Interdisciplinary Applied Mathematics

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12.4 Slip Condition


Marry and collaborators have presented an approach to modeling nanometer-scale electroosmotic flows with slip conditions on the channel walls (Marry et al., 2003). Starting from the Navier-Stokes equation given in Section 12.1, the velocity in the x-direction is given by


ff Fe(z) dz dz


l^Eext


where Fe    is the    force acting    on    the solution    due    to    the    external    electrical


field (Eext),    given    by    Fe    =    c(z)qEext.    If    the    origin    of    the    z-axis    is chosen


to be the middle of the channel (along the width direction), the symmetry of the system results in the relation


u(z) f-w/2 Jo Fe(z) dz dz


Eext    dEext


In the above equation, W is the channel width. The value of a depends on the boundary conditions. In the classical Smoluchowski treatment of electroosmosis (see Chapter 7), the velocity is zero at the surface and a = 0. Furthermore, if we consider the Poisson-Boltzmann equation to compute the concentration profiles, then we get


u(z)    q    cos(az)


Eex t    2ty/zLb    cos(alF/2)’


where LB = q2/4ne0erkBT, a is given by a tan(aW/2) = 2nLB&s/q, and as is the opposite of the surface charge density. As discussed in the previous section, the above expression can be in large error from MD simulations.


A slip boundary condition was presented by (Marry et al., 2003) starting from the work of (Bocquet and Barrat, 1993) and (Bocquet and Barrat, 1994). The component of the hydrodynamic velocity parallel to the interface u(z) is assumed to be proportional to the perpendicular derivative, namely,


du(zp) _ u(z0) dz ~    6


The no-slip boundary condition is recovered if the slip length S satisfies 6 = 0. Here z0 is the hydrodynamic position of the interface, S and z0 can be obtained from the microscopic Kubo relations (Kubo et al., 1991):

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