Biomedical Engineering Reference
In-Depth Information
Z
C
Z
t
0
Z
Z
L
2
j@
t
j
2
f
2
j
u
j
s
e
2
.t/
„ ƒ‚ …
Fluid kinetic energy
2
2
f
.t/
j
D
.
u
/j
C
Cj@
t
j
f
.t/
0
„ ƒ‚ …
Kinetic energy of the structure
„
ƒ‚ …
Fluid dissipation
Z
L
0
j@
x
j
Z
L
0
j@
xx
j
Z
L
0
j@
x
j
Ǜ
1
2
2
Ǜ
2
2
2
2
2
C
C
C
„
ƒ‚
…
Mechanical energy of the structure
Z
t
Z
L
0
j@
x
@
t
j
Z
t
Z
L
0
j@
x
@
t
j
LJ
1
2
LJ
2
2
2
2
C
C
0
0
„
ƒ‚
…
Structure dissipation
Z
t
Z
Z
t
Z
Z
f
2
j
u
0
j
2
D
f
u
C
g @
t
d
C
†
0
f
.s/
0
f
.0/
„
ƒ‚
…
„ ƒ‚ …
Initial energy of the fluid
D
E
f
.0/
Power of the exterior forces
Z
L
Z
L
0
j@
x
0
j
Z
L
0
j@
xx
0
j
Z
L
0
j@
x
0
j
s
e
2
.j
1
j
Ǜ
1
2
2
Ǜ
2
2
2
2
/ C
2
2
2
C
Cj
1
j
C
C
0
„
ƒ‚
…
Initial energy of the structure
D
E
s
.0/
(1.24)
Remark 1.3.
In the case of an elastic or hyperelastic thick structure, the energy takes
the same form except for the mechanical energy of the structure which depends on
the considered material.
The energy balance (
1.24
) implies, by using the Gronwall lemma, that for t T
Z
2
.t/ C
Z
t
0
Z
Z
L
2
j@
t
j
2
.t/
f
2
j
u
j
s
e
2
2
.t/ Cj@
t
j
f
.s/
j
D
.
u
/j
C
f
.t/
0
Z
L
0
j@
x
j
Z
L
0
j@
xx
j
Z
L
0
j@
x
j
Ǜ
1
2
2
Ǜ
2
2
2
.t/ C
2
.t/ C
2
.t/
C
Z
t
Z
L
0
j@
x
@
t
j
Z
t
Z
L
0
j@
x
@
t
j
LJ
1
2
LJ
2
2
2
2
C
C
0
0
e
t
E
f
.0/ C E
s
.0/ C C
Z
t
0
e
t
s
Z
2
!!
: (1.25)
Z
L
0
jgj
2
f
.s/
jf j
C
Note that here we have not used the dissipation coming from the structure to
obtain this energy estimate and this is the reason why we have a constant in time
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