# Interdisciplinary Applied Mathematics

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Up =-Уф,    (18.3)

i

where Уф is the potential gradient across the fluidic channel and Z is the zeta potential of the fluidic channel. The Poisson-Boltzmann equation, which is    used    for    the    full-scale    simulation    of    electroosmotic    flow,    can be

linearized for low values of surface charge density. Then, the Debye-Hiickel theory predicts the following relationship between the zeta potential, Z, and the surface potential, ф0 :

Z = фо exp(-Kx),

where к is the inverse of the Debye length and x is the radius of the counterion. The surface potential can be computed from equation (18.2) using the capacitance model. Thus, from knowing the surface potential, the zeta potential of the channel wall can be computed. The velocity profile across a capillary slit is a function of only the slip velocity and the pressure gradient, i.e.,

u

1 dp 2p dx

y

2

4

+ up,

(18.4)

where x denotes the stream direction of the channel, y denotes the transverse direction of the channel, and h is the channel depth. Since up is given by equation (18.3), solving for the velocity in equation (18.4) is reduced to computing the pressure distribution in the fluidic network. By taking divergence of the momentum equation and applying the continuity condition, we get the expression

V2p = V- F -V- (pf (u • V)u).

In the regions where the flow is fully developed, the convection term (u-V)u is zero. Thus, V • (pf (u • V)u) vanishes. The term corresponding to the divergence of the force must be zero in the fully developed flow regions; otherwise, the flow would not be fully developed due to the nonuniform body    force.    Hence,    for    the    region    where    the    flow is    fully    developed, the

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