Chemistry Reference
In-Depth Information
Summing up all the contributions and nondimensionalizing the following we
found:
{
}
Er
2
3
(
)
t
t
=
ν
**
A
+
ν
*
Q
i
A
*
+
A
*
i
Q
−
Q : A
*
I
1
2
Er
(
[
(
)
+
ν
*
A:QQ
*
+
AAQQQAQ
*
i i
+
i i
*
4
{
} ])
−
(
)
*
*
+
QQA
i i
QQ:A I
i
{
}
32
3
2
3
!
"
(
)
+−
ββ
HHQQHH
−
i
+
i
−
:: Q
I
U
3
2
3
[
(
)
{
(
)
}
]
+
U
β
H:QQ HQQ QHQ QQH
+
i i
+
i i
+
i i
−
QQ:HI
i
3
L
L
!
"
(
)
T
(
)
(
)
T
(
)
+ ∇ ∇
2
*
*
*
*
+
HQ QH
i
−
i
Q:
Q
−∇
i i
Q
∇
Q
U
R
1
(2.37)
where
t
ckT
t
ν
D
ckT
6
ν
D
ckT
6
ν
D
ckT
6
t
1
r
2
r
4
r
(2.38)
t
*
*
*
=
ν
=
ν
=
ν
=
1
2
4
*
*
*
*
The total dimensionless extra stress tensor [Eq. (2.38)] is neither symmetric
nor traceless. In this model there are three viscosity coeffi cients. The term
introduced by
Q
·
H
is the asymmetric
stress, and the last is the purely elastic Ericksen stress.
β
indicates backfl ow. The term
H
·
Q
+
2.2.4.2 Rheological Functions
Projecting the LdG into the LE model,
we fi nd the six Leslie coeffi cients (Grecov and Rey, 2003b, 2004; Rey, 2009):
8
9
8
9 2
S
2
*
2
2
2
αην
=
2
S
−
β
S
−
S
+
1
4
1
3
S
2
2
αη
=−−
S
β
S
2
+−
S
2
2
1
3
S
2
αη
=
S
2
−
β
S
2
+
S
−
3
2
(2.39)
2
3
1
4
9
S
2
αην ν
=
*
−
*
S
+
ν
*
S
2
+
β
2
1
−
S
−
4
1
2
4
3
4
1
3
1
2
1
3
+
αην
=
*
SSSS
+
β
2
+
−
2
β
2
SSS
(
4
−
−
2
)
5
2
1
3
1
2
1
3
+
αην
=
*
SSSS SS
−
β
2
+
−
2
β
2
(
4
−
−
S
2
)
6
2
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