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Steady Navier–Stokes Equations in Planar Domains with Obstacle and Explicit Bounds for Unique Solvability

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Abstract

Fluid flows around an obstacle generate vortices which, in turn, generate forces on the obstacle. This phenomenon is studied for planar viscous flows governed by the stationary Navier–Stokes equations with inhomogeneous Dirichlet boundary data in a (virtual) square containing an obstacle. In a symmetric framework the appearance of forces is strictly related to the multiplicity of solutions. Precise bounds on the data ensuring uniqueness are then sought and several functional inequalities (concerning relative capacity, Sobolev embedding, solenoidal extensions) are analyzed in detail: explicit bounds are obtained for constant boundary data. The case of “almost symmetric” frameworks is also considered. A universal threshold on the Reynolds number ensuring that the flow generates no lift is obtained regardless of the shape and the nature of the obstacle. Based on the asymmetry/multiplicity principle, the performance of different obstacle shapes is then compared numerically. Finally, connections of the results with elasticity and mechanics are emphasized.

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Acknowledgements

This paper was initiated when both the Authors were visiting the Moscow State University in August 2018. The second Author remained in Moscow for a whole term and he is grateful to Professor Andrei Fursikov for the kind hospitality and for fruitful discussions. The first Author is partially supported by the PRIN project Direct and inverse problems for partial differential equations: theoretical aspects and applications and by the GNAMPA group of the INdAM. The final corrections of this paper were done when the second Author was already a post-doctoral researcher at the Department of Mathematical Analysis of the Charles University in Prague (Czech Republic), supported by the Primus Research Programme PRIMUS/19/SCI/01, by the program GJ19-11707Y of the Czech National Grant Agency GAČR, and by the University Centre UNCE/SCI/023 of the Charles University in Prague. The Authors warmly thank two anonymous Referees for their careful proofreading and for several suggestions that led to an improvement of the present work.

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  1. Gianmarco Sperone

    Present address: Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University in Prague, Sokolovská 83, 186 75, Prague, Czech Republic

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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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Gazzola, F., Sperone, G. Steady Navier–Stokes Equations in Planar Domains with Obstacle and Explicit Bounds for Unique Solvability. Arch Rational Mech Anal 238, 1283–1347 (2020). https://doi.org/10.1007/s00205-020-01565-9

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