Biomedical Engineering Reference
In-Depth Information
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
„
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/
Z
t
Z
Z
t
Z
2
.t/
2
2
.t/
2
f
j
u
j
f
j
u
j
u
.t/ n
u
.t/ n
0
in
0
out
„
ƒ‚
…
Flux of fluid kinetic energy at the interfaces
(1.27)
These two last additional terms have an undetermined sign and could not be
easily estimated by the fluid energy. In dimension 2 one could obtain an energy
estimate locally in time and for small initial data but it is not feasible in dimension 3.
We refer to [
9
,
103
,
138
] for existence results in the case of Navier-Stokes in a
given domain with Neumann type boundary conditions. Note that in these papers
the solutions are strong and not weak. Since no energy estimate could be derived, a
way to get rid of this difficulty is to modify the initial problem and impose:
f
u
;pC
2
f
2
j
u
j
n Dp
in
n on
in
;
(1.28)
f
u
;pC
2
f
2
j
u
j
n Dp
out
n on
out
;
(1.29)
f
2
j
u
j
2
. By doing so one obtains
an energy estimate and one can hope to prove existence of solutions in the energy
space (see [
93
,
109
-
111
,
121
,
129
], where such kind of boundary conditions are
considered). These type of boundary conditions will also be used in Sect.
1.2.4
.
Note moreover that imposing the Neumann boundary conditions (
1.5
), (
1.6
)forthe
Navier-Stokes system leads also to numerical issues since the energy entering the
computational domain is not controlled, which may induce numerical instabilities
[
94
,
113
,
126
].
where p is replaced by the total pressure p C
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