A Liouville-type theorem for subsonic flows around an infinite long ramp. (English) Zbl 1292.35065
Summary: We focus on the two-dimensional subsonic flow problem for polytropic gases around an infinite long ramp, which is motivated by a description in Section 111 of R. Courant and K. O. Friedrichs’ book [Supersonic flow and shock waves. New York: Interscience Publ. (1948; Zbl 0041.11302)]. The flow is assumed to be steady, isentropic and irrotational; namely, the movement of the flow is described by a second-order steady potential equation. By the complex methods together with some properties on quasi-conformal mappings, we show that a nontrivial subsonic flow around the infinite long ramp does not exist if the flow is uniformly subsonic.
MSC:
| 35B53 | Liouville theorems and Phragmén-Lindelöf theorems in context of PDEs |
| 35L65 | Hyperbolic conservation laws |
| 35L67 | Shocks and singularities for hyperbolic equations |
| 76N15 | Gas dynamics (general theory) |
Keywords:
potential equation; Riemann metric; isentropic irrotational steady flow; polytropic gases; quasi-conformal mappingsCitations:
Zbl 0041.11302References:
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