Existence of a solution to a fluid–multi-layered-structure interaction problem

Abstract

We study a nonlinear, unsteady, moving boundary, fluid-structure (FSI) problem in which the structure is composed of two layers: a thin layer which is in contact with the fluid, and a thick layer which sits on top of the thin structural layer. The fluid flow, which is driven by the time-dependent dynamic pressure data, is governed by the 2D Navier-Stokes equations for an incompressible, viscous fluid, defined on a 2D cylinder. The elastodynamics of the cylinder wall is governed by the 1D linear wave equation modeling the thin structural layer, and by the 2D equations of linear elasticity modeling the thick structural layer. The fluid and the structure, as well as the two structural layers, are fully coupled via the kinematic and dynamic coupling conditions describing continuity of velocity and balance of contact forces. The thin structural layer acts as a fluid-structure interface with mass. The resulting FSI problem is a nonlinear moving boundary problem of parabolic-hyperbolic type. This problem is motivated by the flow of blood in elastic arteries whose walls are composed of several layers, each with different mechanical characteristics and thickness. We prove existence of a weak solution to this nonlinear FSI problem as long as the cylinder radius is greater than zero. The proof is based on a novel semi-discrete, operator splitting numerical scheme, known as the kinematically coupled scheme. We effectively prove convergence of that numerical scheme to a solution of the nonlinear fluid-multi-layered-structure interaction problem. The spaces of weak solutions presented in this manuscript reveal a striking new feature: the presence of a thin fluid-structure interface with mass regularizes solutions of the coupled problem. 1 arXiv:1305.5310v1 [math.AP] 23 May 20133)

A nonlinear, 3D fluid-structure interaction problem driven by the time-dependent dynamic pressure data: a constructive existence proof

“…Ideas based on the Lie operator splitting scheme were also used by Temam in [49] to prove the existence of a solution to the nonlinear Carleman equation. These results were recently extended by Muha andČanić to a FSI problem with two structural layers [42]. This was a first step toward modeling the FSI between blood flow and arterial walls which are known to be composed of three main layers, separated by thin elastic laminae, each with different mechanical characteristics.…”

“…Similarly as in [43,42], we use semi-discretization in time to define a sequence of approximate solutions to Problem 3.1. This approach defines a time step, which will be denoted by Δt, and a number of time sub-intervals N ∈ N, so that…”

“…However, since the Korn's constant in general depends on the domain, i.e., on the ALE mapping, we will need a result which gives a universal Korn's constant, independent of the family of domains under consideration. Indeed, we use an approach similar to the one presented in [42] (proof of Proposition 6.5, pp. 683-685) to find a universal Korn's constant, independent of the ALE mappings, that can be obtained under certain assumptions on the family of domains, which hold true in our case.…”

Coupling Requirements for Multiphysics Problems Posed on Two Domains

“…The choice (55) and α p c = ρ s (c f −w)/ρc f imply that the conditions (18) and (19) are satisfied, which means that the coupled problem (1)- (2) is well posed.…”

“…The reason for that is the somewhat more unclear nature of coupling conditions compared to boundary conditions [12,17,22,23,26], even though there are similarities. Well-posedness, especially related to the coupling of multiphysics problems, is discussed in [19,20,23,9,15].…”