Biomedical Engineering Reference
In-Depth Information
Tabl e 2. 4
Blood density and
dynamics viscosity
coefficients
Blood density
F
(kg/m
3
)
1:1
10
3
[
133
]
Blood dynamic viscosity
F
(kg/ms)
1;050 [
133
]
Fig. 2.7
Domain sketch and
notation
These equations are defined on the reference domain
W
z
2 .0;L/; R <
p
x
2
3
S
Df.x;y;
z
/ 2
R
C y
2
<RC H/g:
The flow of an incompressible, viscous fluid is modeled by the Navier-Stokes
equations. They are defined on a time-dependent cylindrical fluid domain
F
.t/,
which is not known a priori:
in
F
.t/; t 2 .0;T/;
F
.@
t
u
C
u
r
u
/ Dr;
r
u
D 0;
FLUID
W
(2.60)
where
F
denotes the fluid density;
u
the fluid velocity; Dp
I
C 2
F
D
.
u
/ is
the fluid Cauchy stress tensor; p is the fluid pressure;
F
is the dynamic viscosity
coefficient; and
D
.
u
/ D
u
/ is the symmetrized gradient of
u
. The typical
values of the parameters
F
and
F
for blood are given in Table
2.4
.
We will be working with the fluid equations written in Cartesian coordinates
.x;y;
z
/, while the structure equations will be written in cylindrical coordinates
.r;;
z
/. For any function f given in Cartesian coordinates, we define
1
2
.r
u
Cr
f to be the
corresponding function given in cylindrical coordinates:
f.r;;
z
/ WD f.x;y;
z
/:
For simplicity, in the rest of this chapter, we drop the tilde notation.
The cylindrical fluid domain is of length L, with reference radius r D R.
See Fig.
2.7
. The thin structure, described by Eq. (
2.58
), serves as a fluid-structure
interface. The nonzero inertia term
K
h@
2
=@t
2
indicates that our fluid-structure
interface has mass. This has important implications for the analysis and numerical
simulation of FSI problems, discussed in Sect.
2.7.7
.
For simplicity, in the rest of this chapter, we will be assuming that only the radial
component of the displacement of the thin structure is different from zero, i.e., we
will be assuming
ASSUMPTION
W
D .
r
;
;
z
/ D .
r
;0;0/DW
e
r
;
(2.61)
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