Biomedical Engineering Reference
In-Depth Information
The notation used here is analogous to that used in the proof of Proposition
2.7
.
t
Before we can pass to the limit in the weak formulation of the approximate
problems, there is one more useful observation that we need. Namely, notice that
although
q
are smooth functions both in the spatial variables and in time, the
functions
q
N
are discontinuous at nt because
t
N
is a step function in time.
As we shall see below, it will be useful to approximate each discontinuous function
q
N
in time by a piece-wise constant function,
q
N
,sothat
q
N
.t;:/ D
q
.nt;:/; t 2 Œ.n 1/t;nt/; n D 1;:::;N;
where
q
N
.nt/ is the limit from the left of
q
N
at nt, n D 1;:::;N.Byusing
Lemma
2.6
, and by applying the same arguments in the proof of Lemma
2.5
,weget
q
N
!
q
uniformly on Œ0;T :
Passing to the Limit
To get to the weak formulation of the coupled problem, take the test functions
. .t/; .t// 2
X
W
X
S
as the test functions in the weak formulation of the
structure sub-problem (
2.6.5
) and integrate the weak formulation (
2.6.5
) with
respect to t from nt to .nC1/t. Notice that the construction of the test functions
is done in such a way that . .t/; .t// do not depend on N, and are continuous.
Then, consider the weak formulation (
2.146
) of the fluid sub-problem and take the
test functions .
q
N
.t/; .t// (where
q
N
D
q
ı A
t
N
,
q
2
X
F
). Integrate the fluid
sub-problem (
2.146
) with respect to t from nt to .n C 1/t. Add the two weak
formulations together, and take the sum from n D 0;:::;N 1 to get the time
integrals over .0;T/ as follows:
Z
T
Z
.1 C
t
N
/
@
t
u
N
q
N
C
1
2
.
t
u
N
w
N
/ r
t
N
u
N
q
N
0
F
Z
T
Z
t
N
q
N
u
N
1
2
.
t
u
N
w
N
/ r
1
2
v
N
u
N
q
N
C
0
F
Z
T
Z
Z
T
Z
1
.1 C
t
N
/2
D
t
N
.
u
N
/ W
D
t
N
.
q
N
/ C
(2.172)
C
@
t
v
N
0
F
0
0
Z
T
Z
1
Z
T
Z
Z
T
Z
@
t
V
N
C
C
@
z
N
@
z
C
a
S
.d
N
; /
0
0
0
S
0
S
Z
T
P
in
dt
Z
1
0
Z
T
P
out
dt
Z
1
0
D
q
z
.t;0;r/dr
q
z
.t;L;r/dr;
0
0
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