6.3 Solving the Navier-Stokes equations for electroosmotic flow in the thin double layer limit [electroosmosis top]

The previous section derived the inner solution for electroosmotic flow in terms of the electrical potential
distribution, and built intuition about the nature of the flow in the electrical double layer. The outer solution
describes the flow outside the electrical double layer, which is the part of the flow that we readily observe under a
microscope. In both cases, a basic assumption of this analysis is that the electrical double layer thickness is smallas
compared to the characteristic size of the microchannel.

6.3.1 Outer solution

In the part of the flow where the distance from the wall is much greater than λ_{D} (where φ becomes zero by
definition), substituting φ = 0 into Equation 6.8 leads to the Helmholtz-Smoluchowski equation:

Equation 6.9 is the exterior boundary condition of the inner solution. The outer solution for electroosmotic flow is
found by considering the Navier-Stokes solutions for the bulk fluid, for which the net charge density is zero, and
using Equation 6.9 as the interior boundary condition. Thus, the governing equation for the outer flow
is

and the interior boundary condition is

Equation 6.11 is an example of aVan Dyke matching condition[42]. The idea is that the inner solution describes the
flow in the electrical double layer, the outer solution describes the flow outside the double layer, and the Van Dyke
matching condition matches the two at a point in the middle. The outer solution is invalid in the electrical double
layer, and violates the no-slip condition at the wall, but properly describes the velocity distribution outside the
double layer. For a straight channel of uniform cross-section, the velocity profile outside the electrical double layer
is everywhere equal to this effective slip velocity.

The outer solution above gives the electroosmotic flow distribution outside the electrical double layer, which is
the most important region of the flow in most cases, since the electrical double layer is thin and difficult to
interrogate. In this section, we discuss how general electroosmotic flow problems can be simplified and solved with
relatively simple equations.

6.3.2 Replacing the electrical double layer with an effective slip boundary condition

VIDEO:replacing the EDL with a slip condition.

Thematched asymptotic approach above subsumes the net charge density of the electrical double layer into a
boundary condition for flow outside the electrical double layer. The Van Dyke condition in Equation 6.11 dictates
that the limit of the velocity as the outer solution approaches the wall is a function of the wall potential, fluid
permittivity, fluid viscosity, and the electric field magnitude near the wall. Thus, if we wish to solve only for the
outer solution (where the fluid is electroneutral) the flow can be described by solving the Laplace equation for the
electrical potential:

and solving the Navier-Stokes equations with no source term:

combined with the effective electroosmotic slip velocity:

The electroosmotic slip velocity requires knowledge of the electric field at the wall, which is a result of the solution
for φ. So, to solve this system, we solve the Laplace equation, use the electric field solution to specify the effective
slip boundary conditions, and then use the effective slip boundary conditions to solve the Navier-Stokes
equations.

6.3.3 Replacing the Navier-Stokes equations with the Laplace equation: flow-current similitude

If the flowis isobaric and the surface potential φ_{0} is uniform, the outer solution for the velocity is irrotational, in
which case we need solve only the Laplace equation. This is an enormous simplification that greatly changes our
ability to study electroosmotic flows.

Importantly, the similitude we observe for the 1D flow solution applies in three dimensions. In fact, we can say
that, for a microfluidic system with uniform electroosmotic mobility, uniform permittivity, viscosity, and
conductivity, and no applied pressure gradients, the velocity far away from any wall is proportional to the localelectric field multiplied by the electroosmotic mobility (Figure 6.5):.

(6.15)

This solution is a potential flow and satisfies the Laplace equation (since the electric field satisfies the Laplace
equation). It satisfies the no-penetration condition at the wall, because the electric field solution satisfies the
no-current condition at an insulating wall. It satisfies the electroosmotic slip condition at the wall as well.
Reference [43] goes on to show what is required for this solution to be unique.

Equation 6.15is hugely important, because electroosmotic flows can be calculated rather easily in this limit, and
analytical solutions for potential flows can give immediate physical insight into the flow distributions
in an electroosmotically-driven system. These potential flows are discussed in detail in Chapter 7.

Figure 6.5: Flow-current similitude for electroosmotic flow in a channel with a circular insulating wall. If
the conductivity and surface potential φ_{0} are uniform, the flow velocity is proportional to the local electric
field, and thus (a) electric field lines are the same as streamlines and (b) velocity potential isocontours are
the same as electrical potential isocontours.

Experimental applicability.
Experiments are often designed to satisfy the assumptions required for Equation 6.15 to apply, since system
design is then more straightforward. Uniform conductivity can be achieved by ensuring that the fluid is well-mixed
and that Joule heating is minimal (that is, fluid heating owing to the current conducted through the
fluid) is caused by the application of electric fields. Uniform surface potential can be achieved by
making a device from one material and ensuring that the surfaces are clean. Removing applied pressure
fields typically requires monitoring hydrostatic head and interface curvature at the reservoirs. If care is
taken to satisfy these requirements, the flow generated in these systems is easy to model and often
advantageous for a variety of applications. For example, the potential flow described here leads to
dispersionless transport, which leads to optimal performance for capillary electrophoresis separations (see
Chapter 12).

6.3.4 Reconciling the no-slip condition with irrotational flow

Electroosmosis outside thin double layers is described by an irrotational flow with a velocity proportional
everywhere to the local electric field. While vorticity is generated at the wall, the electrostatic force on the charge
density in the double layer generates avorticity source term (Figure 6.6) which, when integrated over the double
layer, is of precisely the same magnitude but of opposite sign to the vorticity generated at the wall. Thus the
vorticity inside the double layer is large, but the vorticity outside the double layer is identically zero.

Figure 6.6: Vorticitygeneration and cancelation in electrical double layers.