# Interdisciplinary Applied Mathematics

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FIGURE 7.13. Near-wall velocity distribution for time-periodic electroosmotic flow and Stokes’s second problem for к = 0.1 at various times. See the legend of Figure 7.12 for details.

The flow dynamics are determined by a normalized parameter к, which can be    interpreted    as    the    ratio    of    the    EDL    thickness    AD    to

a characteristic    diffusion    length    scale    lD.    Based on    the    values    of    к

and the half channel height, various flow conditions, ranging from the oscillatory plug flow to flows resembling the flat plate oscillating in a semi-infinite domain with velocity uHS sin (Ш) (Stokes’s second problem) are observed.

For large values of к there are similarities between Stokes’s second solution and the analytical solution in the bulk flow region. This leads to identification of the instantaneous Helmholtz-Smoluchowski velocity as the appropriate electroosmotic slip condition for time-periodic flows.

7-4-3 EDL/Bulk Flow Interface Velocity Matching Condition

In this section, we present the velocity matching condition between the EDL and the bulk flow regions. This is important in assessing the interaction of high-vorticity fluid    in    the    EDL    with    the    vorticity    of    the    bulk    flow region.

It seems that the often-used Helmholtz-Smoluchowski velocity (7.24) as the “matching condition”    at    one    Debye    length    (Ad)    away from    the    wall    is

inadequate for the following two reasons:

First, such a matching condition should be implemented at the effective EDL thickness (699Ad), which is considerably larger than the Debye length.

The second limitation arises due to the variation of the bulk velocity across the EDL.

If we    examine    the    velocity    distribution    at    the    edge of    the    EDL in    Fig

ures 7.13 and 7.14, it is clear that the matching velocity changes with the velocity gradient of the bulk flow region. Hence, the appropriate velocity matching condition (umatch)    at    the    edge    of    the    EDL    (y    —    699 Ad )    should

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