# Interdisciplinary Applied Mathematics

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Since the above equations are steady, the independent variable can be changed from time t to angle в to obtain

dx

dy

d6

c

1152 c

[4 — 5x2

xy [З — .

192 y L

2

23y2 + x4 + 8x2y2 + 7y4] ,

y2],

(9.3)

(9.4)

where c = a/в. These equations define the mapping of fluid particles from one cross-section to another (Jones et al., 1989). They produce chaotic solutions for the particle paths for certain ranges of the parameters c and X. For example, for c = 100 the most chaotic pattern is obtained if x = 90°, but as x ^ 180° a regular pattern is achieved. In (Jones et al., 1989), it was shown that transverse chaotic transport occurs when

X . , nc

tan — < smh-,

2 “    192’

which provides a criterion for the design of the twisted pipe or other threedimensional channels in order to realize, at least partially, the effects of chaotic advection on longitudinal dispersion. Finally, we note that the stochastic patterns of the particle paths are similar to the instantaneous ones seen in a corresponding turbulent flow, but with the important difference that in the chaotic advection these paths are stationary in time.

##### 9.3 Micromixers

The first implementation of the twisted pipe concept in microfluidics was presented in (Liu et al., 2000). A three-dimensional serpentine microchannel was designed as    shown    in    the    sketch    of    Figure    9.2.    The basic    building

block is a “C-shaped” section; the planes of two successive such sections are perpendicular to each other. The channel height is 300 pm, and its total length 7.5 mm. The microchannel was fabricated in a silicon wafer using a double-sided KOH wet-etching technique. The mixing streams were phe-nolphthalein and sodium hydroxide solutions, which typically mix fast compared to other large-molecule solutions. In the experiments of (Liu et al., 2000), after ten “C-shaped” sections at least 98% of the maximum intensity of reacted phenolphthalein was observed. This mixing was superior to that in a straight channel (a factor of 16 better) and about 60% better than in a similar square-wave channel that has all “C-shaped” sections on the same plane. However, full chaotic advection was established only at a relatively high Reynolds number, i.e., Re = 70; this value is higher than in typical microfluidic applications.

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