HT Proceedings of Proceedings of the HT ASME Summer Heat Transfer Conference July, 2009, San Francisco, California USA

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1 Proceedings of Proceedings of the HT ASME Summer Heat Transfer Conference July, 2009, San Francisco, California USA HT ON THE DEPENDENCE OF THE RATE OF DECAY IN CONCENTRATION AND/OR TEMPERATURE DISTRIBUTION FOR A PASSIVE SCALAR IN A MICRO-MIXER AND THE FREQUENCY OF THE ADVECTING VELOCITY FIELDS J. Rafael Pacheco, KangPing Chen 1 MAE Department Arizona State University Tempe, AZ , USA rpacheco@asu.edu k.p.chen@asu.edu 4 Arturo Pacheco-Vega 4 Department of Mechanical Engineering California State University, Los Angeles Los Angeles, CA 90032, USA apacheco@calstatela.edu ABSTRACT The mixing of a diffusive passive-scalar driven by the motion of liquid induced by an applied potential across a microchannel is studies numerically. Secondary time-dependent periodic or random electric fields orthogonal to the main stream are applied to generate the cross-sectional mixing. Our investigation focuses on the the mixing dynamics and its dependence on the frequency of the driving mechanism. For periodic flows, we document that the probability density function (PDF) of the scaled concentration settles into a self-similar curve; in this case, the scalar field shows spatially repeating patterns. In contrast, for random flows there is a lack of self-similarity in the PDF for the interval of time considered in this investigation. Our study confirms an exponential decay of the variance of concentration for the periodic and random flows. 1 INTRODUCTION Rapid and efficient mixing for micro-scale flow has become a very active area of research due to the emergence of microfluidic devices for applications in biological and chemical analysis. Conventional methods used in generating mixing in macro-scale fluid flows require sufficiently large Reynolds numbers and they become ineffective when applied to micro-scale flows. Strategies to enhance mixing in a micro-channel include passive meth- Address all correspondence to this author. ASU P.O. Box rpacheco@asu.edu ods and active methods. Passive methods (also known as static mixers) do not require external forces, except for those to deliver the fluid, and the mixing process relies entirely on diffusion or chaotic advection [1, 2]. Active methods use disturbances generated by an external field for the mixing process, need external power sources for their operation, and the structures may require complex fabrication processes. Integration into a microfluidic system can also be challenging and expensive. As electroosmotic flows (EOFs) can achieve higher volumetric flow rates than pressure-gradient flows, and can be controlled using electric fields instead of moving parts, they have proven an attractive method for transporting and manipulating fluids in microdevices, particularly for channels. In many applications the EDL is very thin, allowing to specify a slip-velocity at the solid wall of the micro-channel (known as the Smolouchewski slip-velocity) [3 7]. The slip-velocity is related to the strength of the electric field and the so-called zetapotential, which is the static electric potential difference across the EDL. Several methods have been proposed for enhanced mixing in EOFs of electrolyte solutions in micro-channels. These range from (i) uniform electric fields in rectangular cavities, with uniform and non-uniform zeta-potentials, to drive a twodimensional, time-independent or time-dependent, EOFs [3,4,8]; (ii) using electro-kinetic instabilities in the Stokes flow regime caused by conductivity gradients in the EOFs [6, 7]; and (iii) electric fields transversal to the main axis of the channel with 1 Copyright c 2009 by ASME

2 non-uniform electrical properties of the fluid [9 13]. An alternative idea for generating mixing in electro-osmotic flows in a long rectangular three-dimensional micro-channel was proposed in [14]. The method consists on applying a transverse electric field to promote mixing in the channel by generating periodic and random velocity fields [15, 16] in which the zeta-potential can be controlled via external power supply and energized electrodes. In addition to increasing the strength of the stochasticity or increasing the magnitude of the electric field (which would obviously generate better mixing), it was found that by carefully selecting the intervals of duration for which the electric fields were on and off, mixing was optimal and Taylor dispersion effects were reduced. This process is perhaps more closely related to the cross-channel micro-mixer first suggested by Volpert et al (Proc. ASME Intern. Congress, MEMS, Vol 1, 1999), and addressed in papers such as [17], Dodge et al (Comptes Rendus Physique, 2004) and [18]. The Bottausci paper may be particularly relevant, since it offers a three-dimensional analysis for this problem. Regardless of the physical mechanism that advects the flow to generate the mixing, the fundamental interest for an appropriate design of a micromixer is how well it mixes. The analysis of mixing can be broadly classified in two categories depending on whether the effects of diffusion are considered or not. When the effects of molecular diffusion on the evolution of the scalar are taken into account, the characterization of mixing performance is based on the analysis of a passive tracer that is advected by the velocity field u(x,t). This implies a solution to the advection-diffusion equation t c + (cu) = D m 2 c, (1) where c(x,t) is the tracer field and D m > 0 is the the molecular diffusivity. Because the incorporation of weak diffusion into the analysis, the variance of the tracer field c(x, t), without sources, fluxes or sinks at the boundaries, will decrease in time. The rate of decay of variance is an important property of the system, and measures the quality of mixing [19, 20]. In the absence of diffusion (D m = 0), equation (1) states that the value of the scalar c(x,t) is constant in a moving fluid element. The trajectory of the fluid element is given by the kinematic equation dr/dt = v(r,t) with initial condition r(x,t o ) = x. Computationally, v(r,t) is obtained from a bilinear interpolation from the nodal values of the Eulerian velocity field u(x,t) from which statistical properties of particle trajectories such as Lyapunov exponents, Poincaré maps, probability density functions (PDFs) of finite-time stretching rates are extracted [16, 21, 22]. The study of the trajectories of passive tracer particles without diffusion is important to determine regions of stretching. However, the analysis of trajectories is not sufficient if we require information on the dispersion of the reagent in the flow field. In the former case, studies of two-dimensional fully chaotic flows have shown to exhibit an exponential decay of scalar variance in the long time limit [21, 23, 24]. It has been argued that if the domain scale is significantly larger than the flow scale, the description based on Lyapunov exponents alone can be inadequate for the prediction of decay rates during the final stage of mixing [25, 26]. Accurate experimental measurements of [27] and [28] in chaotic two-dimensional time-periodic flows revealed an exponential decay in tracer variance, with evidence for persistent spatial patterns in the concentration field. These repeating patterns are also known as strange eigenmodes; at the late stages of mixing, the scalar represents a periodic eigenfunction of the linear advection-diffusion equation. The appearance of self-similar asymptotic probability density function (PDF) of the scalar field normalized by its variance, suggests that strange or statistical eigenmodes appear in flows with aperiodic time-dependence and has been examined in detail by [29,30], [24], and [31]. [32] studied the PDF for a decaying passive scalar advected by deterministic velocities. They found a lack of self-similarity in the PDFs with time-periodic flows and self-similar PDFs in steady flows. [30] have shown among other things, that for this advectiondiffusion problem, a finite-dimensional inertial manifold exist, in which the decay of the tracer at the late stages of mixing is governed by the structure of the slowest-decaying mode in the manifold. Motivated by the studies above, we detail a numerical study of an active method for enhancing mixing of a passive tracer in a three-dimensional channel. We solve the exact threedimensional governing-equations, which include the effect of the electric double layers. Our aim is to investigate how the randomization protocols proposed affect the mixing dynamics, to assess if spatially repeating patterns develop for these flows, and to determine if these periodic and random flows generate exponential decays of variance in concentration. The article is organized as follows: section 2.1 briefly describes the governing equations and numerical method. The results from the numerical solver are presented in Section 3. A summary of the results is presented in Section 4, which concludes the paper. 2 Governing equations and the numerical scheme Consider the electro-osmotic flow in a long rectangular micro-channel with height 2H, and a width of 2W as shown in figure 2. The primary steady flow along the channel is driven by a steady electric field along the x-direction, and the transverse flow is driven by a secondary electric field perpendicular to the main stream of the channel. The motion in the axial direction is driven by the zetapotential ˆζ on the top and bottom surfaces ŷ = ±H, have the same value, ˆζ = ˆζ 0. On the side-walls ẑ = ±W, ˆζ = 0. 2 Copyright c 2009 by ASME

3 φ/ y = 0 on the top and bottom walls, and Dirichlet boundary conditions on the lateral walls selected according to { δ2,i if α(t) > 0, φ i = δ 3,i if α(t) < 0. (7) Figure 1. Schematic of the flow apparatus to generate the velocity fields. as The geometry of the EOF micromixer studied by Pacheco and coworkers [14 16] is depicted in Figure 2 relative to the Cartesian coordinate system ( ˆx,ŷ,ẑ) with unit vectors (i, j, k). It consists of a rectangular channel with height 2H, and a width of 2W. The zeta-potential ˆζ on the top and bottom surfaces ŷ = ±H, have the same value, ˆζ = ˆζ 0. On the side-walls ẑ = ±W, ˆζ = Governing equations In the micromixer described in the previous paragraph, the governing equations read ReSt t u = p + 2 u K 2 ψ(c L i φ C 0 ψ),(2) St t c + (cu) = Pe 1 2 c, (3) 2 ψ = K2 ψ, (4) 2 φ = 0. (5) where the unsteady term on the left-hand-side of (2) is significant only for high frequency unsteady flows. In the (2) we have assumed that Re is small so that the convective nonlinear term in the Navier-Stokes equation is negligible. The velocity vector u = u L + u T, where u L = u(y,z)i is the velocity along the longitudinal direction of the channel, and u T = v(y,z,t)j + w(y,z,t)k is the cross-sectional velocity Equations (2-refeq:laplace) are solved subject to no-slip boundary conditions for the velocity on all walls u(±a,±1) = 0, no mass flux normal to the wall c n = 0 (n is the unit normal vector directed into the fluid), where a = 2W/2H is defined as the width-to-height aspect ratio. The prescribed values of the electrical potential ψ on the walls are ψ(±a,z) = 1 and ψ(y,±1) = 0. The electric potential φ is obtained by solving the linear equation 2 φ with linear boundary conditions, thus allowing us to write φ(y,z,t) = φ i (t)f i (y,z). (6) with i = 1,2,3,4. The functions F i (y,z) are solutions of the Laplace equation (5), with Neumann boundary conditions Here, α(t) depends on the manner the electric fields alternate (see [16]), which represents a time-like interval of variable length, i.e., α(t) = sin(2πt/t γ) + εβ. The random variable with normal distribution, β, takes values between [-1,1] and acts as a vertical shift that modifies the values of α(t) above and below zero, whereas γ = arcsin(εβ) is a phase shift. The value of σ is related to the phase shift γ by σ = 1/2 γ/t, and represents the fraction of the interval T in which the value of α(t) > 0. In the proposed scheme, we ensure that within the time interval T, φ 2 and φ 3 are on/off only once. This is achieved by setting the value of the shift 0 ε 1. Note that the smaller the value of ε the smaller the strength of the perturbation. 2.2 Numerical integration Computer simulations of the flow in micro-scale systems enable easy control of the flow parameters and extensive data collection, while physical micro-channel devices enable verification testing. The unsteady governing equations have been considered in a Cartesian coordinate frame and discretized on a staggered mesh by central second-order accurate finite-difference approximations for the viscous terms. The resulting discretized system is then solved by a fractional-step procedure with approximate factorization. The two-dimensional elliptic equations for φ and ψ were solved using the FISHPACK package [35]. The time integration is performed with a third-order low-storage Runge-Kutta scheme [36 38]. The useful feature of this scheme is the possibility to advance in time by a variable time step, without reducing the accuracy or introducing interpolations. The Poisson equation for pressure is inverted with a Multigrid method and introduced at the old time step to simplify the boundary conditions for the intermediate velocity field. The code is written in fortran 77 and uses openmp directives allowing the use of multi-processor shared-memory computers. We found that twelve mesh points inside the Debye boundary layer were sufficient to resolve it, and achieve grid-independent results. During the computations the time step value was set to satisfy the Courant-Friedrichs-Lewy criterion CFL = 2. In order to ensure the correct implementation of the numerical scheme previously mentioned, the velocity fields were compared to those obtained using a a collocated arrangement of the variables on the grid, [39, 40]. It was found that the numerical results using the staggered layout agreed with those using the collocated arrangement. 3 Copyright c 2009 by ASME

4 (a) T = 3. (a) T = 10. (b) T = 10. (b) T = 20. Figure 2. Evolution of the concentration at multiple times showing the convergence to a strange eigenmode. The parameters are: ε = 0, Re = 0.04 and Pe = (c) T = Results We present the locations of 10,000 passive non-diffusive particles for ε = 0,0.5 and 1 and T = 3,10 and 20. The particles in figure 3 are initially located within the circular disk of radius of 0.1 centered at (x, y, z) = (1, 0, 0). When the flow is periodic (ε = 0) there are two regions of poor mixing whose size decreases as T increases from 3 to 20 as shown in the left-hand-side of figures 3(a) 3(c). As the strength of the randomization ε increases from 0 to 1, the particles spread to a wider region (see figure 3), covering nearly the entire yzplane for all the values of T considered. The jittering caused by ε acts like a random agitation for the passive particles and allows KAM tori to break. This random agitation is similar to the diffusion effect, and these cross-sectional diffusive transport intensifies with an increase in both T and ε. This behavior is well known and has been documented for a variety of systems using both, experiments and numerical simulations. If the scalar evolution is self-similar, that is stationary for χ, then the decay rate of the moments will be prescribed by c(x,t) n e υ nt [41] with υ n growing linearly with n. This theoretical prediction is also valid when υ n is a nonlinear function of n, in this case, the moments are not stationary. We verify that the moments decay as predicted by the theory by extracting the slope from the values of υ n using the portion of the curves Figure 3. Influence of ε on the location of 10,000 passive non-diffusive particles projected onto the yz-plane at t = 500. The particles are initially located within a circular disk of radius 0.1 with center at (x,y,z)=(1,0,0). showing linear behavior in the semi-log plot of figure 4. The equations of the lines are plotted in figure 5 along with the values of υ n showing that decay-rates grow linearly with n. The long-time behavior of the re-scaled moments χ n for n = 2,3,4 is depicted in figure 7. Note that the moments become stationary which also implies that υ n is a linear function of n, and the PDF of the re-scaled variable (H(χ)) must be self-similar. Note also that for higher values of T the time at which the moments become stationary decreases. The evolution towards the self-similar stage for H(χ) is shown in figure 7 for T = 10 and 20, with ε = 0. The regions that remain relatively isolated are captured by the bi-modal characteristic of H(χ) shown in figure 7(a). The bi-modal nature of H(χ) tends to disappear due to the reduction in the size of the region of poor mixing, forming a long core and tailed PDF, as shown in figure 7(b). Self similar behavior also appears for other values of Pe, except that for higher values of Pe, the time at which the evolution becomes self-similar increases. The long-time evolution of H(χ) and the moments are shown in figure 8 for T = 20 and ε = 1. Since the flow is fully 4 Copyright c 2009 by ASME

5 (a) T = 10. (a) T = 10. (b) T = 20. Figure 4. Decay of the various moments for ε = 0, Re = 0.04, Pe = 10 4., n = 2;, n = 3;, n = 4. chaotic, H(χ) is unimodal with a Gaussian structure, but never settles down into a self-similar eigenmode as demonstrated by the oscillatory nature of the higher-order moments of figure 8. [31] has argued that in the strange eigenmode regime the maximum scale of variation of the scalar field (l c ) is of the same order as the scale of variation of the velocity field (l v ), and that the exponential decay of moments is also valid when l c l v but in this case, υ n is a nonlinear function of the order of the moment n. For small values of Pe, large-amplitude fluctuations of the rescaled moments were reported. In our study, the periodic flow (Pe = 10 4 ) is in the selfsimilar regime with l c l v and an exponential decay of the (b) T = 20. Figure 5. The extracted values of υ n vs n for ε = 0, Re = 0.04, Pe = The symbols correspond to υ n and the line to υ n = (n 1)υ 2. rescaled moments. For the random flow (ε = 1), the exponential decay of the moments is also exponential, but it does not exhibit self-similarity for the PDFs at least for the times considered in this investigation. Since the introduction of randomization in the process is equivalent to increasing the diffusion, and the results of self-similarity are valid for large Pe, we could expect that a strong modulation of the flow would produce fluctuations with large amplitude in c(x,t) n in the late stages of mixing, preventing the PDFs from becoming stationary. We argue that the 5 Copyright c 2009 by ASME

6 (a) T = 10. (a) T = 10 (b) T = 20. Figure 6. Long-time behavior of the moments χ n ;, n = 2;, n = 3;, n = 4. (b) T = 20 Figure 7. Probability density function of the scaled concentration H(χ) for the periodic flow (ε = 0): t = 0;, t = 900;, t = 1,000. lack of self-similarity of the PDFs is due to the large number of eigenmodes of the advection-diffusion operator that are present throughout the randomization process [30]. Nevertheless, the asymptotic self-similarity of the tracer PDFs remains to be established for our velocity fields with aperiodic time-dependence. 4 Summary We have discussed the enhancement effect on mixing induced by a periodic and random velocities in an electro-osmotic flow of an electrolyte solution flowing inside a three-dimensional micro-channel. We analyzed the effect of the Debye layer on the axial and transverse velocities and its effect on the Taylor dispersion effects. We found that for flows with more effective mixing in the cross-section Taylor dispersion effects are less pronounced. The PDF of the scaled concentration settles into a self-similar curve for the periodic flows in which the scalar field shows spatially repeating patterns. When stochasticity is introduced, the random flow does not enter into the strange eigenmode regime for the interval of time considered in this investigation. We confirm an exponential decay of the variance of concentration for the periodic and random flows. 6 Copyright c 2009 by ASME

7 ACKNOWLEDGMENT Computational resources for this work were provided by the Ira A. Fulton High Performance Computing Initiative at ASU. (a) H(χ): t = 0;, t = 900;, t = 1,000 (b) χ n :, n = 2;, n = 3;, n = 4. (c) c n :, n = 2;, n = 3;, n = 4. Figure 8. Random flow with ε = 1, T = 20 and Pe = REFERENCES [1] Stone, H. A., Stroock, A. D., and Ajdari, A., Engineering flows in small devices: microfluidics toward a lab-on-a-chip. Ann. Rev. Fluid Mech., 36 [], pp [2] Nguyen, N.-T., and Wu, Z., Micromixers-a review. J. Micromech. Microeng., 15 [], pp [3] Qian, S., and Bau, H. H., A chaotic electroosmotic stirrer. Anal. Chem., 74 [], pp [4] Qian, S., and Bau, H. H., Theoretical investigation of electro-osmotic flows and chaotic stirring in rectangular cavities. Appl. Math. Modelling, 29 [], pp [5] Zhao, H., and Bau, H. H., Microfluidic chaotic stirrer utilizing induced-charge electroosmosis. Phys. Rev. E, 75 [], p [6] Oddy, M. H., Santiago, J. G., and Mikkelsen, J. C., Electrokinetic instability micromixing. Anal. Chem., 73 [], pp [7] Lin, H.,, Storey, B. D., Oddy, M. H., Chen, C.-H., and Santiago, J. G., Instability of electrokinetic microchannel flows with conductivity gradients. Phys. Fluids, 16 [], pp [8] Patankar, N. A., and Hu, H. H., Numerical simulation of electroosmotic flow. Anal. Chem., 70 (9) [], pp [9] El Moctar, A. B., Aubry, N., and Batton, J., Electrohydrodynamic micro-fluicic mixer. Lab on a Chip, 3 [], pp [10] Glasgow, I., Batton, J., and Aubry, N., Electroosmotic mixing in microchannels. Lab on a Chip, 4 [], pp [11] Ozen, O., Aubry, N., Papageorgiou, D. T., and Petropoulos, P. G., Electrohydrodynamic linear stability of two immiscible fluids in channel flow. Electrochimica Acta, 51 [], p [12] Li, F., Ozen, O., Aubry, N., Papageorgiou, D. T., and Petropoulos, P. G., Linear stability of a two-fluid interface for electrohydrodynamic mixing in a channel. J. Fluid Mech., 583 [], pp [13] Uguz, A. K., Ozen, O., and Aubry, N., Electric field effect on a two-fluid interface instability in channel flow for fast electric times. Phys. Fluids, 20 (3) [], p [14] Pacheco, J. R., Chen, K. P., and Hayes, M. A., Efficient and rapid mixing in a slip-driven three-dimensional flow in a rectangular channel. Fluid Dynam. Res., 38 (8) [], pp [15] Pacheco, J. R., Chen, K. P., Pacheco-Vega, A., Chen, B., and Hayes, M. A., Chaotic mixing enhancement in 7 Copyright c 2009 by ASME

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