Existence for an Unsteady Fluid-Structure Interaction Problem

Grandmont, Céline; Maday, Yvon

doi:10.1051/m2an:2000159

Cited by 125 publications

(117 citation statements)

References 16 publications

(22 reference statements)

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“…In most of the previous studies the structure velocity is quite regular because of the model or because of the presence of a regularization operator in the equations. The existing results are concerned mainly with rigid body motions [5], [10], [11], [13], [16], [17], [18], [21], [22], [25], [24] or with the motion of a structure described by a finite number of modal functions [12] or a structure with additional "viscous" terms [2], [4], [8]. Recently, a significant breakthrough has been made by D. Coutand and S. Shkoller.…”

Section: Introductionmentioning

confidence: 99%

Existence of Weak Solutions for the Unsteady Interaction of a Viscous Fluid with an Elastic Plate

Chambolle

Desjardins

Esteban

et al. 2005

J. math. fluid mech.

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We consider a three-dimensional viscous incompressible fluid governed by the Navier-Stokes equations, interacting with an elastic plate located on one part of the fluid boundary. We do not neglect the deformation of the fluid domain which consequently depends on the displacement of the structure. The purpose of this work is to study the solutions of this unsteady fluid-structure interaction problem, as the coefficient modeling the viscoelasticity (resp. the rotatory inertia) of the plate tends to zero. As a consequence, we obtain the existence of at least one weak solution for the limit problem (Navier-Stokes equation coupled with a plate in flexion) as long as the structure does not touch the bottom of the fluid cavity.

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Section: Introductionmentioning

confidence: 99%

Existence of Weak Solutions for the Unsteady Interaction of a Viscous Fluid with an Elastic Plate

Chambolle

Desjardins

Esteban

et al. 2005

J. math. fluid mech.

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187

209

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“…This system is a simplified model corresponding to the motion of a rigid body into a viscous incompressible fluid (see [24,10,11,9,19,25,15,17] for some references). In our case, we replace the Navier-Stokes system by the viscous Burgers equation and the rigid body is reduced to a point particle.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Feedback stabilization of a simplified 1d fluid–particle system

Badra

Takahashi

2014

Ann. Inst. H. Poincaré Anal. Non Linéaire

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We consider the feedback stabilization of a simplified 1d model for a fluid-structure interaction system. The fluid equation is the viscous Burgers equation whereas the motion of the particle is given by the Newton's laws. We stabilize this system around a stationary state by using feedbacks located at the exterior boundary of the fluid domain. With one input, we obtain a local stabilizability of the system with an exponential decay rate of order σ < σ0. An arbitrary order for the exponential decay rate can be proved if a unique continuation result holds true or if two inputs are used to stabilize the system. Our method is based on general arguments for stabilization of nonlinear parabolic systems combined with a change of variables to handle the fact that the fluid domains of the stationary state and of the stabilized solution are different.Mathematics Subject Classification (2010): 74F10, 35Q35, 76D55, 93C20, 93D15.

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“…The problem of interaction between a viscous incompressible fluid and a rigid body has been studied intensively in the recent years (see [2], [4], [6], [7], [11], [15], [18], [19], [20], [21], etc.). However, as far as we know, only few results concerning the existence and uniqueness of strong solutions for the problem (1.1), (1.2), (1.4)-(1.10) are available in the case where the system fills the whole space.…”

Section: ∂O(t)mentioning

confidence: 99%