# Interdisciplinary Applied Mathematics

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equivalent to whs. Hence,

• the appropriate slip condition at the wall is the Helmholtz-Smoluchowski velocity whs; even for finite EDL thickness conditions.

For a general numerical algorithm, implementation of slip velocity whs at the walls overpredicts the volumetric flowrate, since the velocity distribution within the EDL is neglected. This flowrate error can be corrected by subtracting 26*/Соф (in nondimensional form) using the EDL displacement thickness S* given in Table 7.2. For engineering applications with a = 1 and в = 10,000, corresponding to a 0.1 mM buffer solution in a 6 pm glass channel, the error in the conservation of mass equation due to

this slip condition is about 4.5%.

With regard to the errors in the momentum equation, neglecting shear stresses due to the velocity distribution within the EDL, given by equation (7.31),    will    be in gross error.    In    the    next    section    we    present    a method

that can be used to predict the extra drag force due to the EDL, in the postprocessing stage of computations.

###### 7.4.5 A Model for Wall Drag Force

For steady incompressible flows, conservation of linear momentum in integral form is given by

J vpv n dr = Fsur + Fbody,    (7.41)

where Fsur and Fbody are the net surface and body forces, respectively. The control surface is shown by Г, and the outward unit normal is denoted by n. D epending on the device geomet ry and the cont rol volume, one must properly apply equation (7.41), which also includes the electrokinetic body force terms. In this section, we propose a method to calculate the additional drag force due to the electrokinetic effects at the postprocessing stage. Hence, we first solve the flow system using the Helmholtz-Smoluchowski slip velocity whs, and then calculate the total drag force by superposition of the calculated hydrodynamic drag with the cumulative effects of the electroosmotic body forces concentrated on domain boundaries.

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