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On Transonic Shocks in a Nozzle with Variable End Pressures

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Abstract

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 p r , 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 p r . When the end pressure p r 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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Authors and Affiliations

  1. Department of Mathematics & IMS, Nanjing University, Nanjing, 210093, P.R. China

    Jun Li & Huicheng Yin

  2. Department of Mathematics and IMS, CUHK, Shatin, N.T., Hong Kong

    Zhouping Xin

Authors
  1. Jun Li
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  2. Zhouping Xin
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  3. Huicheng Yin
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Correspondence to Zhouping Xin.

Additional information

Communicated by P. Constantin

Supported by the National Natural Science Foundation of China (No.10571082) and the National Basic Research Programm of China (No.2006CB805902).

Supported in part by Zheng Ge Ru Foundation, and Hong Kong RGC Earmarked Research Grants CUHK4028/04P, CUHK4040/06P and RGC Central Allocation Grant CA05/06.SC01.

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Li, J., Xin, Z. & Yin, H. On Transonic Shocks in a Nozzle with Variable End Pressures. Commun. Math. Phys. 291, 111–150 (2009). https://doi.org/10.1007/s00220-009-0870-9

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  • DOI: https://doi.org/10.1007/s00220-009-0870-9

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