On an existence theory for a fluid-beam problem encompassing possible contacts
Journal de l’École polytechnique — Mathématiques, Volume 8 (2021), pp. 933-971.

In this paper we consider a coupled system of pdes modeling the interaction between a two-dimensional incompressible viscous fluid and a one-dimensional elastic beam located on the upper part of the fluid domain boundary. We design a functional framework to define weak solutions in case of contact between the elastic beam and the bottom of the fluid cavity. We then prove that such solutions exist globally in time regardless a possible contact by approximating the beam equation by a damped beam and letting this additional viscosity vanish.

Dans cet article, nous considérons un système couplé d’équations aux dérivées partielles modélisant l’interaction entre un fluide visqueux incompressible bi-dimensionnel et une poutre élastique mono-dimensionnelle située sur le bord supérieur du domaine fluide. Après avoir construit un cadre fonctionnel de solutions faibles autorisant les configurations où la poutre est en contact avec le fond de la cavité fluide, l’existence de solutions faibles, globale en temps, est démontrée, que des contacts se produisent ou non. La preuve repose sur l’analyse asymptotique d’un système couplé parabolique-parabolique pour lequel un terme de viscosité est ajouté à la structure, et dont on sait qu’il n’autorise pas les contacts. La limite de viscosité évanescente est alors solution de la formulation faible introduite et autorisant le contact.

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Accepted:
Published online:
DOI: 10.5802/jep.162
Classification: 76D05,  35D30,  35Q35,  74F10,  76D03
Keywords: Incompressible Navier–Stokes equations, fluid-structure interactions, weak solutions, contact issue
Jean-Jérôme Casanova 1; Céline Grandmont 2; Matthieu Hillairet 3

1 CEREMADE, UMR CNRS 7534, Université Paris-Dauphine, PSL Research University Place du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France
2 Inria Paris 75012 Paris, France & Sorbonne Université, UMR 7598 LJLL 75005 Paris, France
3 IMAG, Univ Montpellier, CNRS Montpellier, France
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Jean-Jérôme Casanova; Céline Grandmont; Matthieu Hillairet. On an existence theory for a fluid-beam problem encompassing possible contacts. Journal de l’École polytechnique — Mathématiques, Volume 8 (2021), pp. 933-971. doi : 10.5802/jep.162. https://jep.centre-mersenne.org/articles/10.5802/jep.162/

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