Numerical Analysis and Scientific Computing Preprint Seria A nonlinear moving-boundary problem of parabolic-hyperbolic-hyperbolic type arising in fluid-multi-layered structure interaction problems

Motivated by modeling blood flow in human arteries, we study a fluid-structure interaction problem in which the structure is composed of multiple layers, each with possibly different mechanical characteristics and thickness. In the problem presented in this manuscript the structure is composed of two layers: a thin layer modeled by the 1D wave equation, and a thick layer modeled by the 2D equations of linear elasticity. The flow of an incompressible, viscous fluid is modeled by the Navier-Stokes equations. The thin structure is in contact with the fluid thereby serving as a fluid-structure interface with mass. The coupling between the fluid and the structure is nonlinear. The resulting problem is a nonlinear, moving-boundary problem of parabolic-hyperbolic-hyperbolic type. We show that the model problem has a well-defined energy, and that the energy is bounded by the work done by the inlet and outlet dynamic pressure data. The spaces of weak solutions reveal that the presence of a thin fluid-structure interface with mass regularizes solutions of the coupled problem. This opens up a new area withing the field of fluidstructure interaction problems, possibly revealing properties of FSI solutions that have not been studied before. 1. Motivation. Fluid-structure interaction (FSI) problems arise in many applications. They include multi-physics problems in engineering such as aeroelasticity and propeller turbines, as well as biofluidic application such as self-propulsion organisms, fluid-cell interactions, and the interaction between blood flow and cardiovascular tissue. In biofluidic applications, such as the interaction between blood flow and cardiovascular tissue, the density of the structure (arterial walls) is roughly equal to the density of the fluid (blood). In such problems the energy exchange between the fluid and the structure is significant, leading to a highly nonlinear FSI coupling which is responsible for the instabilities in loosely coupled partitioned algorithms [3]. Despite a significant progress within the past decade [1, 2, 7, 8, 10, 13, 14, 6, 5, 4, 11, 15], 2010 Mathematics Subject Classification. Primary: 74F10, 35Q30, 76D05; Secondary: 74B10, 74K25.