Biomedical Engineering Reference
In-Depth Information
∇·
σ
r
(
r
,t
)=
u
c
(
r
,
u
,t
)
F
h
(
r
,
u
,t
)
l
2
u
·
l
c
(
r
,
u
,t
)
τ
h
(
r
,
u
,t
)
s
.
s
−
(7.32)
u
In the absence of inertial effects, the
total
hydrodynamic force,
F
h
(
r
,
u
,t
), exerted
by the suspension
on the center of mass
of a rod, can be found from the condition
that all forces acting on the rod must balance. The solvent flow field on a given
segment of a rod is calculated using a decoupling approximation where the hydro-
dynamic coupling to other segments of the same rod are treated explicitly within
the Oseen approximation, while the hydrodynamic effects of other rods enter in the
determination of a self-consistent value for the flow velocity of the solvent, yielding,
F
h
(
r
,
u
,t
)=
k
B
T
a
∇
ln
c
+
∇
U
ex
−
F
a
,
(7.33)
where
−k
B
T
a
∇
ln
c
is the Brownian force,
−
∇
U
ex
is the force due to the direct
interaction of the rod with other rods (in this case, via excluded volume), and
F
a
is the active force that can be written as
F
ai
=
ζ
ij
(
u
)
J
ci
/c.
(7.34)
The rod friction tensor
ζ
ij
(
u
) is proportional to the inverse of the rod diffusion
tensor
D
ij
(
u
), with
ζ
ij
(
u
)=
k
B
T
a
D
−
1
(
u
)
ij
=
ζ
u
i
u
j
+
ζ
⊥
(
δ
ij
− u
i
u
j
)
,
(7.35)
with
ζ
=2
πη
0
l/
ln(
l/b
), and
ζ
⊥
=2
ζ
. Similarly, the total hydrodynamic torque is
given by
τ
h
(
r
,
u
,t
)=[
k
B
T
a
R
ln
c
+
R
U
x
−
τ
a
]
×
u
ζ
2
−
uu
(
u
·∇
)
·
v
(
r
)
,
(7.36)
c
/c
the active torque. The last term on the right hand side of Equation
(7.36) is a viscous contribution to the stress proportional to the velocity gradient.
The rod contribution to the stress tensor can now be evaluated explicitly using
the truncated moment expansion for
c
(
r
,
u
,t
), given in Equation (7.29). When eval-
uating the active contributions to the stress tensor, only terms up to first order in
u
1
·
u
2
are retained in the active force
ζ
(
u
1
)
·
v
a
(1; 2) exerted by a motor cluster on
the filament. This approximation only affects the numerical values of the coecients
in the stress tensor, not its general form.
For simplicity, we consider solutions in the presence of a constant velocity gra-
dient,
κ
ij
, and with a uniform mean rate of ATP consumption. We allow for spatial
inhomogeneities in the filament concentration and orientational order parameters
and evaluate the stress tensor up to first order in gradients of these hydrodynamic
fields. The deviatoric part
σ
ij
=
σ
ij
−
(1
/
2)
δ
ij
σ
kk
of the stress tensor of the filaments
is
with
τ
a
=
ζ
r
J
σ
ij
(
r
,t
)=
σ
ij
(
r
,t
)+
σ
ij
(
r
,t
)
,
(7.37)
with
