Incomplete mixing and reactions in laminar shear ﬂopaster/pubs/12_PRE_shear.pdf · Mixing is the physical process that enables this contact to occur. In ﬂowing systems, this is

PHYSICAL REVIEW E 92 012922 (2015)

Incomplete mixing and reactions in laminar shear flow

A Paster

School of Mechanical Engineering Faculty of Engineering Tel Aviv University Tel Aviv 69978 Israel

T Aquinodagger and D BolsterDagger

Department of Civil and Environmental Engineering and Earth Sciences University of Notre Dame Notre Dame Indiana 46556 USA(Received 13 April 2015 published 31 July 2015)

Incomplete mixing of reactive solutes is well known to slow down reaction rates relative to what would beexpected from assuming perfect mixing In purely diffusive systems for example it is known that small initialfluctuations in reactant concentrations can lead to reactant segregation which in the long run can reduce globalreaction rates due to poor mixing In contrast nonuniform flows can enhance mixing between interacting solutesThus a natural question arises Can nonuniform flows sufficiently enhance mixing to restrain incomplete mixingeffects and if so under what conditions We address this question by considering a specific and simple casenamely a laminar pure shear reactive flow Two solution approaches are developed a Lagrangian random walkmethod and a semianalytical solution The results consistently highlight that if shear effects in the system arenot sufficiently strong incomplete mixing effects initially similar to purely diffusive systems will occur slowingdown the overall reaction rate Then at some later time dependent on the strength of the shear the system willreturn to behaving as if it were well mixed but represented by a reduced effective reaction rate

DOI 101103PhysRevE92012922 PACS number(s) 8240Ck 0510Gg 0510Ln 4770Fw

I INTRODUCTION

Chemical reactions driven by mixing are important pro-cesses that occur in a wide variety of engineered and naturalflows Some common examples of practical interest includeatmospheric flows [1] oceanographic flows [2] riverine flows[3] limnologic flows [4] and industrial reactors [5] to mentionbut a few One of the perhaps most obvious but critical featuresto recognize in any reactive system is that in order for reactionsto actually occur the chemicals involved in the reaction mustphysically come into contact with one another Mixing is thephysical process that enables this contact to occur In flowingsystems this is inherently a fluid dynamics problem as flowscan act in such a way as to enhance or indeed suppress mixingand thus chemical reactions

While it is broadly recognized that nonuniform flows canenhance mixing there is increasing evidence that currentmodels do not adequately characterize these mixing processesoften overestimating reaction rates relative to what is observedin field and laboratory experiments [6ndash13] This is likely dueto the fundamental difference between the enhanced spreadingand stretching of solutes that nonuniform flows induce asquantified for example by an effective dispersion coefficientand the true degree of mixing that actually occurs [14]While spreading and mixing are intricately related and indeedhistorically the words have been used interchangeably it isimportant to highlight that they are different processes This isbecause spreading may not account for subscale fluctuationsin concentrations that are critical to understanding mixingThus there is a need for improved models that can accuratelycapture the nature of reactive transport In this work we arguethat Lagrangian-based approaches are naturally conducive to

pastertauacildaggertdecampondeduDaggerdbolsterndedu

capturing these effects To understand this issue in greaterdetail we focus on the irreversible bimolecular reaction A +B rarr P which can be regarded as the fundamental buildingblock of more complex reaction chains (see Ref [15] for adetailed discussion)

A rich body of literature exists exploring the effects ofincomplete mixing on reactions in purely diffusive systems[16ndash21] Consider the following classical experiment wherea domain is initially filled with equal total amounts of A andB such that CA(xt = 0) = CB(xt = 0) = C0 where A andB move by diffusion and react with one another kineticallywith some known rate coefficient k For this setup there isan analytical solution CA(xt) = CB(xt) = C0(1 + kC0t)which at late times scales like inverse time or tminus1 It isimportant to note that this analytical solution relies implicitlyon the assumption that the concentrations are completelyuniform and equal in space or in other words that they arealways well mixed If however there is some stochasticfluctuation of the concentration fields around their mean valuethis late-time scaling will break down For such systemsit has been shown that the late-time mean concentration ofthe species will scale as tminusd4 where d is the number ofspatial dimensions in the system under consideration [1821]Stochastic fluctuations are ubiquitous in real systems and sosuch breakdowns should perhaps be regarded as the normrather than the exception

Specifically for such systems at early times fluctuationsabout the mean concentration are often small enough to bediscarded and the system behaves as is if it were indeedwell mixed However due to the fact that reactions consumemass as the mean concentrations become smaller the relativeinfluence of the fluctuations becomes increasingly importantIndeed at late times isolated pockets or so-called segregatedislands of each reactive constituent emerge and the reactionis limited by how quickly reactants can diffuse across theinterfaces of these islands resulting in the slower tminusd4 scaling[16ndash1821] This phenomenon often goes by the name of

1539-3755201592(1)012922(14) 012922-1 copy2015 American Physical Society

A PASTER T AQUINO AND D BOLSTER PHYSICAL REVIEW E 92 012922 (2015)

Ovchinnikov-Zeldovich segregation and this behavior hasbeen observed experimentally [1920]

This problem has also received a great deal of attention insystems where transport is not by local Fickian diffusion butrather by anomalous dispersive transport including nonlocal inspace superdiffusive systems and nonlocal in time subdiffusivesystems [22ndash24] Such nonlocal transport is common in abroad array of disciplines including transport in porous media[25ndash27] streams [28] groundwater systems [2930] andbiological systems [31] to name a few In all cases similarlate-time scalings for the mean concentration that deviate fromthe well-mixed tminus1 scaling have been predicted and observedThe specific late-time scaling will depend on the nature ofthe transport and the dimensionality of the problem beingconsidered However to our knowledge this problem hasreceived limited attention in the context where transport isby advection and diffusion in a nonuniform velocity field Asnoted above it is well known that nonuniform velocity fieldscan significantly affect the nature of mixing and thus can inprinciple strongly impact mixing-driven reactions

While much focus has been given to mixing in turbulentflows it is important to recognize that mixing in nonuniformlaminar flows can also result in very interesting mixingdynamics [32ndash38] In this work we propose to study thisproblem in one of the simplest forms of nonuniform flownamely a pure laminar shear flow Pure shear flows havereceived a great deal of attention in a variety of applicationsdue to both their simplicity and ability to provide invaluablephysical insight Okubo [3940] studied transport in pure shearflows to better understand solute dispersion in rivers estuarieslakes and oceans Novikov [41] used it as a model to studyturbulent dispersion in streams Others [42ndash44] highlight itas an important case in understanding turbulent dispersionMore recently it was shown [45] that from purely a mixingperspective as quantified by the scalar dissipation rate [4647]or dilution index [48] a pure shear flow is incredibly efficientat mixing The term hypermixing was used emphasizingthat the predicted mixing is even faster than conventionalsuperdiffusion Reference [49] extended this to more complexflows using the Okubo-Weiss metric to quantify enhancementof mixing due to more general locally nonuniform velocityfields

The above discussion leads us the ask the followingquestion Is mixing in a laminar shear flow sufficient toovercome incomplete mixing effects on the evolution of meanconcentration in a reactive system or will incomplete mixingeffects still persist The answer to this question is addressedin this paper with an application to a nonuniform flow of arandom walk particle tracking method designed for reactivetransport It was originally developed for purely diffusivetransport [50] and we extend it further The results of thispaper provide a general understanding of the influence of shearflows on incomplete mixing paving the road for more generalnonuniform velocity fields

The paper is structured as follows In Sec II we presentthe governing equations for flow transport and reaction forthe chosen setup in dimensional and nondimensional formsIn Sec III we discuss the initial conditions focusing on bothdeterministic and stochastic initial conditions which are atthe root of incomplete mixing In Sec IV we describe the

Lagrangian random walk particle tracking method for reactivetransport In Sec V we present and discuss results obtainedwith this method In Sec VI we propose a semianalyticalmodel to interpret observations from Sec V We concludewith a discussion in Sec VII

II GOVERNING EQUATIONS

We consider a bimolecular reactive system that is embeddedin a uniform shear flow Transport of the species is driven bya constant diffusion coefficient and advection according to theuniform shear flow The two constituents in this system A andB react kinetically and irreversibly with one another ie A +B rarr P At this point we are not interested in what happensto the product P but rather on how A and B are consumed sothe fate of P is neglected in this work The flow in the systemis completely independent of the transport ie the flow is notaffected by the concentration of the constituents For an infinitetwo-dimensional space the governing equation for transportis the advection-dispersion-reaction equation (ADRE) givenfor each of the species i = AB by

partCi

partt+ αy

partCi

partx= nabla middot (DnablaCi) minus ri (1)

where Ci(xt) is the concentration [mol Lminusd ] α is the shearrate [Tminus1] D is the dispersion coefficient [L2Tminus1] assumedconstant herein

The sink term in our case is the local rate of reaction andis identical for A and B since they are consumed with a 11stoichiometry ratio ie

rA = rB = r (2)

We assume the law of mass action prevails for our systemand write the reaction rate as

r(CACB) = kCACB (3)

where k [Ld molminus1 Tminus1] is the kinetic rate constant for a givenreaction Thus Eq (1) represents a coupled set

A Nondimensional equations

We consider the following dimensionless variables

Clowast = CC0 tlowast = kC0t αlowast = α(kC0)(4)

xlowast = xl ylowast = yl

where l is a characteristic length associated with the initialfluctuations in concentration (eg a correlation length) Usingthese nondimensional variables our governing equation fortransport becomes

partClowasti

parttlowast+ αlowastylowast partClowast

i

partxlowast = 1

Danablalowast2Clowast

i minus ClowastAClowast

B (5)

where Da = kC0l2

Dis the Damkohler number which quantifies

the relative importance of reaction to transport by diffusion Inother words it quantifies how quickly reactions happen relativeto how quickly diffusion can homogenize a patch of size lαlowast is a dimensionless shear rate that quantifies the relativeimportance of shear to reaction For the sake of conveniencewe drop the stars from here on and all the variables presentedare nondimensional

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INCOMPLETE MIXING AND REACTIONS IN LAMINAR PHYSICAL REVIEW E 92 012922 (2015)

III INITIAL CONDITIONS

The focus of this work is the emergence of incompletemixing of reactants as the system evolves which is known toarise due to stochastic fluctuations in the concentration field[eg 1821ndash23] Thus we focus on the case of a stochasticinitial condition ie the initial concentration comprises amean contribution plus a stochastic or noise term In this casethe exact concentration at a point is unknown and statisticalinformation is available instead

Specifically we assume that for each of the species A andB the stochastic initial concentration field is given by anaverage value plus a white noise contribution Decomposingthe concentration into mean and a perturbation we write fori = AB

Ci(xt) = Ci(xt) + C primei(xt) (6)

where C(xt) is the ensemble mean over all possible realiza-tions The perturbation C prime

i has zero mean The initial conditionfor the mean is given in a general manner by

CA(xt = 0) = CA0(x)(7)

CB(xt = 0) = CB0(x)

Assuming that the initial fluctuation term is Gaussian (orthat higher order moments are irrelevant) the initial conditionfor the perturbation concentration of species i and j is definedby the covariance structure

C primei(x1t = 0)C prime

j (x2t = 0) = μij (x1)δ(x1 minus x2) (8)

where μij is the white noise amplitude and δ is Diracrsquos deltafunction The white noise or delta-correlated initial conditionis tantamount to assuming a very short range correlation inthe fluctuations The delta correlation has been shown to be agood approximation of other short range correlation structuressuch as exponential or Gaussian [5152] and is invoked formathematical convenience

In this work we assume the system is ergodic andstatistically (space) stationary so that the mean concentrationand the correlations depend only on relative position We alsoassume that the species have an identical statistical structureso that the initial mean concentrations are equal and are given(in nondimensional form) by

CA(t = 0) = CB(t = 0) = 1 (9)

Furthermore we assume that the magnitudes of fluctuationvariances are equal and constant in space

μAA = μBB = μ (10)

and assume the species concentration fluctuations are initiallyuncorrelated ie

μAB = 0 (11)

We restrict ourselves to this specific setup for a few reasonsFirst this simple case provides a great deal of physical insightinto the problem at hand Second it is straightforward toimplement this set of initial conditions in a particle trackingalgorithm although more complex structures can readily beincluded Third a semianalytical approximate solution to thissystem for arbitrary dimension (d = 123) when shear is

absent (α = 0) is available from previous work [51] and canbe used for comparison with the new results Additionally thisapproximate solution can at least qualitatively be extended tothe shear problem as will be discussed in the following

IV NUMERICAL PARTICLE TRACKING SIMULATIONS

In this section we describe a Monte Carlo based particletracking approach to finding numerical solutions for the systemand equations presented so far The approach is an extensionof previous works [5153] which were restricted to purelydiffusive transport Here we extend the methodology to thecase of uniform shear flow with the long term goal ofultimately utilizing it to account for more complex generalflows

The fundamental principle behind any particle trackingnumerical method is to represent the concentration field bya cloud of discrete particles representing elementary massesof solute Time is discretized into finite time steps (notnecessarily equal) and the ADRE is applied using the conceptof operator splitting in every time step we first annihilate anyparticles that would react as determined by a physically basedprobabilistic set of laws representing reaction in the systemsecond we move all surviving particles in a manner thatrepresents both advection and diffusion in the system followingclassic random walk principles The annihilation of particlesis determined according to a local reaction probability whichis made up of two components one based on the probabilitythat two particles can collocate which depends entirely ontransport mechanisms and the second based on the probabilitythat reaction occurs given that particles have collocateddetermined entirely by the kinetics of the reaction The detailsare described in the following Note that in this section wefocus on the evolution of the system within a single time stept Hence for simplicity and without loss of generality weuse t = 0 to denote the beginning of the time step

A Advection and diffusion

For the case of a shear flow consider a numerical particlethat at the beginning of a time step is located at x(t = 0) =x0 The combined effect of shear advection and diffusion isexpressed by a random jump of the particle to a new locationat time t We define f as the probability density function (PDF)of the new location neglecting reaction By definition f is thesolution of the advection-diffusion equation

partf

partt+ αy

partf

partx= 1

Danabla2f (12)

for natural boundary conditions (ie f rarr 0 and nablaf rarr 0 asx rarr infin) and initial condition f (xt = 0) = δ(x minus x0) Hencef is the fundamental solution of (12) It is given by themultivariate Gaussian [4045]

f (xt x0) = 1

(2π )d2radic

det[κ]exp

[minus1

2ξT κminus1ξ

](13)

with a covariance matrix given for d = 2 by

κ = κ(t) = 2t

Da

[1 + (αt)23 αt2

αt2 1

] (14)

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0+ 1 2 4αt =

FIG 1 (Color online) Illustration of the temporal evolution ofthe spatial PDF (log scale) and the eigenvectors of κ for a passiveparticle transported by shear flow

where

ξ = x minus (x0 + αy0t x) (15)

is the offset from the mean location y0 = x0 middot y is the initial y

coordinate and x y are the unit vectors in the x y directionsrespectively Isocontours of the distribution are rotated ellipseswith common foci at x0 + αy0t x (see Fig 1) The principalaxes of the ellipses are aligned with the eigenvectors ofκ and rotate with time due to shear one axis is growingsuperdiffusively and the other diffusively Note that whenα rarr 0 the distribution (13) converges to the 2d axisymmetricGaussian (the fundamental solution of the ADE with constantdiffusion) Further details on the implementation of thismethod as well as analytical expressions for the eigenvaluesand eigenvectors κ are provided in Appendix A

B Reaction

Next based purely on transport we wish to determine if twoparticles that are initially separated at t = 0 will collocate andthus potentially react over the next time step Consider twoparticles one of species A and the other of species B that areare located at xA xB initially at t = 0 The temporal density oftheir probability to collocate over some infinitesimal volumedx at time t is obtained by the product of their respectiverandom walk PDFs and dx Thus the temporal density of theirprobability to collocate in any position in space is given by anintegral over this product

v =int

f (xt xA)f (xt xB)dx (16)

where integration is taken over the entire space This expres-sion is a convolution of two Gaussians and is itself equal to aGaussian with the sum of variances

v(st) = 1

2πradic

det[κA + κB]exp

[minus1

2sT (κA + κB)minus1s

] (17)

where s = xA minus xB is the distance between the particles attime t = 0 The convolution in (16) which yields (17) can beperformed in Fourier space via the Faltung theorem [54] SinceκA = κB = κ we can rewrite (17) as

v(st) = Da

8πtradic

1 + (αt)212

times exp

minusDa[|s|2 minus sxsyαt + s2

y (αt)23]

8t[1 + (αt)212]

(18)

where |s| =radic

s2x + s2

y We thus see that the collocationprobability density (18) depends on the following parameters(1) both components of s the initial interparticle distance (2)

the diffusion length scale (2tDa)12 and (3) the characteristicdistance due to shear αt

Assuming that the pair of AB particles has survived overtime t (ie they have not reacted with other particles) theirprobability to react with each other during the infinitesimaltime t prime isin [tt + dt) is given by

mpv(st)dt (19)

where mp is the nondimensional mass carried by a singleparticle ie the number of moles multiplied by l2C0Thus the probability of survival is given by the conditionedprobability

Ps(t + dt) = Ps(t)[1 minus mpv(st)dt] (20)

or in words the probability they are unreacted at t + dt isthe probability they were unreacted at t and did not react witheach other since then Hence

dPsPs = minusmpv(st)dt (21)

and by integration over a time step t we obtain the overallreaction probability of the couple during that time step

Pr (t) = 1 minus Ps(t) = 1 minus exp

[minusmp

int t

0v(st prime) dt prime

] (22)

For the degenerate case α = 0 the integral in (22) yieldsint t

0v(st prime) dt prime = Da

8πE1

(Da

8t|s|2

) (23)

where E1 is the exponential integral Note how this expressiondepends only on the absolute value of the interparticledistance as the system becomes isotropic when α = 0 Forthe general case α = 0 Eq (22) is integrated numerically(Fig 2) However the simplified solution (23) can be usefulfor approximating (22) when (αt)2 12 Hence it can beused if the time steps are small enough (recall that t is thelength of the time step in the framework of this section)

Also with (23) it is easy to see that if Da8t

s2 rarr 0 then theintegral in (22) tends to infinity and Pr rarr 1 This comes asno surprise when the particles are close by or when they aregiven enough time to diffuse their colocation probability isexpected to be close to 1 If on the other hand t rarr 0 whiles = 0 the collocation probability tends to a delta function ins and Pr rarr 0 Again this is expected since particles that aresufficiently far from one another cannot collocate unless theyare given sufficient time to diffuse

From a practical perspective to reduce numerical costs thesearch for neighbors must be limited to a certain distancerange We define this range by the ellipse

sT (2κ)minus1s = 2a2 (24)

where a gt 0 is some predefined constant Choosing larger a

improves the accuracy of the numerical approach but has anincreased computational cost as a rule of thumb the choicea = 2 is typically sufficient as any induced error will beinsignificantly small [51] Note again that when α = 0 theellipse converges to a circle of radius a

radic8tDa

With this cutoff search distance the collocation probability(17) integrates to

η = 1 minus eminusa2 (25)

012922-4


minus08 minus06 minus04 minus02 0 02 04 06 08

minus08

minus06

minus04

minus02

0

02

04

06

08

minus6

minus5

minus4

minus3

minus2

minus3 minus2 minus1 0 1 2 3

minus05

0

05

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

(a) (b)

(c)

sx

s y

sx

s y

sx

s y

FIG 2 (Color online) The integrated nondimensional collocation probabilityint t

0 v(st prime) dt prime (in log scale) for Dat = 1 and αt = 01110in (a) (b) (c) respectively

so the collocation probability needs to be rescaled by the factorηminus1 for consistency to ensure integration over the density to beunity Hence the probability of reaction of a single A particleduring a time step is evaluated by numerical integration of(22) or by the first order approximation

Pr = ηminus1mpt

Nnbsumi=1

v(si t) (26)

where Nnb is the number of B neighbors found within theellipse The probability is compared to a random numbergenerated from a uniform distribution U isin [01] If U gt Pr the A particle under consideration is annihilated (removedfrom the system) and one of the neighboring B particles isannihilated as well The choice between neighbors is donebased on their weighted probability of reaction so that werandomly choose the B particle based on its relative probabilityof reaction Additionally if a naıve search is performed tocalculate collocation probabilities between all product pairs inthe system this can be computationally costly at O(N2) whereN is the number of particles used To speed up this processwe use an algorithm from the data mining literature called thekd tree [55] which accelerates this search to an O(N ln N )process providing significant computational savings

C Numerical domain and finite size effects

The system we consider is idealized as ergodic and infiniteTherefore the total mass in the system is infinite as well Eachparticle represents a finite mass mp Hence the number of

particles needed to simulate the system accurately is infiniteThis is clearly unfeasible since the number of particles in anumerical particle tracking simulation is finite and inherentlyconstrained by the computational resources To overcome thisdifficulty we follow the classical approach of simulating theinfinite system over a finite domain with periodic boundaryconditions At early times of the simulation the finitenessof the domain is expected to have a negligible effect on theresults and the domain can be considered infinite However atlate times boundary effects in the numerical simulation willlead to deviations from this idealization

Indeed previous works on purely diffusive transport[212351] have observed and discussed this effect For thecase of no shear (α = 0) over a domain of nondimensional sized with periodic boundary conditions the finite size effectskick in when the typical size of segregated islands grows toabout half of the domain size This was observed around thetime

tbnd = Da(8)2 (27)

which can be thought of as representing the characteristictime of diffusion of concentration perturbations over the finitesimulation domain Similar if not more restrictive conditionswill arise when α = 0 and will be discussed in greater detailin the following In this study we use a rectangular domainS [0x] times [0y] for the numerical simulation

D Implementation of initial conditions

Implementing periodic boundary conditions in a numer-ical particle tracking method over a rectangular domain is

012922-5


straightforward The initial conditions (9)ndash(11) were imple-mented by randomly spreading N0 particles of each speciesin the domain S The number of particles is not arbitrary butrather is derived from the initial condition Let us define theinitial nondimensional density of particles by

ρ0 = N0(xy) (28)

This particle density is inversely proportional to the magnitudeof the initial white noise μ and given by

ρ0 = 1μ (29)

This expression is a straightforward extension of the onederived by Paster et al [51 Appendix C] To understandthis result consider a system where the initial condition forCi(xt = 0) is deterministic ie

CA(xt = 0) = CA0(x) CB(xt = 0) = CB0(x) (30)

such that μ rarr 0 (zero noise term) By (29) we find ρ0 rarr infinHence even if the domain is finite an exact representationof the initial condition (30) necessitates the use an of infinitenumber of particles which is consistent with classical randomwalk theory [56] For such case a particle tracking algorithmmay be less favorable due the error induced by the finitenumber of particles

In contrast if we are interested in the case of a noisy orstochastic initial condition as is the case with this work thismethod might be regarded as ideal For such cases particletracking methods require a finite number of particles for anaccurate representation of the initial condition When theinitial condition contains a considerable noise the reactiveparticle tracking method may be very efficient with regardto computational resources due to the small number ofinitial particles needed This may be an advantage overother numerical methods such as Monte Carlo simulationsusing Eulerian finite difference volume or element methodsAdditionally random walk methods are known to be muchless prone to numerical diffusion which would lead to greatermixing and is thus an important factor in mixing-drivenreactions [57]

V RESULTS

Using the numerical algorithm described in the previoussection we have performed various simulations to studythe effect of the Dahmkoler number (Da) and shear rate(α) on the evolution of average concentration in the system(Table I) In Fig 3 we show representative results for aspecific Dahmkoler number namely Da = 2 spanning a rangeof α values (0 10minus4 10minus3 10minus2 10minus1 1 10) The figurealso shows two analytical solutions one corresponding tothe well-mixed case and an approximate analytical solution[51] that incorporates noisy initial conditions for α = 0 Aslight discrepancy between the numerical and the approximateanalytical solution for α = 0 is observed As discussed infurther detail in the following section this discrepancy is due

TABLE I Parameters of the numerical simulations To speed upruns the time step size is given by tj = mintmaxt0(1 + ε)j where j is the step number For all simulations tbnd = DaN064 =5 times 104 the number of ensemble realizations was Nsim = 8 thesearch radius factor was a = 2 and domain size was x times y =1 times 4

Da α N0 t0 ε tmax

0 25 times 10minus3 10minus2 5010minus4 25 times 10minus3 001 1010minus3 25 times 10minus3 001 10

2 10minus2 64 times 106 25 times 10minus3 001 110minus1 25 times 10minus3 001 1

1 10minus3 25 times 10minus3 02510 25 times 10minus4 25 times 10minus3 005

0 0025 0025 10010minus4 0025 0025 10010minus3 0025 0025 100

8 10minus2 16 times 106 0025 0025 1010minus1 0025 0025 10

1 0012 0012 510 0012 0012 1

to restrictive assumptions involved in developing the analyticalsolution namely the neglecting of high-order moments of theconcentration fluctuation fields

A Pure diffusion α = 0

First to elucidate some matters let us focus on thepreviously studied case of pure diffusion and no shear (α = 0)The mean concentration curve for this case (see Fig 3) matchesthe well-mixed solution at very early times but clearly divergesfrom the well-mixed solution at relatively early times This isconsistent with the noisy initial condition for concentrationand has been extensively discussed in previous works (egPaster et al [51] and references therein) For this case the onlymixing mechanism is diffusion which strives to homogenizethe concentration field and drive the system to a well-mixedstate At the same time reaction occurs in all locations whereA and B coexist thus annihilating A and B and promotingthe segregation of species destroying mixing in the systemFigure 4 depicts snapshots of particle locations at various timesduring the course of the simulation These reveal that at veryearly times A and B particles largely overlap in space andthe system can be regarded as relatively well mixed At latertimes segregation leads to the formation of the aforementionedislands of single species and reactions become restricted totheir boundaries At times for which island segregation isclearly visible the average concentration in the domain scaleswith time as tminus12 in agreement with previous predictionsand observations As seen in Fig 4 the total number ofislands decreases with time and the average size of an islandgrows respectively Islands that contain significant mass ofone species consume neighboring islands with lower mass ofthe second species which in turn disappear from the systemAt very late time around t tbnd [see Eq (27)] the finitesize of the numerical domain starts to affect the simulationAt this time the area occupied by a single island reaches

012922-6


FIG 3 (Color online) The evolution of mean concentration with time for Da = 2 The approximate analytical solution for α = 0 and thewell-mixed solution are also plotted All parameters are nondimensional

about half the domain area and the infinite-domain behaviorbreaks down As noted above for a detailed discussion on thissee [51]

B Diffusion and shear α = 0

For the cases where shear effects do exist (ie α = 0)we observe a different behavior from the purely diffusivecase highlighting the role that shear plays in this systemUp to four discernible time regime scalings in averageconcentration emerge which will be discussed in detail Firstwe qualitatively describe the results

At very early times in all cases there is a close agreementbetween the zero shear and finite shear simulations at theseearliest times diffusion dominates over shear effects and thelatter are not discernible in the mean behavior of the systemAt these times the solutions break away from the well-mixedcase and other than for the highest simulated shear cases alldisplay the tminus12 regime In the largest shear case (α = 10)shear effects are strong enough to suppress this regime and thesolution closely follows the well-mixed prediction indicatingthat large shear can indeed suppress incomplete mixing effectsThen at some α-dependent time the solutions break awayfrom this scaling and shift to a faster scaling of tminus1 Thelarger α the earlier this transition occurs During this timethe concentration evolves parallel to the well-mixed solutionbut at higher overall concentrations Then again at some α-dependent value the solutions break away from this scalingand transition to another slower scaling of tminus14 This is a finitesize effect associated with the horizontal size of the domainwhich is unavoidable in a numerical study and can be explained

theoretically (see Sec VI) but is not expected in an infinitedomain All solutions collapse together during this regimewith deviations for different α values well within the standarddeviation of the Monte Carlo simulation Not surprisingly weobserved that increasing the number of realizations to calculatethe average concentrations reduced the magnitude of thesedeviations Since we are primarily interested in infinite-domaineffects we will focus on the regimes before the shear- anddiffusion-induced boundary effects take place Nonethelessthe latter warrant some further discussion

Figure 4 shows particle locations at various times duringthe simulations for the Da = 2 case for multiple values ofα What is interesting to note is that at early times when themean concentrations still match the α = 0 case (see Fig 3) thegeneric shapes of the islands look very similar to the α = 0case regardless of the specific value of α However as thebreakaway from this regime occurs to the faster tminus1 scalinga noticeable difference appears whereby the islands take on adifferent shape At these times the islands are more elongatedin one direction which is tilted relative to the natural y axisreflecting the presence of shear in the system and the factthat there is a superdiffusive growth along one axis and adiffusive one along the other This is directly analogous to thefundamental solution of a point source in a diffusive shear flowin Eq (13) It is at these times that the effect of the shear flowis important in the system

Next during the tminus14 regime (the finite size effect scaling)all islands now span the full horizontal extent of the domainIn essence they have tilted all the way such that the superdif-fusively growing axis is now closely aligned with the x axisReactions are now mostly limited by vertical diffusion across

012922-7


α = 0

α = 10minus3

α = 10minus1

x

y

FIG 4 (Color online) Evolution in time of some representative realizations with various shear rates for Da = 2 Note how the islandsform initially with irregular shape Later on for α gt 0 they tilt and become elongated due to the shear and finally lie flat perpendicular to thex axis Compare with Fig 3 Note that in the above figures particle numbers were reduced (randomly) to sim103 to allow a good visualizationof the results and only the bottom half of the domain is shown

islands and the system behaves as if it were one dimensionalwhich is consistent with an incomplete mixing scaling of tminusd4

with d the number of spatial dimensions equal to one Thisis a finite size effect associated with the time it takes for anisland to span the horizontal width of the domain unavoidablein a finite size numerical simulation but not expected in aninfinite domain (see the following section and Appendix B formore discussion and explanation on the matter)

Finally at tbnd [see Eq (27)] another finite size effect thesame as in the cases with no shear kicks in At this pointthe islands occupy half the domain there is no longer anyspace for islands to grow as they would in an infinite domainand a complete breakdown occurs It is interesting to notethat at the latest time in all the shear flow cases the islands arehorizontally elongated again reflecting the role of shear whilefor the pure diffusion case it is equiprobable that they couldbe vertically or horizontally aligned as there is no preferentialdirection for growth in a purely diffusive system

Figure 5 shows the mean concentration against time forthe same Da = 2 as well as Da = 8 This figure serves tohighlight that qualitatively the behavior for both Da is verysimilar and all the observations and scalings discussed abovestill emerge However the breakaways from the well-mixedbehavior and transitions between each of these scaling regimesoccur at different times indicating a Da-dependent behavioralso The emergence of these distinct time scalings is keyto understanding incomplete mixing effects on chemical

reactions but at this point it is difficult to truly quantifythese scalings from these observations alone In the following(Sec VI) the results will be interpreted using a semianalyticalmodel that enables a more mechanistic explanation of eachregime

VI INTERPRETATION OF SIMULATION RESULTSA SEMIANALYTICAL CLOSURE

In this section we present a semianalytical solution basedon methods from previous works that look at purely diffusivetransport to explain each of the time scaling regimes formean concentration that were observed in Sec V Thissemianalytical approach is based on a closure argumentSimilar closures have been invoked in many previous studiesbut it should be noted that it is known to not be exact[1821ndash23] Rather it enables one to predict the emergentscalings with time of mean concentrations but not the exactvalues of mean concentration or time when these occur Even inpurely diffusive cases it is unable to match observations exactlywithout some empirical correction These discrepancies havebeen studied and explained in detail by Paster et al [51] Thisis due to a problematic assumption behind the derivation of theclosure discussed in the following Given these well-knownlimitations our objective here is not to match exactly theobservations from the numerical simulations but to explorethe emergent scalings with such a closure

012922-8


FIG 5 (Color online) Same as Fig 3 with additional solutions for Da = 8 The thin gray lines in the background correspond to the resultsfor Da = 2

A Closure problem for mean concentrations

Averaging over Eqs (1)ndash(3) it can readily be shown that themean concentration of the reactants in this system will evolvebased on the following ordinary differential equations

dCA(t)

dt= dCB(t)

dt= minusCA

2 minus C primeAC prime

B (31)

Here we have used the physical requirement that CA(t) =CB(t) at all times since the initial condition is CA(t = 0) =CB(t = 0) and the reaction stoichiometry is 11 When fluctu-ation concentrations are small relative to mean concentrationsthe second term on the right hand side of (31) will be negligibleand the system will evolve as is if it were well mixedHowever when the fluctuations are not small relative to meanconcentrations this term plays an important role changing theevolution of the system in a meaningful manner This presentsa closure problem as it requires a governing equation or modelfor the C prime

AC primeB term

To close this problem we rely on previous works indiffusive and superdiffusive systems where using the methodof moments it has been shown that a reasonable closure is toassume

C primeAC prime

B = minusχG(x = xpeakt) (32)

where G is the Greenrsquos function for conservative transport inthe specified system xpeak is the location of the peak of theGreenrsquos function and χ is a constant The formal derivation ofclosure (32) can be found in Refs [212358] where exactexpressions for χ are developed Similar closures without

formal derivation have been proposed [1822] However itis important to note that it relies on the assumption thatmoments higher than third order (ie terms that consist ofproducts of three fluctuation concentrations) are negligibleWhile this is a standard assumption in many closure problemsit is inaccurate as highlighted by Paster et al [51] who showedthat in diffusive systems the semianalytical solution and highresolution numerical solutions while qualitatively similar inthe late-time scaling will behave differently Indeed using theldquoexactrdquo values predicted by neglecting higher order moments[2123] can in some instances yield unphysical results (egcreating rather than destroying mass of reactants when χ is toolarge) This mismatch is reconciled by demonstrating that thirdand higher order moments are in fact not negligible but canhave similar structure to second order moments manifesting asa different effective value of χ [51] This justifies the structureof closure (32) but not the specific value of χ predicted Thusfor now we keep χ as a free (constant) parameter in ourclosure

B Greenrsquos function for pure shear flow

The Greenrsquos function for transport in a pure shear flowwithin our dimensionless framework satisfies the followinggoverning equation

partG

partt+ αy

partG

partx= 1

Danabla2G (33)

with natural boundary conditions at infinity and initialcondition

G(xt = 0) = δ(x minus x0) (34)

012922-9


in full analogy with the problem posed by (12) whose solutionis given by (13) Thus

G(x = xpeakt) = 1

2πradic

det[κ]= Da

2πtradic

4 + (αt)23 (35)

Perhaps most notable in this solution is that for small α or smalltimes the leading order behavior of G(x = xpeakt) scales liketminus1 but at sufficiently large times when the α-dependent termdominates in the denominator the leading behavior scales liketminus2 which reflects the hypermixing regime alluded to earlierwhere the plume spreads and the concentration peak decreasessuperdiffusively

C Solutions with the closure

With the proposed closure (32) and (35) our governingequation for the mean concentrations of the reactants (31)becomes

dCA(t)

dt= minusCA

2 + χDa

2πtradic

4 + (αt)23 (36)

While we are not aware of a closed form analytical solutionto this equation for α = 0 it is straightforward to integrate itnumerically Additionally some useful asymptotic argumentscan be made to understand how the solution evolves in timeThese are discussed in the following subsection Then toshow the full emergent behavior and verify our asymptoticarguments we present numerical solutions of this equationNote that for α = 0 Eq (36) is a Riccati equation whosesolution can be expressed in terms of Bessel functions [2123]

Early times At early times if the background fluctuationsin concentration are small relative to mean concentration weexpect the dominant balance in Eq (36) to be between the termon the left hand side and the first term on the right hand sideIn this case the equation can be solved as

CA(t) = 1

1 + t(37)

which is the solution for the well-mixed systemLate times As the concentrations deplete the term dCAdt

in Eq (36) becomes negligible compared to the other termsin the equation Then the dominant balance in the equationshifts and becomes a balance between the two terms on theright hand side Thus

CA =(

Daχ

2πtradic

4 + (αt)23

) 12

=(

Daχ

4π

) 12

(t2 + α2t412)minus14 (38)

Late-time scaling 1 Now if at these times t2 13 α2t412or t lt

radic12α then to leading order the concentration will

decrease as CA sim tminus12 which is the same scaling one wouldobtain for a purely diffusive system at late times in twodimensions Note that for large values of α this intermediatecondition is not necessarily met since it may be violated

when the incomplete mixing effects become important Indeedfor the α = 10 cases presented in Fig 5 this scaling neveremerges

Late-time scaling 2 At larger times when α2t412 13 t2 ort gt

radic12α the concentration will scale as CA sim tminus1 which

is the same scaling as if the system were well mixed Thissuggests that the effect of a pure shear flow can indeed bestrong enough to overcome the effects of incomplete mixingHowever the overall concentrations in the system could beconsiderably larger than the purely well-mixed case due tothe intermediate CA sim tminus12 regime

In a well-mixed system the solution will evolve as 1(1 +t) which at late times approaches tminus1 Thus we can define anasymptotic late-time retardation factor

Rasy =(

2παradic3Daχ

) 12

(39)

such that the concentration converges to (Rasyt)minus1 ie theinfluence of shear is to return the system to behaving in amanner reflective of perfect mixing but at a later time Thisis equivalent to having an effective (retarded) reaction ratekeff = kRasy While the notion of an effective reaction rate atthese late times is useful note that it should only be applied atlate times when t gt

radic12α

Numerical solutions of semianalytical equation

In this section we solve Eq (36) numerically using thenumerical ordinary differential equation solver ODE23 availablein MATLAB From (36) it is evident that CA(t) is a function oftime depending on Daχ and α Results varying and showingthe respective influence of these are shown in Figs 6 and 7

In Fig 6 we consider fixed Daχ and vary α over severalorders of magnitude to demonstrate the influence of shear In allcases at very early times we observe close agreement betweenthe α = 0 incomplete mixing solution and the solutions with

10minus1

100

101

102

103

104

105

10minus4

10minus3

10minus2

10minus1

100

t

C

Perfect Mixingα=0α=1α=1α=1α=1

α=1α=10

tminus1

tminus12

FIG 6 (Color online) Numerical solution of (36) for various αwith fixed Daχ = 5 times 10minus1 At late times concentration scales liketminus12 at even later times if α = 0 it scales like tminus1 The value of α

affects the time of transition between each of the scaling regimes

012922-10

INCOMPLETE MIXING AND REACTIONS IN LAMINAR PHYSICAL REVIEW E 92 012922 (2015)C χ

χχχ

FIG 7 (Color online) Numerical solution of (36) for variousDaχ In this figure Daχ is varied to show how it affects the transitionbetween each of the scaling regimes The solid lines correspond tothe equivalent case with no shear ie α = 0 and the dashed lines toα = 10minus2

shear Then again in all cases the solutions break awayfrom this to a faster tminus1 scaling as anticipated from theasymptotic arguments presented above Larger α means anearlier breakaway to this faster regime Note that the α = 10case never seems to follow the intermediate tminus12 regimeconsistent with the idea that this regime need not occur ifthe conditions discussed above are not met

In Fig 7 we consider fixed α and vary Daχ In this figurethe solutions for each Daχ combination and α = 0 are alsoshown Raising Daχ triggers incomplete mixing effects atearlier times Larger χ suggests larger noise in the initialcondition while larger Da means higher ratio of reactionto diffusion When the product is larger this means thatfluctuations in the mean concentration will play an importantrole at earlier times The effect of shear is to return from thetminus12 to a tminus1 scaling This is clearly seen for all cases andseems to occur at the same time independent of Daχ whichis again consistent with the arguments in the previous sectionsince this transition is determined solely by the value of α

For both Figs 6 and 7 we do not observe the late-time tminus14

scalings observed in the numerical simulations As arguedbefore this scaling is a boundary effect due to the horizontalextent of the numerical domain which an infinite-domainsolution cannot reproduce Since our primary interest here isthe influence of the initial conditions on the late-time scalingof mean concentration in an infinite domain we do not focuson the tminus14 scaling here However to demonstrate that it istruly a finite size effect associated with the limited horizontalextent of the domain we extend the current closure to includefinite boundary effect in Appendix B which consistentlydemonstrates this behavior

VII CONCLUSIONS

In this work we have studied mixing-limited reactions ina laminar pure shear flow to address the question of whethershear effects can overcome the effects of incomplete mixing on

reactions We have considered a simple spatial system initiallyfilled with equal amounts of two reactants A and B subjectedto a background shear flow The average concentrations of A

and B are initially the same but there are also backgroundstochastic fluctuations in the concentrations These can lead tolong term deviations from well-mixed behaviors due to spatialsegregation of the reactants which can form isolated islandsof individual reactants where reactions are limited to the islandinterfaces

To study this system we adapted a Lagrangian numer-ical particle-based random walk model built originally tostudy mixing-driven bimolecular reactions in purely diffusivesystems to the case with a pure shear flow with the longterm goal of developing it for more general nonuniformflows Additionally we studied the system theoretically bydeveloping a semianalytical solution approach by proposinga simple closure argument The results of the two approachesare qualitatively analogous in that the mean concentrations ofreactants over time scale with the same power laws An exactquantitative match however is not obtained it is well knownfrom previous work on diffusive transport that such closuresare not exact but that they can match emergent scalings inmean concentrations which we demonstrate is also true forthe case with shear considered here

With both methods we observed the following behaviorsAt early times when mean concentrations are much largerthan background fluctuations the system behaves as it if werewell mixed Then when the fluctuations become comparablein size to the mean concentrations and the domain becomessegregated incomplete mixing slows down the reactionsThus the mean concentrations in the system evolve witha slower characteristic temporal scaling consistent with adiffusion-limited case where shear is absent Then at latertimes when shear effects begin to dominate the systemreturns to behaving in a manner similar to a perfectly mixedsystem but described by an overall lower effective reactionrate constant If shear is sufficiently strong the diffusivelikeincomplete mixing regime never emerges and the systembehaves as well mixed at all times It is important to notethat this does not mean that the system is actually well mixedas segregated islands still occur but rather that the mixingassociated with the shear flow is sufficiently fast to result in ascaling analogous to a well-mixed system

The system is characterized by two dimensionless num-bers Da is a Damkohler number that quantifies the relativemagnitude of reaction time scales to diffusion time scalesLarge Da means that reactions happen more quickly thandiffusion can homogenize any background fluctuations thussystems with larger Da will amplify initial fluctuations andincomplete mixing patterns can play an important role at latetimes Likewise when Da is small diffusion can homogenizefluctuations more quickly and the system will behave asbetter mixed The second dimensionless number is α adimensionless shear rate that quantifies relative influence ofshear to reaction The Damkohler influences the onset time ofincomplete mixing while α controls the onset of a return towell-mixed type scaling

A key take home message of this work is that while apure shear flow can lead to a behavior that is consistentwith a well-mixed system if the nature and evolution of

012922-11


incomplete mixing in the system is not adequately accountedfor then predictions of reactant concentrations particularly atlate times can be off by as much as orders of magnitude Incontrast if incomplete mixing is accounted for more realisticpredictions can be obtained

ACKNOWLEDGMENTS

PA would like to acknowledge partial funding bythe Startup grant of the VP of research in TAU TAgratefully acknowledges support by the Portuguese Foun-dation for Science and Technology (FCT) under GrantNo SFRHBD894882012 DB would like to acknowledgepartial funding via NSF Grants No EAR-1351625 andNo EAR-1417264 as well as the U S Army ResearchLaboratory and the U S Army Research Office underContractGrant No W911NF1310082 The authors thank thetwo anonymous reviewers for their constructive comments onthe manuscript

APPENDIX A EXPRESSIONS FOR THE EIGENVALUESAND EIGENVECTORS OF κ

The eigenvalues of κ the covariance matrix are given by

λ12 = 12 Tr(κ) plusmn

radic[Tr(κ)]2 minus 4 det κ (A1)

where

Tr(κ) = κ11 + κ22 = 4t

Da

[1 + 1

6α2t2

](A2)

is the trace of κ and

det κ = κ11κ22 minus κ212 = 4t2

Da2

[1 + 1

12α2t2

] (A3)

Substituting (A2) and (A3) into (A1) one finds

λ12 = 2t

Da

1 plusmn

[1

2αt

radic1 + 1

9(αt)2

]+ 1

6α2t2

(A4)

The eigenvectors in turn are aligned with

vprime12 =

[1

minus 13αt plusmn

radic1 + 1

9 (αt)2

](A5)

and the normalized unit vectors v12 are readily obtained bynormalization of vprime

12When αt 1 the eigenvectors are inclined at about plusmn45

relative to the x axis For larger αt the eigenvectors tiltclockwise (see Fig 1) and for αt 13 1 they tend to align withx and minusy or more precisely with [13(2αt)] and (1minus 2

3αt)In practice the new location of a particle can be obtained

by translation of the x coordinate of the particle by αy0t and arandom walk in two dimensions with jumps along the rotatedeigenvectors of κ with magnitudes

x1 = ξ1

radicλ1

(A6)x2 = ξ2

radicλ2

where xi is the walk length in the direction of the itheigenvector and ξi is a random number generated from a

normal distribution with zero mean and unit variance Hencethe new location is given by

x = x0 + αy0t +dsum

i=1

(vi middot x)xi

(A7)

y = y0 +dsum

i=1

(vi middot y)xi

APPENDIX B DOMAIN BOUNDARY EFFECTS

As discussed in Sec VI we wish to demonstrate thatthe tminus14 scaling can be attributed to a boundary effectSpecifically we will argue here that this scaling is associatedwith finite extent of the horizontal boundaries of the domainAs shown in Eq (32) the evolution of the mean concentrationdepends on the peak of the Greenrsquos function for conservativetransport For a finite periodic domain this can be be calculatedby the method of images

G(x = xpeakt) =infinsum

n=minusinfin

1

(2π )d2radic

det[κ]exp

[minus1

2ξTn κminus1ξn

]

(B1)

where

ξn = (nx0) (B2)

and x is the horizontal length of the domain In essence thissolution adds every contribution from point plumes located atequidistant intervals of x along the x axis at y = 0 and sumstheir contribution to the point x = 0 which is where the peakof the Greenrsquos function for a point source located initially at(xy) = (00) will occur It is convenient to take this point asone does not have to account for advection-induced drift inthe peak location but the result is independent of this choiceWe have

G(x = xpeakt) = Da

4πtradic

1 + α2t212

timesinfinsum

n=minusinfinexp

[minus Da(nx)2

4t(1 + α2t212)

](B3)

which at late times converges to

G(x = xpeakt) =radic

3

2π

Da

|α|t2

infinsumn=minusinfin

exp

[minus3Da(nx)2

α2t3

]

(B4)

The term outside of the summation scales like tminus2 as forthe infinite-domain case As t3 13 3Da2

xα2 when the

boundary effects become important the infinite sum can beapproximated by an integral which converges toradic

π

3Da

|α|x

t32 (B5)

Thus the Greenrsquos function converges to

G(x = xpeakt rarr infin) = 1

2

radicDa

π

1

x

tminus12 (B6)

012922-12


10minus1

100

101

102

103

104

105

10minus3

10minus2

10minus1

100

t

C

FIG 8 (Color online) Numerical solution for the average con-centration using the finite-domain Greenrsquos function in Eq (B1)for various α Da = 10 and χ = 001 In this figure α is variedto show how it affects the transition between each of the scalingregimes Note the collapse of all late-time scalings on to the same

line Ci(t) =radic

χ

2x( Da

π)14

tminus14 in agreement with observations from

numerical simulations

Hence the combined effect on the scaling of the peak of theGreenrsquos function is

G(x = xpeakt) sim tminus12 (B7)

Since at late times our dominant balance argument indicatesthat Ci = radic

χG(x = xpeakt) once the horizontal boundaryeffects are felt the mean concentrations will converge to

Ci(t) = radicχG(xpeakt) =

radicχ

2x

(Da

π

)14

tminus14 (B8)

which is independent of the value of α and scales like tminus14These results are in agreement with the observations in theparticle tracking simulations In Fig 8 we plot the evolutionof the average concentration over time by solving Eqs (31)and (32) numerically using the finite-domain Greenrsquos functionin (B3) instead of its infinite-domain counterpart

The results in the figure are very similar to those observedin the numerical simulations particularly in that at late timesall solutions for α = 0 collapse on to the same curve Thisscaling can also be interpreted physically as a boundary effectas follows At the relevant late times the single-reactant islandsspan the full width of the domain and the system essentiallybecomes one dimensional The late-time incomplete-mixingscaling is known to behave as tminusd4 which matches ourfindings [18] This again demonstrates the utility of the simpleproposed closure in interpreting the results observed in thenumerical simulations As a cautionary note we would againlike to highlight that the proposed closure is not exact the goodagreement with simulations is promising but further work isrequired to refine it rigorously

[1] J Seinfeld and S Pandis Atmospheric Chemistry and PhysicsFrom Air Pollution to Climate Change (Wiley Hoboken NJ2006)

[2] I Mezic S Loire V Fonoberov and P Hogan Science 330486 (2010)

[3] C Escauriaza C Gonzalez P Guerra P Pasten and G PizarroBull Am Phys Soc 57 13005 (2012)

[4] P Znachor V Visocka J Nedoma and P Rychtecky FreshwaterBiol 58(9) 1889 (2013)

[5] J Sierra-Pallares D L Marchisio E Alonso M T Parra-Santos F Castro and M J Cocero Chem Eng Sci 66(8)1576 (2011)

[6] W Stockwell Meteorol Atmos Phys 57 159 (1995)[7] L Whalley K Furneaux A Goddard J Lee A Mahajan H

Oetjen K Read N Kaaden L Carpenter A Lewis et alAtmos Chem Phys 10 1555 (2010)

[8] T Michioka and S Komori AIChE Journal 50 2705 (2004)[9] J Pauer K Taunt W Melendez R Kreis Jr and A Anstead

J Great Lakes Res 33 554 (2007)[10] C Gramling C Harvey and L Meigs Environ Sci Technol

36 2508 (2002)[11] M de Simoni J Carrera X Sanchez-Vila and A Guadagnini

Water Resour Res 41 W11410 (2005)[12] P d Anna J Jimenez-Martinez H Tabuteau R Turuban T Le

Borgne M Derrien and Y Mheust Environ Sci Technol 48508 (2014)

[13] P De Anna M Dentz A Tartakovsky and T Le BorgneGeophys Res Lett 41 4586 (2014)

[14] M Dentz and J Carrera Phys Fluids 19 017107 (2007)[15] D Gillespie J Chem Phys 113 297 (2000)[16] A Ovchinnikov and Y Zeldovich Chem Phys 28 215

(1978)[17] D Toussaint and F Wilczek J Chem Phys 78 2642 (1983)[18] K Kang and S Redner Phys Rev Lett 52 955 (1984)[19] E Monson and R Kopelman Phys Rev Lett 85 666 (2000)[20] E Monson and R Kopelman Phys Rev E 69 021103 (2004)[21] A M Tartakovsky P de Anna T Le Borgne A Balter and D

Bolster Water Resour Res 48 W02526 (2012)[22] G Zumofen J Klafter and M F Shlesinger Phys Rev Lett

77 2830 (1996)[23] D Bolster P de Anna D A Benson and A M Tartakovsky

Adv Water Resour 37 86 (2012)[24] R Klages I Radons and Gand Sokolov Anomalous Transport

Foundations and Applications (Wiley Hoboken NJ 2008)[25] D Koch and J Brady J Fluid Mech 180 387 (1987)[26] D Koch and J Brady Phys Fluids 31 965 (1988)[27] D Koch R Cox H Brenner and J Brady J Fluid Mech 200

173 (1989)[28] F Boano A I Packman A Cortis R Revelli and L Ridolfi

Water Resour Res 43 W10425 (2007)[29] B Berkowitz A Cortis M Dentz and H Scher Rev Geophys

44 RG2003 (2006)[30] D Benson and S Wheatcraft Transport Porous Media 42 211

(2001)[31] F Hofling and T Franosch Rep Prog Phys 76 046602 (2013)[32] S Jones and W Young J Fluid Mech 280 149 (1994)

012922-13


[33] C Escauriaza and F Sotiropoulos J Fluid Mech 641 169(2009)

[34] H Aref and S Balachandar Phys Fluids 29 3515 (1986)[35] T H Solomon and J P Gollub Phys Rev A 38 6280

(1988)[36] Z Pouransari M Speetjens and H Clercx J Fluid Mech 654

5 (2010)[37] M Latini and A Bernoff J Fluid Mech 441 399 (2001)[38] P Shankar J Fluid Mech 631 363 (2009)[39] A Okubo J Oceanogr Soc Jpn 24 60 (1968)[40] A Okubo and M J Karweit Limnol Oceanogr 14(4) 514

(1969)[41] E Novikov J Appl Math Mech 22 576 (1958)[42] G Csanady Turbulent Diffusion in the Environmnet Geo-

physics and Astrophysics Monographs Vol 3 (Reidel Dor-drecht 1973) pp 147ndash148

[43] H Tennekes and J L Lumley A First Course in Turbulence(MIT Press Cambridge MA 1972)

[44] A Monin and A Yaglom Statistical Fluid Mechan-ics Mechanics of Turbulence Vol 1 (Dover New York2007)

[45] D Bolster M Dentz and T Le Borgne Water Resourc Res47 W09602 (2011)

[46] S Pope Turbulent Flows (Cambridge University PressCambridge UK 2000)

[47] T LeBorgne M Dentz D Bolster J Carrera J deDreuzy andP Davy Adv Water Resourc 33 1468 (2010)

[48] P Kitanidis Water Resourc Res 30 2011 (1994)[49] F de Barros M Dentz J Koch and W Nowak Geophys Res

Lett 39 L08404 (2012)[50] D A Benson and M M Meerschaert Water Resourc Res 44

W12201 (2008)[51] A Paster D Bolster and D A Benson J Comput Phys 263

91 (2014)[52] D Bolster M Dentz and J Carrera Water Resourc Res 45

W05408 (2009)[53] A Paster D Bolster and D A Benson Water Resourc Res

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Retrieval (Cambridge University Press Cambridge UK 2010)[56] H Risken Fokker-Planck Equation (Springer New York 1984)[57] D Benson D Bolster N Engdahl and T Aquino (unpublished)[58] A Paster and D Bolster Physica A (Amsterdam) 391 4654

(2012)

012922-14


Ovchinnikov-Zeldovich segregation and this behavior hasbeen observed experimentally [1920]

This problem has also received a great deal of attention insystems where transport is not by local Fickian diffusion butrather by anomalous dispersive transport including nonlocal inspace superdiffusive systems and nonlocal in time subdiffusivesystems [22ndash24] Such nonlocal transport is common in abroad array of disciplines including transport in porous media[25ndash27] streams [28] groundwater systems [2930] andbiological systems [31] to name a few In all cases similarlate-time scalings for the mean concentration that deviate fromthe well-mixed tminus1 scaling have been predicted and observedThe specific late-time scaling will depend on the nature ofthe transport and the dimensionality of the problem beingconsidered However to our knowledge this problem hasreceived limited attention in the context where transport isby advection and diffusion in a nonuniform velocity field Asnoted above it is well known that nonuniform velocity fieldscan significantly affect the nature of mixing and thus can inprinciple strongly impact mixing-driven reactions

While much focus has been given to mixing in turbulentflows it is important to recognize that mixing in nonuniformlaminar flows can also result in very interesting mixingdynamics [32ndash38] In this work we propose to study thisproblem in one of the simplest forms of nonuniform flownamely a pure laminar shear flow Pure shear flows havereceived a great deal of attention in a variety of applicationsdue to both their simplicity and ability to provide invaluablephysical insight Okubo [3940] studied transport in pure shearflows to better understand solute dispersion in rivers estuarieslakes and oceans Novikov [41] used it as a model to studyturbulent dispersion in streams Others [42ndash44] highlight itas an important case in understanding turbulent dispersionMore recently it was shown [45] that from purely a mixingperspective as quantified by the scalar dissipation rate [4647]or dilution index [48] a pure shear flow is incredibly efficientat mixing The term hypermixing was used emphasizingthat the predicted mixing is even faster than conventionalsuperdiffusion Reference [49] extended this to more complexflows using the Okubo-Weiss metric to quantify enhancementof mixing due to more general locally nonuniform velocityfields

The above discussion leads us the ask the followingquestion Is mixing in a laminar shear flow sufficient toovercome incomplete mixing effects on the evolution of meanconcentration in a reactive system or will incomplete mixingeffects still persist The answer to this question is addressedin this paper with an application to a nonuniform flow of arandom walk particle tracking method designed for reactivetransport It was originally developed for purely diffusivetransport [50] and we extend it further The results of thispaper provide a general understanding of the influence of shearflows on incomplete mixing paving the road for more generalnonuniform velocity fields

The paper is structured as follows In Sec II we presentthe governing equations for flow transport and reaction forthe chosen setup in dimensional and nondimensional formsIn Sec III we discuss the initial conditions focusing on bothdeterministic and stochastic initial conditions which are atthe root of incomplete mixing In Sec IV we describe the

Lagrangian random walk particle tracking method for reactivetransport In Sec V we present and discuss results obtainedwith this method In Sec VI we propose a semianalyticalmodel to interpret observations from Sec V We concludewith a discussion in Sec VII

II GOVERNING EQUATIONS

We consider a bimolecular reactive system that is embeddedin a uniform shear flow Transport of the species is driven bya constant diffusion coefficient and advection according to theuniform shear flow The two constituents in this system A andB react kinetically and irreversibly with one another ie A +B rarr P At this point we are not interested in what happensto the product P but rather on how A and B are consumed sothe fate of P is neglected in this work The flow in the systemis completely independent of the transport ie the flow is notaffected by the concentration of the constituents For an infinitetwo-dimensional space the governing equation for transportis the advection-dispersion-reaction equation (ADRE) givenfor each of the species i = AB by

partCi

partt+ αy

partCi

partx= nabla middot (DnablaCi) minus ri (1)

where Ci(xt) is the concentration [mol Lminusd ] α is the shearrate [Tminus1] D is the dispersion coefficient [L2Tminus1] assumedconstant herein

The sink term in our case is the local rate of reaction andis identical for A and B since they are consumed with a 11stoichiometry ratio ie

rA = rB = r (2)

We assume the law of mass action prevails for our systemand write the reaction rate as

r(CACB) = kCACB (3)

where k [Ld molminus1 Tminus1] is the kinetic rate constant for a givenreaction Thus Eq (1) represents a coupled set

A Nondimensional equations

We consider the following dimensionless variables

Clowast = CC0 tlowast = kC0t αlowast = α(kC0)(4)

xlowast = xl ylowast = yl

where l is a characteristic length associated with the initialfluctuations in concentration (eg a correlation length) Usingthese nondimensional variables our governing equation fortransport becomes

partClowasti

parttlowast+ αlowastylowast partClowast

i

partxlowast = 1

Danablalowast2Clowast

i minus ClowastAClowast

B (5)

where Da = kC0l2

Dis the Damkohler number which quantifies

the relative importance of reaction to transport by diffusion Inother words it quantifies how quickly reactions happen relativeto how quickly diffusion can homogenize a patch of size lαlowast is a dimensionless shear rate that quantifies the relativeimportance of shear to reaction For the sake of conveniencewe drop the stars from here on and all the variables presentedare nondimensional

012922-2








CA(xt = 0) = CA0(x)(7)

CB(xt = 0) = CB0(x)






CA(t = 0) = CB(t = 0) = 1 (9)


μAA = μBB = μ (10)


μAB = 0 (11)








partf

partt+ αy

partf

partx= 1

Danabla2f (12)


f (xt x0) = 1

(2π )d2radic

det[κ]exp

[minus1

2ξT κminus1ξ

](13)


κ = κ(t) = 2t

Da

[1 + (αt)23 αt2

αt2 1

] (14)

012922-3


0+ 1 2 4αt =


where




B Reaction


v =int



v(st) = 1

2πradic

det[κA + κB]exp

[minus1


] (17)


v(st) = Da

8πtradic

1 + (αt)212

times exp


y (αt)23]

8t[1 + (αt)212]

(18)

where |s| =radic

s2x + s2




mpv(st)dt (19)







[minusmp

int t


] (22)



8πE1

(Da

8t|s|2

) (23)








radic8tDa



012922-4



minus08

minus06

minus04

minus02

0

02

04

06

08

minus6

minus5

minus4

minus3

minus2


minus05

0

05

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

(a) (b)

(c)

sx

s y

sx

s y

sx

s y




Pr = ηminus1mpt

Nnbsumi=1

v(si t) (26)






tbnd = Da(8)2 (27)




012922-5



ρ0 = N0(xy) (28)


ρ0 = 1μ (29)


CA(xt = 0) = CA0(x) CB(xt = 0) = CB0(x) (30)



V RESULTS



Da α N0 t0 ε tmax




0 0025 0025 10010minus4 0025 0025 10010minus3 0025 0025 100


1 0012 0012 510 0012 0012 1




012922-6










012922-7


α = 0

α = 10minus3

α = 10minus1

x

y









012922-8





dCA(t)

dt= dCB(t)

dt= minusCA


B (31)


AC primeB term


C primeAC prime






partG

partt+ αy

partG

partx= 1

Danabla2G (33)


G(xt = 0) = δ(x minus x0) (34)

012922-9



G(x = xpeakt) = 1

2πradic

det[κ]= Da

2πtradic

4 + (αt)23 (35)




dCA(t)

dt= minusCA

2 + χDa

2πtradic

4 + (αt)23 (36)



CA(t) = 1

1 + t(37)



CA =(

Daχ

2πtradic

4 + (αt)23

) 12

=(

Daχ

4π

) 12

(t2 + α2t412)minus14 (38)









Rasy =(

2παradic3Daχ

) 12

(39)


radic12α




10minus1

100

101

102

103

104

105

10minus4

10minus3

10minus2

10minus1

100

t

C


α=1α=10

tminus1

tminus12



012922-10


χχχ






VII CONCLUSIONS








012922-11



ACKNOWLEDGMENTS






where

Tr(κ) = κ11 + κ22 = 4t

Da

[1 + 1

6α2t2

](A2)



Da2

[1 + 1

12α2t2

] (A3)


λ12 = 2t

Da

1 plusmn

[1

2αt

radic1 + 1

9(αt)2

]+ 1

6α2t2

(A4)


vprime12 =

[1

minus 13αt plusmn

radic1 + 1

9 (αt)2

](A5)






x1 = ξ1

radicλ1

(A6)x2 = ξ2

radicλ2




i=1

(vi middot x)xi

(A7)

y = y0 +dsum

i=1

(vi middot y)xi




n=minusinfin

1

(2π )d2radic

det[κ]exp

[minus1

2ξTn κminus1ξn

]

(B1)

where

ξn = (nx0) (B2)


G(x = xpeakt) = Da

4πtradic

1 + α2t212

timesinfinsum

n=minusinfinexp

[minus Da(nx)2

4t(1 + α2t212)

](B3)



3

2π

Da

|α|t2


exp

[minus3Da(nx)2

α2t3

]

(B4)


xα2 when the


π

3Da

|α|x

t32 (B5)



2

radicDa

π

1

x

tminus12 (B6)

012922-12


10minus1

100

101

102

103

104

105

10minus3

10minus2

10minus1

100

t

C


line Ci(t) =radic

χ

2x( Da

π)14








radicχ

2x

(Da

π

)14

tminus14 (B8)


























012922-13





















(2012)

012922-14








CA(xt = 0) = CA0(x)(7)

CB(xt = 0) = CB0(x)






CA(t = 0) = CB(t = 0) = 1 (9)


μAA = μBB = μ (10)


μAB = 0 (11)








partf

partt+ αy

partf

partx= 1

Danabla2f (12)


f (xt x0) = 1

(2π )d2radic

det[κ]exp

[minus1

2ξT κminus1ξ

](13)


κ = κ(t) = 2t

Da

[1 + (αt)23 αt2

αt2 1

] (14)

012922-3


0+ 1 2 4αt =


where




B Reaction


v =int



v(st) = 1

2πradic

det[κA + κB]exp

[minus1


] (17)


v(st) = Da

8πtradic

1 + (αt)212

times exp


y (αt)23]

8t[1 + (αt)212]

(18)

where |s| =radic

s2x + s2




mpv(st)dt (19)







[minusmp

int t


] (22)



8πE1

(Da

8t|s|2

) (23)








radic8tDa



012922-4



minus08

minus06

minus04

minus02

0

02

04

06

08

minus6

minus5

minus4

minus3

minus2


minus05

0

05

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

(a) (b)

(c)

sx

s y

sx

s y

sx

s y




Pr = ηminus1mpt

Nnbsumi=1

v(si t) (26)






tbnd = Da(8)2 (27)




012922-5



ρ0 = N0(xy) (28)


ρ0 = 1μ (29)


CA(xt = 0) = CA0(x) CB(xt = 0) = CB0(x) (30)



V RESULTS



Da α N0 t0 ε tmax




0 0025 0025 10010minus4 0025 0025 10010minus3 0025 0025 100


1 0012 0012 510 0012 0012 1




012922-6










012922-7


α = 0

α = 10minus3

α = 10minus1

x

y









012922-8





dCA(t)

dt= dCB(t)

dt= minusCA


B (31)


AC primeB term


C primeAC prime






partG

partt+ αy

partG

partx= 1

Danabla2G (33)


G(xt = 0) = δ(x minus x0) (34)

012922-9



G(x = xpeakt) = 1

2πradic

det[κ]= Da

2πtradic

4 + (αt)23 (35)




dCA(t)

dt= minusCA

2 + χDa

2πtradic

4 + (αt)23 (36)



CA(t) = 1

1 + t(37)



CA =(

Daχ

2πtradic

4 + (αt)23

) 12

=(

Daχ

4π

) 12

(t2 + α2t412)minus14 (38)









Rasy =(

2παradic3Daχ

) 12

(39)


radic12α




10minus1

100

101

102

103

104

105

10minus4

10minus3

10minus2

10minus1

100

t

C


α=1α=10

tminus1

tminus12



012922-10


χχχ






VII CONCLUSIONS








012922-11



ACKNOWLEDGMENTS






where

Tr(κ) = κ11 + κ22 = 4t

Da

[1 + 1

6α2t2

](A2)



Da2

[1 + 1

12α2t2

] (A3)


λ12 = 2t

Da

1 plusmn

[1

2αt

radic1 + 1

9(αt)2

]+ 1

6α2t2

(A4)


vprime12 =

[1

minus 13αt plusmn

radic1 + 1

9 (αt)2

](A5)






x1 = ξ1

radicλ1

(A6)x2 = ξ2

radicλ2




i=1

(vi middot x)xi

(A7)

y = y0 +dsum

i=1

(vi middot y)xi




n=minusinfin

1

(2π )d2radic

det[κ]exp

[minus1

2ξTn κminus1ξn

]

(B1)

where

ξn = (nx0) (B2)


G(x = xpeakt) = Da

4πtradic

1 + α2t212

timesinfinsum

n=minusinfinexp

[minus Da(nx)2

4t(1 + α2t212)

](B3)



3

2π

Da

|α|t2


exp

[minus3Da(nx)2

α2t3

]

(B4)


xα2 when the


π

3Da

|α|x

t32 (B5)



2

radicDa

π

1

x

tminus12 (B6)

012922-12


10minus1

100

101

102

103

104

105

10minus3

10minus2

10minus1

100

t

C


line Ci(t) =radic

χ

2x( Da

π)14








radicχ

2x

(Da

π

)14

tminus14 (B8)


























012922-13





















(2012)

012922-14


0+ 1 2 4αt =


where




B Reaction


v =int



v(st) = 1

2πradic

det[κA + κB]exp

[minus1


] (17)


v(st) = Da

8πtradic

1 + (αt)212

times exp


y (αt)23]

8t[1 + (αt)212]

(18)

where |s| =radic

s2x + s2




mpv(st)dt (19)







[minusmp

int t


] (22)



8πE1

(Da

8t|s|2

) (23)








radic8tDa



012922-4



minus08

minus06

minus04

minus02

0

02

04

06

08

minus6

minus5

minus4

minus3

minus2


minus05

0

05

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

(a) (b)

(c)

sx

s y

sx

s y

sx

s y




Pr = ηminus1mpt

Nnbsumi=1

v(si t) (26)






tbnd = Da(8)2 (27)




012922-5



ρ0 = N0(xy) (28)


ρ0 = 1μ (29)


CA(xt = 0) = CA0(x) CB(xt = 0) = CB0(x) (30)



V RESULTS



Da α N0 t0 ε tmax




0 0025 0025 10010minus4 0025 0025 10010minus3 0025 0025 100


1 0012 0012 510 0012 0012 1




012922-6










012922-7


α = 0

α = 10minus3

α = 10minus1

x

y









012922-8





dCA(t)

dt= dCB(t)

dt= minusCA


B (31)


AC primeB term


C primeAC prime






partG

partt+ αy

partG

partx= 1

Danabla2G (33)


G(xt = 0) = δ(x minus x0) (34)

012922-9



G(x = xpeakt) = 1

2πradic

det[κ]= Da

2πtradic

4 + (αt)23 (35)




dCA(t)

dt= minusCA

2 + χDa

2πtradic

4 + (αt)23 (36)



CA(t) = 1

1 + t(37)



CA =(

Daχ

2πtradic

4 + (αt)23

) 12

=(

Daχ

4π

) 12

(t2 + α2t412)minus14 (38)









Rasy =(

2παradic3Daχ

) 12

(39)


radic12α




10minus1

100

101

102

103

104

105

10minus4

10minus3

10minus2

10minus1

100

t

C


α=1α=10

tminus1

tminus12



012922-10


χχχ






VII CONCLUSIONS








012922-11



ACKNOWLEDGMENTS






where

Tr(κ) = κ11 + κ22 = 4t

Da

[1 + 1

6α2t2

](A2)



Da2

[1 + 1

12α2t2

] (A3)


λ12 = 2t

Da

1 plusmn

[1

2αt

radic1 + 1

9(αt)2

]+ 1

6α2t2

(A4)


vprime12 =

[1

minus 13αt plusmn

radic1 + 1

9 (αt)2

](A5)






x1 = ξ1

radicλ1

(A6)x2 = ξ2

radicλ2




i=1

(vi middot x)xi

(A7)

y = y0 +dsum

i=1

(vi middot y)xi




n=minusinfin

1

(2π )d2radic

det[κ]exp

[minus1

2ξTn κminus1ξn

]

(B1)

where

ξn = (nx0) (B2)


G(x = xpeakt) = Da

4πtradic

1 + α2t212

timesinfinsum

n=minusinfinexp

[minus Da(nx)2

4t(1 + α2t212)

](B3)



3

2π

Da

|α|t2


exp

[minus3Da(nx)2

α2t3

]

(B4)


xα2 when the


π

3Da

|α|x

t32 (B5)



2

radicDa

π

1

x

tminus12 (B6)

012922-12


10minus1

100

101

102

103

104

105

10minus3

10minus2

10minus1

100

t

C


line Ci(t) =radic

χ

2x( Da

π)14








radicχ

2x

(Da

π

)14

tminus14 (B8)


























012922-13





















(2012)

012922-14



minus08

minus06

minus04

minus02

0

02

04

06

08

minus6

minus5

minus4

minus3

minus2


minus05

0

05

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

(a) (b)

(c)

sx

s y

sx

s y

sx

s y




Pr = ηminus1mpt

Nnbsumi=1

v(si t) (26)






tbnd = Da(8)2 (27)




012922-5



ρ0 = N0(xy) (28)


ρ0 = 1μ (29)


CA(xt = 0) = CA0(x) CB(xt = 0) = CB0(x) (30)



V RESULTS



Da α N0 t0 ε tmax




0 0025 0025 10010minus4 0025 0025 10010minus3 0025 0025 100


1 0012 0012 510 0012 0012 1




012922-6










012922-7


α = 0

α = 10minus3

α = 10minus1

x

y









012922-8





dCA(t)

dt= dCB(t)

dt= minusCA


B (31)


AC primeB term


C primeAC prime






partG

partt+ αy

partG

partx= 1

Danabla2G (33)


G(xt = 0) = δ(x minus x0) (34)

012922-9



G(x = xpeakt) = 1

2πradic

det[κ]= Da

2πtradic

4 + (αt)23 (35)




dCA(t)

dt= minusCA

2 + χDa

2πtradic

4 + (αt)23 (36)



CA(t) = 1

1 + t(37)



CA =(

Daχ

2πtradic

4 + (αt)23

) 12

=(

Daχ

4π

) 12

(t2 + α2t412)minus14 (38)









Rasy =(

2παradic3Daχ

) 12

(39)


radic12α




10minus1

100

101

102

103

104

105

10minus4

10minus3

10minus2

10minus1

100

t

C


α=1α=10

tminus1

tminus12



012922-10


χχχ






VII CONCLUSIONS








012922-11



ACKNOWLEDGMENTS






where

Tr(κ) = κ11 + κ22 = 4t

Da

[1 + 1

6α2t2

](A2)



Da2

[1 + 1

12α2t2

] (A3)


λ12 = 2t

Da

1 plusmn

[1

2αt

radic1 + 1

9(αt)2

]+ 1

6α2t2

(A4)


vprime12 =

[1

minus 13αt plusmn

radic1 + 1

9 (αt)2

](A5)






x1 = ξ1

radicλ1

(A6)x2 = ξ2

radicλ2




i=1

(vi middot x)xi

(A7)

y = y0 +dsum

i=1

(vi middot y)xi




n=minusinfin

1

(2π )d2radic

det[κ]exp

[minus1

2ξTn κminus1ξn

]

(B1)

where

ξn = (nx0) (B2)


G(x = xpeakt) = Da

4πtradic

1 + α2t212

timesinfinsum

n=minusinfinexp

[minus Da(nx)2

4t(1 + α2t212)

](B3)



3

2π

Da

|α|t2


exp

[minus3Da(nx)2

α2t3

]

(B4)


xα2 when the


π

3Da

|α|x

t32 (B5)



2

radicDa

π

1

x

tminus12 (B6)

012922-12


10minus1

100

101

102

103

104

105

10minus3

10minus2

10minus1

100

t

C


line Ci(t) =radic

χ

2x( Da

π)14








radicχ

2x

(Da

π

)14

tminus14 (B8)


























012922-13





















(2012)

012922-14



ρ0 = N0(xy) (28)


ρ0 = 1μ (29)


CA(xt = 0) = CA0(x) CB(xt = 0) = CB0(x) (30)



V RESULTS



Da α N0 t0 ε tmax




0 0025 0025 10010minus4 0025 0025 10010minus3 0025 0025 100


1 0012 0012 510 0012 0012 1




012922-6










012922-7


α = 0

α = 10minus3

α = 10minus1

x

y









012922-8





dCA(t)

dt= dCB(t)

dt= minusCA


B (31)


AC primeB term


C primeAC prime






partG

partt+ αy

partG

partx= 1

Danabla2G (33)


G(xt = 0) = δ(x minus x0) (34)

012922-9



G(x = xpeakt) = 1

2πradic

det[κ]= Da

2πtradic

4 + (αt)23 (35)




dCA(t)

dt= minusCA

2 + χDa

2πtradic

4 + (αt)23 (36)



CA(t) = 1

1 + t(37)



CA =(

Daχ

2πtradic

4 + (αt)23

) 12

=(

Daχ

4π

) 12

(t2 + α2t412)minus14 (38)









Rasy =(

2παradic3Daχ

) 12

(39)


radic12α




10minus1

100

101

102

103

104

105

10minus4

10minus3

10minus2

10minus1

100

t

C


α=1α=10

tminus1

tminus12



012922-10


χχχ






VII CONCLUSIONS








012922-11



ACKNOWLEDGMENTS






where

Tr(κ) = κ11 + κ22 = 4t

Da

[1 + 1

6α2t2

](A2)



Da2

[1 + 1

12α2t2

] (A3)


λ12 = 2t

Da

1 plusmn

[1

2αt

radic1 + 1

9(αt)2

]+ 1

6α2t2

(A4)


vprime12 =

[1

minus 13αt plusmn

radic1 + 1

9 (αt)2

](A5)






x1 = ξ1

radicλ1

(A6)x2 = ξ2

radicλ2




i=1

(vi middot x)xi

(A7)

y = y0 +dsum

i=1

(vi middot y)xi




n=minusinfin

1

(2π )d2radic

det[κ]exp

[minus1

2ξTn κminus1ξn

]

(B1)

where

ξn = (nx0) (B2)


G(x = xpeakt) = Da

4πtradic

1 + α2t212

timesinfinsum

n=minusinfinexp

[minus Da(nx)2

4t(1 + α2t212)

](B3)



3

2π

Da

|α|t2


exp

[minus3Da(nx)2

α2t3

]

(B4)


xα2 when the


π

3Da

|α|x

t32 (B5)



2

radicDa

π

1

x

tminus12 (B6)

012922-12


10minus1

100

101

102

103

104

105

10minus3

10minus2

10minus1

100

t

C


line Ci(t) =radic

χ

2x( Da

π)14








radicχ

2x

(Da

π

)14

tminus14 (B8)


























012922-13





















(2012)

012922-14










012922-7


α = 0

α = 10minus3

α = 10minus1

x

y









012922-8





dCA(t)

dt= dCB(t)

dt= minusCA


B (31)


AC primeB term


C primeAC prime






partG

partt+ αy

partG

partx= 1

Danabla2G (33)


G(xt = 0) = δ(x minus x0) (34)

012922-9



G(x = xpeakt) = 1

2πradic

det[κ]= Da

2πtradic

4 + (αt)23 (35)




dCA(t)

dt= minusCA

2 + χDa

2πtradic

4 + (αt)23 (36)



CA(t) = 1

1 + t(37)



CA =(

Daχ

2πtradic

4 + (αt)23

) 12

=(

Daχ

4π

) 12

(t2 + α2t412)minus14 (38)









Rasy =(

2παradic3Daχ

) 12

(39)


radic12α




10minus1

100

101

102

103

104

105

10minus4

10minus3

10minus2

10minus1

100

t

C


α=1α=10

tminus1

tminus12



012922-10


χχχ






VII CONCLUSIONS








012922-11



ACKNOWLEDGMENTS






where

Tr(κ) = κ11 + κ22 = 4t

Da

[1 + 1

6α2t2

](A2)



Da2

[1 + 1

12α2t2

] (A3)


λ12 = 2t

Da

1 plusmn

[1

2αt

radic1 + 1

9(αt)2

]+ 1

6α2t2

(A4)


vprime12 =

[1

minus 13αt plusmn

radic1 + 1

9 (αt)2

](A5)






x1 = ξ1

radicλ1

(A6)x2 = ξ2

radicλ2




i=1

(vi middot x)xi

(A7)

y = y0 +dsum

i=1

(vi middot y)xi




n=minusinfin

1

(2π )d2radic

det[κ]exp

[minus1

2ξTn κminus1ξn

]

(B1)

where

ξn = (nx0) (B2)


G(x = xpeakt) = Da

4πtradic

1 + α2t212

timesinfinsum

n=minusinfinexp

[minus Da(nx)2

4t(1 + α2t212)

](B3)



3

2π

Da

|α|t2


exp

[minus3Da(nx)2

α2t3

]

(B4)


xα2 when the


π

3Da

|α|x

t32 (B5)



2

radicDa

π

1

x

tminus12 (B6)

012922-12


10minus1

100

101

102

103

104

105

10minus3

10minus2

10minus1

100

t

C


line Ci(t) =radic

χ

2x( Da

π)14








radicχ

2x

(Da

π

)14

tminus14 (B8)


























012922-13





















(2012)

012922-14


α = 0

α = 10minus3

α = 10minus1

x

y









012922-8





dCA(t)

dt= dCB(t)

dt= minusCA


B (31)


AC primeB term


C primeAC prime






partG

partt+ αy

partG

partx= 1

Danabla2G (33)


G(xt = 0) = δ(x minus x0) (34)

012922-9



G(x = xpeakt) = 1

2πradic

det[κ]= Da

2πtradic

4 + (αt)23 (35)




dCA(t)

dt= minusCA

2 + χDa

2πtradic

4 + (αt)23 (36)



CA(t) = 1

1 + t(37)



CA =(

Daχ

2πtradic

4 + (αt)23

) 12

=(

Daχ

4π

) 12

(t2 + α2t412)minus14 (38)









Rasy =(

2παradic3Daχ

) 12

(39)


radic12α




10minus1

100

101

102

103

104

105

10minus4

10minus3

10minus2

10minus1

100

t

C


α=1α=10

tminus1

tminus12



012922-10


χχχ






VII CONCLUSIONS








012922-11



ACKNOWLEDGMENTS






where

Tr(κ) = κ11 + κ22 = 4t

Da

[1 + 1

6α2t2

](A2)



Da2

[1 + 1

12α2t2

] (A3)


λ12 = 2t

Da

1 plusmn

[1

2αt

radic1 + 1

9(αt)2

]+ 1

6α2t2

(A4)


vprime12 =

[1

minus 13αt plusmn

radic1 + 1

9 (αt)2

](A5)






x1 = ξ1

radicλ1

(A6)x2 = ξ2

radicλ2




i=1

(vi middot x)xi

(A7)

y = y0 +dsum

i=1

(vi middot y)xi




n=minusinfin

1

(2π )d2radic

det[κ]exp

[minus1

2ξTn κminus1ξn

]

(B1)

where

ξn = (nx0) (B2)


G(x = xpeakt) = Da

4πtradic

1 + α2t212

timesinfinsum

n=minusinfinexp

[minus Da(nx)2

4t(1 + α2t212)

](B3)



3

2π

Da

|α|t2


exp

[minus3Da(nx)2

α2t3

]

(B4)


xα2 when the


π

3Da

|α|x

t32 (B5)



2

radicDa

π

1

x

tminus12 (B6)

012922-12


10minus1

100

101

102

103

104

105

10minus3

10minus2

10minus1

100

t

C


line Ci(t) =radic

χ

2x( Da

π)14








radicχ

2x

(Da

π

)14

tminus14 (B8)


























012922-13





















(2012)

012922-14





dCA(t)

dt= dCB(t)

dt= minusCA


B (31)


AC primeB term


C primeAC prime






partG

partt+ αy

partG

partx= 1

Danabla2G (33)


G(xt = 0) = δ(x minus x0) (34)

012922-9



G(x = xpeakt) = 1

2πradic

det[κ]= Da

2πtradic

4 + (αt)23 (35)




dCA(t)

dt= minusCA

2 + χDa

2πtradic

4 + (αt)23 (36)



CA(t) = 1

1 + t(37)



CA =(

Daχ

2πtradic

4 + (αt)23

) 12

=(

Daχ

4π

) 12

(t2 + α2t412)minus14 (38)









Rasy =(

2παradic3Daχ

) 12

(39)


radic12α




10minus1

100

101

102

103

104

105

10minus4

10minus3

10minus2

10minus1

100

t

C


α=1α=10

tminus1

tminus12



012922-10


χχχ






VII CONCLUSIONS








012922-11



ACKNOWLEDGMENTS






where

Tr(κ) = κ11 + κ22 = 4t

Da

[1 + 1

6α2t2

](A2)



Da2

[1 + 1

12α2t2

] (A3)


λ12 = 2t

Da

1 plusmn

[1

2αt

radic1 + 1

9(αt)2

]+ 1

6α2t2

(A4)


vprime12 =

[1

minus 13αt plusmn

radic1 + 1

9 (αt)2

](A5)






x1 = ξ1

radicλ1

(A6)x2 = ξ2

radicλ2




i=1

(vi middot x)xi

(A7)

y = y0 +dsum

i=1

(vi middot y)xi




n=minusinfin

1

(2π )d2radic

det[κ]exp

[minus1

2ξTn κminus1ξn

]

(B1)

where

ξn = (nx0) (B2)


G(x = xpeakt) = Da

4πtradic

1 + α2t212

timesinfinsum

n=minusinfinexp

[minus Da(nx)2

4t(1 + α2t212)

](B3)



3

2π

Da

|α|t2


exp

[minus3Da(nx)2

α2t3

]

(B4)


xα2 when the


π

3Da

|α|x

t32 (B5)



2

radicDa

π

1

x

tminus12 (B6)

012922-12


10minus1

100

101

102

103

104

105

10minus3

10minus2

10minus1

100

t

C


line Ci(t) =radic

χ

2x( Da

π)14








radicχ

2x

(Da

π

)14

tminus14 (B8)


























012922-13





















(2012)

012922-14



G(x = xpeakt) = 1

2πradic

det[κ]= Da

2πtradic

4 + (αt)23 (35)




dCA(t)

dt= minusCA

2 + χDa

2πtradic

4 + (αt)23 (36)



CA(t) = 1

1 + t(37)



CA =(

Daχ

2πtradic

4 + (αt)23

) 12

=(

Daχ

4π

) 12

(t2 + α2t412)minus14 (38)









Rasy =(

2παradic3Daχ

) 12

(39)


radic12α




10minus1

100

101

102

103

104

105

10minus4

10minus3

10minus2

10minus1

100

t

C


α=1α=10

tminus1

tminus12



012922-10


χχχ






VII CONCLUSIONS








012922-11



ACKNOWLEDGMENTS






where

Tr(κ) = κ11 + κ22 = 4t

Da

[1 + 1

6α2t2

](A2)



Da2

[1 + 1

12α2t2

] (A3)


λ12 = 2t

Da

1 plusmn

[1

2αt

radic1 + 1

9(αt)2

]+ 1

6α2t2

(A4)


vprime12 =

[1

minus 13αt plusmn

radic1 + 1

9 (αt)2

](A5)






x1 = ξ1

radicλ1

(A6)x2 = ξ2

radicλ2




i=1

(vi middot x)xi

(A7)

y = y0 +dsum

i=1

(vi middot y)xi




n=minusinfin

1

(2π )d2radic

det[κ]exp

[minus1

2ξTn κminus1ξn

]

(B1)

where

ξn = (nx0) (B2)


G(x = xpeakt) = Da

4πtradic

1 + α2t212

timesinfinsum

n=minusinfinexp

[minus Da(nx)2

4t(1 + α2t212)

](B3)



3

2π

Da

|α|t2


exp

[minus3Da(nx)2

α2t3

]

(B4)


xα2 when the


π

3Da

|α|x

t32 (B5)



2

radicDa

π

1

x

tminus12 (B6)

012922-12


10minus1

100

101

102

103

104

105

10minus3

10minus2

10minus1

100

t

C


line Ci(t) =radic

χ

2x( Da

π)14








radicχ

2x

(Da

π

)14

tminus14 (B8)


























012922-13





















(2012)

012922-14


χχχ






VII CONCLUSIONS








012922-11



ACKNOWLEDGMENTS






where

Tr(κ) = κ11 + κ22 = 4t

Da

[1 + 1

6α2t2

](A2)



Da2

[1 + 1

12α2t2

] (A3)


λ12 = 2t

Da

1 plusmn

[1

2αt

radic1 + 1

9(αt)2

]+ 1

6α2t2

(A4)


vprime12 =

[1

minus 13αt plusmn

radic1 + 1

9 (αt)2

](A5)






x1 = ξ1

radicλ1

(A6)x2 = ξ2

radicλ2




i=1

(vi middot x)xi

(A7)

y = y0 +dsum

i=1

(vi middot y)xi




n=minusinfin

1

(2π )d2radic

det[κ]exp

[minus1

2ξTn κminus1ξn

]

(B1)

where

ξn = (nx0) (B2)


G(x = xpeakt) = Da

4πtradic

1 + α2t212

timesinfinsum

n=minusinfinexp

[minus Da(nx)2

4t(1 + α2t212)

](B3)



3

2π

Da

|α|t2


exp

[minus3Da(nx)2

α2t3

]

(B4)


xα2 when the


π

3Da

|α|x

t32 (B5)



2

radicDa

π

1

x

tminus12 (B6)

012922-12


10minus1

100

101

102

103

104

105

10minus3

10minus2

10minus1

100

t

C


line Ci(t) =radic

χ

2x( Da

π)14








radicχ

2x

(Da

π

)14

tminus14 (B8)


























012922-13





















(2012)

012922-14



ACKNOWLEDGMENTS






where

Tr(κ) = κ11 + κ22 = 4t

Da

[1 + 1

6α2t2

](A2)



Da2

[1 + 1

12α2t2

] (A3)


λ12 = 2t

Da

1 plusmn

[1

2αt

radic1 + 1

9(αt)2

]+ 1

6α2t2

(A4)


vprime12 =

[1

minus 13αt plusmn

radic1 + 1

9 (αt)2

](A5)






x1 = ξ1

radicλ1

(A6)x2 = ξ2

radicλ2




i=1

(vi middot x)xi

(A7)

y = y0 +dsum

i=1

(vi middot y)xi




n=minusinfin

1

(2π )d2radic

det[κ]exp

[minus1

2ξTn κminus1ξn

]

(B1)

where

ξn = (nx0) (B2)


G(x = xpeakt) = Da

4πtradic

1 + α2t212

timesinfinsum

n=minusinfinexp

[minus Da(nx)2

4t(1 + α2t212)

](B3)



3

2π

Da

|α|t2


exp

[minus3Da(nx)2

α2t3

]

(B4)


xα2 when the


π

3Da

|α|x

t32 (B5)



2

radicDa

π

1

x

tminus12 (B6)

012922-12


10minus1

100

101

102

103

104

105

10minus3

10minus2

10minus1

100

t

C


line Ci(t) =radic

χ

2x( Da

π)14








radicχ

2x

(Da

π

)14

tminus14 (B8)


























012922-13





















(2012)

012922-14


10minus1

100

101

102

103

104

105

10minus3

10minus2

10minus1

100

t

C


line Ci(t) =radic

χ

2x( Da

π)14








radicχ

2x

(Da

π

)14

tminus14 (B8)


























012922-13





















(2012)

012922-14





















(2012)

012922-14

Documents

Incomplete mixing and reactions in laminar shear ﬂopaster/pubs/12_PRE_shear.pdf · Mixing is the physical process that enables this contact to occur. In ﬂowing systems, this is