# Interdisciplinary Applied Mathematics

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(7.23)

dp d2u    сРф

dx    ^ dy2    1 dry1

This equation is linear, and thus we can decompose the velocity field into two parts:

U = Upois + upo,

where uPois corresponds to the pressure-driven channel flow velocity (i.e., plane Poiseuille flow), and uEO is the electroosmotic flow velocity. In the absence of externally imposed pressure gradients, uPois = 0. Hence, the viscous diffusion terms are balanced by the electroosmotic forces. This leads to the Helmholtz-Smoluchowski electroosmotic velocity uHS (Probstein, 1994):

uhs

(7.24)

From nondimensionalizing equation (7.23), the streamwise momentum equation becomes

(7.25)

where, U = —, P* = —Ц-, and £ = x/h. Here the pressure is norrnal-

’    MHS’    pMHS/h

ized by viscous forces rather than the dynamic head, consistent with the Stokes flow formulation (see Section 2.1).

In the case of zero net pressure gradient we integrate equation (7.25) to obtain (Burgreen and Nakache, 1964)

U(n) = 1 — ф*(п).    (7.26)

Figure 7.2    (right    vertical    axis)    shows    the    velocity    variation    in    pure    elec

troosmotic flows for various values of a and в. As shown in the plot, in the limit of small Debye lengths the electroosmotic potential ф* decays very fast within the thin electric double layer, and a uniform pluglike velocity profile is obtained in most of the channel. The plug flow behavior has been observed in various experiments (Paul et al., 1998; Herr et al., 2000).

For the mixed electroosmotic/pressure-driven flows, the superposition principle for linear equations is used to obtain the following nondimensional velocity profile (Dutta and Beskok, 2001a):

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