Biomedical Engineering Reference
In-Depth Information
interactions between beads on the same rod have been included explicitly, while the
hydrodynamic coupling to other rods is implicitly taken into account in determining
the flow velocity
v
(
r
). The force on bead
m
on the
α−
th rod is therefore given by
8
n
=
m
3
1
|n − m|
f
α
(
m
)=
−ζ
b
[
v
α
(
m
)
−
v
(
r
α
(
m
))]
−
(
δ
+
u
α
u
α
)
·
f
α
(
n
)
.
(7.93)
Now we take the limit
l
b
and introduce the continuous variable
s
=
bm
,where
−
l/
2
≤
s
≤
l/
2, so that
r
α
(
s
)=
r
α
+
s
u
α
. The hydrodynamic force per unit length,
h
α
(
s
), satisfies the equation
F
ds
|s − s
|
3
8
h
α
(
s
)=
−ζ
s
(
v
α
(
s
)
−
v
(
r
α
(
s
)))
−
α
(
s
)
,
(7.94)
h
F
(
δ
+
u
α
u
α
)
·F
|s−s
|≥b
where
ζ
s
=3
πη
0
=
ζ
b
/b
. Approximating
#
1
|s − s
|
$
δ
(
s − s
)
,
1
|s − s
|
≈
(7.95)
where
#
1
|s − s
|
$
=
l
2
1
1
|s − s
|
Θ
(
|s − s
|−b
)
=2ln(
l/
2
b
)
,
(7.96)
s
s
with
Θ
(
x
) the Heaviside function, we obtain
3ln(
l/
2
b
)
4
h
α
(
s
)
−ζ
s
[
v
α
(
s
)
−
v
(
r
α
(
s
))]
.
(
δ
+
u
α
u
α
)
·F
(7.97)
Because
v
α
(
s
)=
v
α
+
s
ω
α
×
u
α
, then integrating Equation (7.97) over
s
,wecan
obtain expressions for the hydrodynamic force,
F
α
=
h
α
(
s
)
s
and torque
τ
α
=
F
s
=
l/
2
h
α
(
s
)
u
α
s
×F
s
at the center of mass of a rod, with
...
−l/
2
ds...
as
1
l
ζ
−
1
(
u
α
)
F
α
=
v
α
−
−
·
v
(
r
α
(
s
))
s
,
(7.98)
ζ
r
τ
α
=
ω
α
− I
−
1
1
1
−
l
u
α
× s
v
(
r
α
(
s
))
s
,
(7.99)
πl
3
η
0
4
πη
0
l
where
ζ
ij
(
u
)=
ζ
⊥
(
δ
ij
− u
i
u
j
)+
ζ
u
i
u
j
,
ζ
⊥
=2
ζ
3ln(
l/
2
b
)
and
I
=
l
2
/
12. Performing a Taylor expansion of the fluid velocity about the center of
mass, we obtain to lowest order in gradients
=
ln(
l/
2
b
)
,
ζ
r
=
I
2
(
u
α
−
ζ
−
1
(
u
α
)
·
F
α
=
v
α
−
v
(
r
α
)
−
·
∇
)
2
v
(
r
α
)+
O
(
∇
4
)
,
(7.100)
1
ζ
r
τ
α
=
ω
α
−
u
α
×
(
u
α
3
)
.
−
·
∇
)
v
(
r
α
)+
O
(
∇
(7.101)
Finally, we require that the hydrodynamic forces and torques be balanced by all
other forces and torques on the rod. This gives
F
α
=
f
α
,
∇
α
U
ex
+
k
B
T
a
∇
ln
c
−
(7.102)
τ
α
=
τ
α
,
R
α
U
ex
+
k
B
T
a
R
α
ln
c
−
(7.103)
