Document Type

Article

Publication Date

12-2014

Department

Mathematics, Statistics, and Computer Science

Keywords

Fluid-structure interaction, 3D linearized Navier-Stokes, Kirchhoff plate, finite element method, Babuška-Brezzi theorem

Abstract

We present qualitative and numerical results on a partial differential equation (PDE) system which models a certain fluid-structure dynamics. Wellposedness is established by constructing for it a nonstandard semigroup generator representation; this representation is accomplished by an appropriate elimination of the pressure. This coupled PDE model involves the Stokes system which evolves on a three dimensional domain O coupled to a fourth order plate equation, possibly with rotational inertia parameter ρ>0. This plate PDE evolves on a flat portion Ω of the boundary of O. The coupling on Ω is implemented via the Dirichlet trace of the Stokes system fluid variable - and so the no-slip condition is necessarily not in play - and via the Dirichlet boundary trace of the pressure, which essentially acts as a forcing term on Ω. We note that as the Stokes fluid velocity does not vanish on Ω, the pressure variable cannot be eliminated by the classic Leray projector; instead, it is identified as the solution of an elliptic boundary value problem. Eventually, wellposedness of the system is attained through a nonstandard variational (``inf-sup") formulation. Subsequently we show how our constructive proof of wellposedness naturally gives rise to a mixed finite element method for numerically approximating solutions of this fluid-structure dynamics.

Comments

This is a pre-print, author-produced PDF of an article accepted for publication in Evolution Equations and Control Theory following peer review. The definitive publisher-authenticated version is available online at:http://aimsciences.org/journals/contentsListnew.jsp?pubID=721

Source Publication Title

Evolution Equations and Control Theory

Publisher

American Institute of Mathematical Sciences

Volume

3

Issue

4

First Page

557

DOI

10.3934/eect.2014.3.557

Download

Included in

Mathematics Commons

Share

COinS