Micromixers based on chaotic advection - Micromixers Fundamentals Design and Fabrication - page 232

FIGURE 6.39
Flow pattern in a microdroplet: (a) flow patterns inside the plugs in a microchannel with repeated turns and
(b) possible mixing pattern inside the plugs.

a microchannel with repeated turns. The repeated turns induce periodic boundary conditions for the
plugs. These periodic boundary conditions, in turn, induce chaotic advection. When a liquid plug moves
in a microchannel, the two counter-rotating vortices stretch and fold the mixing liquids inside the plug.
At each turn, the vortices become asymmetric. One vortex is large and slow, while the other is small and
fast. As a result, the mixing liquids experience a rotation at each turn. The time-periodic folding,
stretching, and rotation lead to chaotic advection inside the plug and consequently faster mixing.
Considering a plug of a length L , a width W , and an initial striation thickness of s (0) z W , the

striation thickness after n steps of repeated stretching, folding, and rotation is [45] :
sðnÞ¼Wl n (6.42)
where l is the Lyapunov exponent of chaotic advection inside the plug. The exponential decrease
of the striation thickness is a sign of a chaotic process. The diffusion time scale across the striation
layer is:

2
2 D ¼

l 2 n
2 D

W 2

sðnÞ

t diff ¼ (6.43)

where D is the molecular diffusion coefficient. Assuming that the plug traveled a distance of nL after
the n steps, the residence time can be estimated as:

nL
u

t res z (6.44)

where u is the velocity of the liquid plug. For complete mixing, the diffusion time should be
approximately the same as the residence time:

l 2 n
2 D z

W 2

nL
u :

(6.45)

Rearranging the above equation leads to:

2 nL l

2 n

zPe (6.46)