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

Abstract

We study asymptotic dynamics of a coupled system consisting of linearized 3D Navier-Stokes equations in a bounded domain and the classical (nonlinear) elastic plate equation for in-plane motions on a flexible flat part of the boundary. The main novelty of the model is the assumption that the transversal displacements of the plate are negligible relative to in-plane displacements. These kinds of models arise in the study of blood flows in large arteries. Our main result states the existence of a compact global attractor of finite dimension. Under some conditions this attractor is an exponentially attracting single point. We also show that the corresponding linearized system generates an exponentially stable C 0 -semigroup. We do not assume any kind of mechanical damping in the plate component. Thus our results mean that dissipation of the energy in the fluid because of viscosity is sufficient to stabilize the system. Copyright © 2011 John Wiley & Sons, Ltd.Keywords: fluid-structure interaction; linearized 3D Navier-Stokes equations; nonlinear plate; finite-dimensional attractorwhere >0 is the dynamical viscosity and Q G is a volume force. We supplement Equations (1)- (3) with the (non-slip) boundary conditions imposed on the velocity field v D v.x, t/:where andwithwhere D D Eh= 1 2 , E is Young's modulus, 0 < < 1=2 is Poisson's ratio, h is the thickness of the plate, is the mass density. The external (in-plane) force F 1 ; F 2 in Equations (5) and (6) consists of two parts,where f 1 .u 1 , u 2 /; f 2 .u 1 , u 2 / is a nonlinear feedback force represented by some potentialˆ(which we specify below)and .T 1 .v/; T 2 .v// is the viscous shear stress exerted by the fluid on the plate, T i .v/ D ..Tn/j , e i / R 3 . Here, T D fT ij g 3 i,jD1 is the stress tensor of the fluid, given by (in the last equality, we use the fact that v 3 .x 1 ; x 2 ; 0/ D 0 for .x 1 ; x 2 / 2 because of the second relation in Equation (4) and henceThus we arrive at the following equations for the in-plane displacement u D u 1 ; u 2 of the plate (below for some notational simplifications we assume that h D 1 and D.1 /=2 D 1):where D .1 C /.1 / 1 is a nonnegative parameter. For the displacement u D u 1 ; u 2 we impose clamped boundary conditions on D @ :Our main point of interest is the well-posedness and long-time dynamics of solutions to the coupled problem in Equations (1)- (4), (7), and (8) for the velocity v and the displacement u D .u 1 ; u 2 / with the initial dataThis problem in the case when Q G Á 0, D 0, and f i .u/ Á 0 was considered in [8] (see also [5,7]) with an additional locally distributed strong (Kelvin-Voight type) damping force applied to the interior of the plate. These papers deal with the existence and asymptotic stability of the corresponding semigroup. In contrast with [5,7,8], we do not assume the presence of mechanical damping terms in the plate component of the system but consider nonlinearly forced model.

Unsteady interaction of a viscous fluid with an elastic shell modeled by full von Karman equations

“…In the case of purely transversal displacements [13] the force exerted on the plate by the fluid contains the pressure only. We also mention that in spite of the model in (1)-(8) is based on rather general shell hypotheses the systems considered in [9,13] are not special cases of (1)- (8). The models in [9,13] involve additional physical type requirements.…”

“…We also mention that in spite of the model in (1)-(8) is based on rather general shell hypotheses the systems considered in [9,13] are not special cases of (1)- (8). The models in [9,13] involve additional physical type requirements. For instance, to arrive at the model of [9] we should (i) neglect by Eq.…”

“…The point is that the surface force T f (v) is not the sum of the corresponding loads in the models [9] and [13]. The models in [9,13] are much simpler in several respects.…”

“…The models in [9,13] involve additional physical type requirements. For instance, to arrive at the model of [9] we should (i) neglect by Eq. (5), (ii) put w = 0 in (6) and w t = 0 in (3).…”

“…We refer to [8,9,13,[19][20][21]24] and the references therein in the case of plates/membranes, to [15] in the case of moving elastic bodies, and to [1][2][3]6,7,16] in the case of elastic bodies with the fixed interface; see also the literature cited in these papers.…”