Sonic-supersonic solutions for the steady Euler equations

Tianyou Zhang, Yuxi Zheng

Research output: Contribution to journalArticle

11 Citations (Scopus)

Abstract

Given a smooth curve as a sonic line in the plane, we construct a local smooth supersonic solution on one side of the curve for the steady compressible Euler system of equations in two space dimensions. Our construction hinges on a new set of coordinates introduced here to handle the inherent degeneracy of the system at the sonic curve. We analyze the streamlines of the solutions to illustrate that the shock-free portion of the solutions may be combined with known results of existence of sonic-subsonic solutions of Xie and Xin [33] on the other side of the curve to form shock-free transonic flows in a channel. The existence result is also a partial generalization of the exact solution of Ringleb [28, ZAMM (1940)] toward a flexible existence.

Original languageEnglish (US)
Pages (from-to)1785-1817
Number of pages33
JournalIndiana University Mathematics Journal
Volume63
Issue number6
DOIs
StatePublished - Jan 1 2014

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Euler Equations
Curve
Shock
Euler System
Transonic Flow
Streamlines
Degeneracy
System of equations
Existence Results
Exact Solution
Partial
Line

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

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Sonic-supersonic solutions for the steady Euler equations. / Zhang, Tianyou; Zheng, Yuxi.

In: Indiana University Mathematics Journal, Vol. 63, No. 6, 01.01.2014, p. 1785-1817.

Research output: Contribution to journalArticle

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