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Non-uniqueness of almost unidirectional inviscid compressible flow. (English) Zbl 1099.76053

Summary: Our aim is to find roots of the non-unique behavior of gases which can be observed in certain axisymmetric nozzle geometries under special flow regimes. For this purpose, we use several versions of the compressible Euler equations. We show that the main reason for the non-uniqueness is hidden in the energy decomposition into its internal and kinetic parts, and their complementary behavior. It turns out that, at least for inviscid compressible flows, a bifurcation can occur only at flow regimes with the Mach number equal to one (sonic states). Analytical quasi-one-dimensional results are supplemented by quasi-one-dimensional and axisymmetric three-dimensional finite volume computations. Good agreement between quasi-one-dimensional and axisymmetric results, including the presence of multiple stationary solutions, is presented for axisymmetric nozzles with reasonably small slopes of the radius.

MSC:

76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35L65 Hyperbolic conservation laws
35Q35 PDEs in connection with fluid mechanics
76H05 Transonic flows
76M12 Finite volume methods applied to problems in fluid mechanics
76N15 Gas dynamics (general theory)
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