Biomedical Engineering Reference
In-Depth Information
small. However, in non-Newtonian fluids there is also a viscous Magnus force that
depends on the microscopic relaxation time
t
m
of the additional components (such
as polymers) in the surrounding liquid [75]. For the two motors that are moving
around their center of mass, the Magnus force leads to an effective repulsion (see
Figure 8.9).
Figure 8.9.
In a system with two rotating mobile motors, each motor is forced
(by the velocity field created by the other motor) on a circular trajectory around
the center of mass. Because the motors perform a rotational and a translational
motion they are repelled from each other by the Magnus-force
F
M
. This repulsion
minimizes the total kinetic energy in the surrounding liquid because as the distance
l
between the motors increases, the region of destructive interference (of the velocity
fields created by the motors) becomes larger.
In systems with
N
rotating mobile motors, this repulsive interaction leads to
the formation of a (Wigner) crystal. More precisely, if hydrodynamic interactions
are strong enough (i.e.,
ω>ω
c
), then the motors arrange on a lattice. For weak
interactions (i.e.,
ω<ω
c
) the lattice melts and the motors form a disordered phase.
The melting point
ω
=
ω
c
can be estimated by comparing two time scales. Namely,
t
fluct
(the typical time scale on which a fluctuation occurs which drives a motor
away from its equilibrium lattice position) and
t
rel
(the time it takes the displaced
motor to reach its initial position again). At the melting point, these two time scales
become comparable (i.e.,
t
rel
=
t
fluct
) [42, 43].
In summary, hydrodynamic interactions can also induce repulsions (or attrac-
tions) between mobile motors giving rise to ordering transitions such as the crys-
tallization of rotating motors. Similar effects have been observed in bacterial layers
[76]. Here, the competition between advection by the bacteria-generated flow and
diffusion can lead to an instability driving bacterial aggregation.
