# Interdisciplinary Applied Mathematics

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concentrations the two EDLs overlap, and while the two zeta potentials Zc and Zb increase, Zc becomes greater than |Zb|. This change in the relative strength of the potentials at the liquid-air interface and at the capillary wall reverses the aforementioned flow reduction effect at the film gap, and

FIGURE 8.16.    Time    history    of    the bubble    velocity    as    a    function    of    the bubble

length. C0 = 10~4 mol/l; V = 42 volts. (Courtesy of H.-C. Chang.)

the bubble moves much slower and eventually stops.

In summary, in transporting air bubbles in micropipes using electroosmosis, asymmetric electric double layers are desirable in order to produce a flow reversal in the annular region around the bubble. This, in turn, will cause a pressure buildup at the back of the bubble with an eventual forward ejection of the bubble. This occurs in a certain range of ionic concentration, which causes significant enhancement of the conductivity of the film. The resulting bubble velocities are of order 1 mm/s, but only for short bubbles; longer bubbles (length to diameter around 5) may travel much slower, at about 1 p,m/s. These results, however, are not valid for noncircular microducts, since corners create complications such as very thick films, which basically are responsible for much of the voltage drop in electrokineti-cally driven flows. Therefore, for microchannels with general cross-sectional shapes, pressure-driven flow may be more efficient in transporting bubbles.

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### Mixers and Chaotic Advection

Mixing is an old subject in fluid mechanics, and it is typically associated with high-speed flows, since it occurs naturally due to turbulent diffusion. For example, in channels with height above 1 cm, water flow is typically turbulent for speeds of 10 cm/s or higher, and mixing occurs rapidly due to turbulent fluctuations. In microchannels, the height is more than an order of magnitude smaller, and thus the flow is laminar. Therefore, transport is    controlled solely    by the    diffusion    coefficient    D    of    the    medium.    In

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