Interdisciplinary Applied Mathematics

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7.5.1 Governing Equations


A detailed formulation of electrophoretic transport can be found in (Ermakov et al., 1998). In this section, we will present a simpler model that is based on the electric neutrality (i.e., pe = 0 ) and uniform electric conductivity (a) assumptions. Using these simplifications, the species transport equation is reduced to (Palusinski et al., 1986; Ermakov et al., 1998)


dn


7ГГ + V(niu + niuEp,i) = AV2nj,    (7.44)


dt


where uEPjj = p,EPjjE is the electrophoretic migration velocity. The electric field (E) is determined using equation (7.21), and the Poisson-Boltzmann equation (7.4) is solved for the electrokinetic potential. The fluid velocity u is found by solution of the incompressible Navier-Stokes equations (7.12), subject to the electrokinetic body force terms (peE) and the no-slip boundary condition on channel surfaces. This formulation ignores interaction of the charged species    with    ions    in    the    EDL region    (where    the    net    electric


charge is nonzero), and possible zeta potential changes.


A simplification of this model neglects the electrokinetic forces in the EDL region, and replaces them with the Helmholtz-Smoluchowski slip condition (7.24) to drive the flow. This approach greatly simplifies the problem of mesh generation and the numerical stiffness associated with the resolution of the thin EDL region. In addition, solution of the Poisson-Boltzmann equation is also not required. Despite these simplifications, the electroosmotic slip velocity (7.24) should be calculated, and imposed as the boundary condition on dielectric surfaces. In flows with complex-geometry, this approach requires imposing spatially varying slip velocity on every grid point on the channel surface, which may not be trivial, depending on the numerical solution methodology.

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