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Boundary Conditions for Planar Stokes Equations Inducing Vortices Around Concave Corners

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Abstract

Fluid flows around an obstacle generate vortices which are difficult to locate and to describe. A variational formulation for a class of mixed and nonstandard boundary conditions on a smooth obstacle is discussed for the Stokes equations. Possible boundary data are then derived through separation of variables of biharmonic equations in a planar region having an internal concave corner. Explicit singular solutions show that, at least qualitatively, these conditions are able to reproduce vortices over the leeward wall of the obstacle.

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Acknowledgements

The Authors are grateful to Andrei Fursikov (Moscow State University) for his valuable comments on a preliminary version of the present paper. The first Author is partially supported by the PRIN project Equazioni alle derivate parziali di tipo ellittico e parabolico: aspetti geometrici, disuguaglianze collegate, e applicazioni and by the Gruppo Nazionale per l’Analisi Matematica, la Probabilitàe le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

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  1. Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133, Milano, Italy

    Filippo Gazzola & Gianmarco Sperone

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  1. Filippo Gazzola
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  2. Gianmarco Sperone
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Correspondence to Filippo Gazzola.

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Gazzola, F., Sperone, G. Boundary Conditions for Planar Stokes Equations Inducing Vortices Around Concave Corners. Milan J. Math. 87, 169–199 (2019). https://doi.org/10.1007/s00032-019-00297-0

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