Biomedical Engineering Reference
In-Depth Information
with
@
@x
;
y
h
@h
@x
@
@ y
1
h
@
@ y
:
@
1
D
@
2
D
With the same notations we define for the viscous term
Z
h
ij
.
D
u
//
D
ij
. /;
..
u
; //
h
D
.
O
(1.103)
D
ij
.
D
u
// D 2.j
D
u
/j/
D
ij
.
.
O
.
O
u
/;
O
(1.104)
1
2
.@
i
.
u
j
/ C @
j
.
u
i
//;
D
ij
.
u
/ D
O
(1.105)
and for the convective term
u
;
z
; / D
f
Z
h
u
1
@
1
z
C
u
2
@
y
z
C
h
2
z
div
h
O
O
b
b
h
.
u
D
Z
1
Z
1
1
2
1
2
R
0
b
u
1
z
1
1
j
x
D
L
C
R
0
b
u
1
z
1
1
j
x
D
0
0
0
Z
L
0
b
u
2
z
2
2
j
y
D
1
:
1
2
(1.106)
Remark 1.11.
Note that the transformed stress tensor
ij
D 2.j
D
u
/j/
D
ij
.
.
O
u
/ from
O
(
1.103
) with .j
D
.
u
/j/ defined in (
1.78
) also satisfies (
1.75
)-(
1.77
).
O
Remark 1.12.
Note that all these terms could be written by means of the notations
introduced in the study of the existence of strong solutions. For instance,
div
h
u
D B
h
Wr
u
.J
h
/
1
;
with B
h
D cof r
h
, J
h
D det r
h
and
h
.x; y/ D .x; yh.x//.
Definition 1.1
(Weak Solution of the Approximated Linearized Problem).
Let
O
u
2 L
p
.0;T IV/ \ L
1
.0;T I L
2
.D//, p 2 L
2
.0;T I H
1
.D// \ L
1
.0;T I L
2
.D//
and 2 L
1
.0;T I
L
2
.0;L// \ L
2
.0;T I H
0
.0;L//: Atriple
w
D .
u
; p;/ is called
a weak solution of the regularized problem (
1.73
)-(
1.89
) if the following equation
holds (for simplicity in what follows all the physical constants are assumed to be
equal to one)
O
Z
T
0
h@
t
.h
u
/; i
O
Z
u
/ h p div
h
Z
T
O
O
O
O
D
D
@
t
h@
y
. y
C
b
h
.
u
;
u
; /
C ..
u
; //
h
0
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