Biomedical Engineering Reference
In-Depth Information
where
e
r
D
e
r
.;
z
/ is the unit vector in the r-direction. This is a common
assumption in the literature on FSI in blood flow. For problems with nonzero radial
and longitudinal displacement
r
;
z
¤ 0, please see [
20
,
21
].
The radius of the deformed domain is equal to R C .t;;
z
/. Thus, the fluid
domain, sketched in Fig.
2.7
,isgivenby
W
z
2 .0;L/;
p
x
2
3
F
.t/ Df.x;y;
z
/ 2
R
C y
2
<.0;RC .t;;
z
//g;
where the lateral boundary of the cylinder corresponds to fluid-structure interface,
denoted by
W
z
2 .0;L/;
p
x
2
3
.t/ Df.x;y;
z
/ 2
R
C y
2
<.0;RC .t;;
z
//g:
The inlet and outlet boundary of the fluid domain will be denoted by
in
and
out
,
respectively.
2.3.2
The Coupling Conditions
Since we have three different physical models describing three different physical
processes which are coupled, we need to describe the physics of the coupling
between all of them. This includes prescribing coupling conditions between the
fluid and structure, and prescribing coupling conditions between the thin and thick
structure.
The coupling between the fluid, the thin structural layer, and the thick structural
layer is achieved via two sets of coupling conditions: the kinematic coupling
condition and the dynamic coupling condition. The kinematic coupling condition
addresses the coupling of kinematic quantities, such as velocity. The dynamic
coupling condition describes balance of forces that occurs at the interface between
different physical models. These two sets of conditions give rise to a well-defined
mathematical problem, while, at the same time, they capture the basic physical laws
of the coupling.
In our problem, the thin structure serves both as a fluid-structure interface, and
as a structure-structure interface. In this chapter we will be assuming that the
kinematic coupling condition is the
no-slip
boundary condition between both the
fluid and thin structure, as well as between the thin and thick structural layers.
Concerning the dynamic coupling condition, since .t/ is a fluid-structure
interface with mass, the dynamic coupling condition is simply the second Newton's
Law of motion. It states that mass times acceleration of the interface is balanced by
the sum of total forces acting on, or within, .t/. This includes the contribution due
to the elastic energy of the structure, and the balance of contact forces exerted by
the fluid and the thick structure onto .t/. More precisely, we have the following
set of coupling conditions written in Lagrangian framework:
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