Biomedical Engineering Reference
In-Depth Information
where
i
.t/ D G
i
.t/;
8 t 2 .0;T/;
(4.6)
!
i
.t/ x D
Q
i
.t/
Q
i
.t/
>
x;
8 x 2
R
d
;
8 t 2 .0;T/:
(4.7)
These identities stand for definitions of the translational (resp. angular) velocity
i
(resp. !
i
)of
B
i
. In the two-dimensional case, there holds !
i
2
R
and
!
i
x D !
i
x
?
;
8 x 2
R
2
;
with ? denoting the rotation with angle =2. In the three-dimensional case !
i
2
R
3
and stands for the usual vector product. To include this dimensional phenomenon,
we shall write !
i
R
d
where d
D 3 for d D 3 and d
2
D 1 for d D 2.
Finally, the body unknowns are .
B
i
.t/;
i
;!
i
/
i
D
1;:::;n
. We note that, given t 7!
B
i
.t/,
we might compute .
i
;!
i
/ through (
4.2
)and(
4.6
)-(
4.7
). Conversely, given our
convention (
4.4
), the identities (
4.6
)-(
4.7
) represent differential equations which
enable to compute G
i
and
Q
i
in terms of
i
and !
i
. Eventually, one might restrict
the set of body unknowns either to .
i
;!
i
/
i
D
1;:::;n
or to .
B
i
.t//
i
D
1;:::;n
.
As for the fluid, we denote
u
f
D .
u
f;1
;:::;
u
f;d
/ the velocity-field and p
f
the
pressure of
. These are the only fluid unknowns. They are defined over the (time-
space) fluid domain
L
[
t
2
.0;T/
ftg
F
Q
F
D
.t/;
where, for all t 2 .0;T/,the
F
.t/ stands for the complement in of the body
domains:
n
[
i
D
1
B
i
.t/:
F
.t/ D n
S
.t/; where
S
.t/ D
Corresponding to the time-space fluid domain, we denote
Q
S
the time-space body
domain:
[
t
2
.0;T/
ftg
S
Q
S
D
.t/:
For simplicity, we assume that the fluid has constant density
f
D 1. We denote
its kinematic viscosity. We deal with newtonian fluids, so that the Cauchy stress
tensor in the fluid
T
is given by Newton law:
T
D
T
.
u
f
;p
f
/ WD 2D.
u
f
/ p
f
I
d
;
(4.8)
where D.
u
f
/ stands for the symmetric part of r
u
f
.
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