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

Abstract

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.

A nonlinear, 3D fluid-structure interaction problem driven by the time-dependent dynamic pressure data: a constructive existence proof

“…Continuous dependence of weak solutions on initial data for a fluid structure interaction problem with a free boundary type coupling condition was studied in [29]. Existence of a weak solution for a FSI problem between a 3D incompressible, viscous fluid and a 2D viscoelastic plate was considered by Chambolle et al in [9], while Grandmont improved this result in [28] to hold for a 2D elastic plate. These results were extended to a more general geometry in [37], and then to the case of generalized Newtonian fluids in [36], and to a non-Newtonian shear dependent fluid in [40].…”

“…It was shown in [9,28,41] that the trace operator γ Γ(t) can be extended by continuity to a linear operator from…”

“…Using the fact that Ω η (t) is locally a sub-graph of a Hölder continuous function we can get the following characterization of the velocity solution space V F (t) (see [9,28]):…”

Steady Flow of a Navier–Stokes Liquid Past an Elastic Body

“…Also in view of the central role that the above problem plays in numerous and diverse engineering [2] and medical [11] applications, more recently mathematicians have begun a systematic study of the interaction of a Navier-Stokes (viscous) liquid with elastic bodies, in both steady [17,12,18] and unsteady [13,4,6,7] cases. In all these works, the liquid occupies a bounded region of the three-dimensional space, that surrounds (or is surrounded by) the elastic structure.…”

“…Our main result is a proof of existence and uniqueness of solutions for the coupled systems (1), (2) and for the coupled systems (1), (4). More precisely, we show that if |v ∞ | is below a certain quantity, depending on Ω 0 and on the material constants characterizing L and B, there is one and only one solution lying in a ball, centered at the origin of a suitable Banach space, whose radius continuously depends on |v ∞ |.…”