Biomedical Engineering Reference
In-Depth Information
Figure 8.3.
Typical snapshot of a Brownian dynamics simulation of an elastic
filament (of length
L
) attached to a no-slip wall. The filaments are modeled as a
collection of (
N
= 20) spherical beads with potential energy given by the worm-like
chain model (with persistence length
ξ
p
/L
=2
10
5
).Themotioniscausedbyan
applied torque consisting of two phases: a phase of small torque
τ
s
=2
·
10
5
k
B
T
[corresponding to the sequence of shapes 1-4 shown in Figure (a)] and a phase of
large torque 4
τ
s
[sequence of shapes 5-8 in Figure (a)]. (b) Flow profile as function of
distance from the wall shown for the conformations 1-8 of (a). (c) Time-dependence
of the induced velocity field. Thus, the main motion of the fluid is caused by the
slower stroke. Figure is reprinted from [40]. Copyright (2006), with permission from
The American Physical Society.
·
1. Again, it is assumed that the force arising from the normal component of
v
12
is balanced by the ciliar filament. In deriving the last equation we have implicitly
assumed that the hydrodynamic interaction alters only the beating velocity but
not the beating pattern of the monocilia. The conditions under which this strong
assumption is justified will be discussed in the next section.
If the no-slip boundary condition on the wall is ignored, then the velocity
v
12
is
that of a stokeslet (given by Equation (8.2)) with strength
s
=3
aR ϕ
2
t
2
/
4where
t
2
is the tangential vector of the circular trajectory of Cilium 2. Then, the equation of
motion of the coupled system becomes
ϕ
1
+
J
(
ϕ
1
,ϕ
2
)
ϕ
2
=2
π,
(8.19)
