Special DE Seminar: Boris Muha

Fluid-structure interaction (FSI) problems describe the dynamics of a multi-physics system involving fluid and solid components. In this mini-course we will discuss questions of well-posedness, design of numerical schemes, and application of FSI problems. First, we will derive an FSI model in the context of continuum mechanics. The fluid flow will be governed by the incompressible Navier-Stokes equations. We will discuss several different structure models: rigid body, 3-D elasticity, plate/shell, composite structure and poroelastic structure. Finally, to close the system we will describe the appropriate coupling conditions at the solid-fluid interface. 

FSI problems are typically defined on a time-changing domain, which is not know a priori, and therefore we need to introduce the appropriate functional framework to define and study weak solutions. In particular, we will prove a generalization of the Aubin-Lions-Simon Lemma, adapted to the settings of the moving boundary problems. We will describe a partitioned scheme called a ``kinematically coupled scheme" and prove its stability and convergence. Furthermore, we will show how the ideas from numerics can be used to demonstrate the existence of a weak solution. Finally, we will discuss some related issues such as weak-strong uniqueness and applications of FSI in biomedicine.