Biomedical Engineering Reference
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whereas the “plate” is clamped at its boundary:
u
.x
0
;t/D n r
0
u
.x
0
;t/D 0.x
0
;t/2 @ .0; 1/;
(3.100)
where r
0
operates only on the x
0
-variable. In view of (
3.97
)
2
and (
3.99
)
2
, the plate
equation (
3.98
) can be simplified. In fact, from (
3.89
)and(
3.99
)
2
deduce
e
3
T.v;p/ e
3
D 2
@v
3
@x
3
p on ;
so that, again by (
3.99
)
2
,and(
3.97
)
2
we conclude
e
3
T.v;p/ e
3
Dp on ;
and (
3.98
) becomes
u
tt
C
2
u
D p C f; in .0; 1/.
(3.101)
We next observe that also this model, as the one discussed in the previous section,
allows for a continuum of steady-state solutions (even) in the case f 0. In fact,
if we take v;pand
u
independent of t,say,v D v
0
.x/;p D p
0
.x/;
u
D
u
0
.x
0
/ we
deduce from (
3.97
)to(
3.99
) (with f 0)thatv
0
satisfies (
3.97
) with homogeneous
boundary conditions on the whole @. Consequently, v
0
0, p
0
D , 2
R
,and
by (
3.101
),
2
u
0
D ; in :
(3.102)
The above observation suggests that the “interesting” dynamics should be
restricted to the space orthogonal to the one-dimensional space characterized by
v
0
D 0 and
2
u
0
D
const: To this end, following [
12
], we introduce the “energy
space”
17
H./ H
0
./ L
2
./g
H
WD f.v;
u
:
u
t
/ 2
where
H./WD fv 2 L
2
./ W divv D 0;v n D 0 on
1
g ;
L
2
./ WD f
u
2 L
2
./ W .
u
;1/
D 0g ;
along with
H
WD f.v;
u
:
u
t
/ 2
H./ H
0
./ L
2
./g;
17
Notice that the requirement
u
t
2
L
2
./ follows from (
3.97
)
2
and (
3.99
)
2
.
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