depthwise averaging approach to cross-stream mixing in a pressure-driven micro channel flow

8/4/2019 Depthwise Averaging Approach to Cross-stream Mixing in a Pressure-driven Micro Channel Flow

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R E S E A R C H PA PE R

Y. C. Lam X. Chen C. Yang

Depthwise averaging approach to cross-stream mixing

in a pressure-driven microchannel flow

Received: 1 August 2004 / Accepted: 18 October 2004 / Published online: 11 May 2005 Springer-Verlag 2005

Abstract Microfluidic devices have many potentialapplications, such as BioMEMs (microelectromechani-cal systems for biomedical applications), miniature fuelcells, and microchannel cooling of electronic circuitry.

One of the important considerations of microfluidicdevices for analytical and bioanalytical chemistry is thedispersion of solutes. In this study, the dispersion ofpassive analyte between two miscible fluids of similarproperties in a side-by-side pressure-driven creeping flowis examined. This study represents a first effort inapplying the lubrication approximation together withthe depthwise averaging method to analyze mass trans-port of passive analyte in a two-stream rectangular mi-crochannel with consideration of the Taylor-Arisdispersion effect.

Keywords Mass diffusion/convection transport

Microfluidics

Pressure-driven flow

Taylor-Arisdispersion

List of symbolsg Viscosity of fluid (Pa s)"u Averaged velocity of fluid (m/s)u Velocity deviation from averaged velocity of

fluid (m/s)p Pressure (Pa)m Kinetic viscosity (m2/s)f Function describing distribution of quantity

of interest in depthwise directionL Length of rectangular slit in microfluidic

device (m)

D Molecular diffusion coefficient of analyte(m2/s)

x, y, z Cross-stream (width), depthwise (height),and axial (length) directions of microfluidic

flow, respectively/ Molar concentration of sample analyte at

any given point in microfluidic device (M)/0 Molar concentration of sample analyte at

inlet stream (M)Deff Effective dispersion coefficient of sample

analyte (m2/s)ux, uy, uz Velocity components of the fluid in the x, y,

and z directions, respectively (m/s)Q Flow rate of the fluid (m3/s)W Width of rectangular slit in microfluidic de-

vice (m)H Height of rectangular slit in microfluidic

device (m)b Half height of rectangular slit in microfluidic

device (m)/* Dimensionless concentration of analyte

/ "//0

z* Dimensionless length of microfluidic device

(z*=z/L)x* Dimensionless width of microfluidic device

(x*=x/W)Pe Peclet number Pe "uW=D

1 Introduction

Research in microfluidic devices and components hasbeen stimulated over the past decades due to itsincreasing importance to analytical and biomedicaltechnologies (Darwin et al. 2002; Pierre et al. 2002).Examples of manipulation of fluids in microfluidic de-vices include dynamic cell separations (Chiu et al. 2000;Li et al. 2000), surface patterning of cells and proteins

Y. C. Lam X. ChenSingaporeMIT Alliance Programme,Nanyang Technological University, 639798, Singapore

Y. C. Lam (&) X. Chen C. YangSchool of Mechanical and Production Engineering,Nanyang Technological University, 639798, SingaporeE-mail: [email protected].: +65-67905866Fax: +65-68627215

Microfluid Nanofluid (2005) 1: 218226DOI 10.1007/s10404-004-0013-8


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(David et al. 2002), delivery modules for mass spec-trometers (Chan et al. 1999; Jacobson et al. 1999), andthe mixing of analytes. In nearly every microfluidicformat, diffusion of analytes or particles is one of thefundamental aspects in microfluidics.

Flows driven by an external electric field based onelectroosmosis (Jeon et al. 2000) and an applied pressuregradient (Tallarek et al. 2000) are two methods com-monly used for continuous flow in microchannel devices.Electroosmotic flow has a wide range of applications asit has a uniform flow velocity profile and no movingparts (Yang et al. 2002). However, it is limited to polarsolvents and may lead to sample damage due to Jouleheating under certain conditions. In diffusional devices,pressure-driven flow is usually employed due to its rel-ative ease, flexibility of fabrication, and insensitivity tosurface contamination, ionic strength, controllable dif-fusion, etc. Such a kind of flow with continuous input ina microfluidic rectangular slit generates additionalcomplexity in the distribution of analytes because of theparabolic velocity gradient across the thickness dimen-sion (Probstein 1994). In addition, the breadth of such a

distribution is decreased or increased by diffusion acrossthe velocity gradient. As such, the distribution is highlydependent on the dispersion mechanism of an analyte.Phenomena specific to pressure-driven side-by-side flowin a microchannel with continuous input are onlybeginning to be investigated (Kamholz et al. 1999;Kamholz and Yager 2001, 2002; Weigl and Yager 1999;Ismagilov et al. 2000; Brody et al. 1997; Holden et al.2003a; 2003b; Virginie et al. 2002; Beard 2001; Dorfmanand Brenner 2001).

One device in which diffusion plays a crucial role isthe Y-shaped structure (Kamholz and Yager 2001;Weigl and Yager 1999; Ismagilov et al. 2000), which

utilizes the interdiffusion of analytes from two or moreinput streams to produce an analyte concentrationchange. The flow is strictly laminar and the masstransport between the analyte solution streams occursside-by-side through interdiffusion because of very lowReynolds numbers. Brody et al. (1997) presented someexamples of the design of a microfluidic device for bio-logical processes. Kamholz et al. (1999) proposed a one-dimensional analytical model to quantitatively describemolecular diffusion in the microchannel of a T-sensor byneglecting axial diffusion. In a subsequent work, a the-oretical analysis of the scaling law in the absence of axisdiffusion of the T-sensor based on molecular diffusion

was presented (Kamholz and Yager 2001). Moleculardiffusion between the two pressure-driven laminar flowsat high Peclet numbers was experimentally and theo-retically presented by Ismagilov (2000). They illustratedthe effects of high aspect ratio microchannel geometryon diffusion in a two-fluid stream flow by using a scalinglaw. A curved shape for the interdiffusion region as flowproceeds was proposed. This is due to increased diffu-sion time near the wall (where the one-third power lawdominates). They also pointed out that, very far downstream, under the conditions of high Peclet numbers, the

dependence of the diffusion layer thickness in thedepthwise direction diminishes (where the one-halfpower law dominates). However, they only presentedtheir findings qualitatively. Holden et al. (2003a, 2003b)developed an approximate model based on the channelstreamwise velocity to predict molecular diffusion be-tween two side-by-side fluids. However, in their model,they neglected the enhanced axial diffusion due toinhomogeneous velocity gradients along the depthwisedirection (i.e., the Taylor-Aris dispersion). In parallelwith the theoretical development, the finite element andfinite difference methods were widely employed to solvethe coupled Navier-Stokes and diffusionconvectionequations to describe the mass transport between fluidflows of two similar liquids (Kamholz and Yager 2002;Virginie et al. 2002; Beard 2001; Dorfman and Brenner2001). However, the main difficulty encountered in thenumerical calculations is the so-called numerical diffu-sion, which is associated with low values of diffusioncoefficients. This is known as pseudo-diffusion, whichintroduces inaccuracy in the computed results. Althoughthe use of very fine meshes can decrease numerical dif-

fusion, these fine meshes would result in severe compu-tational limitations, such as lengthy computational timeand large memory requirements.

Taylor dispersion (Taylor 1953, 1954) is a phenome-non associated with the flow of two miscible fluids.Taylor (1953) showed that, even though the axial dif-fusion is small, the combined effects of axial convectionand lateral diffusion could be important in a circulartube. Taylors results were later verified and extended byAris (1956) and Barton (1983) using the method ofmoment, and by Brenner and Edwards (1993) using themultiple pole expansion. Doshi et al. (1978) obtained anasymptotic value of convection coefficient as the aspect

ratio of rectangular conduit increases. Giddings andSchure (1978) and Dutta Leighton 2001, 2002, 2003)investigated the edge effect of the channel on dispersionreduction. Stone (1989) studied heat and mass transportfrom two-dimensional surface films to a fluid undergo-ing a simple shear flow and analogs to convective anddiffusive transport. However, the analysis is based on anumerical approach.

In this study, we studied analytically the dispersion ofpassive analyte between two miscible fluids of similarproperties in a side-by-side pressure-driven creeping flowin a rectangular microchannel. The so-called depthwiseaveraging method is used to consider the axis dispersion

of a solute along the axial direction. A two-dimensionalanalytical solution in terms of axis dispersion was pro-posed to simulate long duration convection and diffu-sion transport in a pressure-driven creeping flow for arectangular shape slit. The effect of the geometry of themicrofluidic slit was analyzed through the solution of thespecies mass conservative equation. This approach canpredict axial dispersion and does not have the short-comings of numerical methods, such as numerical dif-fusion at the interface between the two fluids and/orsignificant computational requirements.

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2 Mathematical formulations

2.1 Fully developed velocity

Consider two-stream pressure-driven flow side-by-sidebetween two parallel plates of a distance 2b apart (seeFig. 1), with a volumetric flow rate Q. A Cartesiancoordinate system is utilized. In a microfluidic device,

due to the low velocity of the flow and micron-sizedchannels, the Reynolds number (Re) relating the inertialforces to the viscous forces is usually small (Kamholzet al. 1999). This means that viscous forces play adominant role in microchannels rather than inertia. Forsimplicity, the flow down the microchannel is treatedgenerally as if it were flow between infinite parallelplates. This approximation has been justified by thelarge aspect ratio (the ratio of width W to height H) ofthe rectangular cross-section, which is around 13:1 in thepresent study. The difference in the average velocitybetween that with and without side-wall effect is lessthan 5% when the channel aspect ratio is larger than 10

(Doshi et al. 1978; Schlichting 1979). When Re(1, thefully developed velocity profile in the axial direction in arectangular microchannel structure can be derived fromthe Navier-Stokes equation under the lubrication ap-proximation and no-slip condition at the wall, given as(Bird et al. 1960):

uz y @P

@z

b2

2g1

y

b

2 !1

2.2 Mass transport of passive analyte in the microfluidicslit

2.2.1 Concentration distribution of analyte

The molecular diffusion equation, with the associatedCartesian coordinate system, is employed to describe themass transfer of analyte within the microfluidic domain(see Fig. 1). The solution to this equation is derived

using the depthwise averaging method (Bird et al. 1960).The advantage of this method is the elimination of thedepthwise dependence in the molecular diffusion equa-tion by introducing an effective transport coefficient.This effective transport parameter characterizes theconvectivediffusive transport in the microfluidic devicedomain.

In the microfluidic slit, the concentration field / of ananalyte in a homogenous fluid, without bulk reactions, isgoverned by the mass species conservation equation(Aris 1956). The system is modeled using the assump-tions that the solution is homogenous and that theanalyte concentration is dilute. The model also assumesconstant density and diffusivity. In order to apply themass species conservation equation, it is assumed thatthe species move at the local fluid velocity of the fullydeveloped flow. Based on these assumptions, the speciesconcentration equation can be expressed as:

@/

@t uz

@/

@z D

@2/

@x2

@2/

@y2

@2/

@z2

2

2.2.2 Dispersion calculation with depthwise averaging

The interplay of convection and diffusion is crucial inmany microfluidic applications. Due to the Taylor dis-persion effect, the spreading of an injected analyte in apressure-driven Poiseuille flow generally occurs in theaxial direction much more rapidly than that predicted byconsidering molecular diffusion only. Taylor dispersionoccurs at longer times because molecular diffusion al-lows suspended species to cross streamlines that havedifferent speeds. In this study, an average analysis (Birdet al. 1960; Slattery 1981) is used, wherein, it is conve-

nient to work with the depthwise average"f 2b 1

Rbb fdy; where f is a function describing the

distribution of the quantity of interest in the depthwisedirection. The depthwise averaging method can tackleproblems in which the physical variables, e.g., concen-tration, in mass transport are only cross-stream andaxial-dependent, but their variation in the depthwisedirection can be neglected. Thus, the y-dependent con-

Fig. 1 Geometry of Y-shapeanalytical device, with side-by-side two-fluid flow. Diffusion of

analyte occurs across originalinterface (defined by dottedlines) between two fluids.Dimensions for the rectangularslit are: length L=2 mm; widthW=400 lm; half-heightb=15 lm

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centration can be eliminated by the averaged quantity "f:Application of this average to the hydrodynamic veloc-

ity profile gives "u @P@z

b2

3g

:

It is conceptually useful to work in terms of averagesand deviations. Thus, we define uz y "u u

0 y and/ x;y;z; t "/ x;z; t /0 x;y;z; t ; and substitute theminto Eq. 2 to obtain:

@"/

@t @/0

@t "u@"/

@z u0 @

"/

@z "u@/0

@z u0 @/

0

@z

D@2 "/

@x2

@2 "/

@z2

D

@2/0

@x2

@2/0

@y2

@2/0

@z2

3

Depthwise averaging of Eq. 2 leads to the mean con-vectivediffusion equation:

@"/

@t "u

@"/

@z D

@2 "/

@x2

@2 "/

@z2

u0

@/0

@z4

The additional contribution to the effective diffusion ofthe solute arises from the fluctuation generated flux

u0 @/0

@z: u0 y is given as:

u0 y @P

@z

b2

6g1 3

y

b

2 !5

In order to derive the equation governing /, we subtractEq. 4 from Eq. 3 to obtain:

@/0

@t u0

@"/

@z "u

@/0

@z u0

@/0

@z D

@2/0

@x2

@2/0

@y2

@2/0

@z2

u0@/0

@z

6

Due to the long duration characteristic of diffusion, theeffect will only be significant when the flow length isrelatively large, i.e., L ) "ub2

D; and because u0 O "u

while /0 ( "/; then Eq. 6 simplifies to:

u0@"/

@z% D

@2/0

@y27

The form of Eq. 7 was first introduced by Taylor (1953),who showed that the variation of the analyte in thedepthwise (y direction) is proportional to the axialconcentration gradient: a balance between convection inthe axial direction and the diffusion in the depthwise

direction, and it results in a curvilinear-shape in theinterface. Since u is known, the additional contributiondue to velocity variation from the average velocity canbe expressed as:

u0@/0

@z

2

105

@P

@z

2b6

9g2D

" #@2 "/

@z28

For continuous species transport in a microfluidic de-vice, the quasi-steady state and the long channelapproximation are two assumptions that were made

previously, and they will also be assumed here. Thus, thesteady species transport Eq. 4, by considering the Tay-lor-Aris dispersion, is reduced to:

"u@"/

@z D

@2 "/

@x2Deff

@2 "/

@z29

where the effective diffusivity coefficient is defined as1:

Deff D 1 Pe2

210HW

2" #10

The above expression is identical to that previouslyobtained by Wooding (1960), who studied the instabilityof a viscous liquid of variable density distribution in avertical channel, the cross-section of which is a thinrectangle. In his investigation, Taylor-Aris dispersionwas neglected because (PeH/W)2210.

Following the scaling analysis of Ismagilov et al.(2000), under the long duration characteristic, the masstransport in the cross-stream and axial directions issimilar to simple diffusion and gives rise to the one-half

relation of diffusion. However, it should be noted thatthe parameters which support this relationship in theaxial and cross-stream directions is different. In the axialdirection, the parameter is the effective diffusivity, i.e.,the enhanced apparent diffusivity (Eq. 10). In the cross-stream direction, the parameter is the molecular diffu-sivity, which is the same as that in the previous investi-gation of Ismagilov et al. (2000). It provides physicalinsights into the mass dispersion process. The inducedTaylor diffusivity DT "u

2H2=210D, measures the ratioof the axial convection to the intensity of mixing and willproduce the enhanced mixing in the axial direction.

The following initial and boundary conditions areapplied:

B.C.1: / x;z jz0 0; 0 6 x\ W=2 11a

B.C.2 : / x;z z0j /0; W= 2\x 6W 11b

B.C.3 :@/

@zx;zL

0; @/@x

x0;z

0; @/@x

xW;z

011c

By using / "//0

; z z=L ; x x=W ; and

Pe "uW=D ; the dimensionless form of Eq. 9 can bewritten as:

PeW

L @/

@z @2/

@x2 Deff

D

W

L 2 @2/

@z2 12

Similarly, the initial and boundary conditions (Eqs. 11a,11b, 11c) can also be expressed in a dimensionless formas:

1A similar investigation was carried out by Beard (2001). However,the derivation of Eq. 10, and the associated equations, was incor-rect. A detailed discussion can be found in Dorfman and Brenner(2001).

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B.C.1 : / x;z z0j 0; 0 6 x \1=2 13a

B.C.2 : / x;z z0j 1; 1=2\x 61 13b

B.C.3 :@/

@zx;z1

0; @/@x

x0;z

0;@/

@xx1;z

013c

An analytical solution of Eq. 12 with boundaryconditions (Eqs. 13a, 13b, 13c) can be obtained by usingthe separation of variables method:

/ 1

2

2

p

X1n1

1 n1

2n 1 cos

2n 1 px

W

!wn z 14

where:

wn z ek2z k2=k1 e

k2k1 Lk1z

1 k2=k1 e k2k1 L15a

k1;2

Pe

2W

Pe2

4W2

D

Deff

2n 1 2p2

W2" #

1=2

15b

2.3 Two special cases

2.3.1 Without axial diffusion nor Taylor-Aris dispersion

If there is no axial diffusion and Taylor-Aris dispersion(i.e., Deff is zero), the streamwise diffusion is negligible

compared to the cross-stream diffusion (i.e., 2/*/z*22/*/x*2). Thus, Eq. 12 can be reduced to thefollowing form:

Pe WL

@/

@z @

2

/

@x216

Using boundary conditions in Eqs. 13a, 13b, 13c, theanalytical solution to Eq. 16 is:

/ 1

2

2

p

X1n1

1 n1

2n 1 cos

2n 1 px

W

!e 2n1 p

2 zWPe

17

which is the same as that obtained by Holden et al.(2003).

2.3.2 With axial diffusion and without Taylor-Arisdispersion

In this situation, Eq. 12 can be written in the followingform:

PeW

L

@/

@z

@2/

@x2

W

L

2 @2/@z2

18

The solution to Eq. 18 is of the same form as that givenby Eq. 14, with the effective diffusivity Deff replaced bythe mass diffusivity D.

3 Results and discussion

In this study, the Y-shape microfluidic slit, which couldbe made from silica, has a length L=2 mm, widthW=400 lm, and height H=30 lm (see Fig. 1). As anillustration, one could imagine the situation where asyringe pump is used to inject two different fluids withand without a diffusible analyte, respectively, in the two

entrances of the device. The channel on the right-hand-side contains the diffusible analyte with concentration/0, while the channel on the left-hand-side has no dif-fusible analyte. In between, the channel holds a mixingzone. The two working fluids are assumed to have thesame molecular diffusion of D=1010 m2/s. Parametricstudies, including length and height of the device andPeclet number, are carried out to quantitatively predictthe concentration profiles of analyte in the slit.

Figure 2 shows a comparison of concentration pro-files at the slit exit for three different cases: (1) withaxial diffusion and Taylors dispersion (Eq. 12); (2)without axial diffusion and without Taylors dispersion

(Eq. 16); and (3) with axial diffusion and withoutTaylors dispersion (Eq. 18). From Fig. 2, we can ob-serve that the inter-diffusion between the two fluidsdecreases with consideration of the Taylor-Aris dis-persion (Eq. 12). The presence of the Taylor-Aris dis-persion enhances the axial diffusion (or mixing in theaxial direction). However, the Taylor-Aris dispersionreduces the cross-stream diffusion between these twofluids. Thus, it will result in a sharp transition ofconcentration profile between these two fluids in com-parison with the two cases without Taylor-Aris dis-persion, as shown in Fig. 2. The reason is that the

Fig. 2 Concentration profile with and without Taylor-Aris disper-sion and axial diffusion at exit section of right side of device. Pecletnumber (Pe)=400, molecular diffusion coefficient D=1010 m2/s

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actual diffusion is a function of velocity for a givenflow length. For a parabolic velocity distribution in thechannel, the average diffusion is less with the Taylor-Aris dispersion than that without by using the averagevelocity "u as the reference. This is reflected in thecorrection of the diffusivity Deff, which is one order of

magnitude larger than D. In addition, molecular dif-fusion in the axial direction as the flow proceeds haslittle effect on the concentration profile between thetwo fluids, i.e., the results of Eqs. 16 and 18 are similar,as shown in Fig. 2.

Figure 3 shows a plot of the effective diffusivitycoefficients ratio (Deff/D) versus the channels height to

Fig. 3 Plot of effective diffusivity coefficient Deff versus ratio ofchannel height to width ratio (H/W) for different Peclet numbers

Fig. 4 Evolution of analyte concentration profiles at three differentsections along the axial direction with axial diffusion and Taylor-Aris dispersion. Peclet number (Pe)=300, molecular diffusioncoefficient D=1010 m2/s

Fig. 5 Concentration profiles at the slit exit for different Peclet

numbers. Molecular diffusion coefficient D=1010 m2/s, with axialdiffusion and Taylor-Aris dispersion

Fig. 6a, b Concentration evolution of interdiffusion zone withtwo different Peclet numbers. a Pe=800. b Pe=150. Moleculardiffusion coefficient D=1010 m2/s, with axial diffusion andTaylor-Aris dispersion

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width ratio (H/W) for different Peclet numbers (Pe).This plot represents the dependence of the effective dif-fusivity coefficient Deff on Pe and H/W, as given byEq. 10. With increasing values of Pe and H/W, (Deff/D)increases gradually and monotonously. In this casewhere H/W=0.25 and Pe ranges from 1 to 1,000, Deff/Dvaries from 1 to 300 accordingly. With a large ratio of(Deff/D), the inhomogeneous velocity profile contributesmuch more than molecular diffusion to the total analyte

diffusion in the axial direction.We will now focus our attention on the situation

where both axial and Taylors dispersion are consid-ered, namely, Eq. 12. Figure 4 shows the evolution ofthe concentration profile at three different sections(z=0, z=0.25L, z=L) along the axial direction. At theentrance (z=0.0), the fluid on the right side contains adiffusible analyte that is assumed to be uniform. Atposition z=0.25L, one can observe the evolution ofconcentration due to interdiffusion between the twoside-by-side fluids (see Fig. 4). At the exit section(z=1.0), interdiffusion is more pronounced. It also canbe observed that the cross-stream diffusive broadening

is proportional to the distance along the axial direc-tion.

Figure 5 shows the concentration variation along thecross-stream direction. As the Peclet number increases,the reduction in the mixing area of the concentrationprofile across the contact interface can be observed. Inaddition, the evolution of the interdiffusion zone duringflow with different Peclet numbers for the rectangularslit is shown in Fig. 6. As expected, the diffusible analytebroadens further down stream due to the longerresidence time of the diffusible analyte. As the Peclet

number decreases, the diffusible analyte also broadens atthe contact interface of the two streams. This is becausethe residence time of the diffusible analyte with a smallerPeclet number is longer than that with a larger Pecletnumber.

As a bench mark solution for the dispersion of apassive analyte in a microfluidic slit, a solution table forthe three analytical models with and without Taylordispersion and axial diffusion at two different locations,

points A and B (at the middle and the exit sections onthe right wall, respectively, see Fig. 1), under differentPelect numbers is presented in Table 1. The results showthat there is no significant difference (relative errorer3


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4 Conclusions

This study provides an analytical solution for the dif-fusion between two-stream fluids in a rectangular mi-crofluidic slit. Since diffusion dominates cross-streamtransport and convection dominates axial transport, itis reasonable to simplify the three-dimensional situa-tion to two-dimensional mean convectivediffusion

equations for a high aspect ratio system. This isachieved by introducing an effective dispersion coeffi-cient, which can be several orders greater than themolecular diffusion coefficient. Two-dimensional ana-lytical solutions for a pressure-driven side-by-side flowin a rectangular slit are obtained based on convectionand diffusion transport, with specific attention paid tothe importance of the Taylor-Aris dispersion resultingfrom depthwise inhomogeneous velocity gradients. Inaddition, the effects of various parameters on theconcentration profiles of analyte in a microfluidic slitare provided and examined.

References

Darwin R, Dimitri I, Pierre A, Andreas M (2002) Micro totalanalysis systems. 1. Introduction, theory, and technology. AnalChem 74:26232636

Pierre A, Dimitri I, Darwin R, Andreas M (2002) Micro totalanalysis systems. 2. Analytical standard operations and appli-cations. Anal Chem 74:26372652

Chiu DT, Jeon NL, Huang S, Kane RS, Wargo CJ, Choi IS,Ingber DE, Whitesides GM (2000) Patterned deposition of cellsand proteins onto surfaces by using three-dimensional micro-fluidic systems. Proc Natl Acad Sci USA 97:24082413

Li J, Kelly JF, Chernushevich I, Harrison DJ, Thibault P (2000)Separation and identification of peptides from gel-isolatedmembrane proteins using a microfabricated device for com-bined capillary electrophoresis/nanoelectrospray mass spec-trometry. Anal Chem 72:599609

David JB, Glennys AM, Walker GM (2002) Physics and applica-tions of microfluidics in biology. Annu Rev Biomed Eng 4:261286

Chan JH, Timperman AT, Qin D, Aebersold R (1999) Microfab-ricated polymer devices for automated sample delivery of pep-tides for analysis by electrospray ionization tandem mass

spectrometry. Anal Chem 71:44374444Jacobson SC, McKnight TE, Ramsey JM (1999) Microfluidic de-

vices for electrokinetically driven parallel and serial mixing.Anal Chem 71:44554459

Jeon NL, Dertinger KW, Chiu DT, Choi IS, Stroock AD,Whitesides GM (2000) Generation of solution and sur-face gradients using microfluidic systems. Langmuir 16:83118316

Tallarek U, Rapp E, Huang T, Bayer E, Van H (2000) Electroos-motic and pressure-driven flow in open and packed capillaries.Anal Chem 72:22922301

Yang C, Ng CB, Chan V (2002) Transient analysis of electroos-motic flow in a slit microchannel. J Colloid Interface Sci248:524527

Probstein RF (1994) Physicochemical hydrodynamics: an intro-duction. Wiley, New York

Kamholz A, Weigl B, Finlason B, Yager P (1999) Quantitativeanalysis of molecular interaction in a microfluidic channel: theT-sensor. Anal Chem 71:53405347

Kamholz A, Yager P (2001) Theoretical analysis of moleculardiffusion in pressure-driven laminar flow in microfluidic chan-nels. Biophys J 80:155160

Weigl B, Yager P (1999) Microfluidic diffusion-based separationand detection. Science 283:346347

Ismagilov R, Stroock A, Kenis P, Whitesides G, Stone H (2000)Experimental and theoretical scaling laws for transverse diffu-sive broadening in two-phase laminar flows in microchannels.Appl Phys Lett 76:23762378

Brody JP, Kamholz AE, Yager P (1997) Prominent microscopiceffects in microfabricated fluidic analysis systems. In: Proceed-ings of the conference on micro- and nano-fabricated electro-optical mechanical systems for biomedical and environmental

applications, San Hose, California, February 1997, vol 103,pp 103110

Holden MA, Kumar S, Castellana ET, Beskok A, Cremer PS(2003a) Generating fixed concentration arrays in a microfluidicdevice. Sens Actuators B 92:199207

Holden MA, Kumar S, Beskok A, Cremer PS (2003b) Microfluidicdiffusion diluter: bulging of PDMS microchannels under pres-sure-driven flow. J Micromech Microeng 13:412418

Kamholz A., Yager P (2002) Molecular diffusive scaling laws inpressure-driven microfluidic channels: deviation fromone-dimensional Einstein approximations. Sens Actuators B82:117121

Virginie M, Jacques J, Hubert HG (2002) Mixing processes in azigzag microchannel: finite element simulations and opticalstudy. Anal Chem 74:42794286

Beard DA (2001) Taylor dispersion of a solute in a microfluidic

channel. J Appl Phys 89:46674669Dorfman KD, Brenner H (2001) Comment on Taylor dispersion

of a solute in a microfluidic channel [J Appl Phys 89:4667(2001)]. J Appl Phys 90:65536554

Taylor GI (1953) Dispersion of solute matter in solvent flowingslowly through a tube. Proc R Soc Lond A 219:186203

Taylor GI (1954) Conditions under which dispersion of a solute ina stream of solvent can be used to measure molecular diffusion.Proc R Soc Lond A 225:473477

Aris R (1956) On the dispersion of a solute in a fluid flowingthrough a tube. Proc R Soc Lond A 235:6777

Barton NG (1983) On the method of moments for solute disper-sion. J Fluid Mech 126:205218

Fig. 7 Normalized concentration profiles at right and left wallsfor different Peclet numbers. Molecular diffusion coefficientD=1010 m2/s, with axial diffusion and Taylor-Aris dispersion

225


9/9

Brenner H, Edwards DA (1993) Macrotransport processes. But-terworth-Heinemann, Boston, Massachusetts

Doshi MR, Daiya PM, Gill WN (1978) Three dimensional laminardispersion in open and closed rectangular conduits. Chem EngSci 33:795804

Giddings JA, Schure MR (1978) Theoretical analysis of edge effectsin field-flow fractionation. Chem Eng Sci 46:14711479

Dutta D, Leighton DT (2001) Dispersion reduction in pressure-driven flow through microetched channels. Anal Chem 73:504513

Dutta D, Leighton DT (2002) A low dispersion geometry for

microchip separation devices. Anal Chem 74:10071016Dutta D, Leighton DT (2003) Dispersion in large aspect ratio

microchannels for open-channel liquid chromatography. AnalChem 75:5770

Stone HA (1989) Heat/mass transfer from surface films to shearflows at arbitrary Peclet numbers. Phys Fluids A 1:11121122

Schlichting H (1979) Boundary-layer theory. McGraw-Hill, NewYork

Bird R, Steward W, Lightfoot E (1960) Transport phenomena.Wiley, New York

Slattery JC (1981) Momentum, energy, and mass transfer in con-tinua. Krieger, New York

Wooding RA (1960) Instability of a viscous liquid of variabledensity in a vertical Hele-Shaw cell. J Fluid Mech 7:501515

226

depthwise averaging approach to cross-stream mixing in a pressure-driven micro channel flow

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