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Copyright Brian J. Kirby. With questions, contact Prof. Kirby here. This material may not be distributed without the author's consent. When linking to these pages, please use the URL http://www.kirbyresearch.com/textbook.

This web posting is a draft, abridged version of the Cambridge University Press text. Follow the links to buy at Cambridge or Amazon or Powell's or Barnes and Noble. Contact Prof. Kirby here. Click here for the most recent version of the errata for the print version.

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Jump To: [Kinematics] [Couette/Poiseuille Flow] [Fluid Circuits] [Mixing] [Electrodynamics] [Electroosmosis] [Potential Flow] [Stokes Flow] [Debye Layer] [Zeta Potential] [Species Transport] [Separations] [Particle Electrophoresis] [DNA] [Nanofluidics] [Induced-Charge Effects] [DEP] [Solution Chemistry]

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 λD (where φ becomes zero by definition), substituting φ = 0 into Equation 6.8 leads to the Helmholtz-Smoluchowski equation:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

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

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

and the interior boundary condition is

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

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:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

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

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

combined with the effective electroosmotic slip velocity:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

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 φ0 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 local electric field multiplied by the electroosmotic mobility (Figure 6.5):.
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(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.


microfluidics textbook nanofluidics textbook Brian Kirby CornellFigure 6.5: Flow-current similitude for electroosmotic flow in a channel with a circular insulating wall. If the conductivity and surface potential φ0 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.


microfluidics textbook nanofluidics textbook Brian Kirby CornellFigure 6.6: Vorticitygeneration and cancelation in electrical double layers.


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Jump To: [Kinematics] [Couette/Poiseuille Flow] [Fluid Circuits] [Mixing] [Electrodynamics] [Electroosmosis] [Potential Flow] [Stokes Flow] [Debye Layer] [Zeta Potential] [Species Transport] [Separations] [Particle Electrophoresis] [DNA] [Nanofluidics] [Induced-Charge Effects] [DEP] [Solution Chemistry]

Copyright Brian J. Kirby. Please contact Prof. Kirby here with questions or corrections. This material may not be distributed without the author's consent. When linking to these pages, please use the URL http://www.kirbyresearch.com/textbook.

This web posting is a draft, abridged version of the Cambridge University Press text. Follow the links to buy at Cambridge or Amazon or Powell's or Barnes and Noble. Contact Prof. Kirby here. Click here for the most recent version of the errata for the print version.


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