Hemodynamical Flows

In this chapter we review the basic continuum mechanics at the foundation of the technical material in this book. Readers interested in further information are referred to monographs on this subject including [65], [10], [27], [21], [22], [32], [61], and [59].

Hemorheology is the science of deformation and flow of blood and its formed elements. This field includes investigations of both macroscopic blood properties using rheometric experiments as well as microscopic properties in vitro and in vivo. Hemorheology also encompasses the study of the interactions among blood components and between these components and the endothelial cells that line blood vessels.

This chapter describes numerical methods for simulating the interaction of viscous liquids with rigid or elastic bodies.

General examples of fluid-solid/structure interaction (FSI) problems are flow transporting rigid or elastic particles (particulate flow), flow around elastic structures (airplanes, submarines) and flow in elastic structures (hemodynamics, transport of fluids in closed containers). In all these settings the dilemma in modeling the coupled dynamics is that the fluid model is normally based on an Eulerian perspective in contrast to the usual Lagrangian formulation of the solid model. This makes the setup of a common variational description difficult. However, such a variational formulation of FSI is needed as the basis of a consistent Galerkin discretization with a posteriori error control and mesh adaptation, as well as the solution of optimal control problems based on the Euler-Lagrange approach.

In many fluid applications, particularly in multiphase flow problems with liquidsolid interaction based on the incompressible Navier-Stokes equations, the mathematical description and the numerical schemes have to be designed in such a way that quite complicated constitutive relations and interactions between fluid and solids can be incorporated into existing flow solvers in an accurate and robust manner. In the following sections, first of all we describe finite-element discretization strategies and corresponding solver techniques (approximate Newton methods, multilevel pressure Schur complement techniques, operator-splitting approaches) for the resulting discrete systems of equations. The need for the development of robust and efficient iterative solvers for implicit high-resolution discretization schemes is emphasized and the numerical treatment of extensions of the NavierStokes equations (Boussinesq approximation, Κ-ε turbulence model) is addressed which is evaluated by simulation results for prototypical applications including multiphase and granular flows. In the second part, a fully monolithic finite-element approach is described for fluid-structure interactions with elastic materials which is applied to several benchmark configurations. In the third part, the concept of FEM fictitious-boundary techniques, together with operator-splitting approaches for particulate flow, is introduced which allows the efficient simulation of systems with many solid particles of different shape and size.