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Singularities and unsteady separation for the inviscid two-dimensional Prandtl system
- Source :
- Archive for Rational Mechanics and Analysis, Archive for Rational Mechanics and Analysis, Springer Verlag, 2021, 240 (3), pp.1349-1430. ⟨10.1007/s00205-021-01637-4⟩
- Publication Year :
- 2021
- Publisher :
- Springer Science and Business Media LLC, 2021.
-
Abstract
- We consider the inviscid unsteady Prandtl system in two dimensions, motivated by the fact that it should model to leading order separation and singularity formation for the original viscous system. We give a sharp expression for the maximal time of existence of regular solutions, showing that singularities only happen at the boundary or on the set of zero vorticity, and that they correspond to boundary layer separation. We then exhibit new Lagrangian formulae for backward self-similar profiles, and study them also with a different approach that was initiated by Elliott–Smith–Cowley and Cassel–Smith–Walker. One particular profile is at the heart of the so-called Van-Dommelen and Shen singularity, and we prove its generic appearance (that is, for an open and dense set of blow-up solutions) for any prescribed Eulerian outer flow. We comment on the connection between these results and the full viscous Prandtl system. This paper combines ideas for transport equations, such as Lagrangian coordinates and incompressibility, and for singularity formation, such as self-similarity and renormalisation, in a novel manner, and designs a new way to study singularities for quasilinear transport equations.
- Subjects :
- Physics
Mechanical Engineering
010102 general mathematics
Mathematical analysis
Prandtl number
Boundary (topology)
Vorticity
01 natural sciences
Physics::Fluid Dynamics
010101 applied mathematics
symbols.namesake
Flow separation
Lagrangian and Eulerian specification of the flow field
Mathematics (miscellaneous)
Singularity
Inviscid flow
symbols
Gravitational singularity
[MATH]Mathematics [math]
0101 mathematics
ComputingMilieux_MISCELLANEOUS
Analysis
Subjects
Details
- ISSN :
- 14320673 and 00039527
- Volume :
- 240
- Database :
- OpenAIRE
- Journal :
- Archive for Rational Mechanics and Analysis
- Accession number :
- edsair.doi.dedup.....0eb268c5cb62f582e150b7db86ff49f9