Biomedical Engineering Reference
In-Depth Information
u
condition
D 0. Furthermore, since is not a Lipschitz function, the
ALE mapping is not necessarily a Lipschitz function either, and, as a result,
u
is
not necessarily an H
1
function on
F
. Therefore we need to redefine the function
spaces for the fluid velocity by introducing
r
F
Df
u
V
W
u
2
V
F
.t/g;
where
u
is defined in (
2.108
). Under the assumption 1 C .
z
/>0,
z
2 Œ0;1,the
F
:
following defines a scalar product on
V
Z
.1 C /
u
v
D .
u
;
v
/
H
1
.
F
.t//
:
.
u
;
v
/
v
u
D
Cr
Wr
F
V
F
F
,so
F
is
u
is an isometric isomorphism between
Therefore,
u
7!
V
F
.t/ and
V
V
F
.0;T/ and
.0;T/ are defined as
also a Hilbert space. The function spaces
W
W
F
instead
before, but with
V
V
F
.t/. More precisely:
F
.t//;
F
.0;T/ D L
1
.0;T I L
2
.
F
// \ L
2
.0;T I
V
W
(2.136)
F
.0;T/
W
W
.0;T/
W
S
.0;T/ W
u
.t;
z
;1/D @
t
.t;
z
/
e
r
;.t;
z
/ D d.t;
z
;1/g:
W
≡
.0;T/ Df.
u
;;d/ 2
W
(2.137)
The corresponding test space is defined by
F
V
W
V
S
/ W
q
.t;
z
;1/D .t;
z
/
e
r
; .t;
z
;1/D .t;
z
/
e
r
g:
Q
≡
.0;T/ Df.
q
; ; / 2 C
c
.Œ0;T/I
V
(2.138)
.0;T/ is a weak solution of prob-
lem (
2.111
)-(
2.118
) defined on the reference domain
F
,ifforevery.
q
; ; / 2
Q
≡
.0;T/ the following equality holds:
Definition 2.2.
We s a y t h a t .
u
;;d/ 2
W
Z
T
Z
Z
T
b
.
u
;
u
;
q
/C2
Z
T
0
Z
.1 C /
D
.
u
/ W
D
.
q
/
.1 C /
u
@
t
q
C
0
F
0
F
Z
T
Z
Z
T
Z
1
Z
T
Z
1
1
2
.@
t
/
u
q
@
t
@
t
C
@
z
@
z
0
F
0
0
0
0
Z
T
Z
Z
T
@
t
d @
t
C
a
S
.d; /
0
S
0
Z
T
0
hF.t/;
q
i
in=out
C
Z
Z
1
Z
D
u
0
q
.0/ C
v
0
.0/ C
V
0
.0/:
0
0
S
(2.139)
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