124
Q. Wang
• Constitutive equation accounting for the microstructure of the system for σ and
F e . These include: σ and F e , and transport equations for the internal variables
(c , p, Q).
The singular limits of the momentum equation like the Stokes equation for very viscous mixtures are derived by taking the respective singular limits of the full equation.
Consider a mixture of complex fluids, whose micro-structures are described by
the first 3 moments (c, p, Q) of a microstructural distribution function and solvent
with the total mass density ρ and mass averaged velocity v. The mass conservation is
given by (3.107). The number density of the microscopic constituent c in the complex
fluid is described by the following equation
∂ t c + ∇ · (cv + j) = 0,
(3.109)
where j is an extra diffusive flux.
The momentum equation is given by (3.108). The above equations need to be augmented by the constitutive equations. Namely, we need to derive evolution equations
for c, p, Q and constitutive equation for flux j, stress σ and F e .
Constitutive equations:
We assume the fee energy of the mixture system is given by
F = F[c, ∇c, p, ∇p, Q, ∇Q] =
V
f (c, ∇c, p, ∇p, Q, ∇Q)dV, (3.110)
where f is the free energy density per unit volume. The total free energy is defined
by the sum of the kinetic energy and the free energy:
E
total
=
V
[
ρ
2
v
2
+ f ]dV.
(3.111)
The energy dissipation rate at a constant temperature T is calculated as follows
d E total
dt
=
V { 1
2
∂(ρv 2 )
∂t
+
∂ f
∂t }dV =
V [−∇ · (
ρv
2 v 2 ) + v · (∇ · σ + F e ) + μ ∂c
∂t − h ·
∂p
∂t − G :
∂Q
∂t }dV =
V [−∇ · (
ρv
2 v 2 ) + v · (∇ · σ + F e ) + μ( ˙
c − ∇ · (vc)) − h · ( ˙
P − v · ∇p − · p)−
G : (
Q − v · ∇Q − · q + Q · )}dV =
∂ V [−
ρn·v
2 v 2 + v · σ · n]ds +
V [(−∇v : σ + h · · p + G : (( · Q − Q · ))+
(v · F e − v · ∇c + h · v · ∇p + G : v · ∇Q) + μ ˙
c − h · ˙
P − G :
Q − r μ]dV =
∂ V [−
ρn·v
2 v 2 + v · σ · n − n · jμ]ds +
V [−D · σ s + ∇μ · j − h · ˙
P − G :
Q]dV ,
(3.112)
where μ =
δ F
δc
, h = −
δ F
δp
, and G = −
δ F
δQ
are the variation of the free energy with
respect to the three internal variables, σ
a is the antisymmetric part of the stress, σ e
is the Ericksen stress, σ s is the symmetric part of the stress,
Q. Wang
• Constitutive equation accounting for the microstructure of the system for σ and
F e . These include: σ and F e , and transport equations for the internal variables
(c , p, Q).
The singular limits of the momentum equation like the Stokes equation for very viscous mixtures are derived by taking the respective singular limits of the full equation.
Consider a mixture of complex fluids, whose micro-structures are described by
the first 3 moments (c, p, Q) of a microstructural distribution function and solvent
with the total mass density ρ and mass averaged velocity v. The mass conservation is
given by (3.107). The number density of the microscopic constituent c in the complex
fluid is described by the following equation
∂ t c + ∇ · (cv + j) = 0,
(3.109)
where j is an extra diffusive flux.
The momentum equation is given by (3.108). The above equations need to be augmented by the constitutive equations. Namely, we need to derive evolution equations
for c, p, Q and constitutive equation for flux j, stress σ and F e .
Constitutive equations:
We assume the fee energy of the mixture system is given by
F = F[c, ∇c, p, ∇p, Q, ∇Q] =
V
f (c, ∇c, p, ∇p, Q, ∇Q)dV, (3.110)
where f is the free energy density per unit volume. The total free energy is defined
by the sum of the kinetic energy and the free energy:
E
total
=
V
[
ρ
2
v
2
+ f ]dV.
(3.111)
The energy dissipation rate at a constant temperature T is calculated as follows
d E total
dt
=
V { 1
2
∂(ρv 2 )
∂t
+
∂ f
∂t }dV =
V [−∇ · (
ρv
2 v 2 ) + v · (∇ · σ + F e ) + μ ∂c
∂t − h ·
∂p
∂t − G :
∂Q
∂t }dV =
V [−∇ · (
ρv
2 v 2 ) + v · (∇ · σ + F e ) + μ( ˙
c − ∇ · (vc)) − h · ( ˙
P − v · ∇p − · p)−
G : (
Q − v · ∇Q − · q + Q · )}dV =
∂ V [−
ρn·v
2 v 2 + v · σ · n]ds +
V [(−∇v : σ + h · · p + G : (( · Q − Q · ))+
(v · F e − v · ∇c + h · v · ∇p + G : v · ∇Q) + μ ˙
c − h · ˙
P − G :
Q − r μ]dV =
∂ V [−
ρn·v
2 v 2 + v · σ · n − n · jμ]ds +
V [−D · σ s + ∇μ · j − h · ˙
P − G :
Q]dV ,
(3.112)
where μ =
δ F
δc
, h = −
δ F
δp
, and G = −
δ F
δQ
are the variation of the free energy with
respect to the three internal variables, σ
a is the antisymmetric part of the stress, σ e
is the Ericksen stress, σ s is the symmetric part of the stress,