Interdisciplinary Applied Mathematics

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1 dP *


и(л) = ~11-^^-г12) + 1-Ф*(л1    (7-27)


where corresponds to the pressure gradient in the mixed electroos-motic/pressure-driven flow regime. Substituting the solution for ф* from equation (7.9), we obtain an analytical formula for the velocity distribution. In Figure 7.7 velocity profiles under various pressure gradients are shown. The case for = 0 corresponds to a pure pluglike flow, and the cases


< 0 and > 0 correspond to flow with favorable and adverse pressure gradients, respectively.


In order to obtain the mass flowrate, we integrate the velocity and the electroosmotic potential distribution across the channel (see equation


(7.27)). This can be cumbersome in the ^-coordinate system, where ф* is a    function    of    both    a and в,    but    in    the    у-coordinate    system,    ф*    is a


function only of a.    In    (Dutta    and    Beskok,    2001a),    the    electric    double    layer


displacement thickness defined S* was in analogy with the boundary layer displacement thickness in fluid mechanics in the following form:


S* = Г ф* dy,    (7.28)


J 0


where у is a large enough distance to include variations in ф* as observed from Figure 7.3. For example, у « 10 is sufficient to accurately define S*. Typical values of S* as a function of a are presented in Table 7.2.


• The physical meaning of S* is that it expresses the volumetric flowrate defect due to the velocity distribution within the EDL.


Integration of the ф* term in equation (7.27) is performed using equation


(7.28), where


1



0



2



ae J 0



rx    2S*

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