# Interdisciplinary Applied Mathematics

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crossover frequency from attractive to repulsive induced forces is a few MHz, with high frequencies generating negative forces. Therefore, particles are attracted toward the electrodes at low frequencies and are pushed away at higher frequencies. This is the principle that is exploited in the DEP micromixer fabricated in (Deval et al., 2002). A top view of the micromixer test section is shown in Figure 9.9. The chamber dimensions are 200 x 200 x 25 p,m. The electric field is created by a 1-15 MHz, 10 V AC voltage applied between selected pairs of micromachined electrodes located on the walls of the two cavities. Visualizations of the particle motion revealed that as the particles enter the first cavity, positive DEP forcing attracts them into the low-velocity region, while frequency switching repels them back to the main flow. The combined competing motion generates folding and stretching, thereby producing enhanced mixing. Typical results are shown in Figure 9.10, with the background velocity being approximately 420 p,m/s.

##### 9.4 Quantitative Characterization of Mixing

In many experimental and numerical studies on micromixers mixing is characterized only qualitatively by snapshots of a passive tracer or two differently colored fluids. Clearly, crossing of streamlines is an indication of chaotic mixing, but how exactly do we quantify the degree of mixing? The Lyapunov exponent (LE) is a possible accurate measure, since it is related to the stretching rate. It is defined by the equation

1 d(t)    . .

#### л~ = ЙМ1п;щ-    <95>

where d(t) is the distance between two points that are initially very close to each other. An n-dimensional system has at most n LEs, and it is characterized as chaotic if at least one of the LEs is positive. One of the problems in applying the above definition in microfluidic systems is that we have only a finite length and corresponding mixing time. In addition, it is quite expensive to compute the LE from this definition, although it has been applied with success in quantifying mixing in an active micromixer, similar to the SSM presented earlier, in (Lee, 2002).

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