# Interdisciplinary Applied Mathematics

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A passive chaotic mixer for microchannels that exploits the aforementioned “complete mixing” concept and verifies the logarithmic Peclet number dependence was developed by (Stroock et al., 2002). To generate transverse secondary flows, microriblets similar to rifling in a gun barrel were

FIGURE 9.2. Schematic of a serpentine channel used in the experiments of (Liu et al., 2000).

FIGURE 9.3. A: Schematic of microchannel with riblets. B: Optical micrograph showing    the    two    streams flowing    on    either    side    of    a clear    stream.    C: Fluores

cent confocal micrographs at three different cross-sections showing the rotation and distortion    of    a    stream    of fluorescent    solution    that    was    injected    upstream.

h = 70 pm; w = 200 jim, a = 0.2,q = 2n/200 jim-1 = 45°. (Courtesy of H.A. Stone.)

placed on the floor of the channel at an oblique angle в with respect to the long axis y of the channel, as shown in Figure 9.3. They were fabricated using two steps of photolithography. These microriblets, whose height is typically less than 30% of the channel height, present an anisotropic resistance to flow, with less resistance along the main flow direction than in the orthogonal direction. A transverse component of the flow (along the ж-direction in the figure) is then produced that is initiated at the riblet surface, with the flow circulating back across the top of the channel. Optical micrographs    used in    the    experiments    of    (Stroock    et    al.,    2002)    show

that the flow has helical trajectories, as shown in Figure 9.3(A).

A somewhat different design was also tested in the experiments of (Stroock

et al., 2002); it uses V-grooves to make a structured surface, instead of straight riblets, as shown in Figure 9.4. The objective here is to subject the fluid to a repeated sequence of rotations and extensions in order to realize stronger chaotic transport. This is effectively analogous to the baker’s transformation, i.e., a repetitive action involving stretching, cutting, and fusion that can, in principle, achieve the best achievable mixing (Ottino, 1997). In this design the shape of the grooves is changing along the flow in each half-cycle, as shown in Figure 9.4. The efficiency of mixing is primarily controlled by two parameters: the measure of asymmetry and the amplitude of rotation of the fluid in each cycle. For symmetric microgrooves or zero amplitude of rotation the flow becomes deterministic, but optimum values for both parameters have been obtained experimentally. The degree of mixing was quantified using the standard deviation of the intensity distribution in confocal images of the cross-section of the flow. The results suggest that the V-grooves achieve mixing with zero standard deviation only 1 cm downstream, while the microchannel with the straight riblets, although effective, does not quite achieve complete mixing. These results were obtained for a Peclet number range from 2 x 103 to 9 x 105, and they confirmed the theoretical logarithmic dependence of the mixing length on the Peclet number.

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