Oscillatory Electroosmotic Flow of Power-Law Fluids in a Microchannel Rub´ en Ba˜ nos, Jos´ e Arcos, Oscar Bautista, Federico M´ endez Abstract—The Oscillatory electroosmotic flow (OEOF) in power law fluids through a microchannel is studied numerically. A time-dependent external electric field (AC) is suddenly imposed at the ends of the microchannel which induces the fluid motion. The continuity and momentum equations in the x and y direction for the flow field were simplified in the limit of the lubrication approximation theory (LAT), and then solved using a numerical scheme. The solution of the electric potential is based on the Debye-H¨ uckel approximation which suggest that the surface potential is small,say, smaller than 0.025V and for a symmetric (z : z) electrolyte. Our results suggest that the velocity profiles across the channel-width are controlled by the following dimensionless parameters: the angular Reynolds number, Reω, the electrokinetic parameter, ¯ κ, defined as the ratio of the characteristic length scale to the Debye length, the parameter λ which represents the ratio of the Helmholtz-Smoluchowski velocity to the characteristic length scale and the flow behavior index, n. Also, the results reveal that the velocity profiles become more and more non-uniform across the channel-width as the Reω and ¯ κ are increased, so oscillatory OEOF can be really useful in micro-fluidic devices such as micro-mixers. Keywords—Oscillatory electroosmotic flow, Non-Newtonian fluids, power-law model, low zeta potentials. I. I NTRODUCTION M ICRO-FLUIDIC components such as micro-channels, micro-mixers and micro-pumps are commonly implemented in the design of biochips, for example electroosmosis has been well established as a micropumping technique used in many of these devices. Extensive studies about electroosmotic flow in microcapillaries have been reported in the literature, most of them considering the Debye-H¨ uckel approximation [6], [7], some others regarding the time-dependent external electric field in Newtonian fluids [6], [8]. The rheology of the fluids is very important due to the immense application of microfluidic to analyze biofluids, in which may not be treated as Newtonian fluids. In this context, the rheology in electroosmotic flow with constant electric field (DC) has been extensivelly studied [9], [11], [12]. Lately, there are few studies considering both the rheology of the fluid and a time-dependent electrical field. Each effect has been studied in a separately manner, that is why our study consider an oscillatory electrical field which causes an oscillatory electroosmotic flow in the microchannel and the power-law model [10] to obtain the Rub´ en Ba˜ nos is with the ESIME Azcapotzalco, Instituto Polit´ ecnico Nacional, Av. de las Granjas No. 682, Col. Santa Catarina, Azcapotzalco, Ciudad de M´ exico, 02250, Mexico (e-mail: rdbm94@hotmail.com). Jos´ e Arcos and Oscar Bautista are with the ESIME Azcapotzalco, Instituto Polit´ ecnico Nacional, Av. de las Granjas No. 682, Col. Santa Catarina, Azcapotzalco, Ciudad de M´ exico, 02250, Mexico. Federico M´ endez is with the Universidad Nacional Aut´ onoma de M´ exico, Coyoacan, Ciudad de M´ exico 04510, M´ exico. velocity profiles across the microchannel-width as a function of the principal dimensionless parameters involved in the present investigation. II. PROBLEM FORMULATION In Fig. 1 the schematic representation of the OEOF is shown. Consider a two-dimensio-nal microchannel of height 2h and length L, where L  h. The microchannel is filled with a symmetric electrolyte (z : z) solution whose rheological behaviour follows the well-known Ostwald de Waele model. The anode and cathode at the ends of the microchannel provide a time-dependent electrical field which generates a periodically oscillatory flow via the electroosmotic effect. A. Electrical Field The electrical potential in the location (x, y) in the microchannel, given by Φ(x, y) arises by the superposition of the externally electric potential, φ(x, y), and the potential ψ(y) into the electrical double layer (with surface potential ζ ). It is reasonable to assume that the electric potential is given by the linear superposition of the electrical double layer potential and the externally applied potential, which is valid for long microchannels [1]. Therefore, the Poisson-Boltzmann equation for a slit microchannel becomes, d 2 ψ dy 2 = − ρ e  . (1) From the above equation and the boundary conditions dψ/dy =0 at y =0 and ψ = ζ at y = h, the well-known solution for the distribution of the potential, ψ, with a free charge density defined as ρ e = −κ 2 ψ is given by: ¯ ψ = cosh ¯ κ ¯ y cosh ¯ κ (2) where ¯ ψ = ψ/ζ , ¯ κ = κh and ¯ y = y/h. B. Velocity Field To determine the flow field, we consider that α = h/L  1, therefore, the motion can be approximated as unidirectional [2]. The velocity components in the x and y directions are u = u(y,t) and v = 0, for t ≥ 0. Also, neglecting external pressure gradient and gravity effects, the one-dimensional momentum equation is represented by, ρ ∂u ∂t = dτ dy + ρ e E x (t). (3) Equation (3) is subjected to the following boundary conditions. The symmetry boundary condition ( du/ dy=0) was World Academy of Science, Engineering and Technology International Journal of Mechanical and Mechatronics Engineering Vol:12, No:8, 2018 773 International Scholarly and Scientific Research & Innovation 12(8) 2018 scholar.waset.org/1307-6892/10009307 International Science Index, Mechanical and Mechatronics Engineering Vol:12, No:8, 2018 waset.org/Publication/10009307