Interdisciplinary Applied Mathematics

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FIGURE 12.2. A schematic of the channel system under investigation. The two channel walls are symmetrical with respect to the channel center line. Each wall is made    up    of four    layers    of silicon    atoms. The channel    width    W    is    defined    as


the distance between the two innermost wall layers. The dark dots denote water molecules, and    the    shaded    circles    denote    either    Cl_    or    Na+    ions.    For the co


ordinate system chosen, г = 0 corresponds to the central plane of the channel system.


1. Only counterions are present in the channel, and


2.    The flow is fully developed, and there is no externally applied pressure gradient.


Under these assumptions, the mathematical model presented in Chapter 7 can be simplified as


^ = -1~спе-^^квТ    (12.1a)


dz2    €


|(м^)+,гф)£“=0‘    (12ЛЬ)


where ^(z) is the potential induced by the charges on the channel wall and the ions, q is the electron charge (i.e., 1.6 x 10~19 C), z is the valency of the counterion, со is the counterion concentration at the channel center, where the potential is assumed to be zero (note that is true when the double layers do not overlap), is the permittivity of the fluid in the channel, kB is the Boltzmann constant, T is the temperature, u(z) is the velocity of the fluid, p is the dynamic viscosity of the fluid, c(z) is the counterion concentration across the channel, and Eext is the external electric field applied along the channel.

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