Interdisciplinary Applied Mathematics

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For the к = 0.01 case, there is vorticity in the entire bulk flow region (see Figure 7.10). From the к = 0.01 and к = 0.1 cases, we see that the time-periodic electroosmotic flows are rotational when the diffusion length scale is comparable to or less than the channel half height (lD < h). The vorticity is put into the problem on the walls, and its magnitude alternates in time due to the time fluctuations of the external electric field. Unlike the steady electroosmotic flows, vorticity diffuses deeper into the channel, while its

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FIGURE 7.10. The velocity distribution of time-periodic electroosmotic flow for various values of к at time в = n/2. Here, the ionic energy parameter a equals 5.


value on the wall is cyclically changing. The high-frequency excitation case (к = 0.1) presented in Figure 7.11 shows exponentially damped vorticity waves traveling into the bulk flow domain, penetrating the channels as much as 80 Debye lengths. A discussion on vorticity creation mechanisms for unsteady electroosmotic flows is given in (Santiago, 2001).

The velocity profiles presented in Figure 7.11 resemble the classical solution of a flat plate oscillating in a semi-infinite flow domain, also known as Stokes’s second problem. In Figure 7.12, time-periodic electroosmotic velocity profiles are compared with the solution of Stokes’s second problem. Unlike the time-periodic electroosmotic flow, the fluid is driven here by an oscillating plate with velocity uw = ugs sin(tQ), where Q is the frequency of the plate oscillations and uHS is the amplitude of the plate velocity. The governing equation for Stokes’s second problem becomes

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