Interdisciplinary Applied Mathematics

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where Fx is the microscopic friction exerted by the wall on the fluid, and S is the interface area. The position of the fluid-wall interface is given by


8    f+


z° =SkBT/x J0    dt,    (12.11)


with axz the xz component of the microscopic stress tensor.


By introducing the above slip boundary condition, we get the slip solution of the Navier-Stokes equations. The electroosmotic profile is given by equation (12.8) with


a=——— ( f ( Fe(z) dz dz — 8 ( Fe(z)dz. (12.12)


MEext у J-W/2J 0    Jo    J


Note that the slip boundary condition shifts the electroosmotic profiles by adding a constant a that depends on 8 and z0. If z0 is close to the channel walls, it does not influence a significantly. Assuming that z0 « -W/2, which is a reasonable approximation, we get


8    rWj 2


a=———    Fe(z)dz.    (12.13)


Eext J0


The expression in (12.13) implies that the slip boundary condition increases the electroosmotic flow. Using the expression for the force Fe (and an analytical solution for the concentration obtained from the Poisson-Boltzmann equation, c(z) = 2ъ COs“(az)) in equation (12.13), we get


8


CL —    (7 s .


V


The calculation of the slip parameters from the Kubo formula (12.10) and (12.11) cannot be performed accurately from MD simulations. For the sake of simplicity, (Marry et al., 2003) have computed the slip parameters from the electroosmotic profile itself. Since the hydrodynamic limit holds only for large channel widths, in this limit, we can assume that 8 and z0 do not vary a lot and can be taken as constant. Furthermore, we can also assume that z0 is    equal    to    -W/2.    For    the    largest    channel    width    shown in Figure    12.9,

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