Interdisciplinary Applied Mathematics

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where в = Ш is the nondimensional time, x = v/Ad is the nondimensional distance, and U = u/uhs is the nondimensional velocity. Rearranging the dimensional components in equation (7.36) results in the following equation:



dU


~дв



1


re2



d2U


дх2 +



sin(e)    «


-sinnfa^ ) ,


а



(7.37)


where к = /Щ7 v is a normalized parameter that is a function of the Debye length (Ad), the kinematic viscosity, and the electric field excitation frequency.


• The parameter к can be interpreted as the ratio of the Debye length Ad    to a    diffusion    length    scale    (Id),    based    on the    kinematic    viscosity


and the excitation frequency.


This diffusion length scale can be estimated from the unsteady Stokes equations using dimensional analysis as Id « Jn/Q. Hence, the diffusion length scale is related to the Debye length and к in the following form:


Id ~ Ad/к.


Time-periodic solution of equation (7.37) is obtained using the separation of variables technique (Dutta and Beskok, 2001b). Here we summarize the basic results. Time-periodic velocity distributions across the channel for three different к values are shown in Figure 7.10 at time в = п/2. These results are obtained for a channel half-height h = 100Ad. Since the electroosmotic forces are confined within the EDL, the dynamics of fluid motion in the bulk of the channel are mostly determined by an unsteady diffusion process. Therefore, the magnitude of the channel half-height (h) relative to the diffusion length scale (lD) plays an important role in determining the dynamics of the bulk flow region. For the к = 0.001 case, the diffusion length scale (lD = 1000Ad) is an order of magnitude larger than the channel half-height (100Ad). This results in a quasi-steady velocity distribution that resembles the “plug velocity” obtained in steady electroosmotic flows, as shown in Figure 7.10. On the other hand, for к = 0.01 the channel half-height and the diffusion length scales are of the same order of magnitude, and the bulk flow velocity deviates from the plug profile. For к = 0.1 the velocity becomes practically zero in the channel center at any time, as shown in Figure 7.11.

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