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

Abstract

We perform a mathematical analysis of the steady flow of a viscous liquid, L, past a three-dimensional elastic body, B. We assume that L fills the whole space exterior to B, and that its motion is governed by the Navier-Stokes equations corresponding to non-zero velocity at infinity, v∞. As for B, we suppose that it is a St.Venant-Kirchoff material, held in equilibrium either by keeping an interior portion of it attached to a rigid body, or by means of appropriate control body force and surface traction. We treat the problem as a coupled steady state fluid-structure problem with the surface of B as a free boundary. Our main goal is to show existence and uniqueness for the coupled system liquid-body, for sufficiently small |v∞|. This goal is reached by a fixed point approach based upon a suitable reformulation of the Navier-Stokes equation in the reference configuration, along with appropriate a priori estimates of solutions to the corresponding Oseen linearization and to the elasticity equations.

Existence of local strong solutions to fluid–beam and fluid–rod interaction systems

“…In the case of a 3D elastic structure evolving in a 3D viscous incompressible Newtonian flow, we refer the reader to [9] and [4] where the structure is described by a finite number of eigenmodes or to [2] for an artificially damped elastic structure. For the case of the full system describing the motion of a three-dimensional elastic structure interacting with a three-dimensional fluid, we mention [12,10] in the steady state case and [7,8,17,24] for the full unsteady case. In [7,8], the authors consider the existence of strong solutions for small enough data locally in time, whereas, in [17,24], the existence of local-in-time strong solutions is proven in the case where the fluid structure interface is flat and for a zero initial displacement field.…”

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.…”

Existence and regularity of weak solutions for a fluid interacting with a non-linear shell in $3D$

“…Short time or small data existence result in the context of strong solutions for various non-linear fluid structure models have been obtained in [8,9,13,37]. Finally we wish to mention some results in the static case that can be found here [16,19].…”