Existence for a Three-Dimensional Steady State Fluid-Structure Interaction Problem

Abstract

A steady state fluid-structure interaction problem is considered and its solvability is studied. Both the fluid and the structure are three-dimensional. The equations of the viscous fluid motion are set in an unknown domain depending on the structure displacement. The structure is elastic and its deformation depends on the stress coming from the fluid. We prove that, for small enough applied exterior forces, there exists at least one regular solution of this nonlinear coupled problem. Mathematics Subject Classification (2000). 76D03, 74F10, 35Q30, 35Q35.

Weak solutions for a fluid-elastic structure interaction model

“…The purpose of this work is to prove theoretical results for the interaction between an incompressible viscous fluid governed by the Navier-Stokes equations and an elastic structure whose deformation is given by a linear combination of a finite number of modes. The case of a finite number of rigid structures embedded in a fluid was treated in [6], dealing with the incompressible NavierStokes equations as well as the compressible Navier-Stokes equations for isentropic fluids (see also [1] [5] [8] [12] [13] [20]). Following a similar approach, we intend here to prove existence of weak solutions "à la …”

Numerical procedure with analytic derivative for unsteady fluid–structure interaction

“…By conditions (17) and (18), for w ∈ W the following equalities hold: 2 . Now, we can introduce the weak formulation for the fluid equations: for all t ∈ (0, T ), find the velocity v(·, t) ∈ (H 1 ( F t )) 2 verifying the boundary conditions (8), (10), (12), (14) …”

Numerical simulation of a pulsatile flow through a flexible channel

“…The existence results for the fluid structure interaction can be found for example in [1,15] for the steady case and in [2,4,9,16] for the unsteady case. We didn't cited here the results concerning the interaction between a fluid and a rigid solid in rotation or in translation.…”