Biomedical Engineering Reference
In-Depth Information
7.2.3
Pulsed source-sink chaotic advection
Passive mixers based on chaotic advection were described in the previous chapter. Hydrodynamic
instability is also a form of chaotic advection caused by a periodic disturbance. This section dedicates
to a special concept for active generation of chaotic advection: the pulsed source-sink concept.
The original concept proposed by Jones and Aref
[7]
consider a single point source and a single
point sink on an unbounded plane. This system is two-dimensional and can be used for flat microfluidic
systems, which are modelled by the Hele-Shaw flow. The point source and sink represent singularities
that are switched alternately with a fixed time period. Both source and sink have the same magnitude of
flow rate. Thus, the net flow in the system is zero. Using Poincar
´
section and Lyapunov exponents,
Jones and Aref showed that this system creates chaotic advection over a wide range of operation
parameters.
A two-dimensional model can be formulated for a source-sink pair. In a Cartesian coordinate
system, a source and a sink are placed at (
x
þ
,
y
þ
) and (
x
-
,
y
-
), respectively. The strengths of the source
and the sink are
Q
and -
Q
, respectively. For the convenience of mathematical treatment, the coor-
dinates
x
and
y
describing a point on this plane can be replaced by a complex number
z ¼ x þ iy
, where
i
is the complex unit. Considering the positive half of the imaginary axis and the complex position
z
þ
of the source and the complex position
z
-
of the sink, the stream function of the flow system is the
imaginary part of the following complex potential:
h
log
i
h
log
i
Q
2
p
z
*
z
*
W
ð
z
Þ¼
ð
z
z
þ
Þþ
log
ð
z
þ
Þ
ð
z
z
Þ
log
ð
z
Þ
;
(7.22)
where the asterisk denotes the complex conjugate. The real part
f
of the above function is the velocity
potential. The streamlines are the level sets of the imaginary part
j
. Velocity components can be
subsequently evaluated using the advection equation (2.86). Integrating the velocity components over
time results in positions of the fluid particles. Plotting the particle position at regular time intervals
results in Poincar
´
section, which is used for evaluating the degree of chaotic mixing.
The stream function of the system with
z
þ
¼
(-
a
, 0) and
z
-
¼
(-
a
, 0) is the imaginary part of the
following complex potential:
Q
2
p
½
W
ð
z
Þ¼
f
þ
ij
¼
ð
z
þ
a
Þ
ð
z
a
Þ:
log
log
(7.23)
Figure 7.5
shows this source-sink pair and their stream lines. Under a steady condition, the fluid particles
move along these streamlines, which have the forms of circular arcs. This flow system is determined by
two parameters: the distance 2
a
between the source and the sink, and their strength
Q.
The speed of
evolution is controlled by
Q.
Normalizing the time by 2
a
2
/
Q
and integrating the velocity from the
source to the sink along the line (0,
y
) results in the shortest dimensionless transfer time
t
0
¼
p
2/3.
The mixing domain in real applications is bounded inside a mixing chamber. For a circular mixing
chamber with a radius
R
and a center positioned between the source and the sink, the complex potential
transforms to the form
[8]
:
2
p
log
Þ
:
Q
R
2
R
2
W
ð
z
Þ¼
ð
z
þ
a
Þ
log
ð
z
a
Þþ
log
ð
þ
az
Þ
log
ð
az
(7.24)
Streamlines of this system are depicted in
Fig. 7.6
(a). The shortest dimensionless response time is
t
0
z
0.556.
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