Chaotic particle dynamics in viscous flows: The three-particle Stokeslet problem

PHYSICAL REVIEW E SEPTEMBER 1997VOLUME 56, NUMBER 3

Chaotic particle dynamics in viscous flows: The three-particle Stokeslet problem

Imre M. Janosi,1,* Tamas Tel,2 Dietrich E. Wolf,1 and Jason A. C. Gallas1,3

1HLRZ, Forschungszentrum Ju¨lich, D-52425 Ju¨lich, Germany2Institute for Theoretical Physics, Eo¨tvos University, Puskin utca 5-7, H-1088 Budapest, Hungary3Instituto de Fı´sica, Universidade Federal do Rio Grande do Sul, 91501-970 Porto Alegre, Brazil

~Received 17 December 1996!

It is well known, that the dynamics of small particles moving in a viscous fluid is strongly influenced by thelong-range hydrodynamical interaction between them. Motion at high viscosity is usually treated by means ofthe Stokes equations, which are linear and instantaneous. Nevertheless, the hydrodynamical interaction medi-ated by the liquid is nonlinear; therefore the dynamics of more than two particles can be rather complex. Herewe present a high resolution numerical analysis of the classical three-particle Stokeslet problem in a verticalplane. We show that a chaotic saddle in the phase space is responsible for the extreme sensitivity to initialconfigurations, which has been mentioned several times in the literature without an explanation. A detailedanalysis of the transiently chaotic dynamics and the underlying fractal patterns is given.@S1063-651X~97!10408-1#

PACS number~s!: 47.10.1g, 47.15.Gf, 47.52.1j, 05.45.1b

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I. INTRODUCTION

When microscopic objects move in a fluid, the stressethe liquid due to viscosity may be several orders of magtude larger than those due to inertia. A first estimate ofratio

~stress in fluid due to inertia!

~stress in fluid due to viscosity!

is given by the Reynolds number Re as Re5LU%/m, whereL and U denote the characteristic length and speed ofparticle,% andm are the density and the dynamic viscosof the fluid. Collective phenomena involving the motionassemblies of small particles in the low Reynolds numlimit cover a wide range of interest to both scientists aengineers. Probably the most extensively studied classproblem is the passive sedimentation of particles ingravitational field@1#. A closely related system is that osmall charged or polarized particles in a viscous fluid drivby an external electrical field, the dynamics of which is oftreferred to as electrophoretic motion@2#. Similar physics isinvolved in another, more complicated example for transtion in a strongly viscous environment, the swimmingmicroorganisms such as algae and bacteria. In a quiesfluid even their slow motion~typically a couple of meters peday! can result in considerable spatial rearrangements,namic pattern formation, or bioconvection@3# due to hydro-dynamical interactions.

The familiar approximation, which gives reasonably accrate results up to Re'1, is based on the Stokes equations@1#

“–u50, ~1!

2“p1mDu1f50. ~2!

*Permanent address: Department of Atomic Physics, Eo¨tvos Uni-versity, Puskin u. 5-7, H-1088 Budapest, Hungary.

561063-651X/97/56~3!/2858~11!/$10.00

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Hereu5u(r ) is the velocity field,p5p(r ) and f5f(r ) de-note the local pressure and force fields, respectively. Eqtion ~1! expresses the incompressibility of the flow, whiEq. ~2! is the dynamical equation obtained by neglectinginertial terms in the full Navier-Stokes equation due to tslow temporal changes characterizing the process. The mimportant simplification represented by the Stokes equati~1! and~2! is that they are linear and instantaneous@4#. Theseequations are solved subject to boundary conditions moften on the velocityu. Usually the so called ‘‘no slip’’condition is prescribed on solid surfaces, which meansthe relative velocity between the fluid and the body vanison the boundary@1#.

In this work we revisit a classical problem of viscouflows, namely, the dynamics of three microscopic, noBrownian particles driven by constant external forces, lgravity, or electrostatic field. To achieve the highest possresolution in the initial configuration space with acceptacomputational demand, we used the simplest approximacapturing the essence of hydrodynamical interaction,Stokeslet model, described below. The more compStokes, or creeping flow, solutions for a couple of particcan be obtained by a number of special techniques tailorethe viscous limit@1,5–15#. Although these equations of motion are relatively simple, long-range interactions and coplicated boundary shapes usually require further simplifyassumptions@16#. Up to now, lattice Boltzmann technique@17,18# have come closest to realistic simulations of rathlarge systems at wide Re ranges. In a remarkable recent cputation, Ladd@19# treated 32 768 suspended particles at lReynolds numbers, and he could track the trajectoriesabout 500 Stokes times.~A unit Stokes time is defined as thtime needed for an isolated sphere to pass over a distancone-particle radius.! In our Stokeslet model simulations, wfollowed particle trajectories for several thousands Stotimes from about a million initial configurations whicwould have been unfeasible with more sophisticated teniques.

2858 © 1997 The American Physical Society

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56 2859CHAOTIC PARTICLE DYNAMICS IN VISCOUS . . .

The paper is organized as follows. In Sec. II we brieoutline the Stokeslet model forn interacting particles. Nex~Sec. III! the motion of three particles in a vertical planestudied in the framework of this model. Results of high relution numerical simulations are presented in Sec. IV. Wshow that a chaotic saddle existing in the phase spacresponsible for the extreme sensitivity to initial configurtions which has been mentioned several times in the liteture without an explanation. A detailed analysis of the trasiently chaotic dynamics and the underlying fractal patteis given. Section V is devoted to a discussion of the rangevalidity of our results.

II. THE STOKESLET MODEL

The simplest possible treatment of a viscous two-phflow is based on the so-called Stokeslet model@1#, where themoving objects are approximated by pointlike particleHocking @20# first studied the motion of small clusters osuch particles by means of this approximation in orderexplain related sedimentation experiments by JayaweMason, and Slack@21#. Recent analytical treatments forrestricted class of initial configurations add to the interesthe Stokeslet approximation@22#.

The Stokeslet velocity distribution is the solution to E~2! with a little spherical particle of radiusa moving withvelocity v0 in an unbounded fluid which is at rest at infini@16#. A particle momentarily in the origin generates the vlocity field

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Heree is the unit vector in the direction of the self-velociv05v0e, r 5ir i , and terms ofO(a3/r 3) are neglected. In thecase of sedimentation, e.g.,e is the unit vector pointingdownward along the vertical (z) axis. In the following, how-ever, we usually refer toe as directed vertically upward inorder to emphasize the generality of the treatment, and icate the relevance to, e.g., electrophoresis, too.

Let us now considern particles which would all have thesame steady state velocityv0 if they were isolated. In thecase of sedimentation or electrophoresis,v05F%/(6pma)due to Stokes law, whereF is the modulus of the constanexternal force. Then, particlej generates a velocityu(r2r j )at point r . The linearity of Eqs.~1! and ~2! implies in thelowest order approximation that we can use free superption. Thus, the velocity distribution due ton particles is( j 51

n u(r2r j ). At position r i of particle i the backgroundflow generated by the othern21 particles is( j Þ i

n u(r i2r j ).In the lowest order approximation one assumes that thelocity of the particler i at r i in the laboratory frame is thesum of this passive advection velocity andv0

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It is useful to measure the length and the time in units oaand a/v0 ~Stokes time!, respectively. The dimensionlesform of Eq. ~4! then is

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With the given expression ofU @see Eq.~3!#, this defines aclosed set of differential equations for the trajectoriesr i(t) ofthe different particles. Because particles are assumed tpointlike, vectorsr i and r j must not coincide but otherwisthey can take on any values. With a fixed set of initial coditions r i(t50)5r i0 , i 51, . . . ,n, Eq. ~5! has a unique so-lution.

We note that Eq.~5! is invariant under the transformatiot→2t,e→2e, which means the reversal of time and thdriving force. Consequently, it is also invariant under ttransformationt→2t,r i→2r i for i 51,2,3 at afixede. Thesystem is thusreversiblein the sense that there is an involution ~a transformation that composed with itself yields tidentity! in phase space which reverses the direction of ti@23#. In addition, the phase-space volume is conserv( i“ i• r i50, although there is dissipation in the system~it isnot Hamiltonian!. As a consequence of this volume presevation, however, the dynamics cannot have any attractor

It is worth mentioning that this equation is similar in spirto that describing the dynamics of ideal point vortices intwo-dimensional fluid@24,25#. They are also freely advectein the flow field induced by the others. The vortices do nhave self-velocities (v05e50), and the functionU(r ) isthen proportional to 1/r according to the reciprocal distancdependence of the velocity field around a single vortex@24#.

The relativemotions of the particles are unaffected by tconstant termv0 or e. For this motion, an alteration of lengtscale in the configuration is equivalent to a change intime scale, since the remaining terms in Eq.~5! are homoge-neous functions of the positions. Without any loss of genality, any convenient length in the initial configuration cabe chosen as a unit@20#.

III. THREE PARTICLES IN A PLANE

The analytical solution of Eq.~5! for n52 ~the two-particle problem! is not difficult @1#. It corresponds to a parallel displacement of the pair in a direction being, in genedifferent from that of the driving forcee. The first nontrivialcase is the motion of three particles. Although the trajecries from a general initial configuration can be tracked onumerically, useful exact results are known for special cfigurations. Already Hocking recognized a conservation l@20#: The horizontal projection~more generally, a projectionalong e! of the triangle formed by the three particles isconstant area. Thus an initial configuration of zero projecarea leads to a motion in a vertical plane. Periodic solutibelonging to symmetric initial configurations of nonzero prjected area have been discussed by several aut@20,22,26#.

The three-particle problem in a vertical plane has aattracted considerable interest@20,21,6–8,10,15#. Periodicorbits were not found. Typically, after some ‘‘mixing’phase, during which the particles totally change their relatpositions compared to the initial configuration, a coupleformed and the third particle lags behind~see Fig. 1!. ~Acouple moves faster than an isolated particle.! It is also

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2860 56JANOSI, TEL, WOLF, AND GALLAS

known from experiments@21# and simulations@20,7,8# thatthe final configuration dependsvery sensitivelyon the initialone.

We have systematically studied the Stokeslet modelthree particles in a vertical plane. By introducing relaticoordinatesr125r22r1 andr235r32r2, we find for the rela-tive motion

r125U~r23!2U~r121r23!, ~6a!

r235U~r121r23!2U~r12!. ~6b!

Here we have utilized thatU(r ) is an even function. Notethat after the restriction to two-component vector variablEq. ~6! still preserves the phase-space volume. In spite ofand the aforementioned reversibility property, Eq.~6! cannotbe written in a Hamiltonian form@27#. In fact, the systemseems to have no global conserved quantities. We sothese equations with a step-adaptive fourth order RunKutta algorithm@28#. The simplicity of Eq.~6! allowed us touse a much higher resolution than in previous experimeor numerical studies.

The phase space of this system of equations is fourmensional. In order to gain insight into the complex gemetrical structures underlying the dynamics, we fix onethe relative coordinatesr12, at time zero, and monitor thchanges due to varying the other initial coordinater23. The(x,z) plane defined by the two components ofr23 will be

FIG. 1. ~Color! Numerical solution of model~5! from two initialconfigurations for 105 Stokes times. The particles started from tsame height z50. The horizontal locationsx were ~a!~23.9050,21.0,1.0!, and~b! ~23.9048,21.0,1.0!. The color codingfor both cases is~blue, yellow, red!. Note that the particle leftbehind and the length of the mixing phase are very different.

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called the initial condition space, or the phase space, oftest particle~at fixed inital condition for the other two!.

IV. RESULTS

We first show the trajectories obtained by two simulatiowith slightly different initial conditions~see Fig. 1!. The ul-timate separation into a couple and an isolated particleclear in both cases. A tiny change in the relative positioresults, however, in a strong rearrangement. The mixphases, during which all particles stay close to each othave drastically different durations. Note, furthermore, thacompletely different particle is left behind in these two cas

We are now in a position to quantitatively characterithis sensitivity to initial conditions. A perturbationdx(t50)510210 was imposed on only one of the particle~Note that a perturbation on a single particle changes alltrajectories as a consequence of strong hydrodynamicateraction.! We determined the Euclidean distanD5A(xp2xu)21(zp2zu)2 for each of the three particleswhere the subscriptsp and u belong to the perturbed anunperturbed positions at the same time. Figure 2 showsthe divergence has an overall exponential time dependeD(t);exp(lt), and the fit gives an estimate for the first~lo-cal! Lyapunov exponentl50.03860.001. Because of thereversibility property and the continuous time dependenEq. ~6! must have three other~local! Lyapunov epxonentsTwo of these vanish, and the last one is2l @29#. Note thatother values of the first local Lyapunov exponent were otained in the range between 0.0~power-law divergence! and0.1 depending on the initial configurations, but we fouvalues around 0.04 to be representative. The positivity offirst Lyapunov exponent is a clear indication ofchaos. Infact, this kind of chaos is of transient type@30#, and has afinite average lifetime.

This situation is rather similar to what is called chao

FIG. 2. ~Color! The growth of the Euclidean distanceD betweenunperturbed and perturbed trajectories as a function of timet. Theblue, yellow, and red particles started from the same heightz50 atthe horizontal locationsx522.9510, 21.0, and 1.0, respectivelyThe initial perturbation imposed on the blue particle wdx510210. The exponential fit~black! has a slope of 0.03860.001 for each of the particles.

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scattering@31#. The latter occurs in Hamiltonian systemcharacterized by a complicated interaction at short distanbut without any interaction at asymptotically large distancExamples are the motion of a single particle in a nontrivpotential of finite extension or classical models of chemireactions@32#. Incoming particles then undergo a free mtion, later they enter the region of strong interaction adepending on their initital conditions, spend a longershorter time in this region. Ultimately all the particles escato infinity in a free motion. The initial conditions used in th

FIG. 3. ~Color! Numerical solution of model~5! for 106 Stokestimes illustrating a scatteringlike process. The particles started fthe following (x,z) initial positions: Red~2169.697 802, 0.0!,yellow ~2164.658 129, 0.973 979!, and blue ~212.157 925,694.086 9269!. Note that the couple formed after the scatterievent need not coincide with the initial one. In this respectexample shown is special.

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Stokeslet model above correspond to placing the partiinto the interaction region. We can, however, easily fiother initial conditions where the motion starts with a phaof approach. An example is shown in Fig. 3. The analowith a scattering process is clear: The asymptotic statethe interaction region correspond to configurations whenleast one particle is far away from the others, and whenthree particles are close to each other, respectively. In fthe stay in the mixing phase is the analog, in the languagchemical reactions, of the creation of an intermediate coplex of finite lifetime.

The lifetime distribution, i.e., the duration of the staythe interaction region is known to be a rather irregular funtion in chaotic scattering. This irregularity is due to the eistence of an underlyinginvariant chaotic saddle@30#, whichis a globally nonattracting set, and cannot be reached exa

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FIG. 4. ~Color! Escape-time distribution from 640 000 initia(x,z) configurations withr125(2,0). The color scale from redacross yellow, green, and blue to violet indicates increasing timfrom 0 to 10 000 Stokes units. Two cross sections are explicshown at the heights ofz59.0 ~middle! andz53.5 ~bottom!, indi-cated by dotted lines.

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FIG. 5. ~Color! Four sections of the initial condition space of the test particle. Particles 1 and 2 start always from the@x(t50),z(t50)# positions.~a! Top left, ~21,6! and ~1,6!; ~b! top right, ~-1,6! and ~1,6.7!; ~c! bottom left ~21,6! and ~1,7.5!; ~d! bottomright, ~0,25! and~0,7!, respectively. The final configurations from 640 000 initial points are indicated in each figure. Different colors dwhich particle isleft behind. Red, particle 1; yellow, particle 2; blue, test particle. The thin white squares on the upper left panel shregions zoomed with higher resolutions in Fig. 6.

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Particles, however, can come arbitrarily close to it andhibit chaotic motion before escaping. The chaotic saddleself is a fractal with a local structure resembling the dirproduct of two Cantor sets in certain regions. It containsinfinite number of bounded orbits. Furthermore, there exa complicated fractal curve, the so-called stable manifoalong which particles can hit the saddle~although, as men-tioned above, the probability of falling exactly onto thcurve is zero, because it is a fractal!. Points where the life-time distribution takes on infinitely large values correspoto initial conditions falling exactly on the stable manifolNumerically very large values can really be obtained beloing to trajectories starting close to the stable manifold, astaying therefore a long time around the saddle.

In order to measure a lifetime distribution in the system

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three particles, we say that an ‘‘escape’’ of a couple istablished, if itsvertical distance from the third particle lefbehind exceeds a thresholdzc . We found that at separationzc>100.0 the particle which is left behind has a negligibinteraction with the pair, it moves vertically upwards with iown self-velocity. The lifetime or escape timeT dependssensitively on the initial configuration. To illustrate this, wfixed the initial value ofr12 to be (2,0) by letting particles 1and 2 start in (21,6) and (1,6), respectively, and simulateseveral trajectories with initialr235(x,z) values in a broadrange. The lower part of Fig. 4 shows the dependence ofescape time on the test particle’s initialx coordinate at twofixed initial heightsz. The distributions are rather irregulawith regions of very wild changes over several ordersmagnitude in the escape timeT. The color plate was obtaine

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by distributing initial points on the rectangleuxu<7,0,z,14 uniformly. The color coding corresponds to thvalue of the escape time. Note that the escape time isshort close to the other two [email protected]., around the points~21,6! and ~1,6!#, because the velocity field diverges in thvicinity of any particle. Points with escape time on the ordof 10 000 Stokes units~violet! trace out with very good accuracy the chaotic saddle’s stable manifold. The complicawinding and the fractal character of this curve is clear.

The coordinates of the test particle were chosen from(8003800) grid with one of the grid points slightly displaced bydx5dz5331027 from the fixed initial position

FIG. 6. ~Color! Enlarged regions of the phase-space sectshown in Fig. 5~a!. The resolution is proportionally increase640 000 initial configurations are plotted. The color coding is idetical to that of Fig. 5.

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of one of the other two particles. This is because exacsymmetric initial configurations result in particle ‘‘collisions,’’ which show up as divergences in the numericaltegration process.

To obtain a further impression about the intricate phaspace structures, we consider now the same part of the

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FIG. 7. ~Color! Top: escape-time distribution in the forward anbackward dynamics on the (x,z) plane for r125(2,0). ~The back-ward dynamics was evaluated explicitly on the original grid, itnot a mirror image. Hence the agreement with the reflected patis a direct assessment of the quality of the resolution.! The colorscale from red across yellow, green, and blue to violet indicaincreasing time from 0 to 20 000 Stokes units~cf. Fig. 4!. Bottom:the chaotic saddle on the (x,z) plane, i.e., initial configurationsbelonging to long triplet orbits in both temporal directions are incated by the black dots.

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particle’s initial condition space as in Fig. 4, but with a dferent coloring. The colors now mean different escape cfigurations: They correspond to the particle left behind inasymptotic state. Figure 5~a! shows the results, Fig. 6 is magnification of the regions denoted by white squares in F5~a!. It is natural that if the test particle is close to any of tother two, they form a couple and leave behind the partthat was originally furthest away from them. Thus, big ucolored regions appear around the points (21,6) and (1,6)denoted by 1 and 2, respectively. It is less trivial that copact regions of a given color can be found also further awThe most surprising fact is, however, that they are separby regions where different colors can come arbitrarily cloto each other. A comparison with Fig. 4 shows that this taplace along the lines where the escape time is particullarge, i.e., along the stable manifold. In fact this manifold han interesting topological property~just as in other scatteringsystems with at least three different exit modes@33#!: Anyneighborhood of any point on the manifold contains all thcolors~cf. Figs. 5, 6!. This shows that there is a strong miing of the escape modes along the stable manifold whicanother consequence of the sensitivity to initial condition

The dependence of the test particle’s phase-space stures on the initial position of particles 1 and 2 is illustratby Figs. 5~b!–5~d!. Here a sequence ofr12 values were takencorresponding to initial alignments deviating more and mfrom a horizontal one, and ending in a vertical arrangemThe figure indicates that a rotation of the initial vectorr12leads to a smooth deformation of the phase-space patte

It is instructive to construct the chaotic saddle itself, orleast a part of it. Since the saddle contains all the pointsnever escape either forward or backward in time, it canobtained as the intersection of the stable manifolds ofdirect and time inverted dynamics. Due to the reversibiproperty of Eq.~6! mentioned above, the latter is just thmirror image of the former with respect to the axisz56@34#. In other words, we obtain a rather accurate picturethe saddle by plotting points with very large escape timesboth types of dynamics. Figure 7 shows the escape-timetribution of both the forward and backward dynamics on(x,z) plane with color coding. Points that belong simultneously to large values in both dynamics are plotted in blin the lower part of Fig. 7 and correspond to points of tchaotic saddle in the initial condition space of the test pticle. Note the striking direct product structure.

Besides the typical asymptotic states formed by a couseparated from a single particle, the system has more extional asymptotic states in which all three particles areaway from each other and move with their own sevelocities upward. In such three-singlet statesr125 r2350.Our numerical procedure with a long duration of integratiand a large fixedzc provides us with initial points that comafter a long time close to a three-singlet state in bothforward and backward dynamics. In other words, the blapointsr23 of Fig. 7 along withr125(2,0) correspond to midstates of complicated trajectories which can be reachedan infinitely long time from some initial three-singlet cofiguration and which decay towards another three-sinconfiguration. The property that the invariant manifoldssome asymptotic state provide a fractal foliation of the phspace~or initial condition space! has also been observed

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three-degree-of-freedom scattering systems with energyservation@36#. It is generally believed that a full chaotisaddle contains an infinite number of unstable periodicbits, too@30#. This would imply that, in contrast to previoustatements, the three-particle dynamics has unstable persolution as well, although they might be rather unstable aof rather long period@cf. trajectories at around 800,z,3000in Fig. 1~a!#.

As a next step, one can determine some quantitative cacteristics of the saddle@35#. Figures 4 and 5 suggest that thstable manifold in the four-dimensional phase space hafractal dimension 31Dp with 0,Dp,1. Consequently, onthe initial condition plane it appears as a curve of dimens11Dp . Due to the reversibility property, the intersectionthe stable and unstable manifolds in the full space is 212Dpdimensional. The chaotic saddle on Fig. 7 should thus hdimension 2Dp . One of the simplest methods for measurithe dimension of a fractal object is the box counting: Tnumber of two-dimensional boxesN(e) covering the saddleas a function of box sizee scales with a nontrivial exponenDbox, see Fig. 8~a!. The power-law fit gives an estimatioDbox'1.2. Consequently,Dp'0.6 is the partial dimensionof the saddle along the stable~unstable! manifold on the(x,z) plane of Fig. 7.

Figure 8~b! shows the escape-time statisticsP(T), theprobability density for finding an escape timeT in a largeensemble of particles. In the time regime 150,T,350 itobeys an exponential decayP(T);exp(2kT) with a decay

FIG. 8. ~a! Result of the box counting for the chaotic saddshown in Fig. 7. The sizee of the square boxes covering the imagof the fractal set is measured in pixels, the resolution of the whimage was 8003800 pixels. The slope of the power-law fit i21.2460.04. ~b! Normalized escape-time statisticsP(T) for theforward dynamics in the (x,z) phase-space segment shown in F7. The slope of the exponential fit is20.01360.002.

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rate k'0.013. This is an important fingerprint of transiechaos too@30#. There is a fundamental relationship betwethe partial dimensionDp , the decay ratek and the first av-erage Lyapunov exponentl @30#: Dp'12k/l. Our esti-mated parameters are consistent with this relationship ingiven numerical accuracy.

The sharp jump in Fig. 8~b! at aroundT'350 Stokes timeis most probably associated with characteristic shapes intrajectories. In the mixing phase, the number of ‘‘knotswhere the three particles move close to each other, hainteger value. Starting from the initial conditions in thphase-space segment shown in Fig. 5~a!, we find that thejump in P(T) belongs to trajectories with three knots. Orbwith three knots have escape times dominantly aroT5350. This behavior is characteristic only in the largsingle-colored regions in the phase space where the trajeries are not sensitive strongly to small changes in the inconfigurations. At smaller escape times most of the orhave only two knots. Interestingly, neither the single knorbits ~there are not too many!, nor orbits with a larger knotnumber have such a characteristic escape time.~Note thatthere is no direct relationship between the escape time acharacteristic ‘‘meeting time’’ at the last ‘‘knot,’’ becausthe escape time contains an interval during which the coumoves away from the third particle to the critical distanzc5100.!

V. DISCUSSION

We pointed out in Sec. IV that the dynamics of thrparticles is chaotic, and it is associated with an invarichaotic saddle. Since the Stokeslet model of point particwe used is the simplest approximation for describing hyddynamical interactions in the zero Reynolds number limwe should discuss to which extent our results can be valismall but finite Reynolds numbers.

First of all, we compare the numerical results with relatexperiments. Jayaweera, Mason, and Slack@21# performedthe first sedimentation experiment with three particles sting from a line. Later, Ganatos, Pfeffer, and Weinbaum@7#developed a numerical method to calculate particle velociand drag coefficients for systems of identical spheres. Tobtained a general agreement between the experimentanumerical results, however they noted some differencespointed out again the sensitivity to initial configurations. Wshow a comparison between the experiments and our rein Fig. 9. We cannot expect a better agreement, especdue to the differences in the resolutions. In addition,experiments were performed with finite spheres, where rtion, neglected in our treatment, can modify the trajector

As for corrections to the lowest order approximation, whave checked the effect ofO(1/r 3) terms. The simplesStokeslet velocity field Eq.~3! can be replaced in Eq.~4! bythe full solution@4#

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we recover the nondimensional forms~5! or ~6! with U re-placed byU8. Thus we can implement the same methodthe numerical solutions. The result for the initial conditiospace of the test particle forr125(2,0) is plotted in Fig. 10.A comparison with Fig. 5~a! shows that some new featureappeared mostly around the fixed particles, but the ovephase-space structure remained considerably intact. Onealso recognize a slight global shrinkage of the correspondcolored domains.

It is instructive to see the effect of different approximtions considering special trajectories. Durlofsky, Brady, aBossis@8# show the trajectories of three sedimenting spheon a vertical plane starting from a line with a horizont

FIG. 9. ~Color! Comparison of the simulations~top band! withthe experimental results of Jayaweera, Mason, and Slack@21# ~bot-tom band!. Three particles sedimenting in a vertical plane fromhorizontal line. The positions of the red and the yellow particlesfixed, indicated also by crosses in the bands. The blue test parstarts from different initial positions. Color stripes show which paticle is left behind after a long time.

FIG. 10. ~Color! Segment of the initial condition space (x,z) forthree particles of the modified velocity field Eq.~7! @cf. Fig. 5~a!#.Particles 1 and 2 start from the positions~21,6! and ~1,6!, respec-tively, the third one is the test particle. The final configuratiofrom 640 000 initial points are indicated. The color coding is idetical to that of Figs. 5 and 6.

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spacing~25,0,7! „see Ref.@8#, Fig. 5~b! …. Their full hydro-dynamical model also accounted for near-field lubricateffects and the dominant many-body interactions. The sainitial configuration was used for benchmarking by Philli@15#, who implemented also an accurate model, and repduced the trajectories almost identically„see Ref.@15#, Fig.2~a!…. Our simple model~5! gives from the~25,0,7! initialconfiguration trajectories differing significantly from thcited ones. On the other hand, direct examination of morbits gave the result that small changes in the initial cfigurations alter ‘‘smoothly’’ the resulting trajectories in thsingle-colored large regions in the phase space~Fig. 5!. Witha simple trial and error method we located the initial cofiguration ~25,0,7.258! which reproduced the trajectorieshown in Refs.@8# and @15#, see Fig. 11~a!. Based on theobservation of the global shrinkage of the phase-space stures by introducingO(1/r 3) corrections, we found veryquickly the proper initial configuration~25,0,7.156! for themodified field model~7! which also gives almost identicatrajectories@Fig. 11~b!#. It is likely that the inclusion of ad-ditional terms, e.g., of those describing rotation, resultsbetter and better approximations and modifies furtherphase-space structures so that the initial configurationsthe particular trajectories shown in Fig. 11 approach the ‘‘act’’ ones for ~25,0,7!. Nevertheless, we believe that thsimplest Stokeslet approximation with the leadingO(1/r )terms captures already quite well the most essential featof the hydrodynamical interactions.

Here we would like to emphasize that the CPU timeobtaining the exact trajectories with full hydrodynamics;12 h on a HP 715/100 workstation@37#, while the Stokes-

FIG. 11. Numerical solution of model~5! for two different ve-locity fields. The particles started from the same heightz50. Thehorizontal locationsx were ~a! ~25.0,0.0,7.258!, velocity field ~3!;and ~b! ~25,0.0, 7.165!, velocity field ~7!. Axis z is inverted.

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FIG. 12. ~a! Empirical distribution of the magnitudeuuu of theparticles’ instantaneous velocities during the full simulation of E~6! and ~3! with initial conditions in the phase-space segmentFig. 7. The inset shows the same distribution on a double logamic scale. The dotted lines are power-law fits for velocities bsmaller and larger than the self-velocityuuu51.0, the exponents areclose to15 and25, respectively.~b! Distribution of the instanta-neous moving directionsb with respect to the vertical axis~theparticle left behind moves withuuu and b50). The dotted lineshows a Gaussian fit exp(20.03b2). ~c! Correlation diagram be-tween moving directions and velocity absolute values. The enlope of the set can be well fitted with an exponential functiuuumin(ubu);exp(0.26ubu).

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let solution requires only 1.831026 h for a similar solutionplotted in Fig. 11.

We mentioned in the Introduction the problem of swimming microorganisms. Although the hydrodynamical interation between algae or bacteria is certainly weaker@38,39#than that of, e.g., sedimenting particles, simply becauseare driven by internal forces, these interactions may contute to the collective behavior. For example, it is a commobservation that ‘‘individual’’ trajectories tracked in an asembly of microswimmers show a rather noisy charac~see, e.g., Fig. 2 in@39#!. These trajectories can be charaterized by rather wide distributions of swimming directioand velocity. In Fig. 12 we illustrate that the fully determiistic Stokeslet dynamics can also lead to rather widesmooth distributions. Even more, the histogram of the ming directions@Fig. 12~b!# can be well approximated withGaussian, which is usually attributed to ‘‘pure’’ randonoise. In our approach the stochastic behavior is duedeterministic internal dynamics which is strongly chaotUnfortunately there is no well established simple far-fieapproximation for the velocity around a self-driven sm

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‘‘microswimmer,’’ such as Eqs.~3! or ~7!, therefore wecould not repeat the detailed analysis performed inframework of the Stokeslet model.

In summary, we introduced the simplest Stokeslet mofor studying the hydrodynamical interaction between micscopic particles driven by external forces. We illustratedefficiency of the approximation by a detailed descriptionthe three-particle motion in a vertical plane, and pointedthat the sensitivity to initial configurations is associated wa chaotic saddle resulting in a chaotic scattering like dynaics. Our resolution made it possible to locate the saddlequantitatively characterize its dynamical and fractal propties. We argued that corrections do not alter the univephase-space structures drastically, thus even the Stokesleproximation should be able to capture the essential hydronamics in an interacting assembly.

This work was partially supported by NATO under GraNo. CRG960634 and the Hungarian National Science Fodation ~OTKA! under Grant Nos. T17493, T19483, anF014967. Helpful discussions with A. Bringer, Z. Kova´cs,and L. Wiesenfeld are acknowledged.

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st

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io

io

.

ndthe

ect

s.

J.


Lyapunov exponent is zero. Because of the conservationphase-space volume, the sum of Lyapunov exponents muzero and hence the last exponent must be2l.

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ofbe

n

n

(2x,2z) belonging to2r1, 2r2 is on the unstable manifold

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Chaotic particle dynamics in viscous flows: The three-particle Stokeslet problem

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Transcript of Chaotic particle dynamics in viscous flows: The three-particle Stokeslet problem