Biomedical Engineering Reference
In-Depth Information
2.7.4
Weak Formulation of FSI Problem (
2.177
)-(
2.191
)
For t 2 Œ0;T/ introduce the following test function spaces: the fluid velocity space
is defined by
/
1
; ' 2 .H
1
.
F
//
2
;
'
r
j
r
D
0
D 0; 'j
z
D
0;L
D 0g;;
2
V
F
.t/ Df' W
F
.t/ !
R
j ' D' ı .
A
the fluid pressure:
/
1
; q 2 L
2
.
F
/g;;
Q.t/ Dfq W
F
.t/ !
R
j q Dq ı .
A
the test space for the thin structure problem:
2
j 2 .H
0
.//
2
V
K
Df W .0;L/ !
R
g;
and the test space for the thick structure problem
2
j 2 .H
1
.
S
//
2
; j
z
D
0;L
D 0;
z
j
Sext
V
S
Df W
S
!
R
D 0g:
The test space for the coupled FSI problem is given by:
Q
.t/ Df.';; / 2
V
F
.t/
V
K
V
S
j 'j
.t/
D ; D j
g;
(2.199)
The variational formulation of the coupled FSI problem now reads: for t 2
.0;T/,find.
u
;p;;
d
/ 2
V
F
.t/Q.t/
V
K
V
S
such that the kinematic coupling
conditions (
2.189
)and(
2.190
) hold, and such that for all .';; ;q/2
Q
.t/Q.t/
the following equations are satisfied:
F
Z
Z
.
u
r/
u
' C 2
F
Z
@
u
@t
' C
D.
u
/ W D.'/
F
.t/
F
.t/
F
.t/
Z
pr'
Z
qr
u
C
K
h
Z
L
0
@t
2
z
C
K
h
Z
L
0
@
2
z
@
2
r
@t
2
r
F
.t/
F
.t/
C
2
Z
L
0
@
z
z
C C
1
Z
L
0
@
z
C C
0
Z
L
0
r
r
C C
2
Z
L
0
@
r
@
z
@
z
@
z
@
z
@
z
r
(2.200)
C
S
Z
@t
2
C 2
Z
D.
d
/ W D. / C
Z
@
2
d
.r
d
/.r /
S
S
S
C
Z
Z
R
Z
R
d
D
p
in
.t/'
z
j
z
D
0
dr
p
out
.t/'
z
j
z
D
L
dr:
S
0
0
Z
qr
u
D 0:
F
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