Interdisciplinary Applied Mathematics

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A099 fffj w Mhs’


where corresponds to the bulk flow gradient obtained on the wall. The appropriate matching distance is taken to be the effective EDL thickness (699Ad). Here, the first term in (7.40) corresponds to a Taylor series expansion of bulk flow velocity at the edge of the EDL from the wall. Equation (7.40) is analogous to slip velocity in rarefied gas flows given in Section 2.3. It is noteworthy to mention that for finite Debye layers with large bulk flow gradients, the velocity matching using equation (7.40) will give considerable deviations from the Helmholtz-Smoluchowski prediction.

FIGURE 7.14. A magnified view of the velocity distribution in a mixed electroos-motic/pressure-driven flow near a wall for a = 1, в = 10,000. Extrapolation of the velocity using a parabolic velocity profile with constant slip value Uhs on the wall are shown by the solid lines. The analytical solution is shown by the dashed lines.

7-4-4 Slip Condition

The electroosmotic forces are concentrated within the EDL, which has an effective thickness on the order of 1 nm to 100 nm. On the other hand, the microchannels utilized for many laboratory-on-a-chip applications have a typical height of 100 pm to 1 pm This two to five orders of magnitude difference in the EDL and the channel length scales is a great challenge in numerical simulation of electroosmotically driven microflows. Therefore, it is desired to develop a unified slip condition, which incorporates the EDL effects by specifying an appropriate velocity slip condition on the wall. Examining Figures 7.13 and 7.14, and equation (7.40), it is seen that the bulk    velocity    field    extended onto    the    wall    has a    constant    slip    value

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