# Interdisciplinary Applied Mathematics

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pressure calculation is reduced to a Laplace equation,

V2p = 0.    (18.5)

Thus, equation (18.5) decouples the solution of pressure from the solution of velocity.

Integrating the velocity profile given in equation (18.4) across the crosssection of the capillary slit and using equations (18.1), (18.3), and (18.5), we get the following expression for the flowrate per unit width:

Q= (^)Ар+(\$)Аф (186)

For the *th channel in an array of channels, equation (18.6) can be rewritten as:

Qi = HiApi + В^Афъ,    (18.7)

where Hi    is    the    hydraulic    conductance of the    *th    channel,    Ei    is    the    elec

trohydraulic conductance of the *th channel, Api is the pressure drop in the *th channel, and Афi is the electrical potential drop in the *th channel. The expressions    for    Hi    and    Ei    for    the    capillary    slit    are    given    in    equa

tion (18.6). For a cylindrical channel, the hydraulic conductance and the electro-hydraulic conductance are given by

Hi

4

8 Pi Li

and

Ei

piLi

where ri is the inner radius of the *th cylindrical channel. Equation (18.7) is the constitutive relationship, which relates the “through quantity” to the “across quantities” (a combined pressure and electrical potential drop). If the flow is driven by only a pressure gradient, then the second term in equation (18.6) can be neglected. Similarly, if the flow is driven by only an electric field, then the first term on the right-hand side of equation (18.6) can be neglected. Figure 18.4 shows the circuit representations of the fluidic domain for the cross-shaped channel segment shown in Figure 18.3. It is to be noted that the total flow is the sum of the electrokinetically driven flow and the pressure-driven flow.

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