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

摘要

We study a nonlinear, unsteady, moving boundary, fluid-structure interaction(FSI) problem arising in modeling blood flow through elastic and viscoelasticarteries. The fluid flow, which is driven by the time-dependent pressure data,is governed by 2D incompressible Navier-Stokes equations, while theelastodynamics of the cylindrical wall is modeled by the 1D cylindrical Koitershell model. Two cases are considered: the linearly viscoelastic and thelinearly elastic Koiter shell. The fluid and structure are fully coupled (2-waycoupling) via the kinematic and dynamic lateral boundary conditions describingcontinuity of velocity (the no-slip condition), and balance of contact forcesat the fluid-structure interface. We prove existence of weak solutions to thetwo FSI problems (the viscoelastic and the elastic case) as long as thecylinder radius is greater than zero. The proof is based on a novel semi-discrete, operator splitting numericalscheme, known as the kinematically coupled scheme, introduced in \cite{GioSun}to solve the underlying FSI problems. The backbone of the kinematically coupledscheme is the well-known Marchuk-Yanenko scheme, also known as the Liesplitting scheme. We effectively prove convergence of that numerical scheme toa solution of the corresponding FSI problem.