Biomedical Engineering Reference
In-Depth Information
The continuum equations for the polarized state are obtained by assuming that
the alignment tensor relaxes on microscopic time scales to the form given by Equa-
tion (7.77), which is then used to eliminate
Q
ij
in favor of
P
. With the exception
of homogeneous terms such as the
O
((
P
)
3
) term just described, this leads to a high
density renormalization of the various coecients in the continuum equations, but
does not generate any new terms. For the sake of simplicity in the following, we
neglect this renormalization and only keep those terms in the polarization equa-
tion generated by the coupling to the alignment tensor that have a qualitatively
new structure. We also neglect all excluded volume corrections. The equation for
filament concentration is given by
c
v
+
36
m
a
βc
P
+
∂
i
D
ij
(
c
)
∂
j
c
7
p
∂
t
c
=
−
∇
·
2
α m
a
∂
i
c
2
∂
j
(
P
i
P
j
)
,
1
−
(7.78)
with
ij
(
c,
P
)=
3
D
4
α m
a
c
δ
ij
−
α m
a
c
P
i
P
j
+
1
2
δ
ij
P
2
.
p
D
−
(7.79)
The equation for the polarization vector has the form
∂
t
+
v
·
∇
(
c
P
)=
1
2
∇
v
+(
∇
v
)
T
· c
P
2
(
∇
×
v
)
×
(
c
P
)+
λ
P
+
H
(
c,
P
)
.
(7.80)
where
H
(
c,
P
) generalizes the molecular field of equilibrium polar fluids [63] by
including active contributions. It is given by
H
i
(
c,
P
)
−
D
r
− γ
P
m
a
c
+
a
3
P
2
cP
i
+
2
36
m
a
β∂
j
c
2
P
i
P
j
−
2
δ
ij
P
2
1
5
9
m
a
βc∂
i
c−
+
∂
j
K
p
∂
i
(
cP
j
)+
∂
i
K
p
∂
j
(
cP
j
)
+
∂
j
K
p
∂
j
(
cP
i
)
m
a
c
24
∇
p
ijk
(
c,
P
)
∂
k
c
+
γ
P
2
(
cP
i
)
,
−∂
j
D
(7.81)
where
K
p
(
c
)=
D
α m
a
4
8
−
c,
(7.82)
K
p
(
c
)=
5
D
α m
a
4
8
−
c,
(7.83)
3
P
j
δ
ik
(7.84)
The parameter
a
3
determines the value
P
0
of the magnitude of the polarization in
a quiescent (active) suspension, with
P
0
=
a
3
/
[
D
r
(
c/c
IP
−
1)].
In contrast to the case of the nematic, all three active mechanisms of motor-
induced filament dynamics controlled by
α
,
β
,and
γ
P
appear in the hydrodynamic
equations of polarized active suspension. Polarization sorting at a rate
β ∼ u
0
yields
novel convective contributions in the first term on the right-hand side of the equa-
tion for the filament concentration, Equation (7.78). In an equilibrium suspension
the filament concentration is convected with the suspension flow velocity
v
.Inan
ijk
(
c,
P
)=
c
Dv
0
8
3
(
P
i
δ
jk
+
δ
ij
P
k
)+
17
Dv
0
+
α m
a
+
2
α m
a
p
D
8
