Biomedical Engineering Reference
In-Depth Information
Fig. 1.1
Description of the fluid domain
f
.t/
of the fluid domain cannot be neglected and which consequently depends on the
structure displacement. The fluid evolves according to the structure displacement
itself resulting from the fluid force. We restrict ourselves to the two-dimensional
case. Note that the model we will consider in what follows can be viewed as a first
model to describe the blood flow in large arteries [
138
].
Let us start by setting the full nonlinear coupled problem we will study in
what follows. The fluid is described by the unsteady Navier-Stokes equations (or
by incompressible non-Newtonian fluid system), whereas the structure will be a
thin linear elastic structure. We consider
f
.t/ the domain occupied by the fluid
at time t,
f
.t/
R
2
. Furthermore, we ass
ume
t
hat
th
e fl
uid boundary can be
decomposed into four parts: @
f
.t/ D †.t/ [
0
[
in
[
out
,seeFig.
1.1
.
The first part †.t/ denotes the elastic wall. The boundary †.t/ is consequently
the deformed configuration of the structure which is an unknown of the problem
depending on the structure displacement, whereas
0
,
in
,
out
are fixed given
boundaries where different kind of boundary conditions could be applied. We
denote by
O
f
and † the reference configurations of the fluid and of the structure,
respectively. Here we consider that † is flat and equal to .0;L/ fRg and that
O
f
D .0;L/.0;R/. The behavior of the structure is described by the displacement
of each point of the reference configuration. Then each point x of † occupies at
time t the position x.t/ Dx C d.x;t/ where d denotes the displacement of the
structure that satisfies the constitutive equations that describe the structure motion.
Concerning the fluid, most of the descriptions are Eulerian ones. The unknowns
(velocity, pressure) are evaluated at each time and at each point of the physical
domain. The resolution of the fluid part, if one considers a Newtonian flow, then
consists in: finding.
u
;p/defined over
f
.t/ suchthat
f
@
t
u
C
f
.
u
:r/
u
u
Crp D f; in
f
.t/;
(1.1)
div
u
D 0; in
f
.t/;
(1.2)
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