Analytical Solution of Combined Electroosmotic/ Pressure Driven Flows in Two-Dimensional Straight Channels: Finite Debye Layer Effects Prashanta Dutta and Ali Beskok* Microfluidics Laboratory, Mechanical Engineering Department, Texas A&M University, College Station, Texas 77843-3123 Analytical results for the velocity distribution, mass flow rate, pressure gradient, wall shear stress, and vorticity in mixed electroosmotic/ pressure driven flows are pre- sented for two-dimensional straight channel geometry. We particularly analyze the electric double-layer (EDL) region near the walls and define three new concepts based on the electroosmotic potential distribution. These are the effective EDL thickness, the EDL displacement thickness, and the EDL vorticity thickness. We show that imposing Helmholtz-Smoluchowski velocity at the edge of the EDL as the velocity matching condition between the EDL and the bulk flow region is incomplete under spatial bulk flow variations across the finite EDL. However, the Helmholtz- Smoluchowski velocity can be used as the appropriate slip velocity on the wall. We discuss the limitations of this approach in satisfying the global conservation laws. Recent developments in microfabrication technologies have enabled a variety of miniaturized fluidic systems, which can be utilized for medical, pharmaceutical, defense, and environmental monitoring applications. Examples of such applications are drug delivery, 1 DNA analysis/ sequencing systems, 2 and biological/ chemical agent detection sensors on microchips. Along with the necessary sensors and electronic units, these devices include various fluid handling components such as microchannels, pumps, 3 and valves. Utilization of electrokinetic body forces in microfluidic design can revolutionize various fluid handling applications, since it will be possible to build flow control elements with nonmoving components. The electrokinetic effects were first discovered by Reuss 4 in 1809 from an experimental investigation on porous clay, which was followed by experiments of Wiedmann. 5 In 1879, Helmholtz developed the electric double-layer (EDL) theory, relating the electric and flow parameters for electrokinetic transport. The case of EDL thickness being much smaller than the channel dimen- sions was analyzed by von Smoluchowski, who also derived a velocity slip condition for electroosmotically driven flows. 6 Elec- troosmotic flows in thin two-dimensional slits and thin cylindrical capillaries were analyzed by Burgreen and Nakache 7 and Rice and Whitehead, 8 respectively. In 1952, Overbeek proposed the irro- tationality condition of internal electrosomotic flows for arbitrarily shaped geometry. 9 This was followed by the ideal electroosmosis concept of Cummings et al., who showed similarity between the electric and the velocity fields under some specific outer field boundary conditions. 10,11 In the past decade, there have been numerous theoretical, 12,13 numerical, 14-17 and experimental 18,19 studies on electrokinetic microflows. Microchannels are one of the primary components of microf- luidic systems. Motivated by the development of fluid handling devices with nonmoving components, in this paper, we study the combined electroosmotic/ pressure driven flows in straight two- dimensional channels. Our analysis is particularly important for small, yet finite electric double-layer applications, where the distance between the two walls of a microfluidic device is about 1-3 orders of magnitude larger than the electric double layer. This is commonly observed in channel dimensions of 10 µm or less, depending on the ionic concentration. Currently, it is possible to build microchannels with 1 µm or smaller dimensions. For example, Chen et al. 20 recently built an electrokinetic pump with dimensions of 40 mm × 1 mm × 1 µm. Although it may be very difficult to perform pointwise measurements in micrometer and * Corresponding author: (e-mail) abeskok@ mengr.tamu.edu; (phone) (979) 862 1073. (1) Arangoa, M. A.; Campanero, M. A.; Popineau, Y.; Irache, J. M. Chro- matographia 1999 , 50 (3-4), 243-246. (2) Chang, H. T.; Chen, H. S.; Hsieh, M. M.; Tseng, W. L. Rev. Anal. Chem. 2000 , 19 (1), 45-74. (3) DeCourtye, D.; Sen, M.; Gad-El-Hak M. Int J. Comput. Fluid Dyn. 1998 , 10 (1) 13-25. (4) Reuss, F. F. Memoires de la Societe Imperiale de Naturalistes de Moscou 1809, 2, 327. (5) Wiedemann, G. Pogg. Ann. 1852 , 87, 321. (6) Smoluchowski, M. Krak. Anz. 1903 , 182. (7) Burgreen, D.; Nakache, F. R. J. Phys. 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