Interdisciplinary Applied Mathematics

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Ceff ,гФо,г = qst,i = (^T,iAs,i,


or



^0,i



&T,iAs,i


Ceff. ’



(18.2)


where ^0ji is the surface potential on the ith channel and qat}i is the total charge stored in the EDL of the ith channel.

18.1.2 Fluidic Model


For the fluidic transport driven by an electrical field and/or a pressure gradient, the “through quantities” are the flow rates through the channels, while the “across quantities” are the electrical potential differences and the pressure differences imposed on the fluidic channels. In this section we present a derivation of the constitutive equation relating the “through quantities” to the “across quantities” making use of the continuity equation and the steady-state momentum equation for electroosmotic flows (see Chapter 7 for details).


Slip Case


The slip case model can be used when the thickness of the EDL is insignificant compared to the depth or diameter of the channel. The body force, F = PeE (see Chapter 7), is nonzero only within a few Debye lengths from the channel wall, since the potential induced by the zeta potential drops to zero very quickly near the channel wall (Mitchell et al., 2000). In the development of the compact model for the slip flow case, we will assume that the flow is fully developed and the thickness of the EDL is insignificant compared to the thickness or diameter of the channel (this assumption usually holds good for channels larger than 200 nm; see Chapter 7 for details). As a result, the effect of the electrokinetic force can be represented by a slip velocity at the wall given by the Helmholtz-Smoluchowski equation (see section 7.3)

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