Solute dispersion in channels with periodically varying aperturesDiogo Bolster,1,a� Marco Dentz,1,b� and Tanguy Le Borgne2,c�

1Department of Geotechnical Engineering and Geosciences, Technical University of Catalonia (UPC),Barcelona 08034, Spain2Géosciences Rennes, UMR 6118, CNRS, Université de Rennes 1, Rennes 35000, France

�Received 29 October 2008; accepted 20 April 2009; published online 13 May 2009�

We study solute dispersion in channels with periodically varying apertures. Based on anapproximate analytical solution of the flow equation, we study the impact of the geometry andmolecular diffusion on effective solute dispersion analytically using the method of local moments.We also study the problem numerically using a random walk particle tracking method. For transportin parallel shear flow, the effective dispersion coefficient is dependant on the square of the Pecletnumber. Here, when the fluctuation of the channel aperture becomes comparable with the channelwidth, the effective dispersion coefficients show a more complex dependence on the Peclet numberand the pore geometry. We find that for a fixed flow rate, periodic fluctuations of the channelaperture can lead to both a decrease and an increase in effective dispersion. © 2009 AmericanInstitute of Physics. �DOI: 10.1063/1.3131982�

I. INTRODUCTION

The understanding and quantification of the dispersionof dissolved substances in the flow through channels is ofimportance in a series of applications ranging from the de-sign of microfluidic devices,1,2 nutrient transport in bloodvessels,3,4 dispersion in porous media,5–9 and mixing andspreading of contaminants in groundwater aquifers.10–13

Groundwater aquifers are frequently modeled as heteroge-neous porous media whose pore scale structure can be rep-resented by a network of pore channels.14 Fractured geologi-cal media are often represented as networks of discretefractures.15,16 Thus, studying the channel model under con-sideration here gives valuable insight in the pore scale mix-ing and spreading processes which are of paramount impor-tance for the correct modeling of mixing-limited reactivetransport in porous media, as pointed out, for example, byRef. 17.

The volume occupied by a solute dissolved in the fluidflow through a channel expands at short time scales due tomolecular diffusion. Eventually at some intermediate scale,the interaction of molecular diffusion, local flow variability,and the geometry of the channel leads to a mixing andspreading behavior that is different from the one induced bymolecular diffusion. In general, when dealing with solutetransport in inhomogeneous environments, one distinguishesbetween spreading, which consists in increasing the surfacearea of a solute plume, and mixing and dilution, which con-sists in increasing the volume occupied by a dissolved sub-stance in the host fluid.13,18–20 This distinction is of particularimportance at preasymptotic times, that is, at times, forwhich the solute has not sampled the full geometrical andflow variability.13,20–22 At asymptotic times, solute spreadingcan be measured by constant dispersion coefficients that in-

tegrate the interaction of small scale spatial variability anddiffusion on effective spreading and mixing. Manystudies5,23,24 focus on the determination of the coefficient thatdescribe the effective solute dispersion on the large scale,i.e., at length and time scales on which the microscale vari-ability of flow and geometry are homogenized.

The impact of the interaction of molecular diffusion andflow variability on effective solute dispersion in axisymmet-ric channels was studied by Taylor23 and Aris.25 Theyshowed that given enough time for vertical concentrationgradients to be smeared out by diffusion, effective solutetransport is one dimensional and completely defined by themean flow velocity U and the effective Taylor dispersioncoefficient,23

D� = D +U2a2

210D, �1�

where D is the molecular diffusion coefficient and a isthe channel width. This result relies on a constant channelaperture.

In many applications, however, channel apertures are notconstant. The geometry of the channel can have a significantimpact on the effective dispersion behavior and lead to abehavior that is qualitatively and quantitatively differentfrom the one found in channels with constant aperture. Inmicrofluidics, for example, the design of specific mixingproperties in microchannels is based on the manipulation ofthe channel geometry.1 The channel model is also of interestto study solute dispersion in geological media.26–28 Althoughnatural media is in general not periodic, it represents thesimplest system to analyze the effect of the convergence anddivergence of flow lines on dispersion. The latter plays acentral role for transport in heterogeneous porous media. Atfracture scale, the proposed channel model is relevant to un-derstand the role of fracture wall roughness on the transportproperties. Tartakovsky and co-worker29,30 studied the im-pact of small scale roughness on the global dispersion using

a�Electronic mail: diogo.bolster@gmail.com.b�Electronic mail: marco.dentz@upc.edu.c�Electronic mail: tanguy.le-borgne@univ-rennes1.fr.

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a stochastic model. Using an alternate stochastic model31 es-timated the dispersion due to fluctuations in the boundaryand within this framework always observed an increase indispersive effects due to fluctuations. Studying transport inself-affine rough fractures, Drazer et al.32,33 illustrated thatsurface roughness could reduce dispersion effects at very lowPeclet numbers. Smith34 analyzed longitudinal dispersion co-efficients for varying channels. Mercer and Roberts35 inves-tigated effective solute transport in channels with varyingflow properties using center manifold theory. Rosecrans36

studied Taylor dispersion in curved channels using the sameapproach. In the latter paper a heuristic argument is given forthe existence of channel shapes and flow fields that lead to areduction instead of an increase in solute dispersion as it isobserved in channels with constant aperture. Hoagland andPrud’homme37 studied the transport of a passive solute in asinusoidal tube in a pressure driven flow. Other authors�e.g., Refs. 38–40� looked at transport by processes such asthermophoresis and electrophoresis �force, rather thanpressure driven� in sinusoidal geometries and quantified thedependence of transport properties on Peclet number andgeometry.

The geometry of the pore has a significant impact on theflow velocity. Here we focus on flow regimes at small Rey-nolds numbers so that the flow problem is given by Stokesequation. Kitanidis and Dykaar12 derived an analytical ex-pression for the velocity distribution in a channel that con-sists of a connected periodic chain of slowly varying two-dimensional pores, as illustrated in Fig. 1. Their analysisdemonstrates that short wavelengths and large amplitudescan give rise to recirculation zones. Such recirculation zonesrepresent immobile regions that can have a significant impacton effective solute transport depending on the typical masstransfer time scales.41–45 Similar results for the flow velocityin periodic channels were obtained in Refs. 46–48. Specifi-cally Cao and Kitanidis47 illustrated the existence of recircu-lation zones for pores that open rapidly.

The methods devised by Taylor and Aris for the calcula-tion of asymptotic dispersion coefficients are typically re-stricted to parallel flows. In a series of seminal papers a

“generalized” theory of dispersion was developed forpassive5,49,50 as well as reactive flows.51,52 These methodsallow for the calculation of the asymptotic effective disper-sion and reaction coefficients for flow domains having a pe-riodic structure.

Based on this generalized theory of dispersion, Dykaarand Kitanidis53 studied macrotransport of a reactive solute ina porous medium on the basis of the analytical expression forthe pore velocity field given in Ref. 12. They determined thecombined effects of reaction, flow variability. and moleculardiffusion on macrotransport for a particular pore geometrywhich included a recirculation zone.

In this paper, we systematically study the impact of porescale geometry and molecular diffusion on asymptotic solutedispersion using the analytical expression for the flow veloc-ity given in Ref. 12. We investigate analytically and numeri-cally under which conditions solute dispersion can decreaseand probe the impact of recirculation zones on asymptoticdispersion. To this end, we apply the approach by Taylor23

and Aris25 as well as the one devised by Brenner5 to deter-mine the asymptotic longitudinal dispersion coefficient andcompare them to the dispersion coefficients obtained by nu-merical random walk simulations.

II. PORE GEOMETRY AND FLOW VELOCITY

We consider flow in a channel that is two dimensionaland symmetric about a central axis at y=0. The channelwalls fluctuate periodically in horizontal direction,

h�x� = h + h� sin�2�x

L� , �2�

where h is the average channel height and h� is the amplitudeof the fluctuations. A unit cell in the following is denoted by�, its volume by V�. The aspect ratio is defined by

� =2h

L. �3�

The ratio between the amplitude of the aperture fluctua-tions h� and the mean aperture in the following called thefluctuation ratio, is denoted by

a =h�

2h. �4�

The flow rate through the channel Q and the fluid viscosity �define the Reynolds numbers

Re =Q

�. �5�

On the pore scale, Reynolds numbers are typically small12,14

of the order of 10−4–10−1. For Reynolds number of this scaleflow is described by the Stokes equation.

For a slowly varying boundary, i.e., ��1 Kitanidis andDykaar12 derived an analytical solution for the flow velocityusing a perturbation expansion in �. In the following, wegive a brief summary of their solution. For a detailed deri-vation we refer the reader.12

h

h’

L

Q Q

FIG. 1. A schematic of the pore we are considering and a connected periodicnetwork of pores

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The divergence-free flow velocity u= �u ,v�T is given interms of the stream function � by

u =��

�y, v = −

��

�x. �6�

The flow equation for � is biharmonic,12,54

�4� = 0. �7�

At the channel walls no-flux and no-slip boundary conditionsare specified. The solution for � is periodic with period L. Inaddition to mass and momentum balance as expressed by Eq.�7� the solution of the flow problem requires a further equa-tion expressing energy conservation.12 The viscous dissipa-tion energy within a cell of length L is balanced by the workapplied to the fluid in the cell volume,12,55

��

dV�� = − �pQ , �8�

where � is the dissipation function,55 �p is the pressure dropover the unit cell, and Q is the flow rate.

Note that the flow rate in a channel with constant aper-

ture given by the average aperture 2h is described by thecubic law,14,56

Q = −2h3

3�

�p

L. �9�

For the case of sinusoidally varying walls, however, it isknown that the cubic law does not hold.48

Kitandis and Dykaar12 performed a perturbation expan-sion of the stream function and the flow rate Q in powersof �,

� = �i=0

�i�i, Q = �i=0

�iQi. �10�

They determine the contributions up to fourth order, where�1=�3=0 and Q1=Q3=0. Their explicit results for �0, �2,and �4, and Q0, Q2, and Q4 are given in Appendix A forcompleteness. Accordingly, we obtain for the flow velocity,

u�x� = �i=0

�iu�i��x� . �11�

In order to illustrate the different types of flow that canarise within such a geometry, three sets of streamlines calcu-lated using Eq. �10� are shown in Fig. 2. Figure 2�a� showsthe channel flow for �=0. It is given by the well knownHagen–Poiseuille flow. For increasing �, the streamlines be-come more curved, see Fig. 2�a� for �=0.2. For �=0.4, Fig.2�c�, recirculation cells develop at the location of the maxi-mum channel diameter.

The deviation of the flux Q from the cubic law is illus-trated in Fig. 3 for the contributions Q0, Q2, and Q4 normal-

ized by Q, Eq. �9�, which follows the cubic law. Figure 3illustrates how the three quantities vary with varying fluctua-tion ratio a. The pressure drop �P across each pore is iden-tical. Q0 decreases monotonically with a, which is a reflec-tion on the fact that a given pressure drop will cause less

flow for highly fluctuating pores, eventually approachingzero as the thinnest section of the pore approaches zero too�h�→h, i.e., if there is zero cross sectional area no matterhow large the pressure drop no flow can occur�.

Q2 is a negative quantity, which is zero for h�=0 andh�=h. This suggests that second-order effects can cause areduction in effective flow rate and thus perhaps also disper-sion. Finally Q4 is always positive with zero value at h�=0and h�=h.

III. SOLUTE DISPERSION

Transport in the flow field through a two-dimensionalchannel with varying diameter is described by the advectiondiffusion equation. The temporal change in the solute distri-bution c�x , t� is balanced by the divergence of the advective-diffusive solute flux

�c�x,t��t

+ � · �u�x� − D��c�x,t� = 0, �12�

where D is the molecular diffusion coefficient. We investi-gate the asymptotic longitudinal dispersion behavior of a dis-solved substance. The boundary conditions are naturalboundary conditions at x= � and vanishing solute flux atthe periodically fluctuating horizontal boundary. The initialdistribution is given by c�x , t=0�=��y� �x�, i.e., a verticalline source.

The longitudinal flow velocity u�x� is divided into itsaverage value over the unit cell

u =D

V��

�

dxu�x� , �13�

and fluctuations about it,

u�x� = u�1 + u��x�� . �14�

Flow and transport can be characterized by two dimen-sionless numbers; the aspect ratio �, Eq. �3�, and the Pecletnumber Pe, which is defined by

Pe =2hu

D. �15�

The Peclet number denotes the ratio between the advectiveand dispersive time scales

�u =2h

u, �D =

�2h�2

D, �16�

respectively.Solute dispersion here is characterized in terms of the

longitudinal width � of the solute distribution c�x , t�,

��t� =� dxx2c�x,t� − � dxxc�x,t�2

. �17�

Specifically, asymptotic longitudinal dispersion is quantifiedin terms of the effective dispersion coefficient

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De =1

2limt→

d��t�dt

. �18�

Brenner5 determined the following expression for the disper-sion coefficient in periodic flow scenarios, using the methodof local moments:

De =D

V��

�

dx���x� · ���x� + 2���x�

�x , �19�

where the auxiliary function ��x� satisfies

− D�2��x� + � · u�x���x� = − u��x� . �20�

The boundary conditions for ��x� are

� · u�x���x� = 0 �21�

for x���, where �� is the boundary of the unit cell. Fur-thermore, � is periodic and its integral over the unit cell iszero

��

dx��x� = 0. �22�

Using these properties of ��x� it is easy to show by applyingthe Gauss theorem that Eq. �19� can be written as

De =1

V��

�

dx− u��x���x� + 2D���x�

�x . �23�

In order to solve for the asymptotic dispersion tensor, wefirst employ an approximation in the spirit of Taylor’sderivation23 of an effective dispersion coefficient in parallelchannels, second we numerically solve Eqs. �19� and �20�and third, we use numerical random walk simulations tosolve the full transport problem. The results are comparedagainst the numerical simulations.

−0.5 0 0.5 1 1.5 2 2.5 3 3.5 4−1

−0.5

0

0.5

1

−1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5−1

−0.5

0

0.5

1

−0.5 0 0.5 1 1.5 2 2.5 3 3.5 4−1

−0.5

0

0.5

1

FIG. 2. Streamlines for a pore with �=0 and a=0 �top�,�=0.2 and a=0.2 �middle�, and �=0.4 and a=0.4�bottom�.

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A. Taylor approximation

We assume that vertical gradients of ��x� are muchlarger than longitudinal ones. We furthermore assume thatadvective transport vertical to the channel axis can be disre-garded. Thus, Eq. �20� simplifies to

− D�2��x�

�y2 = − u��x� . �24�

This approximation is motivated by the original assumptionsmade by Taylor23 and Aris25 and a desire to know how farthey can take use. It is similar to the Fick–Jacobs approxi-mation in thermophoretic transport.40 As shown by Yariv andDorman �2007�,38 the approximation becomes invalid whenPe �2�O�1�, where the other terms in Eq. �20� make a moresignificant contribution. Nonetheless it is important to studyit to see how far such an approximation can take you andwhat type of error one may expect if it is made as it plays animportant role in the validity of the cubic law and its Taylordispersion interpretation.31,57,14,56 Equation �24� can be inte-grated straightforwardly as outlined in Appendix B. Note thatthe present approximation requires the disregard of horizon-tal gradients of ��x� in expression �19� for the effectiveasymptotic dispersion coefficient, which is equivalent to di-rectly employing Eq. �23�.

1. Effective dispersion coefficient

Using expansion �11� for the flow velocity, we obtain anexpansion in � for �

��x� = �i=0

�i�i�x� . �25�

Inserting the latter into Eq. �19� yields for the effective dis-persion coefficient,

De = �i=0

�iDie, �26a�

Die = �

j=0

iD

V��

�

dx���i−j��x�

�y

���j��x��y

+2D

V��

�

dx���i��x�

�x.

�26b�

Solving Eq. �24� and using Eq. �6� we obtain for the zerothorder contribution to De,

D0e =

1

210

Q02

D+

a2

10

Q02

D. �27�

Note that this term consists of one part that is analog tothe classic Taylor dispersion result with the difference thatQ0 does not follow the cubic law and a second term thatdepends on the fluctuation size a. One might expect thiszero-order analysis to hold for any flow where the boundaryfluctuates slowly over a length scale larger than the averagechannel width as this corresponds to a lubrication theory typeapproach.58

The second-order contribution to the effective dispersioncoefficient is given by

D2e =

1

D1

5a2Q0

2 +1

105Q0Q2 −

2

25a4�2Q0

2

+2

175a2�2Q0

2 . �28�

Similarly at fourth order, we obtain

D4e =

1

D� 71

606 375a2�4Q0

2 +2918

40 425a4�4Q0

2

−548 381

4 036 032 000Q0

2a2�6 −2

875Q0

2a6�4

+4

175Q0Q2a2�2 −

4

25Q0Q2a4�2 +

1

5Q0Q4a2

+1

10Q2

2a2 +1

210Q2

2 +1

105Q0Q4� . �29�

Using the flow rates given in Appendix A we can com-pute the contributions to the effective dispersion coefficientsfrom Eqs. �27�–�29�. The dependence of De on a is illus-trated Fig. 4. Note here that the decrease in D0

e with a is not

0 0.1 0.2 0.3 0.4 0.5−0.5

0

0.5

1

a

Flo

wR

ates

FIG. 3. Normalized Q0 �–�, Q2 �- -�, and Q4 �-.� as a function of the fluc-tuation ratio a.

0 0.1 0.2 0.3 0.4 0.5−1

0

1

2

3

a

Dis

per

sio

nC

oef

fici

ents

FIG. 4. Dispersion coefficients as a function of a. D0e �–�,

D2e �- -�, and D4

e �-.�.

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due to pore geometry but reflects the fact that the flow rateQ0 decreases with increasing a for a constant pressure drop.Note that D0

e decreases more rapidly with a than Q0 didbecause of the quadratic relationship. It is worth noting thatD2

e is negative, reflecting the influence of Q2.

2. Dependence on the pore wall geometry

From Fig. 4 we have seen that due to nonlinear effectsthrough the orders De does not scale with Q in the samemanner as for transport in a parallel channel flow. In order tostudy the dependence of the dispersion properties on the porewall geometry, we choose the flow rate Qtot=Q0+�2Q2

+�4Q4 such that the average flow rate is always the sameregardless of a and � �i.e., we no longer impose a fixedpressure drop, but rather a constant flow rate�. Additionally,in order to compare the relative impact or the curvature, wenormalize all dispersion coefficients by the dispersion coef-ficient for normal parallel flow, i.e.,

De =De

De�a=0,�=0. �30�

De is a dimensionless dispersion coefficient. As the val-ues of Eqs. �27�–�29� are difficult to interpret intuitively dueto the size and complexity of the expressions, a plot illustrat-ing the values of D0

e, D2e, and D4

e for several values of a isshown in Fig. 5. One feature in this figure that should raisecaution is the relatively large values of D4

e compared to D0e

for larger values of a. This suggests that for these largervalues of a the asymptotic expansion in Eq. �25� may be-come invalid and that as such one must be cautious in inter-preting these results. As D4

e is multiplied by �4 these contri-butions can still be relatively small.

A contour plot showing De for various values of a and �is shown in Fig. 6. Now that the effect of decreased flow ratedue to a constant pressure drop has been removed it is evi-dent that the boundary fluctuations could potentially havesignificant influence on the overall enhanced Taylor disper-sion, particularly for large values of a and �, which corre-sponds to shorter pores with large boundary fluctuations.

In order to investigate this further we will focus on fourpoints illustrated in Fig. 6 marked �a�–�d�. Using thisclassical model we would expect an increase in Taylor dis-persion by a factor of �a� 1.87, �b� 2.06, �c� 4.4, and �d� 5.93.The only one of these cases that has a recirculation zone iscase �d�.

Figure 7 depicts a surface plot of the total dispersioncoefficient, normalized by the parallel wall case, over a range0�a�0.45 and 0.01���0.5 for various typical Pecletnumbers. It is worth noting that the behavior for all cases issimilar with any increase in � or a resulting in an increase inthe effective dispersion coefficient. Once the Peclet numberis large the relative changes in behavior become identical asseen for the Pe=100, Pe=1000, and Pe=2000 cases. This issimply because for these cases the enhanced dispersion effectis much larger than the molecular diffusion contribution,which still plays a significant enough role for the Pe=10case.

B. Brenner solution

One of the major drawbacks of a straightforward appli-cation of the Taylor–Aris predictions for dispersion is thefact that it disregards horizontal gradients and advectivemass transfer perpendicular to the mean flow direction. Thistype of approximation is not valid for larger values of a and� as the curvature of the flow increases and particularly in theregimes for which recirculation zones appear. In this case werely on the numerical solution of Eq. �20� for the auxiliaryfunction ��x� and Eq. �19� for the asymptotic dispersion co-efficient. We employ the finite difference methods outlined inRef. 53.

Figure 8 illustrates the normalized dispersion coefficientfor various values of Peclet number. There are several inter-esting and perhaps unexpected features illustrated here, par-ticularly in light of what we saw in the previous section andFig. 7. First, note that the general behavior for each order ofPeclet number is different. That is, the figure for Pe=10 isdistinct from the one illustrating Pe=100, which in turn isdifferent from the Pe=1000 case. With the result obtained bythe Taylor–Aris approximation presented in Sec. III A one

0 0.1 0.2 0.3 0.4 0.5−20

0

20

40

60

80

100

a

No

rmal

ized

Dis

per

sio

nC

oef

fici

ents

FIG. 5. De0 �–�, De2 �- -�, and De4 �-.� for various values of a.

1.011.01

1.011.05

1.051.05

1.11.1

1.11.25

1.251.25

1.51.5

1.52

22

2.52.5

2.53

33

4

44

5

5

5

6

6

6

7

78910

a

ε

(a)

(b)

(c)

(d)

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

FIG. 6. Contour plot of normalized De0+�2De2+�4De4 over various valuesof a and �. The four points marked with an � correspond to the four pointswe will focus our attention on in numerical simulations. They correspond to�a� �=0.2 and a=0.2, �b� �=0.4 and a=0.2, �c� �=0.2 and a=0.4, and �d��=0.4 and a=0.4.

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would not expect this as the effective dispersion is dependanton the square of the Peclet number �i.e., De�D Pe2�.

Based on simple scaling arguments one might expect aneffective dispersion De=�D Pe2, where � is a constant thatwould depends on the geometry, namely, � and a. The resultsin Fig. 8 seem to indicate that this scaling is not accuratewhen fluctuations in the pore wall exist.

In particular, the Taylor–Aris results predict that the onlyeffect the fluctuations of the boundaries can have is to in-crease the effective dispersion. This certainly seems to holdtrue for the Pe=10 case, although the behavior is not mono-tonic in � for all values of a. For the larger Peclet numbercases, Pe=100, Pe=1000, and Pe=2000, we actually predict

that in certain cases the boundary fluctuations decrease theeffective dispersion coefficient, particularly for small valuesof �. Values of the maximum and minimum dispersion coef-ficients and the respective values of a and � for each of thePeclet numbers considered are shown in Table I. A crosssection of Figs. 7 and 8 at Pe=1000 for �=0.2 is shown inFig. 9 to illustrate and compare this effect. Note that whilethe “Taylor–Aris–type” solution increases with a, the fullsolution actually decreases, although the relative magnitudeof this change is much less.

Also, in general, one might expect the influence of theboundaries to play a larger role for the larger Peclet numbercases, as dispersion is an advectively driven phenomenon

FIG. 7. �Color online� Normalized dispersion coeffi-cients for various values of Peclet number, calculatedusing the Taylor–Aris approximation model �Eq. �26a��.

FIG. 8. �Color online� Surface contour plots for thedispersion coefficients for various Peclet numbers overa range of a and � calculated using Brenner’s method�Eq. �23��. All values are normalized with the valuecorresponding to a smooth channel �i.e., a=0�.

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�although diffusion is necessary to activate it�. However, thelargest relative change in dispersion for these cases seems tooccur for Pe=10 for small values of � and larger a. Thissuggests that for lower Peclet numbers long wavelengthswith large amplitudes can increase dispersion. A tentativeexplanation for these observations is given in Sec. III C.

C. Random walk simulations

While experiments for the types of flows presented heredo exist, changing the geometry and accurate measurementscan be elusive. Random walk particle tracking provides anefficient way to determine effective dispersioncoefficients.13,59–63 The random walk simulations presentedin the following are based on the Langevin equation

dx�t�x��dt

= u�x�t�x��� + �1�t� ,

�31�dy�t�x��

dt= v�x�t�x��� + �2�t� ,

where x�t �x�� denotes the tracetory of a solute particle that isinitially located at x�t=0 �x��=x�. The initial position of asolute particle is denoted by x�t=0�=x�, which for a linesource at x=0 is x�= �0,b�t, where b is uniformly distributedover the channel cross section at x=0. The transport descrip-tion in terms of the Langevin Eq. �31� is exactly equivalentto the one in terms of the Fokker–Planck Eq. �12�.64

The ��t� denotes a two-dimensional Gaussian whitenoise, which is defined by its first and second moments

�i�t�� = 0, �i�t�� j�t��� = 2D ij �t − t�� . �32�

The numerical solution of Eq. �31� and the calculation of theeffective dispersion coefficient �18� using random walk par-ticle tracking are outline in Appendix C.

Figure 10 shows particle distributions at a transport timeof 50�u for three different Peclet numbers of �a� Pe=103,�b� 102, and �c� 10 and the fluctuation ratios a=0,0.2,0.4and apertures of �=0.2 and 0.4.

Note that for the classical Taylor problem �a=0�, theasymptotic regime is reached at times larger than the diffu-sion time scale, t��D, i.e., when vertical gradients have dis-appeared due to mixing. For Pe=103 and Pe=102, the dis-played transport time is smaller than �D and the observeddistributions are preasymptotic. Nonetheless, these figures il-lustrate the influence of the boundaries on the dispersionbehavior.

Recall Fig. 2, which illustrates the streamlines of theflow field for different values of the aspect ratio �. For�=0.4 a recirculation zone forms where the channel width ismaximum. This notes both in the center of mass velocity ofthe plume as well as in the width compared to the flow fieldswith the flow fields for smaller aspect ratios. With increasingPeclet number the plume width increases significantly due toa higher residence time of trapped particles in the recircula-tion zones.

Figure 11 compares the predicted dispersion coefficientsfor points �a�–�d� from Fig. 7�b� using the parallel wall Tay-lor solution �i.e., a=0, Eq. �1��, our approximate “Taylor–Aris” solution �Eq. �26a��, the Brenner solution �Eq. �23��and the solutions calculated from the random walk simula-tions. Even for case �a� the agreement for all cases is notgood as the Taylor–Aris approximation overpredicts the ef-fective dispersion coefficient. However, the parallel wall casewith no fluctuations seems to work quite well. In cases �b�and �d� the parallel wall solution underpredicts the measured,while the full higher order solution overpredicts it. For case�c�, where the influence of the boundaries is to decrease thedispersion, both the parallel wall and full Taylor–Aris pre-dictions are unable to capture this decrease. This clearly il-lustrates the limitations of a traditional Taylor–Aris ap-proach, which cannot capture the influence of nonparallelflow. This disagreement, particularly evident at larger Pe isunsurprising as the approximation in Eq. �24� becomes in-valid when Pe �2=O�1� as shown by Ref. 38. Additionally,the asymptotic expansion in Eq. �25� also becomes question-able for the larger values of a. On the other hand, the Bren-ner theory, which does not suffer from this limitation agreesexcellently with all the cases presented in Fig. 11.

While the four cases illustrated here seem to indicatethat Brenner’s theory works at predicting the influence of theboundaries on dispersion we investigated this further by con-sidering the maximum and minimum predicted values for thePe=10, Pe=100, and Pe=1000 as depicted in Fig. 8. In allcases good agreement was found between Brenner’s theoryand the random walk simulations, suggesting further that thetheory works well.

The results of this study can be understood in a Lagrang-ian framework. Pore wall fluctuations affect the distributionof Lagrangian velocities within a single pore and the corre-

0 0.1 0.2 0.3 0.40

1

2

3

4

5

a

D/D

0

FIG. 9. Normalized dispersion coefficients against a using the Taylor–Arisapproximation �Eq. �26a�� �–� and full solution �Eq. �23�� �- -� at Pe=1000for �=0.2.

TABLE I. Values and locations of the maximum and minimum normalizeddispersion coefficients from each of the cases depicted in Fig. 7.

Pe Maximum Location Minimum Location

10 2.1 �=0.02 a=0.45 1 �=0 a=0

100 1.867 �=0.45 a=0.45 0.85 �=0.13 a=0.45

1000 1.46 �=0.46 a=0.08 0.54 �=0.45 a=0.4

2000 1.46 �=0.46 a=0.08 0.52 �=0.45 a=0.4

056601-8 Bolster, Dentz, and Le Borgne Phys. Fluids 21, 056601 �2009�

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lation of Lagrangian velocities from pore throat to porethroat. The increase in the pore wall fluctuations tends towiden the Lagrangian velocity distribution within pores. Forinstance, for large fluctuations of the pore walls, particles

traveling close to the wall may take a long time to leave thepore, in particular, if they remain trapped inside a recircula-tion zone. On the other hand, the increase in the pore wallfluctuations increases the convergence of flow lines at porethroats. The latter implies a decorrelation of Lagrangian ve-locities since it is easier for particles to jump betweenstreamlines by diffusion. The widening of the Lagrangianvelocity distribution implies an increase in the asymptoticdispersion coefficient while the decrease in the Lagrangianvelocity correlation implies a decrease in the asymptotic dis-persion coefficient. Thus, when increasing pore wall fluctua-tions, the changes in the Lagrangian velocity distribution andin the Lagragian velocity correlation have an antagonist ef-fect on the asymptotic dispersion coefficient. The existenceof these two competitive effects implies that pore wall fluc-tuations can either increase or decrease the asymptotic dis-persion coefficient depending on the relative strength ofthese two effects �Fig. 8�.

For small aspect ratio �, i.e., for elongated pores, thedominant effect of the increase in the pore wall fluctuationsis a stronger convergence of flow lines at pore throats andthus a decorrelation of the Lagrangian velocities. This im-plies a decrease in the asymptotic dispersion coefficient. Onthe other hand, for a larger aspect ratio, the increase of thepore wall fluctuations leads to the existence of recirculationzones �Fig. 2�. The trapping of particles in these recirculationzones implies a significant widening of the Lagrangian ve-locity distribution within a pore. Therefore, in this case, thelatter effect is found to dominate over the decorrelation ofLagrangian velocities and the asymptotic dispersion coeffi-cient is increased.

Our Taylor–Aris approximation in essence neglects theeffect of the convergence and divergence of streamlines. Byneglecting this effect this approximation leads to a strongercorrelation of Lagrangian velocities and thus a larger

0 20 40 60 80 100−1

0

1

0 20 40 60 80 100−1

0

1

0 20 40 60 80 100−1

0

1

(a)

0 20 40 60 80 100−1

0

1

0 20 40 60 80 100−1

0

1

0 20 40 60 80 100−1

0

1

(b)

0 20 40 60 80 100−1

0

1

0 20 40 60 80 100−1

0

1

0 20 40 60 80 100−1

0

1

(c)

FIG. 10. Random walk simulations at several Peclet numbers after a time of50T �where 1 T is the advective time scale�. In all cases the top plot corre-sponds to a=0, the middle plot to a=0.2 and �=0.2, and the bottom plot toa=0.4 and �=0.4. �a� Pe=1000, �b� Pe=100, and �c� Pe=10.

10−1

100

101

102

103

0

10

20

Pe

Pe ef

f−

1

ε=0.2 a=0.2

10−1

100

101

102

103

0

10

20

Pe

Pe ef

f−

1

ε=0.4 a=0.2

10−1

100

101

102

103

0

10

20

30

Pe

Pe ef

f−

1

ε=0.2 a=0.4

10−1

100

101

102

103

0

20

40

60

Pe

Pe ef

f−

1

ε=0.4 a=0.4(a)

(b)

(c)

(d)

FIG. 11. Taylor dispersion �inverse effective Peclet number� predicted bythe Taylor theory, Brenner, and measured by random walks for cases �a�–�d�from Fig. 7�b�. Brenner theory �Eq. �23�� �–�, parallel wall Taylor–Aris �Eq.�1�� �..�, full Taylor–Aris �Eq. �26a�� �- -�, random walk simulations ���.Here Peeff

−1 =Deff /Q.

056601-9 Solute dispersion in channels Phys. Fluids 21, 056601 �2009�

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asymptotic dispersion coefficient than predicted by the fullnumerical solution �Fig. 11�. On the other hand, the Brennersolution includes the effect of the convergence and diver-gence of flow lines and thus reproduces the numerical simu-lations satisfactorily.

This is a qualitative interpretation of the presented re-sults. We are currently performing a fully quantitative analy-sis of the Lagrangian velocity properties within such systemusing the methods proposed in Refs. 65 and 66�.

IV. SUMMARY AND DISCUSSION

Using an analytical perturbation solution for the flowfield, we have studied solute dispersion at asymptotic timesin two-dimensional channels with periodic wavy walls. To dothis we used the method of local moments.5 In particular wefocused on the effects of varying the amplitude and wave-length of the boundary fluctuations.

First, we employed an approximation analogous to theTaylor–Aris theory23 �Eq. �26a��. By making this assumptionwe analytically calculated the effective dispersion coeffi-cients. We found that for a fixed flow rate, dispersive effectsincrease monotonically with the amplitude of fluctuation andwith decreasing wavelength, i.e., the shorter and wider thepores the larger the dispersive effects. Additionally, under theTaylor–Aris assumption, dispersion effects scale linearlywith Peclet number, implying that the behavior while quan-titatively different �i.e., at large Peclet numbers dispersiveeffects dominate diffusion, while at smaller values diffusionstill plays a role� is qualitatively similar.

In order to check the validity of these Taylor–Aris-typeassumptions we computed the full solution to the local mo-ment equation numerically �Eq. �23�� and found results thatdiffered significantly. First, we found that the qualitative ef-fects on effective dispersion are quite different when varyingthe Peclet number. We did not find a simple linear scalingthat one might expect. The behavior at Pe=10 was qualita-tively different from that at Pe=100, which in turn was quali-tatively different from that at Pe=1000. For small Reynoldsnumber flows �i.e., Re�1�, as studied here, one expects theflow field to look the same regardless of the flow rate sinceadvective effects are negligible on flow. However, whilethe flow fields at different flow rates, i.e., at different Pecletnumbers, look identical, the transport behavior differssignificantly.

Second, we found cases where increasing the curvature�i.e., increasing the amplitude of fluctuation or decreasing thewavelength� actually led to a decrease in the effective dis-persion. We observed this particularly for the larger Pecletnumber cases. While this may be surprising, especially giventhe results of the “Taylor” approximation, such a possibilityhas previously been predicted heuristically in Ref. 36. Wepresent a tentative explanation for these observation basedon a Lagrangian interpretation, which will be pursued ingreater detail as future work.

Finally, we also found that while increases in dispersiondid occur, particularly for large fluctuations and short wave-lengths, these increases were much smaller than those pre-dicted by our Taylor approximation. This suggests that while

curvature can influence the effective dispersion, its influencemay not be as large as one might expect from a simplifiedanalysis.

In order to validate our analysis we studied the transportof a passive contaminant in such flows using a high accuracynumerical random walk method. We found in all cases thatthe dispersion coefficient predicted by the full “Brenner” so-lution �Eq. �23�� compared very well with these simulations,unlike the Taylor–Aris approximation model �Eq. �26a��,which seriously overpredicted dispersion. For most cases,taking the original Taylor–Aris solution �Eq. �1�� for a chan-nel with no fluctuations provided a better estimate than theapproximate local moment solution, although it was still off.Our Taylor–Aris approximation consisted of two parts, ne-glecting horizontal gradients and advection transverse to themean flow direction. As shown in Ref. 38 this approximationfalls apart when Pe �2=O�1� and these neglected featuresappear to play an important role. Additionally the expansionproposed in Eq. �25� may become invalid for some of thelarger values of a.

In summary, changes in the periodic boundary fluctua-tions in a channel can lead to both increases and decreases ineffective dispersion. This statement may have strong impli-cations for applied fields such as transport through porous/fractured media where curvature in the flow field is common-place, or in microfluidics where such curvature effects maybe exploited to control mixing.

APPENDIX A: THE FLOW STREAMFUNCTIONS

The zeroth, second, and fourth order flow rates, Q0, Q2,and Q4 are given by

Q0 = �1 − 4a2�5/2

1 + 2a2 Q ,

Q2 = −12

5

�1 − 4a2�7/2�2a2

�1 + 2a2�2 Q , �A1�

Q4 = 144

25

�1 − 4a2�9/2�4a4

�1 + 2a2�3

−�1 − 4a2�5/2�4

175�1 + 2a2�2 �7648a6 – 7680a4 + 2406a2

+ 214�1 − 4a2�5/2 − 241�Q .

The zeroth, second, and fourth order contributions to thestream function, given in Ref. 12, are

�0 =Q0

4�3� − �3� , �A2�

�2 =Q2

Q00 +

3Q0

404� dh

d��2

− hdh

d����2 − 1�2, �A3�

056601-10 Bolster, Dentz, and Le Borgne Phys. Fluids 21, 056601 �2009�

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�4 = Q4

Q0− �Q2

Q0�20 +

Q2

Q02 −

Q0

5600�408 + 1800�2�

�� dh

d��4

+ �684 – 1800�2�h� dh

d��2 d2h

d�2 − �270

− 180�2�h2� d2h

d�2�2

− �248 − 240�2�h2 dh

d�

d3h

d�3

+ �19 − 15�2�h3 d4h

d�4���2 − 1�2, �A4�

where

� = �x, � =y

h���and h��� =

1

2+ a sin�2��� . �A5�

APPENDIX B: TAYLOR METHOD

Integrating Eq. �24� twice with respect to y gives

��x� =� dyf�x� + A�x�y + B�x� , �B1�

where f�x�=�dyu��x�. The symmetry of the channel geom-etry implies that � is symmetric about y=0, which yieldsA�x��0. As the average of � is zero, we obtain

B�x� = − F�x� , �B2�

where F�x�=�dyf�x�. Thus, we obtain for ��x�,

��x� = F�x� − F�x� . �B3�

APPENDIX C: RANDOM WALK SIMULATIONS

The transport problem is solved numerically by randomwalk simulations based on the Langevin Eq. �31�. In discretetime, the equation of motion of the nth solute particle readsas

x�n��t + �t�x�� = x�t,x�� + u�x�t�x����t + �2D�t�1,

�C1�y�n��t + �t�x�� = y�t�x�� + v�x�t�x����t + �2D�t�2.

The �i�i=1, . . . ,d� are independently distributed Gaussianrandom variables with zero mean and variance one. Theimpermeable channel walls are modeled as reflectingboundaries.

The ith local moment is given by summation over the ithpower of the particle trajectories of all simulated particlesoriginating from x�,

��i��t�x�� = limN→

1

N�n=1

�x1�n��t�x���i. �C2�

The global moments are obtained by summation over allinitial positions x�, which are distributed according to ��x��,

m�i��t� = limM→

1

M �m=1

M

��i��t�x��m�� . �C3�

The asymptotic effective dispersion coefficient then is givenby

De = limt→

1

2

d

dt�m2�t� − m�1��t�2� . �C4�

The simulations presented release N particles from eachinitial position. The line source is represented by M equidis-tantly positioned point sources along the cross section of thechannel at x=0.

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