Laval nozzle described by Courant and Friedrichs: Given the appropriately large receiver pressure p_r, if the upstream flow is still supersonic behind the throat of the nozzle, then at a certain place in the widening part of the nozzle a shock front intervenes, and the gas is compressed and slowed down to subsonic speed. The position and the strength of the shock front are automatically adjusted so that the end pressure at the exit becomes p_r. We study this problem for the inviscid steady potential equation. In this case, the transonic shock is a free boundary dividing the hyperbolic region and the elliptic region in the nozzle. One main result is that for a general class of nozzles, such a transonic shock solution is unique if the shock exists and is assumed to pass through a fixed point. We also construct a class of de Laval nozzles such that the transonic shock phenomena do not occur for the generally given large pressures at the exit for the potential flow model.

Keywords

transonic flow, ill-posedness, well-posedness, potential equation, multidimensional shock wave, nozzle

Mathematical Subject Classification

Primary: 35L67, 35L65, 35L70

Secondary: 76N15

Authors