# Interdisciplinary Applied Mathematics

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FIGURE 7.9. Schematic diagram for AC electroosmosis. The tangential electric field    (Et)    near    the    neighboring    electrodes    interacts    with induced    charges    (qe)

on the electrodes and creates electroosmotic force and velocity in the horizontal direction, shown by    the    qeEt    vector.    Changing    the    electrode    polarity    does    not

alter the flow direction. The figure is adopted from (Morgan and Green, 2003).

were measured 5 p,m away from the electrode edge at 100 Hz, 2 kHz, and 20 kHz, respectively (Green et al., 2000). Numerical modeling of these cases resulted in quantitatively consistent predictions only after the numerical results were corrected by a scaling factor of 0.24 (see Figure 8.9 in Morgan and Green, 2003). Green and coworkers also reported that for increased conductivity of the fluid, the AC electroosmosis can be overwhelmed by the increased dielectrophoretic effects.

Unlike the steady electroosmotic flows, time-periodic or unsteady electroosmosis has been addressed in relatively few publications. For example, numerical results for impulsively started electroosmotic flow have been reported in (Dose and Guiochon, 1993). Analytical solutions of starting electroosmotic flows for a number of geometries, including the flow over a flat plate and two-dimensional microchannel and microtube flows, have been presented in (Soderman and Jonsson, 1996). Effects of a sinusoidally alternating electric field superimposed onto a steady electroosmotic flow have been reported in (Barragan and Bauza, 2000). Time-periodic electroosmotic flows can be combined with steady electroosmotic or pressure-driven flows to induce temporal and spatial flow modifications. This has the po-

tential to produce continuous-flow chaotic mixers (see Chapter 9).

In the rest of this section, we study channel flows driven by time-periodic axial electric fields. Although we are essentially dealing with an AC electric field, we assume that the external electric field obeys the laws of electrostatics. Furthermore, the ion distribution in the EDL region is determined by the zeta potential and the ion density of the buffer solution, which are both constants. Hence, the electrokinetic potential ф is not affected by the oscillatory external electric field. Neglecting the channel entry and exit effects, the flow is fully developed. Hence, we do not expect any streamwise velocity gradients. This enables us to neglect the inertial terms in the Navier-Stokes equations (7.12), resulting in the unsteady Stokes equation. In the absence of externally imposed pressure gradients, the flow is driven purely due to the electrokinetic effects. Following (Dutta and Beskok, 2001b), the momentum equation can be simplified as

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