an electroosmotic slip velocity is often used to simplify the flow without the calculation of the
flow field in the EDL.74, 76, 78, 80
While EOF is continuously exploited in a variety of applications, channels with non-
uniform geometry have increasingly been designed for different microfluidic elements, including
pumps,82 dispensers,83 and mixers.20 As a result, there is a growing need for a fundamental
understanding about the effects of the geometry on the fluid motion and the associated transport
properties. Although there are some efforts to address this need, most of them used commercial
or in-house software to search for the solution of EOF. Little work focuses on obtaining
analytical solutions; one example is Ghosal's study on the flow in a channel with slowly varying
cross-sectional areas.78 The solution, however, requires the wall variation small enough to
validate the lubrication approximation and perturbation expansion method.
At the same time, a vast amount of studies are found in classic fluid mechanics studies for
the creeping motion in periodic channel geometries.84-99 Various analytical and numerical
methods have been employed in these studies. However, most of these creeping flows are either
Poiseuille or Couette flows, in which no slip conditions are used at the boundary. They cannot
be used for studying EOF.
In this chapter, an exact solution to an EOF in a microchannel with a periodically varying
cross-sectional area is described, with a focus on a channel confined by a flat wall and a
sinusoidal wall. The governing equations for the EOF are simplified using the slip velocity
approximation.76, 100 An exact solution to the electric potential in the microchannel is obtained
by solving a Poisson equation using complex function formulation and boundary integral
method. The EOF is then solved by applying the same approach to a biharmonic equation of the
stream function. Using the exact solutions of the electric potential and EOF flow, the effects of