Biomedical Engineering Reference
In-Depth Information
Z
L
s
e@
tt
b
1
C Ǜ
1
Z
L
0
@
x
@
x
b
1
C LJ
1
Z
L
0
C
@
x
@
t
@
x
b
1
0
Z
L
s
e@
tt
b
2
C
Z
L
0
@
xx
@
xx
b
2
C Ǜ
2
Z
L
0
@
x
@
x
b
2
C LJ
2
Z
L
0
C
@
x
@
t
@
x
b
2
0
Z
Z
Z
L
Z
L
D
T
f
b C
f v C
g
1
b
1
C
g
2
b
2
;
†
f
.t/
0
0
x D .x;R/ 2 †;v D 0 on
0
:
8.v;b/;b.x/ D .b
1
.x/;b
2
.x//
T
for
Taking into account the Neumann boundary conditions on
in
and
out
we get
Z
Z
f
.
u
r/
u
vC
Z
Z
f
@
t
u
v C
f
.t/
D
.
u
/ W
D
.v/
p divv
f
.t/
f
.t/
f
.t/
Z
L
s
e@
tt
b
1
C Ǜ
1
Z
L
0
@
x
@
x
b
1
C LJ
1
Z
L
0
C
@
x
@
t
@
x
b
1
Z
L
s
e@
tt
b
2
C
Z
L
0
0
@
xx
@
xx
b
2
C Ǜ
2
Z
L
0
@
x
@
x
b
2
C LJ
2
Z
L
0
C
@
x
@
t
@
x
b
2
Z
Z
Z
Z
0
D
f v
p
out
v n
p
in
v n C
g b;
†
f
.t/
out
in
8.v;b/ such that v D 0 on
0
;
b.t; x/ D .b
1
.t;x/;b
2
.t;x//
T
x D .x;R/ 2 †:
(1.22)
D v.t; x C d.t; x// for
Energy Estimates
In order to obtain the energy estimates satisfied by .
u
;p/ we choose the fluid
velocity
u
and the structure velocity @
t
d as test functions in the previous variational
formulations (
1.21
)or(
1.22
). These test functions are admissible since they satisfy
the kinematic condition (
1.15
). Thanks to the fluid incompressibility, in the case
of homogeneous Dirichlet boundary conditions on
f
and since the interface †.t/
moves at the structure velocity which is equal to the fluid velocity, we note that due
to the Reynolds transport theorem it holds
Z
Z
Z
f
2
j
u
j
d
dt
2
:
f
@
t
u
u
C
f
.
u
r/
u
u
D
(1.23)
f
.t/
f
.t/
f
.t/
Consequently the kinetic energy of the fluid appears leading to the following energy
balance:
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