Er ^
Q
à ¼ Er
2
3
βA
Ã
þ β A
Ã
Á Q þ Q Á A
Ã
À
2
3
A
Ã
: Q
ð
ÞI
!
À
1
2
β A
Ã
: Q
ð
ÞQ þ A
Ã
Á Q Á Q þ Q Á A
Ã
Á Q þ Q Á Q Á A
Ã
À Q Á Q
ð
Þ: A
Ã
f
gI
½
!
À
3R
1 À
3
2 Q : Q
À
Á 2 1 À
1
3
U
Q À UQ Á Q þ U Q : Q
ð
ÞQ þ
1
3
Q : Q
ð
ÞI
&
' !
þ
3
1 À
3
2 Q : Q
À
Á 2 ∇
Ã2 Q þ
1
2
L
Ã
2 ∇
Ã
∇
Ã
Á Q
ð
Þþ ∇
Ã
∇
Ã
Á Q
ð
Þ
f
g
T À
2
3
tr ∇
Ã
∇
Ã
Á Q
ð
Þ
f
gI
! !
(15)
Equation 15 shows a dimensionless form of the Q tensor dynamics, with the
following dimensionless variables: t
Ã
¼ _
γt , A
Ã
¼ A= _
γ , W
Ã
¼ W= _
γ , ∇
Ã
= H∇,
L 2
Ã
= L 2 /L 1 , _
γ is a characteristic shear rate, and β is a molecular shape parameter. The
terms linear in A denote the flow-induced orientation, the terms involving Q and its
products only denote the phase ordering, and the ones including gradients of Q are
the elastic terms (Grecov and Rey 2003a-c; Rey 2007, 2009, 2010; Rey and HerreraValencia 2012; Rey et al. 2014). The total extra stress tensor t
t for liquid crystalline
materials is given by the sum of symmetric viscoelastic stress tensor t
s , antisymmetric stress tensor, and Ericksen stress tensor t
Er (Grecov and Rey 2004):
t
t
¼ t
s
þ t
a
þ t
Er
(16)
Summing up all the contributions and non-dimensionalizing the following is
found:
~ t
t ¼
Er
R
ν
Ã
1 A
Ã
þ ν
Ã
2 Q Á A
Ã
þ A
Ã
Á Q À
2
3
Q : A
Ã
ð
ÞI
&
'
þ
Er
R
ν
Ã
4 A
Ã
: Q
ð
ÞQ þ A
Ã
Á Q Á Q þ Q Á A
Ã
Á Q þ Q Á Q Á A
Ã
À Q Á Q
ð
Þ: A
Ã
f
gI
½
À
Á
þ 3 À
2
3
β H À β H Á Q þ Q Á H À
2
3
H : Q
ð
ÞI
&
' !
þ
3
2
β H : Q
ð
ÞQ þ H Á Q Á Q þ Q Á H Á Q þ Q Á Q Á H À Q Á Q
ð
Þ: H
f
gI
½
þ 3 H Á Q À Q Á H
ð
Þ þ
3
R
À∇
à Q : ∇
à Q
ð
Þ
T À
L 2
L 1
∇
Ã
Á Q
ð
ÞÁ ∇
à Q
ð
Þ
T
!
(17)
with the following definitions for the dimensionless variables:
~ t
t ¼
t
t
ckT
à ,ν
Ã
1 ¼
ν 1 6D r
ckT
à ,ν
Ã
2 ¼
ν 2 6D r
ckT
à ,ν
Ã
4 ¼
ν 4 6D r
ckT
Ã
(18)
288
A. D. Rey et al.
Q
à ¼ Er
2
3
βA
Ã
þ β A
Ã
Á Q þ Q Á A
Ã
À
2
3
A
Ã
: Q
ð
ÞI
!
À
1
2
β A
Ã
: Q
ð
ÞQ þ A
Ã
Á Q Á Q þ Q Á A
Ã
Á Q þ Q Á Q Á A
Ã
À Q Á Q
ð
Þ: A
Ã
f
gI
½
!
À
3R
1 À
3
2 Q : Q
À
Á 2 1 À
1
3
U
Q À UQ Á Q þ U Q : Q
ð
ÞQ þ
1
3
Q : Q
ð
ÞI
&
' !
þ
3
1 À
3
2 Q : Q
À
Á 2 ∇
Ã2 Q þ
1
2
L
Ã
2 ∇
Ã
∇
Ã
Á Q
ð
Þþ ∇
Ã
∇
Ã
Á Q
ð
Þ
f
g
T À
2
3
tr ∇
Ã
∇
Ã
Á Q
ð
Þ
f
gI
! !
(15)
Equation 15 shows a dimensionless form of the Q tensor dynamics, with the
following dimensionless variables: t
Ã
¼ _
γt , A
Ã
¼ A= _
γ , W
Ã
¼ W= _
γ , ∇
Ã
= H∇,
L 2
Ã
= L 2 /L 1 , _
γ is a characteristic shear rate, and β is a molecular shape parameter. The
terms linear in A denote the flow-induced orientation, the terms involving Q and its
products only denote the phase ordering, and the ones including gradients of Q are
the elastic terms (Grecov and Rey 2003a-c; Rey 2007, 2009, 2010; Rey and HerreraValencia 2012; Rey et al. 2014). The total extra stress tensor t
t for liquid crystalline
materials is given by the sum of symmetric viscoelastic stress tensor t
s , antisymmetric stress tensor, and Ericksen stress tensor t
Er (Grecov and Rey 2004):
t
t
¼ t
s
þ t
a
þ t
Er
(16)
Summing up all the contributions and non-dimensionalizing the following is
found:
~ t
t ¼
Er
R
ν
Ã
1 A
Ã
þ ν
Ã
2 Q Á A
Ã
þ A
Ã
Á Q À
2
3
Q : A
Ã
ð
ÞI
&
'
þ
Er
R
ν
Ã
4 A
Ã
: Q
ð
ÞQ þ A
Ã
Á Q Á Q þ Q Á A
Ã
Á Q þ Q Á Q Á A
Ã
À Q Á Q
ð
Þ: A
Ã
f
gI
½
À
Á
þ 3 À
2
3
β H À β H Á Q þ Q Á H À
2
3
H : Q
ð
ÞI
&
' !
þ
3
2
β H : Q
ð
ÞQ þ H Á Q Á Q þ Q Á H Á Q þ Q Á Q Á H À Q Á Q
ð
Þ: H
f
gI
½
þ 3 H Á Q À Q Á H
ð
Þ þ
3
R
À∇
à Q : ∇
à Q
ð
Þ
T À
L 2
L 1
∇
Ã
Á Q
ð
ÞÁ ∇
à Q
ð
Þ
T
!
(17)
with the following definitions for the dimensionless variables:
~ t
t ¼
t
t
ckT
à ,ν
Ã
1 ¼
ν 1 6D r
ckT
à ,ν
Ã
2 ¼
ν 2 6D r
ckT
à ,ν
Ã
4 ¼
ν 4 6D r
ckT
Ã
(18)
288
A. D. Rey et al.