A global attractor for a fluid–plate interaction model accounting only for longitudinal deformations of the plate

✍ Scribed by Igor D. Chueshov

• Publisher: John Wiley and Sons
• Year: 2011
• Tongue: English
• Weight: 217 KB
• Volume: 34
• Category: Article
• ISSN: 0170-4214
• DOI: 10.1002/mma.1496

✦ Synopsis

Paper 1. Introduction

We consider a coupled (hybrid) system, which describes the interaction of a homogeneous viscous incompressible fluid, which occupies a domain O bounded by the (solid) walls of the container S and a horizontal boundary on which a thin (nonlinear) elastic plate is placed. The motion of the fluid is described by the linearized 3D Navier-Stokes equations. To describe the deformations of the plate, we involve a general (full) Kirchhoff-Karman type model (see, e.g., [1][2][3] and the references therein) with additional hypothesis that the transversal displacements of the plate are negligible relative to in-plane displacements. Thus we only consider longitudinal deformations of the plate and take account of tangential shear forces, which the fluid exerts on the plate. These kinds of models arise in the study of the problem of blood flow in large arteries (see, e.g., [4] and also [5] for a more recent discussion and references).

We note that the mathematical studies of the problem of fluid-structure interaction in the case of viscous fluids and elastic plate/bodies have a long history. We refer to [5-9] and the references therein for the case of plates/membranes, to [10] in the case of moving elastic bodies, and to [11][12][13][14][15][16] in the case of elastic bodies with a fixed interface; see also the literature cited in these references.

Our mathematical model is formulated as follows:

Let O R 3 be a bounded domain with a sufficiently smooth boundary @O. We assume that @O D [ S, where fx D .x 1 ; x 2 ; 0/ : .x 1 ; x 2 / 2 R 2 g and S is a smooth surface, which lies under the plane x 3 D 0. The exterior normal on @O is denoted by n. We have that n D .0; 0; 1/ on . We consider the following linear Navier-Stokes equations in O for the fluid velocity field v D v.x, t/ D v 1 .x, t/; v 2 .x, t/; v 3 .x, t/ and for the pressure p.x, t/:

div v D 0 in O .0, C1/, ( 2 ) v.x, 0/ D v 0 .x/ in O, ( 3 )