Interdisciplinary Applied Mathematics

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FIGURE 7.3. Electrosmotic potential distribution within the electric double layer (left) and its logarithmic scaling (right) as a function of the inner-layer scale X = иу’.


7.3 Governing Equations


Assuming incompressible Newtonian fluid with constant viscosity, the bulk fluid motion is governed by the incompressible Navier-Stokes equations




where p is the pressure, u is a divergence-free velocity field (V- u = 0) subject to the no-slip boundary conditions on the walls, pf is the fluid density, and fEK is the electrokinetic body force. A general equation for the electrokinetic force per unit volume is given by (Stratton, 1941)


fEK = РеE —    • Ee0Ve + |v [pf^EE) ,    (7-13)


where E is the externally applied electric field. The last term shows permittivity variations with density, and it is especially important at liquid/gas interfaces as well as in ionized gas flows (Stratton, 1941). For our purposes, we will assume an incompressible medium with constant electric permittivity. Hence we consider only the contribution of the first term (peE).


The species conservation equation for a multicomponent fluid, in the absence of chemical reactions, can be expressed as


^+V.j, = 0,    (7.14)


where щ is the concentration of the ith species flux, given by


ji    DiVni + ni [u + ^EK,iE] ?


where Di is the diffusion coefficient and mEK is the electrokinetic mobility. The first term on the right-hand side corresponds to molecular diffusion flux due to the concentration gradient, while the second term corresponds to convection due to bulk fluid motion with velocity u. The following term represents transport due to the electrokinetic effects. In general, the electrokinetic mobility (mEK) includes both the electroosmotic and electrophoretic effects (Cummings, 2001; Ermakov et al., 1998). Mobility is related to the electrokinetic migration velocity uEK by

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