Existence of a weak solution to a fluid–elastic structure interaction problem with the Navier slip boundary condition

Abstract

We study a nonlinear, moving boundary fluid-structure interaction (FSI) problem between an incompressible, viscous Newtonian fluid, modeled by the 2D Navier-Stokes equations, and an elastic structure modeled by the shell or plate equations. The fluid and structure are coupled via the Navier slip boundary condition and balance of contact forces at the fluid-structure interface. The slip boundary condition might be more realistic than the classical no-slip boundary condition in situations, e.g., when the structure is "rough", and in modeling FSI dynamics near, or at a contact. Cardiovascular tissue and cell-seeded tissue constructs, which consist of grooves in tissue scaffolds that are lined with cells, are examples of "rough" elastic interfaces interacting with an incompressible, viscous fluid. The problem of heart valve closure is an example of a FSI problem with a contact involving elastic interfaces. We prove the existence of a weak solution to this class of problems by designing a constructive proof based on the time discretization via operator splitting. This is the first existence result for fluid-structure interaction problems involving elastic structures satisfying the Navier slip boundary condition.

A generalization of the Aubin–Lions–Simon compactness lemma for problems on moving domains

“…Property A: Strong bounds. These follow directly from the uniform energy estimates in Proposition 4 in [25].…”

“…More general boundary conditions on the rigid part of the boundary were considered in [25]. The initial data must satisfy the compatibility conditions:…”

“…To deal with the motion of the fluid domain we again consider the ALE mapping A η , given by (20) in [25]. This ALE mapping defines the fluid domain velocity w η = dA η /dt.…”

“…The weak formulation of problem (4.57)-(4.60), written in ALE formulation, is then given by the following (see (43) in [25] pp. 8566): find (u, η) such that…”

“…The following convergence results were proved in [25] (see Theorem 5 and Corollaries 2 and 3) for the geometric quantities:…”