Chemistry Reference
In-Depth Information
advective transport become extremely critical at this gap midpoint point, and as one
shrinks the scale to the nano regime the interplay becomes ever more critical.
A surprising aspect of the forces that determine bumping and the competition
with diffusion can be seen in Figure 3.7 and Figure 3.8 which show the force densities
(extracted from the COMSOL simulation of the full NS equation) acting on the water
elements as they move through the structure. The viscous damping forces are dissipative
and always act in the opposite direction from the local velocity, while the “inertial” forces
are sensitive to the gradients in the velocity and thus can locally change sign as the
gradients flip direction. This effect can be seen in Figure 3.8(B). The gap
g
is shown, and a
bundle of streamlines that pass out of the gap to the left represent the “reset” streamline
bundle β
g
shown in Figure 3.4.
We can estimate the size of various forces we expect in this system based on
some simple arguments. The viscous force density is
ηυ
= 10
6
N/m
3
. Here, the
f
vis
≈
D
2
characteristic size
D
~10
-6
m corresponds to the gap between posts. Note that
as
f
≈
f
p
vis
2
ρυ
expected for viscid flow. These are to be compared to the inertial force density
=
f
iner
≈
D
10
3
N/m
3
. That is, the forces involved with moving the fluid around posts are about three
orders of magnitude smaller than the pressure and inertial forces. It should also be noted
that the inertial force density is dominated by the acceleration and de-acceleration of the
fluid elements in the direction of the motion and that the centripetal acceleration, which
points along the radial direction of the circular motion around the post, is not visible in
Figure 3.8. However, while a purely laminar low
R
e
flow in a plane would be expected to
have zero curl in the velocity stream, the full N-S equation through the inertial terms does
in fact allows non-zero curl even at low
R
e
, as was demonstrated in Equation 3.20. Of
course, our COMSOL simulation also yields non-zero vorticity for the flow at those
regions of the flow field where the centripetal acceleration is greatest, as can be seen in
Figure 3.8(C). These vortices will rotate objects that are embedded in the flow, which
means that non-spherical objects can have quite different flow properties in
micro/nanoarrays then simple spherical ones, although calculation of these dynamics given
the complexities of the flow even at low
R
e
can be quite daunting.
We close our discussion with an illustration of the complexities of low
R
e
flow in
complex, asymmetrical structures. One of the great “mysteries” in the bump array work
was hidden in the original publication.
12
Figure 3.9(A) shows the remarkable separation
that was achieved using the ideas outlined in this review. However, Figure 3.9(B) shows a
strange phenomenon: as the flow rate (and hence the Péclet number) decreases, we would
expect the jet profiles of the bumped particles to broaden due to diffusion, and the sharp
edge of the angle versus particle diameter to broaden on both sides of the critical diameter.
But, this is not what happens. Particles with diameters greater than the critical diameter,
D
c
continue to bump, while those with a diameter less than the critical diameter do start to
move at smaller angles. However the width of the streams does not appreciably grow as
you would expect to happen in diffusional broadening. That is, as the Péclet number
decreases the dispersion of the device changes, becoming less binary with decreasing
speed. However the resolving power of the device remains roughly the same. A small
random movement across a stall line will mean that a particle of sufficient size will
undergo a steric displacement to either one side of the streamline or the other that will be
unrecoverable (on average) by diffusion when the displacement is sufficiently large.
Clearly this effect is particle size, time and velocity dependent in complex ways.
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