3 Generalized Onsager Principle and It Applications
127
M anti =
0 −A 1
A 1 0
,
(3.126)
A 1 = a[Q αk δ βl + δ αk Q βl ],
a ∈ [−1, 1] is a rate parameter. The governing system of equations is given by
Q − a[D · Q + Q · D] −
1
λ
G = 0,
∇ · (v) = 0,
ρ(
∂v
∂t
+ v · ∇v) = −∇ p + ∇ · (σ
s
− σ
e
− σ
a
),
(3.127)
where ρ is a constant density.
The energy dissipation is given by
d
dt
E = −
[2ηD : D +
1
λ
G : G]dx,
(3.128)
where E =
[
ρ
2
v
2
+ f (Q)]dx is the total energy. If a quadratic free energy density
is given, the Oldroyd B model is recovered [2],
f (Q) =
γ
2
tr(Q
2
),
(3.129)
where γ is the elastic modulus. If a cubic free energy density is given, the Giesekus
model is obtained
f (Q) =
γ
2
tr(Q
2
) +
γ 2
3
tr(Q
3
),
(3.130)
where γ 1,2 are elastic moduli. If we choose the free energy density as
f (Q) = γ 1 tr(Q
2
) + γ 2 (tr(Q))
2
,
(3.131)
the linear Phan-Thien Tanner model is recovered.
Example 3.4.3 (Model for nematic liquid crystal solutions) For liquid crystal solutions, we use the polarity vector p and the nematic order tensor Q. The generalized
Onsager principle yields the following constitutive equation.
(σ
s
,
Q, ˙
P, j)
T
= [M sym + M anti ] · (D, G, h, −∇μ)
T
M sym =
⎛
⎜
⎜
⎝
A 0
0
0
0
0
1
γ 2
δ αk δ βl
χ 1
2
A
χ 2
2
A
0 χ 1 p β δ αk δ βl
1
γ 1
δ αk λδ αk
0 χ 2 p β δ αk δ βl λδ αk γ δ αk
⎞
⎟
⎟
⎠ ,
(3.132)