Leslie Ericksen Nematodynamics (LE)
The LE equations consist of the linear momentum balance and director torque
balance with additional constitutive equations for stress tensor T, elastic torque Γ
e ,
and viscous torque Γ
v (Larson and Doi 1991; Rey 2001, 2007, 2009, 2010;
Murugesan and Rey 2010; Rey and Herrera-Valencia 2012; Rey et al. 2014). The
mass and linear momentum balance equations read:
∇ Á v ¼ 0
(7a)
ρ _
v ¼ ∇ Á T þ f
(7b)
where f is the body force per unit volume, v is the linear velocity, and a superposed
dot represents the time material derivative of the velocity field. The constitutive
equation for the total stress tensor T is given by:
T ¼ ÀpI À
@F g
@∇n
ð
Þ
T Á ∇n þ α 1 nn : A
ð
Þnn þ α 2 nN þ α 3 Nn þ α 4 A þ α 5 nn Á A
þ α 6 A Á nn
(8)
2A ¼ ∇v þ ∇v
ð Þ
T
, N ¼ _
n À W Á n, 2W ¼ ∇v À ∇v
ð Þ
T
(9)
where p is the pressure, I is the unit tensor; α i , i = 1,. . .,6 are the six Leslie viscosity
coefficients; A is symmetric part of the velocity gradient tensor denominated the rate
of deformation tensor, N is the Zaremba-Jaumann time derivative of the director, and
W is the vorticity tensor (Rey 2007, 2009, 2010; Rey and Herrera-Valencia 2012;
Rey et al. 2014). The Frank elastic energy density f g is given by:
2f g ¼ K 11 ∇ Á n
ð
Þ
2 þ K 22 n Á ∇ Â n
ð
Þ
2 þ K 33 n  ∇  n
ð
Þ
2
(10)
where {K ii , ii = 11, 22, 33} are the temperature-dependent three elastic constants for
splay, twist, and bend, respectively. Anisotropies and thermal dependence of the
elastic constants are discussed elsewhere (Larson and Doi 1991; de Gennes and Prost
1993; Rey and Denn 2002). The director torque balance equation is given by the sum
of the viscous Γ
v and the elastic Γ
e torque:
Γ
v
þ Γ
e
¼ 0,Γ
v
¼ n  h
v
Àn  γ 1 N þ γ 2 A Á n
ð
Þ Γ
e
¼ n  h
e
Àn Â
@f g
@n
À∇ Á
@f g
@ ∇n
ð Þ
T
!
(11)
where h
v is the viscous molecular field, h
e is the elastic molecular field, γ 1 = α 3 À α 2
is the rotational viscosity, and γ 2 = α 6 À α 3 = α 3 + α 2 is the irrotational torque
coefficient.
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