On an existence theory for a fluid-beam problem encompassing possible contacts

Jean-Jérôme Casanova,Matthieu Hillairet,Céline Grandmont

doi:10.5802/jep.162

Jean-Jérôme Casanova, Matthieu Hillairet + Show 1 more

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https://doi.org/10.5802/jep.162

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Abstract

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.

Highlights

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
The fluid is described by the Navier–Stokes equations set in an unknown domain depending on the structure displacement that is assumed to be only transverse and that satisfies a beam equation
In [3, 11, 12] existence and uniqueness of a strong solution locally in time is proved in case additional viscosity is added to the structure equation

Summary

Publié avec le soutien du Centre National de la Recherche Scientifique

Publication membre du Centre Mersenne pour l’édition scientifique ouverte www.centre-mersenne.org. We 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. In [3, 11, 12] existence and uniqueness of a strong solution locally in time is proved in case additional viscosity is added to the structure equation (so that the structure displacement satisfies a damped Euler–Bernoulli equation). In [7], the authors establish existence of a global-in-time strong solution in the 2D-1D case when the structure is governed by a damped Euler–Bernoulli equation This global-in-time result is a consequence of a no contact one: it is proved therein that, for any T > 0, the structure does not touch the bottom of the cavity.

The deformed elastic configuration is denoted by

This implies

The other terms are estimated directly and we obtain

Findings

Ωi ai