Interdisciplinary Applied Mathematics

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FIGURE 7.22. Numerical simulation results of electrophoretic transport in a microchannel under electroosmotic flow. Figures (a-c) show concentration contours at various times, while (d) shows the sample distribution along the channel center, and (e) shows the streamwise velocity distribution for “pure electroosmotic flow.” The velocity is normalized with the Helmholtz-Smoluchowski velocity (7.18), and simulations are performed for Re = 0.03, Pe = 500, and X/h = 0.01 conditions.



for details):



dn



dn



+ u— = D,



dt



dx



d2n


‘ dx2



(7.45)


where n is the cross-section-averaged species concentration, U is the channel-averaged velocity, and De is the effective diffusion coefficient. For pressure-driven cylindrical capillary flows, the effective diffusion coefficient becomes


D



e





where Pe = Ua/D is the Peclet number based on the capillary radius (a), and D is the species-molecular diffusion coefficient. This relation is valid specifically for Pe ^ 7. For moderate Pe values the streamwise diffusion cannot be neglected to obtain the “Taylor-Aris dispersion coefficient”






We must emphasize that the aforementioned results are valid for pressure-driven capillary flows. For the pure electroosmotic flow shown in Figure 7.22, the effective diffusion coefficient can be written as


De = D(1 + a0 Pe2),


where ao depends on the channel geometry, ratio of the EDL thickness to the channel hydraulic diameter, and the channel surface zeta potential. For pure electroosmotic flow in a tube or a two-dimensional channel, the value of ao is approximately given by (Griffiths and Nilson, 1999)

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