Paper no. 306An analytical model is presented for electro-osmotic flow through a wide rectangular microchannel rotating about an axis perpendicular to its own. The flow is driven by a steady electric field applied along the channel axis, where the upper and lower walls are charged with uniform but possibly disparate zeta potentials. The aim is to understand the interaction between Coriolis force, pressure gradient, viscous force, and the Lorentz force.postprin

Ng, CO

Qi, C

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Title Electro-osmotic flow through a rotating microchannelAuthor(s) Qi, C; Ng, COCitationThe 2015 World Congress on Mechanical, Chemical, and MaterialEngineering (MCM 2015), Barcelona, Spain, 20-21 July 2015. InConference Proceedings, 2015, p. 306-1-306-5Issued Date 2015URL http://hdl.handle.net/10722/214851Rights Creative Commons: Attribution 3.0 Hong Kong LicenseProceedings of the World Congress on Mechanical, Chemical, and Material Engineering (MCM 2015) Barcelona, Spain – July 20 - 21, 2015 Paper No. 306 306-1  Electro-osmotic Flow Through a Rotating Microchannel  Cheng Qi, Chiu-On Ng Department of Mechanical Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong U3001746@hku.hk; cong@hku.hk   Abstract –An analytical model is presented for electro-osmotic flow through a wide rectangular microchannel rotating about an axis perpendicular to its own. The flow is driven by a steady electric field applied along the channel axis, where the upper and lower walls are charged with uniform but possibly disparate zeta potentials. The aim is to understand the interaction between Coriolis force, pressure gradient, viscous force, and the Lorentz force.   Keywords: electro-osmotic flow; rotating microchannel; secondary flow; EDL-Ekman layer.   1. Introduction  Microfluidic devices have been popular in the fields of biomedicine and micro-electromechanical system (MEMS), in which one advanced method for fluidic transport is to make use of externally applied voltages, known as electro-osmotic (EO) pumping.  In recent applications, a microfluidic system may be mounted on a rotating frame, such as centrifuges often used for flow control or mass separation given the advantages that centrifugation may relieve the problem of Joule heating and dispel bubbles of gas (Duffy et al., 1999). Both EO flow and rotating flow have been extensively studied in the literature either theoretically or experimentally, but EO flow in a rotating environment, however, has received less attention than it deserves.  Chang and Wang (2011) pioneered an analytical study on rotating EO flow. They studied rotation-induced secondary EO flow and its effects under various system rotation speeds. In the present problem, the lateral confinement due to side walls, which is ignored in Chang and Wang (2011), is taken into account, and as a result a transverse pressure gradient is induced so as to give a zero net flux in the transverse direction.  We look into effects on the primary and secondary flows as functions of the Debye parameter, the rotating speed, and the zeta potentials on the upper and lower walls. A special boundary layer structure is exhibited when the rotational speed and the Debye parameter are sufficiently large (i.e., when the electric double layer (EDL) and the Ekman layer are both thin layers near each of the two horizontal walls).  Inside such a composite layer, called the Ekman-EDL layer, the Coriolis force interacts with the electrokinetics.  Under fast rotation, the flow is essentially geostrophic in the core region of the channel, where the viscous and electric forces vanish.  2. Problem Formulation  As shown in Fig. 1, the x-axis is defined to be along the longitudinal centerline of the microchannel. The microchannel, whose cross section is rectangular with its width much larger than its height (i.e., a very small height to width aspect-ratio h/b), rotates about the z-axis at a constant rotational speed Ω. The upper and lower walls of the channel are charged with uniform zeta potentials denoted by 𝜁1and 𝜁2 , respectively. An electric field E is applied purely along the x-axis.  Considering a micro length scale, a very low Reynolds number is assumed. Ignoring gravity, for an incompressible, steady and fully developed flow, the governing equations in a rotating frame are  ∇ ∙ 𝐮 = 0, (1)   306-2 2𝜌𝛀 × 𝐮 = −∇𝑃 + 𝜇∇2𝐮+ 𝜌e𝐄, (2)   Fig. 1. Schematic diagram of the problem: EO flow through a wide rectangular channel of height 2h and width 2b rotating about the z-axis with rotational speed Ω with zeta potentials 𝜁1 and 𝜁2 on its upper and lower walls. A steady electric field is applied along the x-axis.   where ∇  is the gradient operator, 𝐮 = (𝑢, 𝑣, 0)  is the velocity vector seen in the rotating frame, 𝛀 = (0,0, Ω), 𝐄 = (E, 0,0), 𝜌 is the fluid density, 𝜇 is the dynamic viscosity of the fluid, 𝑃 is the reduced pressure, and 𝜌e is the free charge density in the electrolyte. By electrostatics, the potential distribution 𝜓(𝑧) in the EDL is related to the free charge density 𝜌e given by the Poisson equation as  d2𝜓d𝑧2= −𝜌e𝜀, (3)   where 𝜀  is the dielectric permittivity of the liquid electrolyte. With the further assumptions of a Boltzmann distribution for the ions in the EDL and a symmetrical electrolyte of valence Z, the free charge density is given by  𝜌e = −2𝑍𝑒𝑛∞ sinh (𝑍𝑒𝜓𝑘𝐵𝑇), (4)   where 𝑒 is the fundamental charge, 𝑛∞ is the bulk concentration, 𝑘𝐵 is the Boltzmann constant, and 𝑇 is the absolute temperature. For a very small potential, combining Eqs. (3) and (4) gives the linearized Poisson-Boltzmann equation  d2𝜓d𝑧2− 𝜅2𝜓 = 0, (5)   where 𝜅 = (2𝑍2𝑒2𝑛∞/𝜀𝑘𝐵𝑇)1/2 is called the Debye parameter. With the given wall potentials, the potential 𝜓 can be found as  𝜓(𝑧) = 𝑎cosh⁡(𝜅𝑧)cosh⁡(𝜅ℎ)+ 𝑏sinh⁡(𝜅𝑧)sinh⁡(𝜅ℎ), (6)   where 𝑎 = (𝜁1 + 𝜁2)/2 and 𝑏 = (𝜁1 − 𝜁2)/2. By introducing the following normalized variables and parameters (distinguished by an overhead caret): ?̂? = 𝑧/ℎ, ?̂? = 𝜅ℎ, (?̂?, 𝜁1, 𝜁2, ?̂?, ?̂?) = (𝜓, 𝜁1, 𝜁2, 𝑎, 𝑏)/𝜁0, where 𝜁0 is a reference potential, the potential can be then normalized as  ?̂?(?̂?) = ?̂?cosh⁡(?̂??̂?)cosh⁡(?̂?)+ ?̂?sinh⁡(?̂??̂?)sinh⁡(?̂?), (7)  306-3  Based on the assumptions made above, the governing equations in a dimensionless form reduce to  −𝜔𝑣 =d2?̂?d?̂?2+ ?̂?2?̂?, (8)  𝜔?̂? = ?̂?𝑦 +d2?̂?d?̂?2, (9)   where (?̂?, 𝑣) = (𝑢, 𝑣)/𝑈  with the scale 𝑈 = −𝜀𝜁0𝐸/𝜇 , ?̂?𝑦  is the pressure gradient along the y-direction, ω = 2𝜌𝛺ℎ2/𝜇  is the dimensionless rotation parameter. The solutions (?̂?, 𝑣)  can be readily found as  ?̂?(?̂?) =?̂?𝑦𝜔− ?̂?𝐴cosh(?̂??̂?)cosh(?̂?)− ?̂?𝐴sinh(?̂??̂?)sinh(?̂?)+ 𝑒𝜂?̂?[𝐶1 cos(𝜂?̂?) + 𝐶2 sin(𝜂?̂?)] + 𝑒−𝜂?̂?[𝐶3 cos(𝜂?̂?) + 𝐶4 sin(𝜂?̂?)], (10)  ?̂?(?̂?) = −?̂?𝐵cosh(?̂??̂?)cosh(?̂?)− ?̂?𝐵sinh(?̂??̂?)sinh(?̂?)+ 𝑒𝜂?̂?[𝐶1 sin(𝜂?̂?) − 𝐶2 cos(𝜂?̂?)] − 𝑒−𝜂?̂?[𝐶3 sin(𝜂?̂?) − 𝐶4 cos(𝜂?̂?)], (11)   where 𝐴 = ?̂?4/(?̂?4 +𝜔2) and 𝐵 = 𝜔?̂?2/(?̂?4 +𝜔2). The four coefficients 𝐶1,2,3,4 can be determined by the no-slip boundary condition, while the transverse pressure gradient ?̂?𝑦 can be determined by the condition of zero transverse net flux.  3. Discussion  It is found that when both ?̂?  and 𝜔  are sufficiently large the velocity profile tends to exhibit a boundary-layer structure. This has led us to examine more closely the flow structure in the vicinity of two horizontal walls. We assume here: 𝜂 ≫ 1 and ?̂? ≫ 1 (the ratio of these two large parameter is defined as 𝜅∗ = ?̂?/𝜂, a modified non-dimensional Debye parameter). In this sense, the EDL and Ekman layer are consolidated into a single thin layer, which we may call an Ekman-EDL layer.  We define two stretched local coordinates: ?̂?1 = 𝜂(1 − ?̂?), for the layer near the upper wall; ?̂?2 = 𝜂(1 + ?̂?), for the layer near the lower wall.  In terms of these newly defined local coordinates, the expressions of velocity can be rearranged as  ?̂?1(?̂?1) = ?̂?(𝐴∗ − 𝐶∗) − (?̂? + ?̂?)𝐴∗𝑒−𝜅∗?̂?1 + 𝑒−?̂?1[(?̂?𝐶∗ + ?̂?𝐴∗) cos(?̂?1) + (?̂? + ?̂?)𝐵∗ sin(?̂?1)], (12)  𝑣1(?̂?1) = −(?̂? + ?̂?)𝐵∗𝑒−𝜅∗?̂?1 − 𝑒−?̂?1[(?̂?𝐶∗ + ?̂?𝐴∗) sin(?̂?1) − (?̂? + ?̂?)𝐵∗ cos(?̂?1)], (13)  ?̂?2(?̂?2) = ?̂?(𝐴∗ − 𝐶∗) − (?̂? − ?̂?)𝐴∗𝑒−𝜅∗?̂?2 + 𝑒−?̂?2[(?̂?𝐶∗ − ?̂?𝐴∗) cos(?̂?1) + (?̂? − ?̂?)𝐵∗ sin(?̂?2)], (14)  𝑣2(?̂?2) = −(?̂? − ?̂?)𝐵∗𝑒−𝜅∗?̂?2 − 𝑒−?̂?2[(?̂?𝐶∗ − ?̂?𝐴∗) sin(?̂?2) − (?̂? − ?̂?)𝐵∗ cos(?̂?2)], (15)   with the definitions of three parameters 𝐴∗ , 𝐵∗  and 𝐶∗ , given as 𝐴∗ = 𝜅∗4/(𝜅∗4 + 4) , 𝐵∗ =2𝜅∗2/(𝜅∗4 + 4) and 𝐶∗ = (𝜅∗2 − 4𝜅∗)/(𝜅∗4 + 4).  Fig. 2 shows axial and transverse velocity profiles in the vicinity of two horizontal walls for different zeta potential distributions. From Fig. 2 (a) and (b), it is obvious that the axial velocity has an overshoot near the walls and decays to a constant value ?̂?∞ in the core region. Theoretically we can obtain this asymptotic limit: lim?̂?1→∞ ?̂?1 = lim?̂?2→∞ ?̂?2 = ?̂?(𝐴∗ − 𝐶∗), and hence  ?̂?∞ = ?̂?(𝐴∗ − 𝐶∗).   It is obvious that the asymptotic axial velocity ?̂?∞ only depends on ?̂? and 𝜅∗, and is independent on ?̂?. The overshoot of axial velocity is denoted by ?̂?m, which may either be positive or negative, and is collectively affected by the zeta potentials on both walls. In Fig. 2 (a), although the zeta potential on the upper wall for all three cases is the same, a larger zeta potential on the lower wall leads to a larger  306-4 positive ?̂?m. In Fig. 2 (b), although the lower wall is not charged for the non-symmetric case (dashed line), there is still an overshoot which may be caused by the zeta potential on the upper wall; the positive 𝜁2 leads to a larger positive ?̂?m and the negative 𝜁2 leads to a negative ?̂?m as the effect of the negative 𝜁2 overweighs that of the positive 𝜁1.  As shown in Fig. 2 (c) and (d), the transverse velocity oscillates about the zero value within the Ekman-EDL layer before it decays to zero in the unsheared core region (it can be readily proved that lim?̂?1→∞ 𝑣1 = lim?̂?2→∞ 𝑣2 = 0), and its integral across the z-direction gives a zero net flow rate. The transverse velocity within the Ekman-EDL layer is affected directly by the rotation-induced Coriolis force and the induced transverse pressure gradient due to electrokinetic effects.   Fig. 2. Velocity profiles within an Ekman-EDL layer formed near the two horizontal walls: (a) axial velocity profile near the upper wall; (b) axial velocity profile near the lower wall; (c) transverse velocity profile near the upper wall; (d) transverse velocity profile near the lower wall, where 𝜅∗ = 1. In each, three cases are shown: (1) 𝜁1 = 𝜁2 = 1 (solid lines); (2) 𝜁1 = 1 and 𝜁2 = 0 (dashed lines); (3) 𝜁1 = 1 and 𝜁2 = −1 (dash-dot lines).  4. Conclusion  An analytical model for rotating EO flow in a very wide rectangular channel has been developed. We have examined the velocity profiles in the Ekman-EDL layer, which is an electrokinetically modulated Ekman boundary layer that occurs when the Debye and rotation parameters are both large.  Acknowledgements  This work was supported by the Research Grants Council of the Hong Kong Special Administrative Region, China, through Project No. HKU 715510E, and also by the University of Hong Kong through the Small Project Funding Scheme under Project Code 201309176109.  References Chang, C., & Wang, C. (2011). Rotating Electro-Osmotic Flow Over A Plate Or Between Two Plates. Phys. Rev. E., 84, 056320.  306-5 Duffy, D.C., Gillis, H.L., Lin, J., Sheppard, N.F., & Kellogg, G.J. (1999). Microfabricated Centrifugal Microfluidic Systems: Characterization And Multiple Enzymatic Assays. Anal. Chem., 71, 4669-4678. 

Electro-osmotic flow through a rotating microchannel

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Electro-osmotic flow through a rotating microchannel

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