Existence of Weak Solutions for the Three-Dimensional Motion of an Elastic Structure in an Incompressible Fluid

Abstract

We study here the three-dimensional motion of an elastic structure immersed in an incompressible viscous fluid. The structure and the fluid are contained in a fixed bounded connected set Ω. We show the existence of a weak solution for regularized elastic deformations as long as elastic deformations are not too important (in order to avoid interpenetration and preserve orientation on the structure) and no collisions between the structure and the boundary occur. As the structure moves freely in the fluid, it seems natural (and it corresponds to many physical applications) to consider that its rigid motion (translation and rotation) may be large.The existence result presented here has been announced in [4]. Some improvements have been provided on the model: the model considered in [4] is a simplified model where the structure motion is modelled by decoupled and linear equations for the translation, the rotation and the purely elastic displacement. In what follows, we consider on the structure a model which represents the motion of a structure with large rigid displacements and small elastic perturbations. This model, introduced by [15] for a structure alone, leads to coupled and nonlinear equations for the translation, the rotation and the elastic displacement.

A regularity result for a solid–fluid system associated to the compressible Navier–Stokes equations

“…The local existence of a strong solution is proved in [6]. In [4] and [3] (with variable density), the authors proved the global existence of a weak solution.…”

Existence of Global Strong Solutions to a Beam–Fluid Interaction System

“…Fewer studies consider the case of an elastic structure evolving in a viscous incompressible Newtonian flow. We refer the reader to [12] and [4] where the structure is described by a finite number of eigenmodes or to [3] for an artificially damped elastic structure while, for the case of a three-dimensional elastic structure interacting with a three-dimensional fluid, we mention [24,16] in the steady state case and [8,7,34,43] for the full unsteady case. In the latter, the authors consider the existence of strong solutions for small enough data locally in time.…”

Well-Posedness Analysis for a Linearization of a Fluid-Elasticity Interaction

“…The ubiquity of this interaction type has led to a rapidly growing research interest in this model. Its mathematical solvability [5,9,15,17,22,21,26,32,33,34,37,38,49,55,52], numerical approximations [23,29,39,54,53,44,46], and stability [14,27,28,31] have been intensively studied.Mathematically, this interaction is described by a partial differential equation (PDE) system that couples the parabolic (fluid) and hyperbolic (elasticity) phases, where the key issue is that the traces of the elastic component at the energy level are not defined via the standard trace theory. The loss of regularity induced by the hyperbolic component left the basic question of existence open until recently.…”