Interdisciplinary Applied Mathematics

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(7.38)


du    d 2u


Ps~dt=P~d^1


and the system is subjected to the following boundary conditions:


u(y = 0,t) = uhs sin(tQ),


u(y ^ <x>, t) = 0.


These equations yield the following similarity solution (Panton, 1984):


U (x, 0) = exp



кх



sin



(7.39)


where U is the velocity normalized with uHS, similar to our solution of time-periodic electroosmotic flow. The solution for both equations is practically the same for x > S99 (See Figure 7.12). Therefore, the bulk flow dynamics are adequately described by the Stokes solution for a flat plate oscillating with frequency Q and amplitude uHS. However, the velocity distribution within the    effective    electric    double    layer    (x <    S99)    differs from    the    Stokes


solution significantly, since the velocity on the wall needs to obey the noslip condition    at    all    times.    A    zoomed    view    of the    velocity distribution    of


Figure 7.12 is presented in Figure 7.13.


We note that the Stokes solution in the bulk flow and the analytical solution outside the effective EDL thickness match without any phase lag. This enables us to conclude that the Stokes solution with a prescribed wall velocity of uHS sin(tQ) can be used to describe the bulk flow region for sufficiently large к values, showing that


• the instantaneous Helmholtz-Smoluckowski velocity is the appropriate wall slip condition for time-periodic electroosmotic flows.


We must note that this claim is also valid for low-frequency (small к) flows, as can be deduced from Figure 7.10 by extending the velocity profiles from the bulk flow region on to the wall.


In summary:

FIGURE 7.12. Comparisons between the time-periodic electroosmotic velocity (TE) and Stokes’s second problem (SS) for к = 0.1 at various instances in time. The results are obtained for ionic energy parameter a = 5.0 and times n/4 and 3n/4.

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