Active micromixers - Micromixers Fundamentals Design and Fabrication - page 250

A similar approach can be used to drive the complex potential with source and sink at (- a ,- a ) and
( a ,- a ) as:

2 p log

log R 2 z log R 2 z :

Q

W ð z Þ¼ ð z þ a þ ia Þ log ð z a þ ia Þþ þ a ð 1 i Þ a ð 1 þ i Þ

(7.25)

Streamlines of this system are depicted in Fig. 7.6 (b). The shortest dimensionless response time is
t 0 z 0.51.

Another common geometry of a mixing chamber is the rectangle. Considering a rectangle with
width of 2 a and a length of 2 aa , where a is the aspect ratio of the rectangle. The mixing domain is
bounded by the four corners (

a , aa ). The complex potential of a source-sink pair inside this

rectangle is [9] :

2 p log

log s

z þ Þ

log s

z Þ :
(7.26)

Q

½ s ð z Þ s ð z þ Þ þ ð z Þ s * ð ½ s ð z Þ s ð z Þ ð z Þ s * ð

W ð z Þ¼ log

where

sn h z

ia K k i s ð z Þ¼

a þ

ð k Þ;

(7.27)

and sn( z , k ) is the Jacobian elliptic sine, K ( k ) is the complete elliptic integral of the first kind. The
variable k is determined from the aspect ratio by the following relation [9] :

K

p

k 2 1

a ¼ : (7.28)

2 K ð k Þ

Mixing protocols based on pulsed source-sink chaotic advection are determined by the way the
fluid is injected and rejected from the source and the sink, respectively. The fluid can be collected from
the sink and transferred to the source according to “first in, first out” or “first in, last out” protocols.
Figure 7.7 illustrates the Poincar´ section of two pairs of pulsed source-sink. The source-sink pairs are
centered at the origin of circular boundary. At a short pumping time of t * ¼ 0.15 the Poincar´ section
indicates the presence of several non-mixing islands, where fluid particles never pass. Best mixing
appears to be at t * ¼ 0.5. This concept can be implemented using external or integrated pumps at the

sources and sinks.
Figure 7.8 shows the Poincar´ sections of two source-sink pairs in a rectangular mixing chamber
with an aspect ratio of 21/71 [9] . The parameter for good mixing is the switching time t switch
between the source-sink pairs. At a fixed source-sink strength of Q , the switching time is
normalized by the maximum pumping time T c needed for emptying the mixing chamber. The
influence of this dimensionless switching time t switch ¼ t switch / T c on the Poincar´ section is illustrated

in Fig. 7.8 . The Poincar´ sections are created by plotting the position of a liquid particle after every
pulsing period or 2 t switch . The section shows areas not passed by the particle, also called elliptic
island. Particles in these areas will experience a periodic regular transport, and consequently cannot
be mixed.