March 7, 2009 10:31

Proceedings of Proceedings of the HT 20092009 ASME Summer Heat Transfer Conference

July , 2009, San Francisco, California USA

HT2009-88626

ON THE DEPENDENCE OF THE RATE OF DECAY IN CONCENTRATION AND/ORTEMPERATURE DISTRIBUTION FOR A PASSIVE SCALAR IN A MICRO-MIXER AND

THE FREQUENCY OF THE ADVECTING VELOCITY FIELDS

J. Rafael Pacheco∗, KangPing Chen1MAE Department

Arizona State UniversityTempe, AZ 85287-6106, USA

Email: rpacheco@asu.eduEmail: k.p.chen@asu.edu

4Arturo Pacheco-Vega4Department of Mechanical EngineeringCalifornia State University, Los Angeles

Los Angeles, CA 90032, USAEmail: apacheco@calstatela.edu

ABSTRACTThe mixing of a diffusive passive-scalar driven by the motion

of liquid induced by an applied potential across a microchannelis studies numerically. Secondary time-dependent periodic orrandom electric fields orthogonal to the main stream are appliedto generate the cross-sectional mixing. Our investigation focuseson the the mixing dynamics and its dependence on the frequencyof the driving mechanism. For periodic flows, we document thatthe probability density function (PDF) of the scaled concentra-tion settles into a self-similar curve; in this case, the scalar fieldshows spatially repeating patterns. In contrast, for random flowsthere is a lack of self-similarity in the PDF for the interval oftime considered in this investigation. Our study confirms an ex-ponential decay of the variance of concentration for the periodicand random flows.

1 INTRODUCTIONRapid and efficient mixing for micro-scale flow has become

a very active area of research due to the emergence of microflu-idic devices for applications in biological and chemical analysis.Conventional methods used in generating mixing in macro-scalefluid flows require sufficiently large Reynolds numbers and theybecome ineffective when applied to micro-scale flows. Strate-gies to enhance mixing in a micro-channel include passive meth-

∗Address all correspondence to this author. ASU P.O. Box 6106. Emailrpacheco@asu.edu

ods and active methods. Passive methods (also known as staticmixers) do not require external forces, except for those to deliverthe fluid, and the mixing process relies entirely on diffusion orchaotic advection [1, 2]. Active methods use disturbances gen-erated by an external field for the mixing process, need exter-nal power sources for their operation, and the structures may re-quire complex fabrication processes. Integration into a microflu-idic system can also be challenging and expensive. As electro-osmotic flows (EOFs) can achieve higher volumetric flow ratesthan pressure-gradient flows, and can be controlled using elec-tric fields instead of moving parts, they have proven an attrac-tive method for transporting and manipulating fluids in micro-devices, particularly for channels.

In many applications the EDL is very thin, allowing to spec-ify a ‘slip-velocity’ at the solid wall of the micro-channel (knownas the ‘Smolouchewski’ slip-velocity) [3–7]. The slip-velocity isrelated to the strength of the electric field and the so-called zeta-potential, which is the static electric potential difference acrossthe EDL.

Several methods have been proposed for enhanced mix-ing in EOFs of electrolyte solutions in micro-channels. Theserange from (i) uniform electric fields in rectangular cavities,with uniform and non-uniform zeta-potentials, to drive a two-dimensional, time-independent or time-dependent, EOFs [3,4,8];(ii) using electro-kinetic instabilities in the Stokes flow regimecaused 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

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 wasproposed in [14]. The method consists on applying a trans-verse electric field to promote mixing in the channel by gener-ating periodic and random velocity fields [15, 16] in which thezeta-potential can be controlled via external power supply andenergized electrodes. In addition to increasing the strength ofthe stochasticity or increasing the magnitude of the electric field(which would obviously generate better mixing), it was foundthat by carefully selecting the intervals of duration for which theelectric fields were on and off, mixing was optimal and Taylordispersion effects were reduced. This process is perhaps moreclosely related to the cross-channel micro-mixer first suggestedby Volpert et al (Proc. ASME Intern. Congress, MEMS, Vol1, 1999), and addressed in papers such as [17], Dodge et al(Comptes Rendus Physique, 2004) and [18]. The Bottausci papermay be particularly relevant, since it offers a three-dimensionalanalysis for this problem.

Regardless of the physical mechanism that advects the flowto generate the mixing, the fundamental interest for an appropri-ate design of a micromixer is how well it mixes. The analysis ofmixing can be broadly classified in two categories depending onwhether the effects of diffusion are considered or not.

When the effects of molecular diffusion on the evolution ofthe scalar are taken into account, the characterization of mixingperformance is based on the analysis of a passive tracer that isadvected by the velocity field uuu(xxx, t). This implies a solution tothe advection-diffusion equation

∂tc+∇ · (cuuu) = Dm∇2c, (1)

where c(x, t) is the tracer field and Dm > 0 is the the moleculardiffusivity. Because the incorporation of weak diffusion into theanalysis, the variance of the tracer field c(x, t), without sources,fluxes or sinks at the boundaries, will decrease in time. The rateof decay of variance is an important property of the system, andmeasures the quality of mixing [19, 20].

In the absence of diffusion (Dm = 0), equation (1) states thatthe value of the scalar c(xxx, t) is constant in a moving fluid ele-ment. The trajectory of the fluid element is given by the kine-matic equation drrr/dt = vvv(rrr, t) with initial condition rrr(xxx, to) = xxx.Computationally, vvv(rrr, t) is obtained from a bilinear interpolationfrom the nodal values of the Eulerian velocity field uuu(xxx, t) fromwhich statistical properties of particle trajectories such as Lya-punov exponents, Poincare maps, probability density functions(PDFs) of finite-time stretching rates are extracted [16, 21, 22].The study of the trajectories of passive tracer particles withoutdiffusion is important to determine regions of stretching. How-ever, the analysis of trajectories is not sufficient if we requireinformation on the dispersion of the reagent in the flow field. In

the former case, studies of two-dimensional fully chaotic flowshave shown to exhibit an exponential decay of scalar variancein the long time limit [21, 23, 24]. It has been argued that if thedomain scale is significantly larger than the flow scale, the de-scription based on Lyapunov exponents alone can be inadequatefor the prediction of decay rates during the final stage of mix-ing [25, 26].

Accurate experimental measurements of [27] and [28] inchaotic two-dimensional time-periodic flows revealed an expo-nential decay in tracer variance, with evidence for persistentspatial patterns in the concentration field. These repeating pat-terns are also known as strange eigenmodes; at the late stages ofmixing, the scalar represents a periodic eigenfunction of the lin-ear advection-diffusion equation. The appearance of self-similarasymptotic probability density function (PDF) of the scalar fieldnormalized by its variance, suggests that strange or statisticaleigenmodes appear in flows with aperiodic time-dependence andhas been examined in detail by [29,30], [24], and [31]. [32] stud-ied the PDF for a decaying passive scalar advected by determin-istic velocities. They found a lack of self-similarity in the PDFswith time-periodic flows and self-similar PDFs in steady flows.[30] have shown among other things, that for this advection-diffusion problem, a finite-dimensional inertial manifold exist,in which the decay of the tracer at the late stages of mixing isgoverned by the structure of the slowest-decaying mode in themanifold.

Motivated by the studies above, we detail a numerical studyof an active method for enhancing mixing of a passive tracerin a three-dimensional channel. We solve the exact three-dimensional governing-equations, which include the effect of theelectric double layers. Our aim is to investigate how the random-ization protocols proposed affect the mixing dynamics, to assessif spatially repeating patterns develop for these flows, and to de-termine if these periodic and random flows generate exponentialdecays of variance in concentration.

The article is organized as follows: section 2.1 briefly de-scribes the governing equations and numerical method. The re-sults from the numerical solver are presented in Section 3. Asummary of the results is presented in Section 4, which con-cludes the paper.

2 Governing equations and the numerical schemeConsider the electro-osmotic flow in a long rectangular

micro-channel with height 2H, and a width of 2W as shown infigure 2. The primary steady flow along the channel is drivenby a steady electric field along the x-direction, and the transverseflow is driven by a secondary electric field perpendicular to themain stream of the channel.

The motion in the axial direction is driven by the zeta-potential ζ on the top and bottom surfaces y = ±H, have thesame value, ζ = ζ0. On the side-walls z =±W , ζ = 0.

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Figure 1. Schematic of the flow apparatus to generate the velocity fields.

as The geometry of the EOF micromixer studied by Pachecoand coworkers [14–16] is depicted in Figure 2 relative to theCartesian coordinate system (x, y, z) with unit vectors (iii, jjj,kkk). Itconsists of a rectangular channel with height 2H, and a width of2W . The zeta-potential ζ on the top and bottom surfaces y =±H,have the same value, ζ = ζ0. On the side-walls z =±W , ζ = 0.

2.1 Governing equationsIn the micromixer described in the previous paragraph, the

governing equations read

ReSt ∂tuuu = −∇p∗+∇2uuu−K2

ψ(CLiii−∇⊥φ−C0∇⊥ψ) ,(2)St∂tc+∇ · (cuuu) = Pe−1

∇2c, (3)

∇2⊥ψ = K2

ψ, (4)∇

2⊥φ = 0. (5)

where the unsteady term on the left-hand-side of (2) is signifi-cant only for high frequency unsteady flows. In the (2) we haveassumed that Re is small so that the convective nonlinear termin the Navier-Stokes equation is negligible. The velocity vectoru = uL +uT , where uL = u(y,z)i is the velocity along the longi-tudinal direction of the channel, and uT = v(y,z, t)j + w(y,z, t)kis the cross-sectional velocity

Equations (2-refeq:laplace) are solved subject to no-slipboundary 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 normalvector directed into the fluid), where a = 2W/2H is defined as thewidth-to-height aspect ratio. The prescribed values of the electri-cal 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) = φi(t)Fi(y,z). (6)

with i = 1,2,3,4. The functions Fi(y,z) are solutions ofthe Laplace equation (5), with Neumann boundary conditions

∂φ/∂y = 0 on the top and bottom walls, and Dirichlet boundaryconditions on the lateral walls selected according to

φi ={

δ2,i if α(t) > 0,δ3,i if α(t) < 0.

(7)

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 nor-mal distribution, β, takes values between [-1,1] and acts as a ver-tical 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 relatedto the phase shift γ by σ = 1/2−γ/T , and represents the fractionof the interval T in which the value of α(t) > 0.

In the proposed scheme, we ensure that within the time in-terval T , φ2 and φ3 are on/off only once. This is achieved bysetting the value of the shift 0≤ ε≤ 1. Note that the smaller thevalue of ε the smaller the strength of the perturbation.

2.2 Numerical integrationComputer simulations of the flow in micro-scale systems en-

able easy control of the flow parameters and extensive data col-lection, while physical micro-channel devices enable verificationtesting. The unsteady governing equations have been consideredin a Cartesian coordinate frame and discretized on a staggeredmesh by central second-order accurate finite-difference approxi-mations for the viscous terms. The resulting discretized systemis then solved by a fractional-step procedure with approximatefactorization. The two-dimensional elliptic equations for φ andψ were solved using the FISHPACK package [35]. The time inte-gration is performed with a third-order low-storage Runge-Kuttascheme [36–38]. The useful feature of this scheme is the possi-bility to advance in time by a variable time step, without reducingthe accuracy or introducing interpolations. The Poisson equa-tion for pressure is inverted with a Multigrid method and intro-duced at the old time step to simplify the boundary conditions forthe intermediate velocity field. The code is written in fortran 77and uses openMP directives allowing the use of multi-processorshared-memory computers. We found that twelve mesh pointsinside the Debye boundary layer were sufficient to resolve it, andachieve grid-independent results. During the computations thetime step value was set to satisfy the Courant-Friedrichs-Lewycriterion CFL = 2. In order to ensure the correct implementa-tion of the numerical scheme previously mentioned, the velocityfields were compared to those obtained using a a collocated ar-rangement of the variables on the grid, [39, 40]. It was foundthat the numerical results using the staggered layout agreed withthose using the collocated arrangement.

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(a) T = 10.

(b) T = 20.

Figure 2. Evolution of the concentration at multiple times showing theconvergence to a strange eigenmode. The parameters are: ε = 0, Re =0.04 and Pe = 104.

3 ResultsWe present the locations of 10,000 passive non-diffusive

particles for ε = 0,0.5 and 1 and T = 3,10 and 20. The parti-cles in figure 3 are initially located within the circular disk ofradius of 0.1 centered at (x,y,z) = (1,0,0).

When the flow is periodic (ε = 0) there are two regions ofpoor mixing whose size decreases as T increases from 3 to 20 asshown in the left-hand-side of figures 3(a)–3(c). As the strengthof the randomization ε increases from 0 to 1, the particles spreadto a wider region (see figure 3), covering nearly the entire yz-plane for all the values of T considered. The jittering causedby ε acts like a random agitation for the passive particles andallows KAM tori to break. This random agitation is similar tothe diffusion effect, and these cross-sectional diffusive transportintensifies with an increase in both T and ε. This behavior is wellknown and has been documented for a variety of systems usingboth, experiments and numerical simulations.

If the scalar evolution is self-similar, that is stationary forχ, then the decay rate of the moments will be prescribed by〈|c(xxx, t)|n〉 ∼ e−υnt [41] with υn growing linearly with n. Thistheoretical prediction is also valid when υn is a nonlinear func-tion of n, in this case, the moments are not stationary. We verifythat the moments decay as predicted by the theory by extractingthe slope from the values of υn using the portion of the curves

(a) T = 3.

(b) T = 10.

(c) T = 20.

Figure 3. Influence of ε on the location of 10,000 passive non-diffusiveparticles projected onto the yz-plane at t = 500. The particles are initiallylocated 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. Theequations of the lines are plotted in figure 5 along with the val-ues of υn showing that decay-rates grow linearly with n. Thelong-time behavior of the re-scaled moments 〈|χ|n〉 for n = 2,3,4is depicted in figure 7. Note that the moments become station-ary which also implies that υn is a linear function of n, and thePDF of the re-scaled variable (H(χ)) must be self-similar. Notealso that for higher values of T the time at which the momentsbecome stationary decreases.

The evolution towards the self-similar stage for H(χ) isshown in figure 7 for T = 10 and 20, with ε = 0. The regionsthat remain relatively isolated are captured by the bi-modal char-acteristic of H(χ) shown in figure 7(a). The bi-modal nature ofH(χ) tends to disappear due to the reduction in the size of theregion of poor mixing, forming a long core and tailed PDF, asshown in figure 7(b). Self similar behavior also appears for othervalues of Pe, except that for higher values of Pe, the time at whichthe evolution becomes self-similar increases.

The long-time evolution of H(χ) and the moments areshown in figure 8 for T = 20 and ε = 1. Since the flow is fully

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(a) T = 10.

(b) T = 20.

Figure 4. Decay of the various moments for ε = 0, Re = 0.04, Pe =104. ——, n = 2; –◦–, n = 3; –4–, n = 4.

chaotic, H(χ) is unimodal with a Gaussian structure, but neversettles down into a self-similar eigenmode as demonstrated bythe oscillatory nature of the higher-order moments of figure 8.

[31] has argued that in the strange eigenmode regime themaximum scale of variation of the scalar field (lc) is of the sameorder as the scale of variation of the velocity field (lv), and thatthe exponential decay of moments is also valid when lc� lv butin this case, υn is a nonlinear function of the order of the momentn. For small values of Pe, large-amplitude fluctuations of therescaled moments were reported.

In our study, the periodic flow (Pe = 104) is in the self-similar regime with lc ∼ lv and an exponential decay of the

(a) T = 10.

(b) T = 20.

Figure 5. The extracted values of υn vs n for ε = 0, Re = 0.04, Pe =104. The symbols correspond to υn and the line to υn = (n−1)υ2.

rescaled moments. For the random flow (ε = 1), the exponentialdecay of the moments is also exponential, but it does not exhibitself-similarity for the PDFs at least for the times considered inthis investigation. Since the introduction of randomization in theprocess is equivalent to increasing the diffusion, and the resultsof self-similarity are valid for large Pe, we could expect that astrong modulation of the flow would produce fluctuations withlarge amplitude in 〈|c(xxx, t)|n〉 in the late stages of mixing, pre-venting the PDFs from becoming stationary. We argue that the

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(a) T = 10.

(b) T = 20.

Figure 6. Long-time behavior of the moments 〈|χ|n〉; —–, n = 2; –◦–,n = 3; –4–, n = 4.

lack of self-similarity of the PDFs is due to the large number ofeigenmodes of the advection-diffusion operator that are presentthroughout the randomization process [30]. Nevertheless, theasymptotic self-similarity of the tracer PDFs remains to be es-tablished for our velocity fields with aperiodic time-dependence.

4 SummaryWe have discussed the enhancement effect on mixing in-

duced by a periodic and random velocities in an electro-osmoticflow of an electrolyte solution flowing inside a three-dimensionalmicro-channel. We analyzed the effect of the Debye layer on

(a) T = 10

(b) T = 20

Figure 7. Probability density function of the scaled concentration H(χ)for the periodic flow (ε = 0): � t = 0;4, t = 900; ◦, t = 1,000.

the axial and transverse velocities and its effect on the Taylordispersion effects. We found that for flows with more effec-tive mixing in the cross-section Taylor dispersion effects are lesspronounced. The PDF of the scaled concentration settles into aself-similar curve for the periodic flows in which the scalar fieldshows spatially repeating patterns. When stochasticity is intro-duced, the random flow does not enter into the strange eigenmoderegime for the interval of time considered in this investigation.We confirm an exponential decay of the variance of concentra-tion for the periodic and random flows.

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(a) H(χ): � t = 0;4, t = 900; ◦, t = 1,000

(b) 〈|χ|n〉: —–, n = 2; –◦–, n = 3; –4–, n = 4.

(c) 〈|c|n〉: —–, n = 2; –◦–, n = 3; –4–, n = 4.

Figure 8. Random flow with ε = 1, T = 20 and Pe = 104.

ACKNOWLEDGMENTComputational resources for this work were provided by the

Ira A. Fulton High Performance Computing Initiative at ASU.

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