Biomedical Engineering Reference
In-Depth Information
the pressure
Π
and the rods' contributions to the stress tensor
σ
r
on polarization and
alignment tensor. The derivation of the constitutive equations for these quantities
was described in Section 7.5.
7.7.1 Isotropic State
In an isotropic suspension, the only hydrodynamic variable describing the filaments
is the concentration,
c
. Its dynamics is governed by a nonlinear convection-diffusion
equation
∂
t
c
+
∇
·
v
c
=
∇
·D
(
c
)
∇
c,
(7.60)
where
D
(
c
) is an effective (concentration-dependent) diffusion coecient, softened
by active processes. It is given by
(
c
)=
3
D
4
D
(1 +
v
0
c
)
−
α m
a
c,
(7.61)
with
m
a
=
m
a
b
2
. The first term on the right-hand side of Equation (7.61) is the
diffusion coecient of long thin rods, with
D
=
D
=2
D
⊥
, including excluded
volume corrections, with
v
0
=2
l
2
/π
. The second term on the right-hand side of
Equation (7.61) arises from filament bundling driven at the rate
α
given in Equation
(7.26) and promotes density inhomogeneities. Equation (7.60) for the concentration
couples to the Stokes equation, Equation (7.57), with
Π
r
(
c, μ
)=
k
B
T
a
c
1+
2
c
π
+
m
a
α
5
k
B
T
a
432
D
c
2
,
(7.62)
and
ij
=
2
η
0
+
k
B
T
a
96
D
c
u
ij
.
σ
r,I
(7.63)
In an isotropic active suspension there are no active contributions to the deviatoric
part of the stress tensor, which has the form usual for passive rods [61]. There is,
however, an active contribution to the pressure corresponding to the second term
on the right-hand side of Equation (7.62). The first term on the right-hand side of
Equation (7.62) is standard for passive rods.
The homogeneous isotropic state in a quiescent suspension is characterized by
v
= 0 and
c
=
c
0
. As discussed in the literature [30, 33, 36], the homogeneous state
becomes unstable at high filament and motor concentration due to contractile effects
generated by motor-induced filament bundling. Bundling is the main mechanism re-
sponsible for the instability of both isotropic and ordered homogeneous states in
quiescent suspensions. It is therefore instructive to explicitly display the details of
this instability for the simple isotropic case. In examining the dynamics of fluctua-
tions in the isotropic state we let
c
=
c
0
+
δc
and
v
=
δ
v
in Equation (7.60) and only
keep terms of first order in the fluctuations. Incompressibility requires
∇
· δ
v
=0,
and the linearized equation for
δc
is simply
2
δc .
(7.64)
Expanding
δc
in Fourier components,
δc
(
r
,t
)=
"
k
=
c
k
(
t
)
e
i
k
·
r
, one finds immedi-
ately that the relaxation of the Fourier amplitudes,
c
k
(
t
)=
c
k
e
−z
c
(
k
)
t
, is controlled
by a diffusive mode
∂
t
δc
=
D
(
c
0
)
∇
