[Existence de solution autorisant d’éventuels contacts pour un problème d’interaction fluide-structure]
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.
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.
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Keywords: Incompressible Navier–Stokes equations, fluid-structure interactions, weak solutions, contact issue
Mot clés : Équations de Navier-Stokes incompressible, interaction fluide-structure, solutions faibles, modélisation du contact
@article{JEP_2021__8__933_0,
author = {Casanova, Jean-J\'er\^ome and Grandmont, C\'eline and Hillairet, Matthieu},
title = {On an existence theory for a fluid-beam problem encompassing possible contacts},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {933--971},
publisher = {Ecole polytechnique},
volume = {8},
year = {2021},
doi = {10.5802/jep.162},
language = {en},
url = {http://www.numdam.org/articles/10.5802/jep.162/}
}
TY - JOUR AU - Casanova, Jean-Jérôme AU - Grandmont, Céline AU - Hillairet, Matthieu TI - On an existence theory for a fluid-beam problem encompassing possible contacts JO - Journal de l’École polytechnique — Mathématiques PY - 2021 SP - 933 EP - 971 VL - 8 PB - Ecole polytechnique UR - http://www.numdam.org/articles/10.5802/jep.162/ DO - 10.5802/jep.162 LA - en ID - JEP_2021__8__933_0 ER -
%0 Journal Article %A Casanova, Jean-Jérôme %A Grandmont, Céline %A Hillairet, Matthieu %T On an existence theory for a fluid-beam problem encompassing possible contacts %J Journal de l’École polytechnique — Mathématiques %D 2021 %P 933-971 %V 8 %I Ecole polytechnique %U http://www.numdam.org/articles/10.5802/jep.162/ %R 10.5802/jep.162 %G en %F JEP_2021__8__933_0
Casanova, Jean-Jérôme; Grandmont, Céline; Hillairet, Matthieu. On an existence theory for a fluid-beam problem encompassing possible contacts. Journal de l’École polytechnique — Mathématiques, Tome 8 (2021), pp. 933-971. doi : 10.5802/jep.162. http://www.numdam.org/articles/10.5802/jep.162/
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