Single clusters in a vibro-fluidized granular gas in N connected compartments
become unstable at strong shaking. They are experimentally shown to collapse
very abruptly. The observed cluster lifetime (as a function of the driving
intensity) is analytically calculated within a flux model, making use of the
self-similarity of the process. After collapse, the cluster diffuses out into
the uniform distribution in a self-similar way, with an anomalous diffusion
exponent 1/3.Comment: 4 pages, 4 figures. Figure quality has been reduced in order to
  decrease file-siz

A. Kudrolli

D. van der Meer

Detlef Lohse

Devaraj van der Meer

G. Barenblatt

H. Jaeger

I. Goldhirsch

J. Eggers

K. van der Weele

Ko van der Weele

L. Kadanoff

L. Landau

S. McNamara

T. Witelski

Y. Du

English

arXiv

Single clusters in a vibro-fluidized granular gas in N connected compartments become unstable at strong shaking. They are experimentally shown to collapse very abruptly. The observed cluster lifetime (as a function of the driving intensity) is analytically calculated within a flux model, making use of the self-similarity of the process. After collapse, the cluster diffuses out into the uniform distribution in a self-similar way, with an anomalous diffusion exponent 1/3

van der Meer, Roger M.

van der Weele, J.P.

Lohse, Detlef

NARCIS 

Sudden collapse of a granular cluster

University of Twente Research Information

VOLUME 88, NUMBER 17 P H Y S I C A L R E V I E W L E T T E R S 29 APRIL 2002174302-1Sudden Collapse of a Granular ClusterDevaraj van der Meer, Ko van der Weele, and Detlef LohseDepartment of Applied Physics and J. M. Burgers Centre for Fluid Dynamics, University of Twente,P.O. Box 217, 7500 AE Enschede, The Netherlands(Received 6 July 2001; revised manuscript received 26 November 2001; published 15 April 2002)Single clusters in a vibro-fluidized granular gas in N connected compartments become unstable atstrong shaking. They are experimentally shown to collapse very abruptly. The observed cluster lifetime(as a function of the driving intensity) is analytically calculated within a flux model, making use of theself-similarity of the process. After collapse, the cluster diffuses out into the uniform distribution in aself-similar way, with an anomalous diffusion exponent 13.DOI: 10.1103/PhysRevLett.88.174302 PACS numbers: 45.70.–n, 02.60.Lj, 05.45.–aOne of the key features of a granular gas, making it fun-damentally different from ordinary molecular gases, is itstendency to spontaneously separate into dense and diluteregions [1]. This clustering originates from the dissipativenature of the particle collisions. It is an unwanted effect inmany applications where granular material is brought intomotion. Therefore we study (within a simple geometry)how declustering occurs. We find that the breakdown of acluster can be very abrupt, making declustering very dif-ferent from clustering in reverse time order.The experimental system (see Fig. 1) consists of a rowof N equal compartments, separated by walls of heighth  25.0 mm and filled with a few hundred steel beads ofdiameter 3.0 mm. We start out with all the particles in themiddle compartment and bring them into a gaseous stateby shaking the system vertically. For weak shaking thecluster is stable: after some initial spilling, a dynamicalequilibrium is established between the outflux of slow par-ticles from the cluster and the influx of fast particles fromoutside [2–4]. For sufficiently strong shaking, however,the cluster breaks down.Two different regimes are observed: (i) At very strongshaking the breakdown occurs immediately, and the clusterspreads out over the boxes with its profile widening as t13(instead of the standard t12 diffusion law). (ii) At lessviolent shaking, the cluster seems to remain stable for along time, showing only a small leakage to its neighbors.But suddenly it collapses and subsequently diffuses overall boxes. The sudden death of the cluster is depicted inFig. 1.One thing this figure shows is that the breakdown of acluster is strikingly different from the reverse process ofcluster formation, which is known to take place graduallyand (for all N $ 3) via transient states showing clusters inseveral boxes [3,4]. This lack of time reversibility is yetanother consequence of the dissipation in the system.The abruptness of the collapse allows us to define acluster lifetime t [via n̈cl t  0; see Fig. 1e]. In Fig. 2the measured lifetimes are plotted as a function of theinverse shaking strength B̃ [Eq. (2)] for various values ofN . The data lie on a universal envelope curve, until at somecritical value B̃c,N (which grows with N) they diverge.0031-90070288(17)174302(4)$20.00All of the above experimental observations can be ac-counted for quantitatively by means of the flux model ofRefs. [2–4]. At the heart of this model is a flux functionFnk, describing the outflow from the kth box to each ofits neighbors. It is a nonmonotonic function of nk (theparticle fraction in the box): Fnk first increases with nk ,but beyond a certain value of nk it decreases again, as theincreasingly frequent inelastic collisions slow the particlesdown, so that they cannot make it over the wall to theneighboring compartments anymore. The precise form ofFnk is not very critical, as long as it is a one-humpedfunction. We will use Eggers’ flux function [2]Fnk  CpB̃ n2ke2B̃n2k , (1)withB̃ ~ghr41 2 e2P2S2af2. (2)The driving parameter B̃ depends on the total numberof particles P and their properties (radius r, restitutioncoefficient e of the particle collisions), on the geometryof the system (height h of the walls, ground area S ofeach box), and on the frequency f and amplitude a of theFIG. 1. (a)–(d) Four images from a five-box experiment, atdriving parameter B̃  8.2. The cluster is clearly present untilt  42 s, then suddenly collapses, leaving no trace one secondlater. (e) The time evolution of the cluster fraction nclt, evalu-ated from the flux model.© 2002 The American Physical Society 174302-1VOLUME 88, NUMBER 17 P H Y S I C A L R E V I E W L E T T E R S 29 APRIL 2002FIG. 2. (a) Lifetime t vs driving parameter B̃, determinedfrom experiments with N  3, 5 boxes (dots with error bars),and from numerical evaluation of the flux model with N 3, 5, 10, 20, 40, 80 boxes (empty symbols). The solid curves areanalytical solutions for N  3 [9] and for the envelope curve,which goes roughly as exp 32B̃ [cf. Eq. (13)]. (b) Enlarge-ment of (a), showing the experimental results for N  3 and 5in more detail. Every point is based on 15 repetitions of theexperiment; the vertical error bars denote the maximal deviationfrom the average t measured, the horizontal ones represent theaccuracy in B̃. (c) Bifurcation diagram for N  5, showing thecritical value B̃c,5. Declustering occurs to the left of this value.shaking. The factor C determines only the absolute rate ofthe flux, and can be incorporated in the time scale.The equation of motion for the fraction in box k is [5]dnkdt Fnk21 2 2Fnk 1 Fnk11 , (3)where k  1, 2, . . . , N . Here we assume a nearest neigh-bor interaction, and a cyclic arrangement of the boxes (k N 1 1 equals k  1). We further impose particle conser-vation,Pk nk  1.The numerical results shown in Figs. 1e and 2 have beenobtained using the above flux model, starting out with allparticles in one box (labeled cl). They quantitatively agreewith the experimental observations. The decaying clustergoes through three different stages.The starting stage is a very short one, in which both ncland Fncl display a jump compared to ni and Fni in thesurrounding boxes, i  1, 2, . . . (we have to consider oneside only because of the symmetry in the system).In the second stage, the flux has become continuous butthe particle fraction remains discontinuous. However, itslow-density counterpart n0 [defined by Fn0  Fncl]does continuously connect to n1. We will use this fact174302-2later in the analysis of the envelope curve. The flux gradu-ally grows, and eventually Fncl reaches its maximumvalue. This is accompanied by rapid density changes andthe sudden death of the cluster at the lifetime t.In the third and last stage, both ncl and Fncl fit con-tinuously to the other boxes (see Fig. 3a). The remains ofthe cluster diffuse over the whole system until the uniformdistribution is reached.In what follows we will analytically solve the fluxmodel. First we focus on the third stage. We rewrite theproblem into its continuum version, by setting nx, t nkt (x  kw by definition, where the box width w willbe incorporated in the x scale). Equation (3) then becomes≠tn  ≠xxFnx, t CpB̃ ≠xxnx, t2e2B̃nx,t2 , (4)and the conservation condition takes the formR`2` nx, t dx  1.For very strong shaking [regime (i), where t is vanish-ingly small] the diffusive stage sets in almost immediately.Here B̃ ! 0, and Eq. (4) reduces to≠tn  CpB̃ ≠xxn2  2CpB̃ ≠xn2 1 n≠xxn , (5)which is known as the porous media equation [6,7]. Thedecay of the cluster in this limit is depicted in Fig. 3a. Itis self-similar: all curves in Fig. 3a fall onto a single curveif we properly rescale the axes (Fig. 3b). The originalpartial differential equation (PDE) can thus be broughtback to an ordinary differential equation (ODE) in termsof the self-similarity variable h  xCB̃12t13. Withnx, t  HhCB̃12t13, Eq. (5) now takes the form≠hhH2 113 ≠hhH  0 . (6)Its symmetric solution is Hh  H0 2 112h2 [withthe constant H0  3134  0.361 determined byR`2` Hh dh  1]. This inverted parabola, depicted inFig. 3b, represents in one curve all the stages of Fig. 3a.The scaling of the axes shows that the height of the clusterdecreases as t213, and its width grows as t13. Thisanomalous diffusion (with exponent 13) is also found inporous media [6,7]. The slowed down diffusion of theFIG. 3. (a) The diffusing profile at successive times t (in thelimit B̃  0). (b) The function Hh onto which all the curvesin plot (a) collapse by a proper rescaling of the axes (H ~ nt13and h ~ xt213).174302-2VOLUME 88, NUMBER 17 P H Y S I C A L R E V I E W L E T T E R S 29 APRIL 2002front originates from the quadratic n dependence in ≠xxn2[Eq. (5)]. In the context of our granular model this reflectsthat particles diffuse to neighboring boxes only throughtwo-particle collisions: In the strong shaking limit theabsence of particles slows down further diffusion, and thepresence enhances it.For nonzero B̃ [regime (ii), where the diffusive stagehas to wait until after the sudden death] we have an addi-tional dimensionless variable x  B̃Ct 2 t. Its influ-ence diminishes with time and the solutions converge tothe inverted parabola of the case B̃ ! 0.Next we turn to the second, semicontinuous stage. Formoderate shaking, B̃  B̃c,N , this stage can take quite along time. At the critical point B̃c,N the lifetime t even di-verges to infinity and the cluster becomes stable. Startingout from the initial . . . , 0, 0, 1, 0, 0, . . .	 distribution, thesystem first very slowly approaches a distribution in whichncl is close to the cluster density at the saddle-node bifur-cation (see Fig. 2c) and all other boxes contain equal frac-tions nk  1 2 nclN 2 1. Only when it has passedthis phantom equilibrium (i.e., when ncl is below the levelof the saddle-node bifurcation), the system quickens itspace and the sudden collapse occurs.This means that t is the time it takes to pass the phantomequilibrium. It can be calculated either numerically or ana-lytically [by integrating the Taylor expansion of Eq. (3)],with the result t ~ B̃c,N 2 B̃212. So t diverges as theinverse square root of the distance to the critical point,which is the common (mean field) power-law behavior neara second order phase transition as we have here for t [8].Finally, we calculate the lifetime away from the criticalpoint, i.e., for B̃ considerably smaller than B̃c,N . This willgive us an analytical expression for the envelope curve inFig. 2a. For these B̃ values, the collapse occurs before theparticles leaking out of the cluster have had time to fillthe outermost boxes to any significant level. Therefore, thebehavior does not depend on the value of N : The systemdoes not feel its finite size during the cluster’s lifetime, sothe number of boxes can be taken to be infinite.The time evolution of the cluster is described by Eq. (3),with Fn21  Fn1:dncldt 22Fncl 1 2Fn1  22Fn0 1 2Fn1 .(7)This equation contains n1, which is governed by a simi-lar equation of motion [Eq. (3)] containing n2, etc. So wehave to deal with an infinite number of coupled nonlinearODE’s [9]. This is a problem that cannot be solved di-rectly, so we attack it in five steps.Step 1.—We first rewrite the problem into its continuumversion, and replace the cluster density ncl by its low-density counterpart n0, t  n0t  ncl exp2B̃n2cl2.Thus, without influencing the fluxes [since Fn0 Fncl] we make nx, t continuous in x  0. The densitynx, t obeys Eq. (4), plus a conservation condition saying174302-3that the increase of material into the rest of the systemequals the influx from x  0:≠tZ `0nx, t dx  2≠xFnx, tx0 . (8)Step 2.—The cluster-equation Eq. (7) now becomesdncldt 2≠xFnx, tx0 2F0n0, t ≠xnx, tx0 .(9)Since F 0 can be derived directly from Eq. (1), the problemreduces to determining ≠xnx, t at x  0.Step 3.— In order to do so, we observe that changes inn0, t happen on a much longer time scale than in thesurrounding boxes [10], so the cluster acts as a constantreservoir spilling granular material. This approximationis illustrated in Fig. 4a: the profile in the system buildsup while n0 remains constant. In fact, this buildup takesplace in a self-similar way (see Fig. 4b). So, as before,the problem for nx, t can be formulated in terms of onevariable j  xn0CB̃12t12. Setting nx, t  n0Gj,Eq. (4) becomes an ODE for Gj, and also the accompa-nying conservation condition [Eq. (8)] contains j only:12j≠jG  2≠jjG2e2bG2 ,≠jGj0  2eb41 2 bZ `0Gj dj ,(10)where b  B̃n20.Step 4.—The slope ≠xnx, tx0 can now be approxi-mated by n0t ≠jGjj0≠xjx0, where we have re-vived the (slow) time dependence in n0t. With ≠xj n0tCB̃12t212, Eq. (9) becomesdncldt 2pn0tpCB̃12tF 0n0t ≠jGjj0 . (11)All quantities on the right hand side are tractable. Wereexpress n0t in terms of nclt, derive F 0 from Eq. (1),and solve Eq. (10) to determine ≠jGjj0. This laststep still requires some work, because Eq. (10) does notallow an analytical solution and moreover contains b [andFIG. 4. (a) The density profile in the boxes surrounding thecluster, at 20 consecutive (equidistant) moments in time, withinthe constant n0 approximation; n0  0.223 and B̃  3.00.(b) The function G  nn0 onto which all curves in plot (a)collapse by a proper rescaling of the axes (j ~ xt212).174302-3VOLUME 88, NUMBER 17 P H Y S I C A L R E V I E W L E T T E R S 29 APRIL 2002hence n0t] explicitly. Since b is small, however, we mayexpand Gj and Eq. (10) in terms of b, and solve numeri-cally. In leading order we find ≠jGjj0  2K 20.3138. Inserting all this, Eq. (11) becomesdncldt 24Kn32cl e234B̃n2clpCB̃12t1 2 B̃n2cle2B̃n2cl  . (12)This is an ODE for ncl in closed form, which replaces theoriginal problem [Eq. (7)] consisting of an infinite numberof coupled ODE’s.Step 5.—Finally, we integrate Eq. (12) over the clusterdensity (dropping the suffix cl) and find an analytical ex-pression for the lifetime t away from the critical points:t "Z 1nthrpCB̃12 n232e34B̃n2dn8K1 2 B̃n2e2B̃n2#2. (13)Here nthr is the value of n at which the sudden deathoccurs. For the evaluation of the solid curve in Fig. 2awe used nthr  0.5, but this value is not too critical(cf. Fig. 1e). The only free parameter is the constant C:if this is adjusted properly, the analytical t curve agreeswith the measured data over the whole range of B̃ values.The above expression shows that t roughly increases asexpB̃32. Recalling that B̃ is the inverse shaking strength,this underlines the experimental observation that even asmall reduction in the shaking strength causes a tremen-dous increase of the cluster lifetime.In conclusion, in the studied compartmentalized systemclusters break down very abruptly, in contrast to their slowformation. As clustering itself, the lack of time reversibil-ity originates from the dissipative nature of the particle col-lisions: The breakdown of the unstable cluster is delayedbecause most of the energy input is dissipated through col-lisions in the cluster. The dynamics is quantitatively de-scribed by a remarkably simple flux model, which can beanalytically solved.174302-4We thank Ernst van Nierop for measuring the experi-mental lifetimes. This work is part of the research pro-gram of the Stichting FOM, which is financially supportedby NWO.[1] I. Goldhirsch and G. Zanetti, Phys. Rev. Lett. 70, 1619(1993); S. McNamara and W. Young, Phys. Rev. E 50, R28(1994); Y. Du, H. Li, and L. Kadanoff, Phys. Rev. Lett. 74,1268 (1995); H. Jaeger, S. Nagel, and R. Behringer, Rev.Mod. Phys. 68, 1259 (1996); A. Kudrolli, M. Wolpert, andJ. Gollub, Phys. Rev. Lett. 78, 1383 (1997); L. Kadanoff,Rev. Mod. Phys. 71, 435 (1999).[2] J. Eggers, Phys. Rev. Lett. 83, 5322 (1999).[3] K. van der Weele, D. van der Meer, M. Versluis, andD. Lohse, Europhys. Lett. 53, 328 (2001).[4] D. van der Meer, K. van der Weele, and D. Lohse, Phys.Rev. E 63, 061304 (2001).[5] Statistical fluctuations, which would add a noise term toEq. (3), are left out of this (mean field) description.[6] G. Barenblatt, Scaling, Self-Similarity, and IntermediateAsymptotics (Cambridge University Press, New York,1996).[7] G. Barenblatt, M. Bertsch, A. Chertock, and V. Pros-tokishin, Proc. Natl. Acad. Sci. U.S.A. 97, 9844 (2000);T. Witelski and A. Bernoff, Stud. Appl. Math. 100, 153(1998).[8] L. Landau and E. Lifshitz, Statistical Physics, Part 1(Pergamon, Oxford, 1980), 3rd ed.[9] The case N  3 is contained in a single ODE, which canbe solved exactly: t R1nthrdn2F1 2 n2 2 Fn	(solid curve in Fig. 2), with nthr determined from n̈t 0. An analogous expression holds for general N close tothe critical point, where all the surrounding boxes are filledto the same level nk  1 2 nclN 2 1.[10] A cluster change Dncl (flowing into the surrounding boxes)corresponds to a much smaller change Dn0 in its low-density counterpart: dncldn0  1 2 B̃n2cl21eB̃n2cl2 1 2 B̃21eB̃2 since ncl  1. So for most B̃ values ofinterest (excluding the strong-shaking regime B̃ ! 0)jDnclj Pifi0 Dni ¿ Dn0.174302-4

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Sudden Collapse of a Granular Cluster

Sudden Collapse of a Granular Cluster

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