Analytical Solution of Time Periodic Electroosmotic Flows: Analogies to Stokes’ Second Problem Prashanta Dutta ² and Ali Beskok* ,‡ Microfluidics Laboratory, Mechanical Engineering Department, Texas A&M University, College Station, Texas, 77843-3123, and School of Mechanical and Materials Engineering, Washington State University, Pullman, Washington 99164-2920 Analytical solutions of time periodic electroosmotic flows in two-dimensional straight channels are obtained as a function of a nondimensional parameter K, which is based on the electric double-layer (EDL) thickness, kinematic viscosity, and frequency of the externally applied electric field. A parametric study as a function of K reveals interesting physics, ranging from oscillatory “pluglike” flows to cases analogous to the oscillating flat plate in a semi-infinite flow domain (Stokes’ second problem). The latter case differs from the Stokes’ second solution within the EDL, since the flow is driven with an oscillatory electric field rather than an oscillating plate. The analo- gous case of plate oscillating with the Helmholtz -Smolu- chowski velocity matches our analytical solution in the bulk flow region. This indicates that the instantaneous Helmholtz -Smoluchowski velocity is the appropriate electroosmotic slip condition even for high-frequency excitations. The velocity profiles for large K values show inflection points very near the walls with localized vorticity extrema that are stronger than the Stokes layers. This have the potential to result in low Reynolds number flow instabilities. It is also shown that, unlike the steady pure electroosmotic flows, the bulk flow region of time periodic electroosmotic flows are rotational when the diffusion length scales are comparable to and less than the half channel height. Electroosmosis is one of the major electrokinetic phenomena in which ionized liquid flows with respect to a charged surface in the presence of an external electric field. Although this phenom- enon was first observed experimentally in the early nineteenth century, only recently has it received considerable attention from engineers and chemists due to the emergence of microfluidic devices. The pressure-building ability of electroosmosis made it very suitable as a pumping mechanism in microchannel geom- etries for electronic cooling 1 and bioanalytical systems. 2,3 Time periodic electroosmotic flow is also known as A/ C electroosmosis, and it is driven by an alternating electric field. Although time periodic electroosmotic flow has potential in biotechnology and separation science, it has been addressed in only a relatively few publications. For example, Dose and Guio- chon reported numerical results for impulsively started electroos- motic flow, 4 and Soderman and Jonsson published an analytical work on starting problems of electroosmotic flows for a number of geometries including the flow over a flat plate and two- dimensional microchannel and microtube flows. 5 Green et al. investigated A/ C electroosmosis on planner microelectrodes using steady and unsteady electric fields. 6 In a later study, they analyzed the same problem based on the linearized Debye layer theory. 7 Most recently, Barragan and Bauza experimentally studied the effects of a sinusoidally alternating electric field, superimposed onto a steady electroosmotic flow. 8 Microscale combustion and power generation applications, as well as the “laboratory on a microchip” devices require develop- ment of reliable micromixers. Mixing in microfluidic systems is difficult due to the negligible inertial effects. Also, small molecular diffusion coefficients require longer convective/ diffusive mixing length, time scales, or both. Moreover, microstirrers with moving components are usually difficult to build, and they are prone to mechanical failure. Currently, several ideas including peristaltic membrane motion and chaotic advection are being utilized to develop various micromixers. 9 Most recently, Oddy et al. have utilized pulsating electrostatic flows to promote and enhance mixing for low Reynolds number flows 10 ( Re > 0.1) and built a micromixer that is capable of stirring two fluid streams continu- ously or intermittently. 11 Motivated by the potential of pulsating flows in promoting mixing, we study here the time periodic electroosmotic flows in straight microchannels by changing the * Corresponding author: (e-mail) abeskok@ mengr.tamu.edu; (phone) (979) 862-1073. † Washington State University. ‡ Texas A&M University. (1) Chen, C. H.; Zeng, S.; Mikkelsen, J. C.; Santiago, J. G. Proc. ASME 2000 , MEMS 1, 523-528. (2) Chang, H. T.; Chen, H. S.; Hsieh, M. M.; Tseng, W. L. Rev. Anal. Chem. 2000 , 19 (1), 45-74. (3) Anderson, G. P.; King, K. D.; Cuttiono, D. S.; Whelan, J. P.; Ligler, F. S.; MacKrell, J. F.; Bovais, C. S.; Indyke, D. K.; Foch, R. J. Field Anal. Chem. Technol . 1999 , 3 (4-5), 307-314. (4) Dose, E. V.; Guiochon, G. J. Chromatogr. 1993 , 652, 263-275. (5) Soderman, O.; Jonsson, B. J. Chem. Phys . 1996 , 105, 10300-10311. (6) Green, N. G.; Ramos, A.; Gonzalez, A.; Morgan H.; Castellanos A. Phys. Rev. E 2000 , 61 (4), 4011-4018. (7) Gonzalez, A.; Ramos, A.; Green, N. G.; Castellanos, A. Phys. Rev. E 2000 , 61 (4), 4019-4028. (8) Barragan, V. M.; Bauza, C. R. J. Colloid Interface Sci. 2000 , 230, 359-366. (9) Yi, M.; Bau, H. H.; Hu, H. Proc. ASME 2000 , MEMS 1, 367-374. (10) Oddy, M. H.; Santiago, J. G.; Mikkelsen, J. C. Anal. Chem., submitted. (11) Oddy, M. H.; Mikkelsen, J. C.; Santiago, J. G. Proc. ASME, in press. Anal. Chem. 2001, 73, 5097-5102 10.1021/ac015546y CCC: $20.00 © 2001 American Chemical Society Analytical Chemistry, Vol. 73, No. 21, November 1, 2001 5097 Published on Web 09/29/2001