ARTICLES Screening mechanisms in sedimentation€¦ · ing sedimentation, and extracted the...

PHYSICS OF FLUIDS VOLUME 11, NUMBER 4 APRIL 1999

ARTICLES

Screening mechanisms in sedimentationMichael P. BrennerDepartment of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

~Received 28 April 1998; accepted 30 December 1998!

This paper considers a mixture of sedimenting particles at low Reynolds numbers and volumefractions. Simple theoretical arguments have long suggested that for a random suspension ofparticles in an infinite system, the fluctuations in the velocity about the mean should diverge withsystem size. On the other hand, experiments have shown no such divergence. The primary goal ofthis paper is to examine the effect of side walls on the predicted divergence in fluctuations, throughtheory, scaling arguments, and numerical simulations. Side walls lead to important modifications ofthe standard arguments. A scaling argument~based on wall effects! is presented to rationalize recentexperiments by Segre´ et al. @Phys. Rev. Lett.79, 2574~1997!#. The paper also briefly discusses therole of inertia in screening fluctuations in infinite systems when the particle Reynolds number isvery low, and also the coupling between the velocity fluctuations and the mean sedimentingvelocity. A physical argument suggests that in some circumstances the fluctuations give a leadingorder correction to the mean sedimenting velocity as a function of volume fraction. ©1999American Institute of Physics.@S1070-6631~99!02104-2#

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I. INTRODUCTION

The sedimentation rate of a mixture of monodispesolid spheres in a liquid is a classical problem in flumechanics.1–4 For slowly sedimenting particles, long rangehydrodynamic interactions lead to a correction to the Stosettling velocity5 Us of a single sphere. In the limit of lowvolume fractionsf, theories predict the sedimentation veloity as3,4

[email protected]~f2!#. ~1!

The O(f) correction is due to the presence of a fluid bacflow arising from the sedimenting particles.4 Another elegantphysical argument for thisO(f) correction is given inHinch,6 who considers it arising fromO(f) corrections ineffective properties of the medium around a test particlehomogeneous fluid. The main assumptions in theories leing to the prediction that the correction toUs is O(f) are:~i!the neglect of inertia, since the particle Reynolds numberp

is small;~ii ! consideration of only two body interactions btween the spheres;~iii ! the particle distribution is random inthe limit of smallf; and~iv! the system size is infinite in thdirection transverse to the settling. Experimental fits ofaverage settling velocity as a function of volume fractishow a roughly linear dependence onf in the limit of smallf, although with a prefactor which is systematically smalthan 6.55.7,1

The consistency of this theory was called into questby Caflisch and Luke,8 who pointed out that the assumptionlisted above imply that the velocity fluctuations of the fludiverge with increasing system size. A physical scalinggument for the divergence in fluctuations was givenHinch.9 Experiments and computer simulations have giv

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contradictory evidence regarding the existence of this divgence: On one hand, computer simulations by Ladd10,11 sup-port the conclusion, finding an increase in the size ofvelocity fluctuations with system size.

On the other, two different types of experiments habeen performed, both finding an independence of the fluctions on system size: Ham and Homsy12 and Nicolaiet al.13,14 studied the diffusion of a colored test particle duing sedimentation, and extracted the effective diffusion cstant of the particle. Systematic studies14 demonstrate thathe diffusivity does not vary with system size when tsmallest dimension of the cell is varied by a factor of 4fixed f. A second type of experiment was recently peformed by Segre´, Herbolzheimer, and Chaikin,15 who usedparticle image velocimetry to record the velocity field in thcenter of the experimental cell at a fixed time. They fouthat by increasing the largest dimension of their cell, the sof the velocity fluctuations saturated, with an explicit depedence of fluctuations on the solid volume fractionf overthree orders of magnitude inf. Their principal results arethat: ~a! the velocity fluctuations saturate at a scale of ordUsf

1/3; ~b! the correlation length is of order 30af21/3,wherea is the particle radius; and~c! experiments with vary-ing system size demonstrated that the velocity fluctuatisaturate for systems larger than approximately ten timescorrelation length. Each correlated region in the experimcontains of order 3000 particles, indicating that the phenoenon is a many particle effect, breaking assumption~ii !.

A single theoretical argument has been put forth toplain the independence of the fluctuations on system sKoch and Shaqfeh argued that screening of the velocity fltuations results from correlations in the partic

© 1999 American Institute of Physics

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755Phys. Fluids, Vol. 11, No. 4, April 1999 Michael P. Brenner

distribution,16 violating assumption~iii !. Their central idea isthat the particle distribution arranges itself precisely to ccel out the divergence in the fluctuations. By explicit coputation, they found a single particle distribution which hthis property. The distribution is characterized by a net dcit of exactly one particle surrounding any test particle. Ttheory predicts that the velocity fluctuations scale likeUs

~independent of volume fraction! and that the correlationlength scales likeaf21, in contrast to the experiments.

The primary goal of this paper is to examine the effectside walls on arguments leading to the prediction of diveing velocity fluctuations with system size. Although it is weknown that side walls greatly modify the velocity field ofsingle particle falling in a container17 ~e.g., in a cylinder thevelocity far from a particle decays exponentially on scafarther from the particle wall separation, instead of liker 21),this effect appears to have been ignored in theoretical dissions of sedimentation, which typically assume4 the systemsize is infinite transverse to the settling direction. We presthe analogue of the Caflish–Luke argument for divergvelocity fluctuations with side walls present, and find thapredicts the velocity fluctuations should vary across a cThe consequences of side walls are explored through scaarguments and numerical simulations. It is argued thaleast two different regimes of sedimentation should exist ibox with walls: a weakly interacting regime, whereCaflish–Luke-like law holds, and a strongly interacting rgime, where particle interactions modify this behavior. Coputer simulations and scaling arguments are presented toplore the strongly interacting regime. It is argued thexperiments have not yet completely ruled out the possibof the divergence of the fluctuations with system size,though if the divergence exists it must be weaker thanCaflish–Luke prediction.

We also present another screening mechanismshould apply to much larger cells or higher particle Reynonumber than those given in current experiments. It is baon the fact that the divergence of velocity fluctuations occbecause of the assumption that momentum diffusion afrom test particles is instantaneous. However, if the velocfluctuations diverge, then the particle diffusivity also dverges~see, for example, Koch18!; eventually, the particlediffusivity will be large enough that particle diffusion beamomentum diffusion. In this limit, it is no longer valid tassume that momentum diffusion is instantaneous, whleads to a self-regulation for the size of fluctuations. Whetthe side-wall-dominated or inertia-dominated screenmechanism applies depends on the relative size of thetainer and the particle Reynolds number.

Finally, it is pointed out that a natural consequencethese arguments is that the fluctuations lead to apositivecorrection to the mean sedimenting velocity, whose formindependent of the screening mechanism. In general, wegue that there is a correction to the mean settling velocityorderDU/ANblob, whereDU is the size of the fluctuationsand Nblob is the number of particles region with correlatevelocity field. From the Segreet al. experimental data, thisimplies a leading orderf1/3 correction to the mean velocityCombined with our suggestion that the saturation mechan


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in the Segre´ et al. experiments is geometry dependent, thleads to the conclusion that the mean sedimenting velocan depend on the container shape,19 as previously suggesteby Tory et al.20

II. STATEMENT OF PROBLEM

Consider a large group of particles falling together inviscous fluid of infinite extent. Neglecting interparticle inteactions, each particle falls according to the Stokes dragF56prnUsa, whereF5Drgvp is the applied force. HereDr is the density difference between the particle density afluid density r, n is the fluid kinematic viscosity,g is thegravitational acceleration, and the particle volumevp

54/3pa3. The fluid velocity solves the Navier–Stokes equtions

r~] tu2Us]zu1u–“u!5rn“2u2“p1f, ~2!

where u is the fluid velocity relative toUsz. The flow isincompressible and the boundary condition is that eachticle moves with the fluid. An effective approximation4,21

eliminating the boundary condition is to consider the limitpoint particles. These are represented by a body force in~2!, of the form f5Snf nd(z2zn) z, with the locationszn

coinciding with the particle positions. Each forcef n is theArchimedian buoyancy forceF described above.

We begin with a brief review of the argument developfor computing the average velocity of the suspension4,6 in aninfinite system: The first step is to neglect the inertial terin Eq. ~2!, since the particle Reynolds number Rep is small.Then, Eq.~2! is averaged over all configurations in whichsingle test particle is fixed at the spatial locationr 1.

6 Denot-ing this conditional average by an overbar yields

n“2u5“p* 1Fzd~r2r 1!, ~3!

where n and F are the ‘‘renormalized’’ viscosity and forceseen by the fixed test particle, representing the average eof the other particles in the suspension. Detailed computions of n andF are described by Batchelor4 and Hinch;6 theupshot is that both containO(f) corrections to the values foa homogeneous fluid. In deriving Eq.~3!, it is necessary touse Batchelor’s renormalization of the pressurep5p*1Drvpfgz, accounting for the uniform backflow arisinfrom the bottom boundary.4 The conclusion of these arguments is that the velocity field surrounding a test particlethe same functional form~decaying asr 21 at large distancesfrom the particle! as a particle in a homogeneous fluid, albwith O(f) corrections to coefficients.

This formalism immediately suggests that, for a randosuspension, the velocity fluctuations diverge with systsize. Caflisch and Luke’s argument8 points out that the vari-ance in the velocity fluctuations is given by

^u2&5E dPu2, ~4!

wheredP represents the probability measure for particle psitions. If the particle distribution is uniform~implying thatin the frame of the mean flow, particles sample their accsible space uniformly! the probability measure isdP


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756 Phys. Fluids, Vol. 11, No. 4, April 1999 Michael P. Brenner

5f d3r/vp . Since the solutionu;r 21, as described above,follows that the integral diverges linearly with system siz^u2&;fr .

A more physical version of this argument was givenSegreet al.,15 in the spirit of Hinch:9 Consider a blob of sizel of fluid. The number of particles in the blob isNblob

5 l 3f/vp . Random statistics impliesANblob fluctuations inthe number of particles and hence fluctuationsANblobDrgvp in the blob mass. This must balance the Stokdrag 6prn lDU, implying

DU

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Figure 1 shows a visualization of the divergence: 10 0particles were placed randomly in a square box with sidelength 200, surrounded by an infinite fluid. Figure 1~a! showsthe velocity field produced by these particles~inside the box

FIG. 1. Visualization of the velocity fluctuations from a random distributiof 104 particles.~a! shows the velocity field in the lab frame, and~b! showsthe velocity field in the frame moving with the particles. The velocity fiewas computed using the fundamental solution to Stokes equation in afinite system~see Sec. III!. The large scale swirl that forms in~b! indicatesthat the velocity fluctuations depend on the system size. The mean paspacing in this figure is approximately 20.


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containing the particles!, and Fig. 1~b! shows the velocityfield relative to the mean. The flow relative to the mean i‘‘swirl’’ on the scale of the box size. The counterintuitivfact is that the motion of a single particle relative to the meis not controlled by its nearest neighbors, but instead bcollective effect of the entire flow field. In Fig. 1, the meaparticle spacing is about 20, which is a factor of 10 smathan the size of the swirl.

The central argument has been that this divergencevelocity fluctuations for infinite systems~or the dependenceof the fluctuations on system size in a finite system! is un-physical, and should be cut off at some scale. That is to sthe ‘‘swirls’’ apparent in Fig. 1 should exist on a scale musmaller than the size of the system. This issue is notacademic: The effective diffusivity in a sedimenting mixtuis controlled by the nature of the velocity fluctuations thform.

A screening mechanism is a physical effect which reders integral~4! convergent. There are essentially two posbilities for how this can occur: the first possibility, as prposed by Koch and Shaqfeh,16 does not change theu;r 21

law predicted by Eq.~3! but instead relies on a nontriviaparticle distribution to force the convergence of the integrKoch and Shaqfeh demonstrated that this only works fovery special class of particle distribution functions: The dtribution of particles around a single test particle mustways have a net deficit~compared with a random distribution! of precisely one particle. This screening mechanismobservable in practice only if this special configurationparticles is stable.

The second possibility for screening is to keep the prability distribution essentially random and makeu decayfaster thanr 21. Both of the mechanisms discussed hereinof this latter type.

III. WALL EFFECTS

According to the arguments presented above, the sizthe velocity fluctuations diverges for an infinite system, a

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is set by the shape of the container for a finite system. Thare two different subtleties that are associated with contawalls. The first is exposed by the following simple questioConsider a container with widthW, depthD, and heightH.Which of these scales provides the cutoff for the divergein the fluctuations? The answer depends on theboundaryconditionson the fluid velocity field; see Fig. 2.

If the boundaries do not exert forces on the flow fiethe divergence is cut off by thelarger of W, D, andH. Thereason for this is that if theu;r 21 law extends to all boundaries, then the distance to the furthest boundary dominthe far field of the integral. On the other hand, if the bounary conditions on the wall are no-slip, the container waexert a drag on the particles, and this conclusion is faExact solutions demonstrate that close to the wall, theu;r 21 law changes to a more rapid decay, generally fasthan r 22 ~see below!. The consequence of this is that in aactual experiment with rigid walls, the distance to thenear-estwall provides the cutoff for the fluctuations.

The nature of the decay of the velocity field aroundsingle particle then depends on the shape of the contaFor a particle translating parallel to a rigid wall a distand away, Blake22,23 and Lorenz24,25 demonstrated that for !d, the Stokes solutionu;r 21 holds. However, farenough away, the decay law transitions tou;r 22. The exactformula for the velocity field is complicated, and containssuperposition of Stokeslets and higher order correctionspedagogical description of the details is givenPozrikidis.25 Exact formulae also exist22,24,25 for particlestranslating perpendicular to a wall; in this case the crossois from r 21 to r 23.

Solutions for point forces moving in the vicinity oboundaries have been tabulated for a number of diffegeometries; two important geometries for the present dission are parallel plates26,27 and circular cylinders.28,29 Forparallel plates, Liron and Mochon constructed the solutiwhich demonstrates26 r 22 decay for motion parallel to theplates. Motion perpendicular to the plates decays expontially in the far field. Liron and Mochon also estimated covergence of the two plate problem to the single plate soluin the limit where the plate spacing becomes infinite, anoted that the influence of the second plate is importwhen the distanced of the particle from the closest platemore thanD/8, whereD is the distance between the plateThe circular cylinder differs qualitatively from the examplwith plates in that the walls surround the particle on all sidThe qualitative consequence of this is that the walls execonstant drag force per unit length of the cylinder. Hencedemonstrated by Blake28 and Liron and Shahar,29 far enoughfrom the point force the velocity field decays exponentialThe crossover betweenr 21 decay and exponential decay ocurs at a scale of order the radius of the pipe.

The second subtlety associated with container wallsvolves the notion3,4 that the particle distribution is randomand homogeneous. This idea is based on the assumptioergodicity: In the reference frame falling with the sedimethe particles trajectories are supposed to explore spaceformly. In the presence of side walls and strong particleteractions, this assumption is false. The reason is that


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walls make the velocity fluctuations, and hence the partdiffusivity, depend on the distance between the particle athe wall. The no-slip boundary condition requires that tdiffusivity vanishes at a wall, so that the diffusivity reachesmaximum on the midplane of the cell. The consequencenonuniform diffusivity is that a collection of particles initially uniformly distributed in the cell will migrate towardthe side walls. This assertion will be demonstrated througcomputer simulation of a model sediment in the next secti

The nonuniformity of the velocity fluctuations in a random sediment follows directly from formula~4!, except us-ing the correct Green’s function for a particle in a cell wisolid boundaries:

DU2~r !UB5fI 5fEcell

u~r2r 8!cell2 d3r 8

vp. ~6!

Here,ucell is the velocity field of a single particle in the celand the integral is over the entire cell. As discussed abosinceucell decays at least as fast asr 22 asymptotically, thisintegral converges. However, the value of the integralpends on the location ofr in the cell, and in particular on thedistance from a side wall. When sampled at a wall,u50 sothe fluctuations vanish;DU is maximum in the center of thecell. Since in generalucell;Usar21 for r ,O(d) ~if d is thedistance from the particle to the nearest container wall!, thisintegral will in general beI 5cUs

2d/a. The constantc is ageometrical factor, and can be much larger than unity~c.f. inthe planar case, where the transition tor 22 behavior occursat 12d). Estimates forc in a random suspension as a functioof distance from a single wall is given in Appendix A.

The physical reason for the dependence of the size ofvelocity fluctuations in a random suspension with distanacross the cell is that particles closer to the wall interact wfewer other particles than particles in the center of the cellparticular, at a distancex from the wall of a cell, a particleinteracts via theu;r 21 law with N(x)5fx3/vp particles,wheref is the local volume fraction. The fluctuations abothis number lead to a particle excess or deficit of6AN(x);Afx3/vp particles, which leads to a fluctuation of sizDU(x);6(6prnx)21AN(x)Drgvp . Hence,

DU~x!;UsAfAx

a. ~7!

Therefore, in a random suspension fluctuations should vwith distance to the wall with via aAx law. Any deviation tothis formula measured in an experiment would reflect a nuniformity in the particle distribution across a cell. A conceptually useful way of expressing the formula for the flutuations is to express

DU2~x!

Us2 ;f

cx

a5fS V0~x!

vpD 1/3

.

V0(x)5x3 is theinteraction volumeof a particle a distancexfrom the wall—the volume of space around a single partiwhere theu;r 21 law holds.

The notion of interaction volume immediately suggethat there should be at least two different regimes forvelocity fluctuations in a finite cell. Cartoons of the two r


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gimes are depicted in Fig. 3. Around each particle, we hdrawn a circle with radius the distance of the particle towall, depicting the interaction volume for this particle. Insiof this volume, the particle interacts strongly~i.e., by theu;r 21 law! and outside the volume the interactions aweaker. In theweakly interacting regime@Fig. 3~a!# the in-teraction volumes surrounding different particles do noverlap. In thestrongly interacting regime@Fig. 3~b!# theinteraction volumes of different particles overlap signicantly.

In the weakly interacting regime, particles interact weach other with velocity laws which decay much faster thr 21. In principle, what happens in this case could dependthe cell geometry, since different cells have different intaction laws. However, the simplest expectation is thatinitially random distribution of particle distribution remainrandom for all times. This would lead to a dependencefluctuations onf and system size that is directly predictedEq. ~6!: Namely,DU/Us5c(W/a)AfAD/a, where the con-stant c is geometry dependent. In the limit of low volumfractions, theAf law must be obeyed, because the time scof interactions between different particles grows signcantly: for example, in Appendix B it is shown that the exasolution of Liron and Mochon26 implies that, for two par-

FIG. 3. Cartoons of the weakly interacting and strongly interacting regimof sedimentation. Around every particle~small dark circles! is the spheredenoting the interaction volume of that particle; within this sphere the pticle interacts with other particles via theu;r 21 law. In the weakly inter-acting regime, the interaction volumes around different particles~defined tobe the volume of space where theu;r 21 law holds! do not overlap. In thestrongly interacting regime, the interaction volumes overlap significantlytransition from weakly interacting to strongly interacting occurs when eitincreasing the number density of particles at fixed gap width~top figure! orincreasing the gap width at fixed number density~bottom figure!.


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ticles falling between two plates, the time scale for motiperpendicular to the plates grows exponentially with thseparation distancer parallel to the plates, whenr.D/2.

In the strongly interacting regime, what happens depeon the result of particle interactions. In this regime the nouniformity in DU across the cell implies that the particle flutoward the side walls is strongest in the center of the cellthat over time the center of the cell will be depleted of pticles. Heuristically, the side walls are kinetic traps for paticles, since once a particle diffuses toward a wall it takincreasingly long for it to escape. Simulations describedlow confirm this conclusion. The consequence of particbeing pushed away from the center of the cell is thattypical size of the velocity fluctuations in the strongly inteacting regime will besmaller than that predicted by Eq.~6!.

When is the transition between the weakly and stroninteracting regime? At fixed volume fractionf, the transitionfrom weakly to strongly interacting occurs by increasing tdistanceD between the walls. Hence, there is a critical dtanceDcrit ~for fixed f! or alternatively a critical volumefraction fcrit ~for fixed D! above which particle interactioncan strongly modify the fluctuations from that of a randodistribution. A lower bound forfcrit is that the particles fall‘‘single file’’ through the cell, occurring atf* 5(2a/D)3.

Before proceeding with numerical simulations to tethese ideas, we first analyze themaximumsize of a blob thatcan occur in a cell of depthD ~with W@D) that behavesprecisely like a blob in an infinite system. Each of the pticles in such a blob must interact with each other viau;r 21 law. This requires that every particle in the blobcloser to the others in the blob than to its nearest wall. Tmaximum blob for which this constraint holds is centeredthe midplane between the two side walls, and has sizeD/3 inthe direction along the depth, and a sizeD/2 along the widthand the height. Any blob which is larger than this bounecessarily has some of its particles interacting more weathan they would in the infinite system limit. It should bemphasized that these bounds follow directly from the prerties of the single particle Green’s functions to Stokes eqtion, and have nothing to do with complications that occwhen many particles interact.

A. Numerical simulations of a model sediment

To substantiate the effect of wall interactions, this susection presents numerical simulations of a model sedimconfined between two infinite rigid walls. We will assume,above, that the particle Reynolds number (Usa/n) is smallso that the fluid velocity solves the Stokes equations. Weinterested in the dynamics at very low particle volume frations, where the average interparticle spacing is much lathan a particle size. In this limit, it is appropriate to considthe limit of point particles, each of which exerts a localizforce on the fluid.

1. Formulation of model sediment

Denoting the particle positions byxn , the equations ofmotion are

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Here,Un is the velocity of the fluid at the location of thenthparticle if this particle were absent,f56pmaUsz is the lo-calized force each particle exerts on the fluid, andS(xi ,x)represents the Green’s function of the Stokes equation fpoint force located atxi , obeying the appropriate boundaconditions at the side walls. Formula~8! is a consequence oFaxen’s first law,5 which states that a particle sitting inflow Un and feeling a forcef moves as the sum of the Stokevelocity (6pma)21f and Un . Given a formula for theGreen’s functionS these equations solve for both the potion and velocities of the particles in the sediment.

To proceed with simulations of a sediment trappedtween two infinite plates~located atx50 andx5L) we needan approximation for the Green’s functionS obeying theboundary conditions. We seek an approximate formula foSwith three important features:~1! It must have the correcnear-field asymptotics, i.e.,r 21 decay closer to the particlthan the nearest wall;~2! It must have the correctr 22 decaylaw on scales farther from the particle than the nearest wand finally ~3! It must exactly satisfy the correct boundaconditions on the plates. This last property is especially ccial because the major point of these simulations is to invtigate the consequences of a vanishing diffusivity nearparticle walls.

There are several different approaches that could belowed for findingS. In the point particle limit, the solutionfor a particle falling between two walls can be representeda infinite sum of image Stokeslets~reflecting the particlepositions about the two walls!, plus a correction. The correction is necessary because, although the sum of image Stolets lead to a vanishing velocity parallel to the walls, tcomponent of the velocity perpendicular to the walls donot vanish. Liron and Mochon26 found the correction by expressing the sum of image Stokeslets in terms of an infiseries of Bessel functions, and solving for the correctioneach mode explicitly. One possibility is to use a truncationthis series solution as an approximation forS, as it convergesvery rapidly farther from the particle than the wall spacinHowever, the convergence of this series is very slow innear field, closer to the particle than the wall spacing. Talternative idea of basing a numerical method on the sumimage stokeslets has the defect that the convergence isslow in the far field.@The best way to do this calculation is tcompute the Green’s function numerically and then uslookup table for efficiently simulating the particle dynami~H. Stone, private communication!. This is planned for afuture work.#

For the present simulations, we chose a balance ofposing evils which is both computationally efficient and rspects important features mentioned above. DenoSp(x,x8) the Stokeslet for a single particle in an infinite mdium for a particle located atx8, the formula forS that weuse is


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where rn5(x822nD,y8,z8) and Rn5(2(x812nD),y8,z8). The functiong(x)51 for the components ofS par-allel to the side walls, andg(x)5sin(px/D) for the x com-ponent ofS. This ensures that the boundary conditions onwalls are obeyed exactly for all components ofS, while re-specting the symmetries ofS with respect to interchange ofxandx8. ~See Appendix B for a discussion of this point.! Thenumber of image StokesletsN taken in the sum is choseindependently for each simulation: in particular, we verthat N is large enough that the numerical results conveand are independent ofN. Typically in the simulations thatfollow this requires keepingN'10. In writing the explicitformula forS we will nondimensionalize all length scales ba and all timescales byaUs

21.Another subtelty that needs to be addressed is that

single particle Stokeslet

Sp51

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ur u3Dhas a singularity atr5(x,y,z)5(0,0,0). Hence, if a pair ofparticles in the simulation happen to overlap, the settlvelocity predicted by Eq.~8! will be unrealistically large dueto the large velocity induced by each particle on the othThis effect could produce large spurious effects on thelocity fluctuations in a simulation. Previous studies haavoided this problem by either using a more accurate resentation of the hydrodynamic forces than the point partapproximation allows,30 or by including a short ranged repusive force between the particles.31 Here, we will simply cutoff the divergence of the single particle Stokeslet by mofying it to be

Sp51

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where r 5(x21y21z211)1/2. This cuts off the unphysicadivergence while maintaining the important asymptotic proerties mentioned above. Also, two overlapping particmove at exactly twice the velocity of a single particle, whiis essentially correct. It should be remarked that this cutofthe Stokeselet divergence is the only place that the parsize enters explicitly into the formula forS.

The last element of the simulations is that we needcorrect the particle mobility due to the presence of the swalls: Particles near walls fall more slowly than particlesthe center of the cell. We account for this by taking tparticle velocity to beU5(6prna)21fc(x), where c(x)512a(2l )21 with l the distance of the particle to the cloest wall.

We emphasize that this formulation is amodel for themotion of the particles in the presence of walls, utilizinseveral approximations which are physically reasonablenot necessarily mathematically well controlled.@The mostegregious approximation in the model isg(x), which implies


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that the velocity field is not divergence free. In principle thcould lead to systematic errors in computed probability dtributions across the cell.#

The goal of the calculations is to show the qualitatieffect of walls on the particle motion. Quantitative compasons between numerical results and experiments requirmore accurate representation of the Green’s function.

2. Blob breakup

The physical argument for computing the size of tfluctuations in an infinite system relied on the dynamics‘‘blobs,’’ i.e., regions of particle density higher or lower thaaverage. In an infinite homogeneous system, a group ofticles clumped together in a spherical clump fall together ispherical clump for long times. Evidence for this long timcoherence is given by Nitsche and Batchelor31 in a numericalsimulation of a falling blob. The simulation demonstratthat particles shed from the blob extremely slowly: e.g., thsimulation demonstrates that in a time of order 200aUs

21

only about 1% of the particles escape from the blob.The lifetime of a blob is important in understanding t

size of the velocity fluctuations during sedimentation; bloare constantly being created and destroyed, and the sizwhich they can grow depends on how long they survive.understand how the presence of container walls modifiesconclusion, we simulate the evolution of an initially randodistribution of particles between two walls. The initial distrbution is finite, and surrounded by clear fluid. Figureshows a simulation of the dynamics of 100 particles confinin a gap of width 10a. The initial distribution of particles israndom, filling the gap as well as a region of size 2a320a, so the initial volume fraction isf0'0.01. The dif-ferent rows represent snapshots att515, 30, 100, and 200respectively~in units of aUs

21). The arrows represent thmagnitude of the velocity field in the laboratory frame. Tleft column shows the particle distribution perpendicularthe plane of the walls; the velocity field is evaluated onmidplane of the cell, halfway between the two walls. Tright column shows the evolution within the thin gap, withe velocity field evaluated aty510.

There are several important features that should be nfrom this simulation, which differentiate the evolution of‘‘blob’’ in a confined geometry from that of an infinite system:

~1! Particles in the center of the cell~near the midplane!move faster because they have a larger interaction voland thus are able to interact with more particles. This caua ‘‘stretching’’ of the initial blob: After a time 30aUs

21, theextent of the blob in the vertical direction has nearly tripleIn contrast, the simulation of Nitsche and Batchelor31 showsthat ~in the absence of walls! the blob size stays roughlconstant during this time.

~2! There is a swirling motion of the particles in the gabetween the two walls. This can be most easily seen throan animated movie of the blob disintegrating~which unfor-tunately cannot be included in this paper!. Another indicationis shown in Fig. 5, which plots a typical particle path fallinthrough the cell. The particle moves back and forth acr


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the smallest dimension of the cell. At long times, the partistops at the right wall.

In fact, essentially every particle in our simulation evetually ends up stuck to the side walls. The reason for thispartly an artifact of our simulation procedure, and partlyreflection of the actual dynamics: on one hand, we makeattempt to accurately model lubrication forces when a pticle approaches a wall, so that the evolution of the partidynamics near the wall is incorrect. On the other hand, Apendix B shows that fortwo particle interactions, when theparticles are sufficiently far from each other~of orderD/2) inthe direction parallel to the wall generically one of the paticles will segregate to the wall. Hence, there is an ovetendency for particles to force each other toward the walthough the details of what happens when the particleproaches the wall are not well accounted for here.

~3! The maximal fluctuations in this cell are localizethree dimensional ‘‘blobs,’’ in spite of the fact that the initiadistribution of particles had a size much larger than the gwidth. Hence, the presence of a thin gap doesnot make thesystem sediment two dimensionally. Evidence for this asstion is shown in Fig. 6, which show a ‘‘front view’’ of theflow field around the group of particles in the above simution which results in the maximal velocity att510. Theradius of this group of 25 particles is of order 5a; the flowfield is very similar to that of the same blob falling in ainfinite three dimensional fluid: The measured falling veloity of the group of particles is'4.5Us , whereas the pre-dicted velocity for a blob ofN particles and radiusR sur-rounded by an infinite fluid isN(a/R)Us54.8Us .

~4! Correspondingly, the magnitude and nature of tfluctuations in the simulation depends on the gap widthD.To illustrate this, Fig. 7 shows the maximal velocity fluctution at t510 in a simulation with the same initial volumfraction as that of Fig. 6, but with double the gap widthD520a. The maximal fluctuation is a blob of 54 particles wiradius'7.

~5! Particles near a wall have a much smaller fluctuatvelocity than a particle in the center of the cell. Becausethis, it takes much longer for a particle near the wall to moto the center of the cell than for a particle in the center ofcell to move to a wall. At long times this results in segregtion of particles to the walls. We believe that in a continously fed sediment, this process must reach a steady sWhen the particle density near the wall increases, the fltuations there will also increase. Eventually the fluctuationear the wall will be of order that in the center of the ceand the system will reach a steady state. Our simulationsnot able to currently capture this steady state because ofour inadequate modeling of the particle-wall interaction athe fact that we use too few particles.

B. Comparison with previous experiments

We now re-examine previous experiments with wall efects in mind. From this viewpoint, the two types of expements that have been performed to date represent veryferent measurements. The particle tracking algorithm of Hand Homsy12 and Nicolai and Guazzelli13,14 follow a single


t

Green’s


FIG. 4. Snapshots of the simulation of the falling blob. Successive rows show times 15, 30, 100, and 200 in units ofaUs21. The left column represents a fron

view, looking down on the sediment perpendicular to the walls. The right column represents a side view, with the walls~hatched regions! located atx50 andx510. The velocity vectors show the relative magnitudes of the velocity field produced by the particle distribution, according to our approximatefunctionS. For the left column the velocity field is evaluated at the midplane of the cell; for the right column the velocity field is evaluated aty510, the initialmidplane of the blob. Note that the scales on the plot in thez direction differ in the different frames.

Downloaded 07 Jun 2004 to 128.103.60.225. Redistribution subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp


Fig. 4~b! ~Continued.!

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marked particle in its path through the cell. If we assume tthe particle distribution is essentially random, then particspend longer near walls; the fluctuations near the wallsthen dominate the long time average. On the other handSegreet al. experiments measure the velocity field at fixtime on the midplane of the cell, where the fluctuationshould be maximal.

The Segre´ et al. experiments consider both rectangucells with W3D varying from 3 mm30.3 mm to30 mm310 mm, and a 0.5 mm radius cylinder. Most of thdata obeying theDU;f1/3 scaling laws have 10 mm31 mmcells. The experiments image the velocity field in the medplane of the cell with a depth of field of approximately60.5mm. On increasing the cell widthW the correlation length

FIG. 5. Top and side view of a particle path in the blob disintegratsimulation shown above. The scales in they and x direction are the same(10a), whereas the scale in the fallingz direction is compressed. Note thathe fluctuations in the trajectory in the plane perpendicular to the side wis of the order of the fluctuations in the trajectory between the walls.


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saturates; the surprising feature of this measurement isthe saturation occurs with thelargestdimension transverse tothe falling direction, contradicting simple expectations olined above for an experiment with rigid walls. Moreovethe measured correlation lengths show a nontrivial volufraction dependencel'30af21/3.

The observed correlation lengths vary between 1 mand 5 mm, and are alwayslarger than the distance to theclosest wallsD/2. As an example, the particle image velocmetry images in Fig. 1 of Segre´ et al.have cell half-depths ofD/251 mm for thef51024 data, andD/250.5 mm for thef50.03 data. The measured correlation lengths~in the di-rections parallel to the nearest plates! are on the order of 4mm and 1 mm, respectively. This means that the walls mbe affecting the results. However, the reported dependeon volume fraction givesDU;Usf

1/3, which differs from

lls

FIG. 6. Closeup of the particles falling the fastest in the above simula~front view!. Twenty-four particles are clumped in a ‘‘blob,’’ of radiuapproximately 5a;D/2. The falling velocity of this blob is measured to b'4.5Us , which is close to the predictionN(a/R)Us54.8Us for a threedimensional blob.


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the volume fraction dependence of the weakly interactregime. TheDU;Usf

1/3 law occurs for the data wherthere is no dependence of velocity fluctuations on syswidth W. This implies that for these data, an additional dnamical process must occur in the experiments.

We hypothesize that the dynamical effect is particle mtion toward the walls. As shown above, fluctuations makparticle initially on the center line of the plate~in the imag-ing window! move off the center line because of a compnentDU of its velocity directed toward the wall. As emphasized above, this limits the size of the fluctuations that cform in this system as it limits the number of particles thcan participate in a correlated velocity fluctuation; see Fig

We now present a scaling argument incorporating tphysical idea, which reproduces the scaling laws measuby Segre´ et al. By dimensional analysis, a particle initiallon the center plane will reach the wall after a timet;D/2DU21, whereDU is the characteristic size of the velocity fluctuations transverse to the walls.@We remark thatsedimentation is unlike the ‘‘blob disintegration’’ calculatioshown above because the characteristic blob size~which de-terminesDU) is set dynamically.# At this point, the velocityfield around the particle no longer decays liker 21 and thusdoes not affect many other particles. As demonstrated bysimulations of the preceding section, the particle no lonparticipates in a correlated velocity fluctuation with othparticles. Hence,t is the correlation time. The distance thparticle falls during this process is of order

l;Ust5D

2

Us

DU. ~12!

This distance represents the maximum size of a regioncorrelated velocities that can form in this experimental

FIG. 7. Closeup of the particles falling the fastest in a simulation withD520a, but with the same initial volume fraction as that of Fig. 6. Fifty-foparticles are clumped in a ‘‘blob,’’ of radius approximately 7a;D/2. Thefalling velocity of this blob is measured to be'8.2Us , which is close to thepredictionN(a/R)Us57.7Us for a three dimensional blob.


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ometry: A larger ‘‘blob’’ does not have time to form withoudifferent parts of the blob becoming uncorrelated.

To determine thef dependence ofl andDU, we need tocombine formula~12! with an estimate with the Caflisch anLuke result Eq.~5! for how DU depends on the size annumber of particles in a correlated region. The combinresult is

l;a~D/2a!2/3f21/3, ~13!

DU

Us;~D/2a!1/3f1/3, ~14!

Nblob;S D

2aD 2

. ~15!

The Segre´ et al. experiments15 operate with a single particlesize~7.8 mm! and a range of cell sizes. The data forDU aremainly taken withD51 mm cells, so thatD/2a'64. Thisimplies Nblob;4100, l;17af21/3, and DU;4Usf

1/3, ingood agreement with the experiments, for whichNblob

53000, l i'11af21/3, and DU i'2Usf1/3. ~Here l i and

DU i denote the correlation length and velocity fluctuatialong the sedimenting direction.!

It therefore follows that particle motion controlled by thgap thicknessD can lead to scaling laws which are consistewith those measured by Segre´ et al. However, the presenauthor wishes to make no secret of the fact that these scalaws equations@~12!–~14!# were derived to reproduce expermental findings which were known to the author at the timof the derivation. Therefore, they do not in any way repsent a ‘‘first principles’’ theory. The substantive conclusiowhich should be drawn from the scaling analysis is that

FIG. 8. Side view of thin experimental cell. The dotted lines denotescenter line, which is where the image plane is located in the Segre´ et al.experiments. The thin solid line represents the proposed particle path.


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experiments areconsistentwith dynamics controlled by thegap thickness. Moreover, this conclusion is consistent wthe simulations described above, which also suggests thadynamics should be influenced by the gap.

The seriousness of this conclusion is that both our geral arguments and formula~13! imply that the size of thevelocity fluctuations is still controlled by the size of the ceThe growth of fluctuations with system size is weaker ththat implied by the Caflish–Luke prediction@Eq. ~5!# al-though it still diverges with increasingD. Although it is notpossible to directly test the form of this divergence quanttively with our current simulations, we have verified thqualitative dependence onD andf: On increasingD keepingf0 fixed, the size of the velocity fluctuations increase; simlarly on increasingf0 keepingD fixed the fluctuations in-crease.

The scaling laws in Eqs.~12!–~14! will only be validwhen the assumption that the three dimensional lawDU;ANblob/ l applies to the fluctuations is valid; if the sizethe blob is large enough relative to the cell size, wall drwill limit the size of the fluctuations. The crossover betwethese regimes occurs when the velocity fluctuations impby Eq. ~13! is of order the upper bound implied by Eq.~6!.SettingDU from Eq. ~14! to be of order, the upper bounimplies a transition at the critical volume fraction

fcrit;2

c6

a

D5f* S D

2aD 2

, ~16!

wheref* is the volume fraction based on a single particper interaction volume. This formula has the physical intpretation that when the volume fraction is small enough tfewer thanNblob particles exist in an interaction volume, thparticle dynamics cannot be described by formulas~12!–~13!. As described in the previous subsection, abovethreshold the particle interactions are weak and henceDU;UsAf law should hold.

A rough estimate for the critical volume fraction is otained in Appendix A, which computes the upper bound foparticle is a distanceh from a single wall, ignoring all otherwalls. Applying this formula for experiments withD/(2a)550 impliesfcrit;531026. As discussed in Appendix Athis estimate overestimates the true answer, and hencecritical volume fraction far underestimates for the correfcrit @note thec6 power in Eq.~16!#. Taking into account allof the side walls in the cell will decrease the interactivolume.

Since we do not have the Green’s function for all expemental geometries, it is not possible to use this argumencomputefcrit for realistic experimental geometries. In thSegreet al. experiments, the ratioD/a is smallest in the3 mm30.3 mm cell and the 0.5 mm cylindrical cells, sthese are the most likely candidates for being in theAfregime.~It is also true that for these two cells the imaginwindow spans the entire width of the cell, which implies thboth sets of walls are probably important for the particbeing sampled. This will further decrease the interaction vume, and hence increasefcrit for these cells.! With this inmind, we consider the scaling of fluctuations with chang


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system sizeW measured by Segre´ et al. Examining Fig. 4 ofSegreet al. for the size of the fluctuations as a functionsystem sizeW, the data naturally separate into two differecategories: Thesaturateddata ~where the velocity fluctua-tions are independent ofW! occur for the cell sizes10 mm31 mm, 20 mm32 mm, and 30 mm310 mm. Thescaling laws for the velocity fluctuations and correlatilengths discussed above in Eqs.~13!, ~14!, and ~15! all re-ferred to the cells in the saturated regime. On the other hathe data which show an increase in the velocity fluctuatiowith increasingW are for the 3 mm30.3 mm cell and the 0.5mm round cell.

This suggests the interpretation that these smaller chave fewer thanNblob particles per interaction volume, anhence their velocity fluctuations obeyDU;Af. If DU/Us

;Af, then whenDU/f1/3 is plotted againstW/af1/3, thegraph will have positive slope, so that it will appear that tfluctuations are increasing with system size. To test thispothesis, Fig. 9 replots the data for 3 mm30.3 mm cells fromFig. 4 of Ref. 15 asDU/Us vs f. It is seen that the data arconsistent with theDU/Us;Af law. Hence, the proposathat the data consist of two different regimes~with differentscaling laws forDU) is a consistent interpretation of thdata.

The other major set of experiments is by Nicolai aGuazzelli,13 who use a very different procedure: insteadthe ‘‘Eulerian’’ procedure of Segre´ et al. they follow thepaths of single particles meandering through the cell. Frthese data, they extract the effective diffusion constant.imaging procedure projects the position of the particle onttwo dimensional plane perpendicular to the thinnest directof the cell. Since the depth of field in the experiments isdepth of the cell,13 no information is obtained about wherthe particles are located relative to the walls parallel toimaging window. Although the cells used by Nicolai anGuazzelli are larger than those of the Segre´ et al. experi-ments~dimensions ranging fromD3W520mm3100 mm to80 mm3100 mm), their particle sizes are also correspon

FIG. 9. Replotting of the data from Segre´ et al. for the 3 mm30.3 mm cell,considering it as the dependence of fluctuations on volume fraction insof system size dependence. The solid lines denotes theDU/Us;Af law.


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ingly larger~394mm radius spheres!, so that their ratio of thesmallest cell dimension to particle size ranges fromD/2a525→100, in precisely the same range as that of Ref. 1

We now compare the ideas formulated above to the msurements of Nicolai and Guazzelli:14 The above argumensuggested that the correlation time should beD/2DU21

5a/Us(D/2a)2/3/f1/3. Comparing with Nicolai and Guazzelli’s values for cells ranging fromD520 mm→80 mm im-plies predictionst;23aUs

21→58aUs21. In contrast, the

measured value is about 17aUs21 for all the cells.

The independence oft on the smallest scale of the container is a contradiction between the experiment and theoretical ideas presented herein. There are several posreasons for this discrepancy:

~1! The argument posed above implicitly assumes tonly one set of walls is important and hence that the aspratio of the cell is large. Most of the Segre´ et al.experimentshaveW/D510, whereas the Nicolai and Guazzelli14 experi-ment varies fromW/D55→1.25. When the interaction volume is limited by both sets of particle walls the prefactorsthe scaling laws will depend on both dimensions transveto the settling.

~2! The ratio of cell size to mean particle spacinD/(af21/3) is larger by about a factor of 5 in the NicolaiGuazzelli experiment than in those of Segre´ et al. In prin-ciple there could be transitions in the flow as this dimensiless parameter increases.

~3! Finally, and most importantly, the presence of siwalls can cloud the interpretation of diffusivities extractfrom single particle measurements. As emphasized abfor a random suspension,DU ~and hence the particle diffusivity! is not homogeneous across the cell. However,measured diffusion constant from single particle trackingflects anaverageof the diffusivity over the path of the particle. In the experiments, particle paths do not include infmation about the distance from a particle to its nearest whence, the measured diffusivities reflect a trajectory averover the cell thickness. Since the particle will spend mtime closer to a wall, where the fluctuations are smallest,average weights smaller fluctuations more than larger on

To illustrate the effect of this averaging, we analyzetypical particle trajectory in the blob disintegration simultions. This is not equivalent to analyzing particle trajectorin a simulation or experiment on sedimentation, as theredistribution of particles across the side walls presumareaches some steady state whose properties are presentknown. Figure 5 already showed the path of a typical partfalling in a blob of width 10a; Fig. 10 shows the path of thsame particle falling through a cell with width 20a. Theinitial volume fraction in both simulations is identicalf0

50.1. In both sets of simulations it is seen that the partwiggles around the cell, as it does in the Nicolaiet al.experiments.13,14 The scale of the wiggling in the two simulational figures is clearly set by the cell width; in both setssimulations when viewed from the top the scale of the wgling is aboutD/2. Similarly, in the Nicolaiet al. experi-ments~e.g., Fig. 1 of Ref. 13!, the scale of the wiggling isabout 40a whereas the cell half-width is about 50a. In addi-tion, the time it takes for the particle to collide with the wa


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is about double in theD520a cell than theD510a cell.Figure 11 compares thex and y components of the velocityfor the two cells. It is seen that, although initially the velocifluctuations for the particle in theD520a cell are larger thanthose in theD510a cell, eventually they both settle down tvelocity fluctuations in the range of about 0.1Us .

We do not want to interpret these results too literally,there are serious differences between our simulationsedimentation, as noted above. However, this set of simtions clearly suggests another possible resolution of the ctradiction between our arguments about the importancewall effects and the Nicolai–Guazzelli experiments’ i.e., thit is possible that long time measurements of single partdiffusivities do not sample the maximal velocity fluctuatioin the cell.

This latter interpretation also provides a consistent ranalization for the more recent measurements of PeyssonGuazzelli,32 who verified that in theD540 mm cell the sizeof the fluctuations and the correlation time are independof the width of the cell, when it is varied from 4→10 cm,with the depth fixed at 4 cm. They also demonstrated thatsize of the fluctuations is independent of the position alothe width of the cell, except for layer of width'0.2W nearthe side walls~bounding the width!. These measurements aconsistent with our arguments: The size of the fluctuatiowill be controlled by theshortestdimension of the cell.Therefore varying the width~whenW@D) should have littleeffect on the size of the fluctuations. Correspondingly, sinthe measurements represent averages across the cell dthere should not be any variation of the fluctuations acrthe width of the cell until the particles sampled are of ordD/252 cm from the side walls~bounding the width!. In Fig.2 of Ref. 32 the fluctuations vary within a layer 0.2W

FIG. 10. Top and side view of a particle path in the blob disintegratsimulation shown above. The scales in they and x direction are the same(20a), whereas the scale in the fallingz direction is compressed. Note thathe fluctuations in the trajectory in the plane perpendicular to the side wis of the order of the fluctuations in the trajectory between the walls.


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52 cm from the side wall, as expected from this simplegument.

C. Summary of this section

To summarize the results of this section, we have psented theory, numerical simulations, and scaling argumillustrating how confinement by rigid walls can produce notrivial scaling laws for the correlation length and velocifluctuations of a sedimenting flow at low volume fractionBased on these ideas, we have argued that current exments do not definitively exclude the divergence of thelocity fluctuations with system size, although they do impthe divergence is weaker than the Caflish–Luke law sgests.

Our arguments suggest at least two different regimesDU(f), depicted picturally in Fig. 3. These regimes impthat the velocity fluctuations should have the qualitativehavior sketched in Fig. 12. Forf,Nblobf* 5fcrit , the par-ticle interactions are weak enough that theDU;Af law is

FIG. 11. Time dependence of the components of the particle velocdepicted in Figs. 5~upper figure! and 10 ~lower figure!. The size of thevelocity fluctuations is initially larger in theD520a cell ~lower figure! butat long times the size of fluctuations for the two cells is essentially identi


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obeyed. Abovefcrit thef1/3 law is obeyed. The critical vol-ume fractionf* has only a single particle per interactiovolume. The volume fractionfcrit hasNblob particles per in-teraction volume, and controls the crossover in the expments. Both of these numbers depend strongly on the shof the container. To our knowledge, the transition betwethese two regimes in a single cell has not yet been obserWe propose that the smallest cells used in Ref. 15 lie infirst Af regime, whereas the larger cells lie in the secoregime. Besides varyingf at fixed cell size, this generapicture could be directly tested by studying the scaling lafor the velocity fluctuations as a function of distance frothe wall of the cell: close to the wall, theAf law should holdand a transition should appear when the sampling volumfar enough from the walls. Finally, we remark that sinceof our arguments are based on the long ranged part ofparticle greens function, their applicability at higher volumfractions is unclear.

D. Inertial screening

We now turn to a brief description of another mechnism for screening the velocity fluctuations. Although wenot believe that this mechanism applies to the present expments, it illustrates another example of a screening argumwhich relies on a cutoff of the slow decay of the Osetensor, instead of a structural transition in the particles dtribution. We also consider this mechanism as a plauspossibility for the infinite system size limit, where wall efects are not important.

Consider a sedimenting mixture in a box large enoughthat the wall effects discussed above do not apply. TCaflisch–Luke–Hinch argument suggests that the velofluctuation diverges with system size, and hence that theticle diffusivity D diverges with system size. The ideainertial screening is that, eventually, the particle diffusiconstantD will become of order of the momentum diffusioconstant

s

l.

FIG. 12. Sketch of the different scaling regimes proposed, on a dologarithmic plot. Forf,fcrit the velocity fluctuations should scale likeAf.For fcrit,fDU;f1/3.


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D;n. ~17!

At this point, particles will diffuse faster than the momentuthey are releasing into the fluid, and momentum transpaway from particles is no longer effective, so that theu;r 21 law will not apply. This provides a cutoff for the sizof the velocity fluctuations.

Scaling laws for the dependence of the correlation lenand velocity fluctuations on both the particle Reynolds nuber Rep and f can then be constructed by balancingD;nand using the Caflisch–Luke formula~5! as above. Details ohow this works depend on how one estimates the diffusconstantD. For illustrative purposes, we list the resultssuch an estimate here.~Although we emphasize that, in thabsence of a detailed theory for the diffusivity, these formlae are speculative.! In general, the diffusivity is D;DU2t, wheret is the correlation time. If the correlatiotime is set by the time for a particle to move across a blthent; l /DU and thenD;DUl . SettingD;n and combin-ing with the Caflisch–Luke argument as above givesscaling laws

l 5a~Rep!22/3f21/3, ~18!

DU

Us5Rep

1/3f1/3, ~19!

Nblob;Rep22. ~20!

Hinch9 previously gave another argument for these scalaws by proposing there might be a finite Reynolds numinstability of a falling blob. The argument leading to Eq.~17!gives a more precise criterion for when inertia is necessaimportant during sedimentation. The correct scaling lawsl andDU would follow from this approach by determiningDself consistently.

Another approach toward inertial screening has preously been described by Koch,33 for a sediment falling atmoderate Reynolds numbers Rep'1. His arguments utilizethe single particle Oseenlet to cut off the divergence. It tuout that this is not quite enough: in the Oseen solutionvelocity decays liker 22 everywhere except a small wakwhich causes a logarithmic divergence inDU. Koch arguesthat this divergence is cancelled by the relative motiontwo particles out of each others wake by a lift force. Tscaling laws derived by Koch differ from those given abovA difference between his treatment and the argument gihere is that the present is designed to work in the limitRep→0, whereas his argument works at Rep;1.

We are confident that inertia is not playing a roleeither of the two sets of experiments discussed herein;reason for this assertion is simply that all previous expments are either~a! in a regime where simple estimates idicate that wall effects are important, or~b! very close to theregime where wall dominated effects are important. Hoever, in the limit of infinite system size, inertially dominatescreening is a plausible theoretical possibility.


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-

IV. COUPLING TO MEAN VELOCITY

We now return to the question of the dependence ofmean sedimenting velocityUsediment on f. The formula ofBurgers and Batchelor3,4 ignores the velocity fluctuationsand only counts two particle interactions; it is thus naturaask whether corrections arise from the fluctuations. Rethat a blob ofNblob6ANblob particles moves downward avelocity Us7DU. Since more particles fluctuate downwathan upward, blobs contribute a net downward volume flof particlescANblobDUvp / l 3, wherec is a constant of orderunity. Hence, there is a correction toUsediment ofANblobDUvp / l 3/f5cDU/ANblob, so

Usediment5UsS 126.55f1c

ANblob

DU

Us1¯ D . ~21!

The central consequence of this formula is that, dependon the screening mechanism, fluctuation induced transcould dominate the backflow correction to the sedimentati

We now discuss the implications of this result for thSegreet al.experiments. The confinement induced screenmechanism implies that the correction to the sedimentvelocity is of order (2a/D)2/3f1/3. There are two interestingfeatures of this result: first, it indicates that in the limitf→0 the fluctuation contribution to the sedimenting velocdominates; second, it demonstrates that the sedimentinglocity depends~albeit weakly! on the shape of the containePrevious studies19 addressing whether the sedimentation vlocity depends on the container shape have examinedthe backflow contribution.

The crossover between the backflow contribution athe fluctuation contribution toUsediment occurs whenf;0.06(2a/D);1023 in Segreet al. This volume fraction isso small as to be irrelevant for most experiments. Howevexperiments1,7 typically measure a coefficient of theO(f)term which is systematically smaller in magnitude than 6.Writing Usediment5Us(12(6.552c(2a/D)2/3f22/3)f)shows that a linear fit to the data with 2a/D51022 and f51022 will give a coefficient of approximately6.55– 1024/331014/355.55, which is in the range of what inormally observed. It should also be remarked that thisgument provides a rationalization for why different expements tend to observe different values for the coefficienttheO(f) terms: The contribution of the velocity fluctuationto the mean velocity implies that the different formulae couapply to different cell geometries. The other argument thausually invoked for explaining the systematically smalmagnitude of theO(f) term than Batchelor’s argument suggests is that a real system has some degree of polydispe

V. DISCUSSION AND CONCLUSIONS

The most basic question in sedimentation is to link t‘‘microscopic’’ description of many particles interacting hydrodynamically in a Stokes flow with a macroscopic descrtion, as envisioned by Kynch:34

] tf1]z~U~f!f!5¹•~D~f!¹f!, ~22!


in

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r fen

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ctalioheonm

gw

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lesio

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ellse ofsedlyz-

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where heref is the local volume fraction,U(f) is the localadvection velocity, andD(f) is the local diffusivity. Thistype of description is enormously successful in describthe local properties of systems~like non-Newtonian fluids!where thermal fluctuations are important, so the diffusivitydominated by the Stokes–Einstein relation. The fundamequestion is to determine whether this type of effective hyddynamic description still applies in the limit where thermfluctuations become irrelevant.

Classical theories of sedimentation2–4 aimed to predictU(f). The original theoretical difficulty was that simple etimates ofU(f) led to divergent answers. This problemdue to a real physical effect: The velocity a blob of sedimat constant volume fraction surrounded by an infinite fluincreases with increasing blob size according toU;usfR2. Batchelor4 realized that in an actual experimencontainer boundaries force the backflow to flow throusediment, which imposes a constraint which cuts off thevergence. This constraint leads to a convergent answeU(f), which Batchelor finds to be essentially independof the properties of the container.

The screening of velocity fluctuations is the analoguethis same problem for determining the effective diffusivD(f). The principal goal of this paper was to explore teffects of container walls on controlling the size of the vlocity fluctuations. Previous theories of sedimentation haassumed that the system is infinite and homogeneous indirection transverse to the settling, and therefore neglethe dynamical effect of container walls. The principle weffect is due to the well known fact that there is a transitin the flow field around a particle at a distance from tparticle of order the particle-wall distance. This transitiimplies that, when a particle moves around a cell, the nuber of particles with which it is effectively interactinchanges. Through scaling arguments and simulationshave argued that side walls lead to a number of imporconsequences:~1! There are at least two different regimessedimentation, i.e., the weakly interacting regime andstrongly interacting regime. The transition between theseregimes occurs at a critical volume fraction~for fixed cellgeometry! or at a critical cell depth~at fixed volume frac-tion!. ~2! The size of the velocity fluctuations acquiresdependence on the distance from the wall. This could rein a nonuniform particle distribution across the cell, and aimplies a variation in the effective properties of the sedimacross the cell.~3! Side walls even affect two particle inteactions, by breaking the usual symmetry that implies thaan infinite system two sedimenting particles maintain a cstant distance from each other. On the basis of our analwe have argued that current experiments have not detively excluded the dependence of the velocity fluctuatioon the size of the cell, although they have demonstratedif the divergence exists it must be weaker than the CafliscLuke prediction.

The sedimentation experiments that we are awaretypically visualize the sediment by projecting the particmotion onto a plane perpendicular to the shortest dimenof the cell. The smallest dimension of experimental cellstypically of order 50 particle diameters, which is sma


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enough that it is still possible to transmit light through tsystem. From the arguments presented herein, it appeaus that at these dimensions the wall effects will be crucialdetermining critical features of the flow. Examples of thisbeyond that of monodisperse sedimentation of spheres:example, Batchelor and van Rensberg’s study35 of clumpinginstabilities in bidisperse suspension uses a cell width of 3mm, with particles of size about 0.3 mm. From their phographs the initial scales of the instabilities they observe afew millimeters across. Similarly, Herzhaftet al.’s study36 ofclumping instabilities in fiber suspensions take place in cwith D/a'40 as the observed clump sizes are in the rangD/2. It is unclear to us whether a theoretical treatment baon an infinite homogeneous system is appropriate for anaing either of these experiments.

One of the major concerns in trying to derive an effetive theory for the sediment a la Kynch is that it is cructhat there be a scale separation between the region wbulk equations@like Eq. ~22!# apply and the region where aeffective boundary conditions apply. In non-Newtonian flids this scale separation is ensured by thermal fluctuatiothe Brownian motion of an object very far from a wall esures that the effect of the wall on the motion is incohereIn the present problem, the interactions of single particbothwith each other and with the boundaries are long ranso that whether a natural scale separation exists is uncThis issue is perhaps a purely theoretical concern; e.g.,rivos and collaborators~see, e.g., Ref. 37! have successfullyapplied phenomenological slip boundary conditions to diffent types of sedimenting flows, with excellent agreementtween theory and experiments.

Our interest in these general questions was initiatedrecent experiments by Segre´ et al.15 and Nicolai andGuazzelli.14 A simple scaling theory for the firsmechanism—based on confinement in cells of high aspratio—was introduced and shown to be in reasonable agment with the recent experiments of Segre` et al.;15 the ideaspresented herein appear to be in contradiction to the consions of Nicolai and Guazelli,14 and several possible resolutions are proposed. Several of our predictions could be teexperimentally.

Finally, our study gives a simple scaling argument whisuggests that in general there is a coupling between thetuations and the mean settling velocity. This fluctuationduced correction to the mean velocity has the opposite sas the backflow contribution, and could provide anotherplanation for why experiments7 systematically observe alower coefficient of theO(f) correction than that predicteby Burgers and Batchelor.3,4 Coupled with the variousscreening mechanisms proposed herein, it also suggeststhe dependence on the fluctuations on cell geometry orticle Reynolds number should be reflected in the meantling velocity.

ACKNOWLEDGMENTS

I am grateful to Howard Stone for many important dicussions, to John Hinch for stimulating criticisms, to PSegre for discussions about his experiments, and to Jo


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Crocker, Daniel Fisher, Don Koch, and Marteen Rutgershelpful comments. Acknowledgment is made to the A.Sloan Foundation, the National Science Foundation, anthe Donors of The Petroleum Research Fund, administeby the American Chemical Society, for partial support of thresearch.

APPENDIX A: UPPER BOUNDS FOR VELOCITYFLUCTUATIONS

The goal of this appendix is to estimate the upper boufor the size of the velocity fluctuations in the vicinity ofsingle wall, assuming that the particle configuration is radom. Our aim in putting forth this calculation is not matematical rigor, but instead to provide a ball park estimateassess when walls are important in a given experimentalfiguration. The result of the calculation is shown in Fig. 1

We consider an experiment which samples the flucttions a distanceh from a rigid wall, where the sedimentindirection is parallel to the wall, and determine an estimatethe upper bound onDU as a function ofh. The solution forthe problem of a point force near a plane rigid boundary wfirst written down by Lorenz,24 and later by Blake.22 Thisproblem differs slightly from a particle translating nearrigid wall; a spherical particle differs from a point force bcorrections of order (a/r )2, wherea is the particle radius,and r the distance from the particle. We will see in the folowing that these terms give only higher order correctionsthe upper bound, and so can be neglected at leading ord

Consider a coordinate system (x1 ,x2 ,x3) with a planewall located atx150, and a particle moving in thex3 direc-tion located atr05(w1 ,w2 ,w3). We seek the solution to

m“

2u5“p1Fx3d~r2r0!, ~A1!

“–u50, ~A2!

FIG. 13. Comparison of the theoretical upper bound~thick solid line!, givenby Eq.~A7! to existing experimental data. The circles are the data from3 of Ref. 15, for large cells. The squares are the data from3 mm30.3 mm cells of Ref. 15. The diamonds are the smallest volufraction data from Ref. 13. The experiments are systematically belowupper bound.


r.toed

d

-

on-.-

r

s

or.

with m the fluid viscosity,F56pmUsa is the force in thex3

direction, and the fluid velocity is required to vanish at twall. Without the boundary condition on the wall, the soltion to this problem is

u5Fz

8pm•S d i , j

r1

r j r k

r 3 D , ~A3!

whered i , j is the identity matrix,r 5ur2r0u, and r j are thecomponents ofr2r0 . The boundary condition of the wall isaccounted for using image singularities. Blake’s solution

u5Fz

8pm•S d i , j

r1

r j r k

r 3 2I

R2

RjRk

R3

1F2r 1~dk,ada,l2dk,1d1,l !]

]RlH r 1Rj

R3 2d j ,1

R1

RjR1

R3 J G D ,

~A4!

whereRi are the components ofR5(x1 ,x2 ,x3)1r0 , the po-sition of the image. The part of the formula containedsquare brackets represents higher order dipolar correctiThese are necessary because the difference between thimage Stokeslets does not cancel out perfectly on the plaboundary. Notice that this formula has the feature that,from the particle, the image Stokeslet cancels out the leadorder r 21 decay of the force, and hence the decay is lr 22.

As remarked upon above, this solution representsfluid velocity produced by a point force, which is not quithe same as the velocity field produced by a sphere. Thare higher order correctionsO(r 22) to the point force solu-tion arising from the boundary condition on the spheHence, although this solution has the correct qualitatproperties of the flow around a sphere falling near a wall,correct formula will contain further dipolar corrections, sthat the terms in the square brackets in Eq.~A4! will bemodified. However, these terms only contribute anO(a/h)correction to the upper bound.

The velocity fluctuations sampled a distanceh from thewall follows from

DU253f

8pa3 E d3r 8u~ r2r 8!2, ~A5!

wherer5(h,0,0), and the integral is over all space. By recaling r 8→hw, and usingF56pmUsa, this integral be-comes

DU25Us2 27f

64

h

aI ,

whereI is given by the integral

I 5E d3w~8pFu~w2w!!2, ~A6!

where w5(1,0,0) and again the integral is over all spacHence, determining the upper bound requires determinthe value of the single integralI. This integral was evaluatedusing the numerical integration procedures in Mathemato be I 537.3. This implies the upper bound

.eeis



FIG. 14. Phase plane of the of the particle trajectories in thex12x2 plane of two particles with locationsx5x1 andx5x2 . The particles are falling betweentwo solid plates located atx50 andx51. The upper left picture denotesr50.1, the upper right denotesr50.5, the bottom leftr51 and the bottom rightr55.

er

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alls

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es

theowentheswn

DU

Us'4AfAh

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Figure 13 compares this upper bound to existing expmental data. In the comparison, we take the value ofh to bethe half the shortest dimension of the container.

The most severe approximation of this calculation is tit only accounts for asingle wall, whereas multiple wallsaffect the experiments. The influence of a second wall cansurprisingly strong. For a particle falling between two parlel plates of spacingD an exact solution was found by Liroand Mochon.26 In comparing the solution for the two plateto the solution for a single plate, they noted that therequantitative discrepancies in the single wall calculationless the particle is closer than a distanceD/8 to one of thewalls.

APPENDIX B: TWO POINT FORCES BETWEEN TWOWALLS

Section III demonstrates through direct numerical simlation that side walls can have a dramatic effect on a smenting flow. Here, we examine the much simpler questof how sidewalls can influence the trajectories oftwo par-


i-

t

e-

e-

-i-n

ticles ~represented as point forces! falling relative to eachother, and demonstrate that even in this case the side whave an important effect.

A basic fact about sedimentation is that two particfalling in infinite space do not move relative to each othThe reason for this follows directly from symmetry: Thinfluence of the first particle on the second is exactlysame as the influence of the second particle on the firsformulas, if we represent each particle by a point force,fluid velocity isu5Sp(r2r 8), whereSp is the single particleStokeslet, defined in Sec. III. The velocity of the particlcan then be expressed as

x15Usz1Sp~x12x2!, ~B1!

x25Usz1Sp~x22x1!. ~B2!

Subtracting Eq.~B1! from ~B2! implies thatx12x2 is a con-stant.

The presence of side walls breaks the symmetry indirection perpendicular to the walls. Here we address hthis symmetry breaking affects the relative motion betwetwo particles. To do this, we need a representation ofStokesletS for a particle falling between two walls. For thipurpose it is efficient to use the exact formulas written do


nlar

o

is

och


by Liron and Mochon~Ref. 26!. ~Comparing the results fromthe exact formula to the model formula used in the maparticle simulations above, it turns out that the two formugive qualitatively similar answers. This provides an impotant check on the validity of our model formula.!5 If thewalls are located atx50 and x5L, and the particles fallalong z, the solution demonstrates that bothS(x1,x2)• z andS(x1,x2)• y are symmetric with to respect to interchangex1 andx2. This implies that both the relativey and z sepa-

si

pa

o

io

n-

thahernin

on

-

io


ys-

f

ration of the two particles isfixedduring the particle motion.However, the Liron–Mochon solution shows that th

symmetry is broken forS• x, so that there will be relativemotion along thex direction. Here, we use their solution tcharacterize the orbits of the two particles relative to eaother along thex direction. Their solution26 for the fluidvelocity atx2 given a particle atx1 is

S~x1,x2!• x521

4p

z22z1

rIm (

m51

` zmH1~1!~rzm!

A11zm2 21

~x1x2zm~sinh~zm~x22x1!!2zm cosh~zm~x11x2!!

1A11zm2 sinh~zm~x11x2!!!1zm~x2 cosh~x2zm!sinh~x1zm!2x1 sinh~x2zm!cosh~x1zm!!

1sinh~x1zm!sinh~x2zm!~~x12x2!A11zm2 2~x11x22L !!!. ~B3!

J.

in a

at

d of

of

on-

is-low

y-

len-

re-

en-

th

g

ape

is

s,’’

lip

id

us-and.

Here r5A(y22y1)21(z22z1)2 and zm are the complexroots of sinh(z)25z2. We have expressed the particle potions x1 andx2 in units of the plate spacingl 0 . The relativeparticle motion of the particles is then governed by

x15S~x2,x1!• x, ~B4!

x25S~x1,x2!• x. ~B5!

The nature of the dynamics depends on the relativeticle spacing parallel to the plates,r. Figure 14~a!–~d! showsthe phase planes of the trajectories in thex12x2 plane forr50.1, 0.5, 1, andr55. In each case, the phase planes shthat the particles tend to move relative to each otherx12x2

plane. It is important to note that the time scale of the motdepends on bothr and (z22z1), and is logically unrelated tothe characteristic time scale for fallingaUs

21. The time scalescales roughly like (z22z1)/r2 whenr,1, so that whenr;1 it is of orderaUs

21(D/a). When r@1 the time scaleincreases exponentially withr. Physically, when one particleis pushed to a wall the other can fall unimpeded.

An important feature of these phase planes is the chain the structure asr→0. This corresponds to taking the infinite system limit, and making the distance betweenwalls much larger than the distance between the two pticles. We know that in this limit, the distance between ttwo particles must become fixed. This implies that evepoint in thex12x2 plane must become a fixed point wher→0. The beginnings of this transition are apparentFig. 14.

1R. H. Davis, inSedimentation of Small Particles in a Viscous Fluid, editedby E. M. Tory ~Computational Mechanics Publications, Southampt1996!.

2M. Smoluchowski, ‘‘On the practical applicability of Stokes’ law,’’ Proceedings of the 5th International Congress on Mathematics2, 192 ~1912!.

3J. M. Burgers, ‘‘On the influence of the concentration of a suspensupon the sedimentation velocity,’’ Proc. Kon. Nederl. Akad. Wet.44,1045 ~1942!.

-

r-

w

n

ge

er-

y

,

n

4G. K. Batchelor, ‘‘Sedimentation in a dilute dispersion of spheres,’’Fluid Mech.52, 245 ~1972!.

5G. K. Batchelor,An Introduction to Fluid Dynamics~Cambridge Univer-sity Press, Cambridge, 1967!.

6E. J. Hinch, ‘‘An averaged-equation approach to particle interactionsfluid suspension,’’ J. Fluid Mech.83, 695 ~1977!.

7R. H. Davis and A. Acrivos, ‘‘Sedimentation of noncolloidal particleslow Reynolds numbers,’’ Annu. Rev. Fluid Mech.17, 91 ~1985!.

8R. E. Caflisch and J. H. C. Luke, ‘‘Variance in the sedimentation speea suspension,’’ Phys. Fluids28, 259 ~1985!.

9E. J. Hinch, inDisorder in Mixing, edited by E. Guyonet al. ~KluwerAcademic, Dordrecht, 1988!, p. 153.

10A. J. C. Ladd, ‘‘Hydrodynamic screening in sedimenting suspensionsnon-Brownian spheres,’’ Phys. Rev. Lett.76, 1392~1996!.

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12J. M. Ham and G. M. Homsy, ‘‘Hindered settling and hydrodynamic dpersion in quiescent sedimenting suspensions.’’ Int. J. Multiphase F14, 533 ~1988!.

13H. Nicolai and E. Guazzelli, ‘‘Effect of the vessel size on the hydrodnamic diffusion of sedimenting spheres,’’ Phys. Fluids7, 3 ~1995!.

14H. Nicolai, B. Herzhaft, E. J. Hinch, L. Oger, and E. Guazzeli, ‘‘Particvelocity fluctuations and hydrodynamic self-diffusion of sedimenting noBrownian spheres,’’ Phys. Fluids7, 12 ~1995!.

15P. N. Segre, E. Herbolzheimer, and P. M. Chaikin, ‘‘Long ranged corlations in sedimentation,’’ Phys. Rev. Lett.79, 2574~1997!.

16D. L. Koch and E. S. G. Shaqfeh, ‘‘Screening in sedimenting suspsions,’’ J. Fluid Mech.224, 275 ~1991!.

17J. Happel and H. Brenner,Low Reynolds Number Hydrodynamics, wiSpecial Applications to Particulate Media~Prentice Hall, EnglewoodCliffs, NJ, 1965!.

18D. L. Koch, ‘‘Hydrodynamic diffusion in a suspension of sedimentinpoint particles with periodic boundary conditions,’’ Phys. Fluids6, 2894~1994!.

19C. W. J. Beenakker and P. Mazur, ‘‘Is sedimentation container shdependent?’’ Phys. Fluids28, 3203~1985!.

20E. M. Tory, M. T. Kamel, and C. F. Chan Man Fong, ‘‘Sedimentationcontainer-size dependent,’’ Powder Technol.73, 219 ~1992!.

21P. G. Saffman, ‘‘On the settling speed of free and fixed suspensionStud. Appl. Math.52, 115 ~1973!.

22J. R. Blake, ‘‘A note on the image system for a stokeslet in a no sboundary,’’ Proc. Cambridge Philos. Soc.70, 303 ~1971!.

23J. Blake, ‘‘A model for the micro-structure in ciliated organisms,’’ J. FluMech.52, 1 ~1972!.

24H. A. Lorenz, ‘‘Ein allgemeiner satz, die bewegung einer reibenden flsigkeit betreffend, nebst einigen anwendungen desselben,’’ AbhTheor. Phys.~Leipzig! 1, 23 ~1907!.


d

ar

a

’ J

s

ng

in-

ns

in

ri-n,’’

ng


25C. Pozrikidis,Boundary Integral and Singularity Methods for LinearizeViscous Flow~Cambridge University Press, Cambridge, 1992!.

26N. Liron and S. Mochon, ‘‘Stokes flow for a stokeslet between two pallel flat plates,’’ J. Eng. Math.10, 287 ~1976!.

27N. J. De Mestre, ‘‘Low Reynolds number fall of slender cylinders neboundaries,’’ J. Fluid Mech.58, 641 ~1973!.

28J. Blake, ‘‘On the generation of viscous toroidal eddies in a cylinder,’Fluid Mech.95, 109 ~1979!.

29N. Liron, ‘‘Stokes flow due to a Stokeslet in a pipe,’’ J. Fluid Mech.86,727 ~1978!.

30L. J. Durlofsky and J. F. Brady, ‘‘Dynamics simulation of bounded supensions of hydrodynamically interacting particles,’’ J. Fluid Mech.200,39 ~1989!.

31J. M. Nitsche and G. K. Batchelor, ‘‘Break-up of a falling drop containidispersed particles,’’ J. Fluid Mech.240, 161 ~1997!.


-

r

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34Kynch, ‘‘A theory of sedimentation,’’ Trans. Faraday Soc.48, 166~1951!.35G. K. Batchelor and R. W. Janse Van Rensburg, ‘‘Structure formation

bidisperse sedimentation,’’ J. Fluid Mech.166, 379 ~1986!.36B. Herzhaft, E. Guazelli, M. Mackaplow, and E. S. G. Shaqfeh, ‘‘Expe

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