Existence of a weak solution to a nonlinear fluid-structure interaction problem modeling the flow of an incompressible, viscous fluid in a cylinder with deformable walls

TL;DR

This paper proves the existence of weak solutions for a nonlinear fluid-structure interaction model involving incompressible viscous flow in elastic or viscoelastic arteries, using a novel numerical scheme and convergence analysis.

Contribution

It introduces a new semi-discrete operator splitting scheme and proves its convergence for solving coupled fluid-structure interaction problems with elastic and viscoelastic arterial walls.

Findings

Existence of weak solutions for both elastic and viscoelastic cases.

Convergence of the proposed numerical scheme to the weak solution.

Applicability to blood flow modeling in arteries.

Abstract

We study a nonlinear, unsteady, moving boundary, fluid-structure interaction (FSI) problem arising in modeling blood flow through elastic and viscoelastic arteries. The fluid flow, which is driven by the time-dependent pressure data, is governed by 2D incompressible Navier-Stokes equations, while the elastodynamics of the cylindrical wall is modeled by the 1D cylindrical Koiter shell model. Two cases are considered: the linearly viscoelastic and the linearly elastic Koiter shell. The fluid and structure are fully coupled (2-way coupling) via the kinematic and dynamic lateral boundary conditions describing continuity of velocity (the no-slip condition), and balance of contact forces at the fluid-structure interface. We prove existence of weak solutions to the two FSI problems (the viscoelastic and the elastic case) 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, introduced in \cite{GioSun} to solve the underlying FSI problems. The backbone of the kinematically coupled scheme is the well-known Marchuk-Yanenko scheme, also known as the Lie splitting scheme. We effectively prove convergence of that numerical scheme to a solution of the corresponding FSI problem.