Conditions for similitude between the fluid velocity and electric field in electroosmotic flow Page: 6 of 13
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show that this condition, along with the necessary
conditions above, is sufficient for global similitude
between the fluid velocity and electric field.
Equations Describing Electroosmotic Flow
To derive the conditions sufficient for simili-
tude, we consider a simply or multiply connected
fluid volume, bounded in its entirety by a mix of
two surface types. The surfaces Si describe the in-
terface between the fluid and an impermeable insu-
lating solid, while the surfaces S2 describe inlet or
outlet boundaries.
The electric potential within this volume is gov-
erned by the Poisson equation relating the diver-
gence of the electric field to the local charge density,
V . (e VO) = -Pe, (1)
where e is the dielectric constant of the fluid and pe
is the local charge density. The local charge density
may be related to the electric potential through the
Boltzmann distribution or similar relations.
Boundary conditions for the electric poten-
tial on the charged surface may be specified either
through a prescribed density of the surface charge
in conjunction with the governing equation (1) or,
alternatively, as a prescribed wall potential with re-
spect to the potential of the adjacent neutral fluid.
The latter is generally preferred since the wall or
zeta potential is often known or can be obtained
from simple experiments. Finally, the applied elec-
tric field is generally specified by prescribed poten-
tials on inlet and outlet boundaries.
Restricting our attention to liquid flows, the
working fluid may be considered incompressible and
the continuity equation reduces to
V-u=0. (2)
Under the additional assumptions that the flow is
steady and that the fluid viscosity, /., is constant,
the momentum equation becomes
p (u V)u= -Vp+ PeVo+ pV2u. (3)
where p is the uniform fluid density. Using the vec-
tor identities of Eqs. (Al) and (A7), the momentum
equation can also be written as
-pux (Vxu) = -V(p+ P u-u)
2
+peVq - pV x(Vx U). (4)This form of the equation is again generally applica-
ble only to an incompressible fluid having constant
viscosity.
Boundary conditions on the fluid velocity fol-
low directly from the nature of the fluid-solid inter-
face. Since no flow crosses the impermeable solid
boundaries,ufi = 0 on Si,
(5)
where fi is a unit vector locally normal to the inter-
face. The velocity tangential to these impermeable
boundaries must also obey the no-slip condition at
the fluid-solid interface,ui- = 0 on S1,
(6)
where I is any unit vector lying in the plane of the
boundary. We will next consider an alternate form
of Eq. (6) describing the fluid velocity at the outer
edge of the Debye layer.
Thin Debye Layer Limit
Assuming that the Debye layer thickness, A,
is much smaller than the channel width, a, a
boundary-layer approximation may be used to se-
quentially solve for the velocity fields in the regions
within and outside the Debye layer. Under all fore-
seeable conditions, the Reynolds number based on
the Debye layer thickness is extremely small, so the
inner Debye-layer solution can be constructed by
considering only the balance between electric and
viscous forces. The maximum velocity at the outer
edge of the layer is then used as a boundary condi-
tion in calculating the larger-scale outer flow field.
Sequential solution procedures of this type are best
justified using the formalism of matched asymptotic
expansions [9]. Here we present only an outline of
the matching procedure for first-order terms.
To identify nonessential terms, the governing
equations are first rewritten in terms of normal and
tangential coordinates rescaled by the Debye layer
thickness and the geometric length scale, respec-
tively. The electric potential is then split into an
applied field 0a(,r) having no normal gradients and
an intrinsic component O_(?r) having no tangential
gradients or at most a tangential gradient induced
by linear polarization that is proportional to the
applied field. In the latter case, polarization of the
intrinsic field can be accounted for by a suitable
choice of the dielectric constant. After dropping
those terms of order A/a and smaller, the resulting5
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Cummings, E. B.; Griffiths, S. K.; Nilson, R. H. & Paul, P. H. Conditions for similitude between the fluid velocity and electric field in electroosmotic flow, report, April 1, 1999; Albuquerque, New Mexico. (https://digital.library.unt.edu/ark:/67531/metadc706210/m1/6/: accessed April 23, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.