# Interdisciplinary Applied Mathematics

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(i.e., Ф*с = ^=o).

(Burgreen and Nakache, 1964) obtained an analytical solution of (7.7) in terms of an elliptic integral of the first kind. Their work presents the potential distribution as a function of the Debye length Ad and the ionic energy parameter    a. It    was    shown    by    (Dutta and    Beskok,    2001a)    that    for

a > 1 and Ad ^ h the electric potential in the middle of the channel is practically zero. Hence, as ФС ^ 0 the last term in equation (7.8) is simplified, and using the identity cosh(p) = 2 sinh2 (p/2) + 1, equation (7.8) can be integrated once more, resulting in the solution

tanh ( — J exp ( — /~af3 rfj

(7.9)

4

ф*(г]*) = — tanh a

where nc is the distance from the wall (i.e., nc = 1 — Ini).

In Figure 7.2, a numerical solution of the electroosmotic potential distribution as a function of various a and в values is presented. The left and right figures show the potential distributions for a =1 and a = 10, respectively, for various values of в. For a =1 and в < 100 the EDL is quite thick, and it covers the entire channel. As the value of в is increased, the electric double layer is confined to a zone near the channel walls, resulting in sharp variations in the electric potential. Comparisons of a =1 and a =10 cases at the same value of в show faster decay of the electroosmotic potential for increased values of a.

###### 7.2.1 Near-Wall Potential Distribution

The potential distribution in equation (7.9) can also be represented as a function of    the    near-wall    parameter    у =    yu>,    where    y    = h — y is    the

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