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molecular dynamics are taken into account. Finally, the presence of shear fl ow
of rate
γ
introduces a fl ow time scale
τ
f
and a fl ow length scale
f
:
1
D
LH
2
L
orient
(2.12)
τ
=
=
D
=
=
f
f
orient
γ
γ
3
τ
η
e
where
D
orient
is the characteristic orientation diffusivity (length
2
/time) in the
system. With regard to the values of Deborah numbers we have two processes:
(a) Orientation process (De << 1): The time scale ordering is
τ
e
, the
orientation processes dominate the rheology, and the scalar order parameter
is close to its equilibrium value. In this regime the fl ow affects the eigenvectors
of
Q
but does not affect the eigenvalues of
Q
. Since LCs are anisotropic, shear
thinning, nonmonotonic stress growth, and fi rst normal stress differences are
possible. (b) Molecular process (De
τ
i
<
τ
f
<
τ
e
,
and the fl ow affects the eigenvectors and eigenvalues of
Q.
The dimensionless
Ericksen number Er and Deborah number De are given by (Larson and Doi,
1991; Rey, 2007, 2009, 2010; Rey and Denn, 2002; Tsuji and Rey, 1998):
>
1): The time scales ordering is
τ
f
<
τ
i
<
2
2
2
3
τ
τ
γη
H
L
Er
τ
τ
γ
=
==
Er
e
e
i
i
(2.13)
==
De
==
R
6
6
D
f
f
f
f
r
To characterize the degree of ordering in the LC phase, a dimensionless con-
centration
U
3
c
/
c
* is used. Hence the most general parametric space for
LdG nematodynamics is given by (1/U,
=
, Er), while for the LE nematodynam-
ics it is Er.
2.2.3
Leslie-Ericksen Nematodynamics
2.2.3.1 Bulk and Interfacial Equations
The LE equations consist of the
linear momentum balance and director torque balance with additional consti-
tutive equations for stress tensor
T
, elastic torque
Γ
e
, and viscous torque
Γ
v
(Larson and Doi, 1991; Murugesan and Rey, 2010; Rey, 2001a,b, 2007, 2009,
2010 ).
The mass and linear momentum balance equations are
∇=
i
v
0
ρ
v
=+∇
f
i
T
(2.14)
where
f
is the body force per unit volume,
v
is the linear velocity, and a super-
posed dot represents the material time derivative of the velocity. The constitu-
tive equation for the total stress tensor
T
is given by
∂
∂∇
F
g
TI
=−
p
−
⋅∇
n
+
α
(
nn : Ann
)
+
α
nN
1
2
(
n
)
T
(2.15)
+
α
Nn
+
α
A
+
α
nn
A
+
α
A
nn
i
i
3
4
5
6
i
2
Av
=∇ +∇
(
(
v
))
T
NnWn Wv
= −
2
=∇ −∇
(
(
v
))
T
(2.16)
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