The Helmholtz-Smoluchowski (H-S) velocity is known to be an accurate and useful formula for estimating the electro-osmotic (EO) flow rates in a simple micro-channel with a thin electric-double layer. However, in case the channel cross section is not so simple, the usefulness of H-S velocity could be sharply limited. A case of fundamental interest representing this situation is a rectangular channel (comprising parallel plates) with built-in vertical gratings, in which the surfaces inside the channel may develop different normalized zeta potentials α (on the gratings) and β (on the side walls). In this study, analytical solutions are pursued under the Debye-Hückel approximation to obtain EO pumping rates in a rectangular channel with vertical gratings. In particular, we identify the conditions under which the H-S formula can be properly applied and investigate how the EO flow rates may deviate from those predicted by the H-S velocity with varying physical parameters. Moreover, a diagram of the optimal EO pumping rates on the α-β plane is introduced that accounts for the general features of the analysis, which is consistent with a mathematical model and may serve as a convenient guide for engineering design and applications.

This paper presents a continued study to our previous work on electroosmotic (EO) flow in a channel with vertical baffle plates by further investigating EO flow through an array of baffle plates arranged in parallel to the channel walls. The flow may be driven either in the direction along or in the direction transverse to the plates, thus distinguishing the longitudinal EO pumping (LEOP) and the transverse EO pumping (TEOP). In both types of EO pumping, it is more interesting to examine the cases when the baffle plates develop a higher zeta potential (denoted by α) than that on the channel walls (β). This semi-analytical study enables us to compare between LEOP and TEOP in the pumping efficiency under similar conditions. The TEOP case is more difficult to solve due to the higher order governing partial differential equations caused by the induced non-uniform pressure gradient distribution. In particular, we examine how the EO pumping rates deviate from those predicted by the Helmholtz-Smoluchowski velocity and illustrate the general trend of optimizing the EO pumping rates with respect to the physical and geometric parameters involved.