Fluid–Structure Interaction in Hemodynamics: Modeling, Analysis, and Numerical Simulation - Fluid-Structure Interaction and Biomedical Applications - page 79

2.1 Introduction

Fluid-structure interaction (FSI) problems arise in many applications. The widely
known examples are aeroelasticity and biofluids. In aeroelasticity, where the
structure (wing of an airplane) is much heavier than the fluid (air), it is sometimes
of interest to study small vibrations of the structure in which case linear coupling
between the fluid and the structure may be sufficient to capture the main features
of the solutions. In that case the fluid domain remains fixed in the FSI model, and
only the location of the structure is computed based on the fluid loading (one-way
coupling). In biofluidic applications, such as the interaction between blood flow and
cardiovascular tissue where the density of the structure (arterial walls) is roughly
equal to the density of the fluid (blood), the coupling between the fluid and the
relatively light structure is highly nonlinear . In that case the fluid domain is not fixed
in the FSI model, and its location is determined by the location of the structure. The
elastodynamics of the structure influences the motion of the fluid through the contact
force exerted by the structure onto the fluid, while the structure location is computed
based on the fluid loading expressed through the contact force exerted by the fluid
onto the structure (two-way coupling). It has recently been shown that classical
“partitioned” time-marching numerical algorithms, which are based on subsequent
solutions of the fluid and structure sub-problems, are unconditionally unstable in
problems in which the density of the structure and of the fluid are comparable [ 30 ].
The exchange of energy between the moving fluid and structure is so significant, that
a mismatch between the energy of the discretized problem and the energy of the
continuous problem causes instabilities in classical “loosely coupled” partitioned
schemes. The difficulties associated with the significant energy exchange and the
high geometric nonlinearity of the fluid-structure interface are reflected not only
in the design of numerical schemes but also in the theoretical studies of existence
and stability of solutions to this class of problems. A comprehensive study of these
problems remains to be a challenge due to their strong nonlinearity and multi-
physics nature.
In the blood flow application, the problems are further exacerbated by the fact
that arterial walls of major arteries are composed of several layers, each with
different mechanical characteristics. The main layers are the tunica intima, media,
and adventitia. They are separated by the thin elastic laminae, see Fig. 2.1 . Recent
developments in ultrasound speckle tracking methods revealed significant shear
strain between the different layers in high adrenaline situations [ 2 , 39 , 40 ]. It was
noted that the consequences of this phenomenon on cardiovascular disease are yet
to be explored! An example of a disease which is associated with a pathophysiology
of the aortic wall layers is aortic dissection: tears in the intimal layer result in
separation of the aortic wall layers causing blood to flow within the aortic wall.
Until recently, there have been no FSI models or computational solvers of arterial
flow that take into account the multi-layered structure of arterial walls. In this
chapter we take a step in this direction by studying a benchmark problem in fluid-
multi-layered-structure interaction in which the structure consists of two layers, a