On Transonic Shocks in a Nozzle with Variable End Pressures

In the book, Courant and Friedrichs (Supersonic Flow and Shock Waves. New York: Interscience Publishers, 1948) described the following transonic shock phenomena in a de Laval nozzle: Given the appropriately large receiver pressure pr, if the upstream flow is still supersonic behind the throat of the nozzle, then at a certain place in the diverging 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 pr. When the end pressure pr varies and lies in an appropriate scope, in general, it is expected that a curved transonic shock is still formed in a nozzle. In this paper, we solve this problem for the two-dimensional steady Euler system with a variable exit pressure in a nozzle whose divergent part is an angular sector. Both existence and uniqueness are established.

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