Mathematical and Numerical Analysis of Some FSI Problems - Fluid-Structure Interaction and Biomedical Applications - page 4

One could also consider simplified reduced models that represent the part of the
fluid-structure domain that have been cut. These models will then be coupled to the
Navier-Stokes system. We refer to [ 103 ] for a theoretical study of the steady and
unsteady Navier-Stokes system with a given prescribed flux, or average pressure.
We also refer to [ 69 - 71 , 138 , 140 , 157 ] for the case of blood flows and 3D=0D
coupling or 3D=1D coupling and to [ 9 ] for the case of air flow modeling.
Concerning the structural part, we consider here a thin elastic structure whose
displacement d is decomposed into two components: the longitudinal one and
the radial or transversal one. For the sake of simplicity we assume that and
satisfy two linear decoupled equations, which correspond to a rod or beam model.
One could consider also more complex model such as shell models (Koiter, Nagdhi,
etc.) where and are coupled, see, e.g., Bukaˇcetal.[ 21 ], Raghu et al. [ 142 ], Canic
et al. [ 24 ], and the references therein.
The considered equations satisfied by d.t;x;R/ D ..t;x/;.t;x// T

in the

reference configuration .0;L/ are:

s e@ tt Ǜ 1 @ xx LJ 1 @ xx @ t D.T f / 1 C g 1 ; on .0;L/; (1.10)

s e@ tt C @ x Ǜ 2 @ xx LJ 2 @ xx @ t D.T f / 2 C g 2 ; on .0;L/;

(1.11)

where s denotes the density of the structure (assumed to be constant), e the
thickness of the elastic wall. The positive constants , Ǜ i ;i D 1;2 denote
mechanical constants depending on the elastic properties of the media and the
LJ i ;i D 1;2 some additional viscous damping. Two types of load act on the
structure: the load T f coming from the fluid, that will be defined later on, and a
given external surfacic load denoted g. Note that one could add to ( 1.11 )theterm

2

3 s e 3 @ xx @ tt that represents the inertia of rotation.
These equations have to be completed with the initial conditions:

.t D 0;/ D 0 ./;.t D 0;/ D 0 ./
@ t .t D 0;/ D 1 ./;@ t .t D 0;/ D 1 ./ on .0;L/;

(1.12)

and by the boundary conditions. We can assume here that the rod is clamped and
thus prescribe homogeneous Dirichlet boundary conditions, even if these type of
conditions are not reflecting reality in the case of blood flow modeling. Thus we
have, for instance

.0/ D .L/ D 0 and .0/ D .L/ D 0 (1.13)

@ x .0/ D @ x .L/ D 0; whenever >0: (1.14)

The next question that we aim to study is:
What are the coupling conditions between the fluid and the structure?
First, we will assume that the fluid sticks to the elastic boundary and conse-
quently the fluid velocity and the structure velocity are equal at the interface. That