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Regular subsonic-sonic flows in general nozzles. (English) Zbl 1458.35321

Summary: This paper concerns subsonic-sonic potential flows in general two dimensional nozzles. For finitely long symmetric nozzles, we formulate the subsonic-sonic flow problem by prescribing the flow angle at the inlet and the outlet. It is shown that this problem admits a unique Lipschitz continuous subsonic-sonic flow, and the sonic points of the flow must occur at the wall or the throat. This is the first result on the well-posedness for general subsonic-sonic flow problems. More importantly, the location of sonic points is classified completely. Indeed, it is shown that there exists a critical value depending only on the length and the geometry of the nozzle such that the flow is sonic on the whole throat if the height of the nozzle is not greater than this critical value, while the sonic points must be located at the wall if the height is greater than this value. Furthermore, the critical height is positive iff the nozzle is suitably flat near the throat. As a direct application of this theory, we can obtain conditions on whether there is a smooth transonic flow of Meyer type whose sonic points are all exceptional in de Laval nozzles.

MSC:

35Q31 Euler equations
35J70 Degenerate elliptic equations
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
76G25 General aerodynamics and subsonic flows
35B65 Smoothness and regularity of solutions to PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
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References:

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