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

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 [32] to numerically 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.

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

“…Muha andČanić recently proved the existence of weak solutions to a class of FSI problems modeling the flow of an incompressible, viscous, Newtonian fluid flowing through a 2D cylinder whose lateral wall was modeled by either the linearly viscoelastic, or by the linearly elastic Koiter shell equations [43], assuming nonlinear coupling at the deformed fluid-structure interface. The fluid flow boundary conditions were not periodic, but rather, the flow was driven by the dynamic pressure drop data.…”

“…The fluid flow boundary conditions were not periodic, but rather, the flow was driven by the dynamic pressure drop data. The methodology of proof in [43] was based on a semi-discrete, operator splitting Lie scheme, which was used in [30] to design a stable, loosely coupled partitioned numerical scheme, called the kinematically coupled scheme (see also [7]). 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.…”

“…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…”

“…This model problem is a good approximation of blood flow in elastic human arteries. We prove the existence of a weak solution to this 3D FSI problem by using nontrivial extensions of the ideas based on an operator splitting approach that has been implemented in computational solvers applied to 2D radially symmetric cases (see e.g., [30,7,43,6,31,40]). This is an extension of our earlier results in which the existence of a weak solution to a 2D problem respecting radial symmetry was proved [43].…”

“…We prove the existence of a weak solution to this 3D FSI problem by using nontrivial extensions of the ideas based on an operator splitting approach that has been implemented in computational solvers applied to 2D radially symmetric cases (see e.g., [30,7,43,6,31,40]). This is an extension of our earlier results in which the existence of a weak solution to a 2D problem respecting radial symmetry was proved [43]. The novelties of the present work include dealing with a Koiter shell model in which the shell displacement depends on both the axial, as well as the azimuthal variable, dealing with the lower regularity of the fluid-structure interface which is not necessarily Lipschitz, making sense of the trace of the fluid velocity at the fluid-structure interface, and using appropriate compactness arguments in 3D that provide the existence of a weak solution.…”