A transition from local to global shear zones is reported for granular flows
in a modified Couette cell. The experimental geometry is a slowly rotating drum
which has a stationary disc of radius R_s fixed at its bottom. Granular
material, which fills this cell up to height H, forms a wide shear zone which
emanates from the discontinuity at the stationary discs edge. For shallow
layers (H/R_s < 0.55), the shear zone reaches the free surface, with the core
of the material resting on the disc and remaining stationary. In contrast, for
deep layers (H/R_s > 0.55), the shear zones meet below the surface and the core
starts to precess. A change in the symmetry of the surface velocities reveals
that this behavior is associated with a transition from a local to a global
shear mode.Comment: 4 pages, 7 figures, submitte

Denis Fenistein

Jan-Willem van de Meent

Martin van Hecke

R. Nedderman

English

arXiv

We report a novel transition to core precession for granular flows in a split-bottomed shear cell. This transition is related to a qualitative change in the 3D flow structure: For shallow layers of granular material, the shear zones emanating from the split reach the free surface, while for deep layers the shear zones meet below the surface, causing precession. The surface velocities reflect this transition by a change of symmetry. As a function of layer depth, we find that three qualitatively different smooth and robust granular flows can be created in this simple shearing geometry.Biological and Soft Matter Physic

Fenistein, D.

Meent, J.W. van de

Hecke, M.L. van

Leiden University Scholary Publications

PRL 96, 118001 (2006) P H Y S I C A L R E V I E W L E T T E R Sweek ending24 MARCH 2006Core Precession and Global Modes in Granular Bulk FlowDenis Fenistein, Jan-Willem van de Meent, and Martin van HeckeKamerlingh Onnes Lab, Leiden University, Post Office box 9504, 2300 RA Leiden, Netherlands(Received 23 July 2005; published 21 March 2006)0031-9007=We report a novel transition to core precession for granular flows in a split-bottomed shear cell. Thistransition is related to a qualitative change in the 3D flow structure: For shallow layers of granularmaterial, the shear zones emanating from the split reach the free surface, while for deep layers the shearzones meet below the surface, causing precession. The surface velocities reflect this transition by a changeof symmetry. As a function of layer depth, we find that three qualitatively different smooth and robustgranular flows can be created in this simple shearing geometry.DOI: 10.1103/PhysRevLett.96.118001 PACS numbers: 45.70.n, 83.50.Ax, 83.85.CgFIG. 1. Core precession in a split-bottomed geometry.(a) Schematic side-view of our split-bottomed shear cell, show-ing the layer height H (indicated by dotted line), a stationarybottom disk of radius Rs ( gray), the bottom ring (striped),and outer cylinder of radius 105 mm that rotate with rate .(b), (c), (d), (e), (f) Series of snapshots of top views of oursetup (for Rs  95 mm, H  60 mm, and rotation rate  0:15 rad=s), where colored particles sprinkled on the surfaceillustrate the core precession for t  0 s (b), t  10 s (c),t  102 s (d), t  103 s (e), and t  104 s (f).Slowly sheared granular matter usually organizes intorigid regions separated by narrow shear bands where thematerial yields and flows [1–5]. Such grain flows appear toprohibit successful continuum modeling—not only be-cause of the steep gradients in velocity, but also becausesubtle microscopic characteristics of the granulate can alterthe flow in a qualitative manner [3].We recently showed that the formation of narrow shearbands can be avoided by driving the granulate from adiscontinuity in the bottom support of the grain layer[Fig. 1(a)], which effectively pins a wide shear zoneaway from the sidewalls [6–8]. The resulting grain flowsare smooth and robust, with both velocity profiles and thelocation of the shear zones exhibiting simple, grain inde-pendent properties. These flows should be amenable to acontinuum description, which forms an important motiva-tion for their detailed study [6–8]. The two crucial pa-rameters of this system are the location Rs of the slip in thebottom support, and the thickness of the granular layer H[Fig. 1(a)]. When H=Rs is small, the core material rests onthe stationary center disc, and these flows have been char-acterized recently [6,7]. With increasing H, the width ofthese shear zones grows continuously, and their locationmoves inward toward the central region [6]. This impliesthat for deep layers new phenomena can be expected tooccur.Figures 1(b)–1(f) illustrate the novel flow patterns thatare characteristic for deep layers. The most striking featureis that the core now precesses with a constant rate—hencematerial in the central part of the surface no longer rests onthe disc. Precession is not simply the consequence of theoverlap of two opposing shear zones, since before beingeroded by shear, the inner core rotates as a solid blob for anappreciable time [Figs. 1(b)–1(f)]. A more intricatemechanism is at play here.In this Letter we will characterize this transition toprecession. We first address how the precession rate growswith layer depth, and relate precession to a qualitativechange in the 3D structure of the flow field. We thenshow how this change is reflected in a change of the06=96(11)=118001(4)$23.00 11800symmetry of the surface velocity profiles. The location ofthe shear zone at the surface for deep layers is compared toan empirical scaling law [6] and a recent theory [7].Finally, we employ the rich structure of the shear zonesfor deep layers to establish that a crucial characteristicmesoscopic length scale, which sets the width of the shearzones, grows as a nontrivial power law with layer depth.Setup.—Our setup is a modified version of the diskgeometry described in earlier work [6], and consists of astationary bottom disk of radius Rs, a rotating bottom ring,and an outer cylinder of radius 105 mm [Fig. 1(a)]. Thedisc radius Rs can be varied from 45 mm to 95 mm. Thecell is filled to a height H with a polydisperse mixture ofspherical glass beads with diameters ranging from 0.6 to0.8 mm; a layer of grains is glued to the side walls andbottom rings to obtain rough boundaries. No segregationcould be observed, and the free surface remains essentiallyflat. The surface velocities are recovered by a variant of1-1 © 2006 The American Physical SocietyPRL 96, 118001 (2006) P H Y S I C A L R E V I E W L E T T E R Sweek ending24 MARCH 2006particle image velocimetry [6]. Transients are short andaverage velocities are azimuthal and proportional to thedriving rate . We represent the velocity profiles as !r,the ratio of the average angular surface velocity, and ,where r is the radial coordinate and  is fixed at0:15 rad=s. Because ! is a ratio of a velocity and r, datataken for r < 20 mm is unreliable and omitted from theanalysis.Precession.—We define the precession rate !p as thelimit of !r for r going to zero. As explained above, !cannot be measured reliably for small r. Nevertheless,!rlevels off for small radius, and in practice we fit a smoothfunction to !r to estimate !p—various functions gavevery comparable results (see below). For various slip radii,the onset height for precession grows with Rs [Fig. 2(a)],and the data for !p collapses when plotted as a function ofH=Rs [Fig. 2(b)]. Note that when H=Rs becomes of orderone, the whole surface rotates rigidly and!p!1. Our datais consistent with recent results in a similar setup where thedisc was rotated and the outer cylinder kept fixed [8].3D flow structure.—The flow in the bulk has beenprobed by putting patterns of lines of colored tracer parti-cles at given height in the bulk, adding more material,rotating the system for a short period (8 s), and recover-ing the deformed pattern by carefully removing the upperlayers of grains. In Fig. 3, two examples of the 3D flowpatterns illustrate that for shallow layers the shear zonesthat emanate from the split in the bottom reach the freesurface, thus leading to a stationary core, while for deeplayers the shear zones (partially) meet in the bulk, leadingto precession of the core.In a recent theory by Unger et al., the shear zones aremodeled as infinitely thin sliding sheets. This model pre-dicts that for shallow layers the shear sheet reaches the freesurface but that for deep layers the shear sheet closes in thebulk of the material, and that a hysteretic transition be-tween these two states occurs for H=Rs  0:65 [7].Figure 3 illustrates that this change in the shear zoneswith increasing height is also observed experimentally.However, the model cannot capture the smooth velocityfields observed experimentally and predicts a hystereticjump between !p  0 and !p  1, while our data showsa smooth, nonhysteretic crossover.10 H (mm) 900ωp 1(a)0.5 H/Rs 0.90ωp 1(b)FIG. 2. (a) Precession rate !p as a function of H forRs  45 mm (diamonds), Rs  65 mm (	) andRs  95 mm (circles). (b) Data collapse when !p is plotted asa function of H=Rs.11800Recent MRI studies and numerical simulations evidencethat in the crossover regime where 0 !p  1, the 3Dflow structure can be conceived as having two shearzones—one reaching the free surface and one closing inthe bulk [8]. The relative amount of shear across these twozones is set by the layer depth—for shallow layers thevertical shear zones dominate and !p is small, for deeperlayers the horizontal shear zones increase in strength and!p increases accordingly. Extending the sliding sheetmodel [7] by taking disorder into account, Kertész et al.have reached a similar conclusion [9].Surface velocities.—In the remainder of this Letter, wewill focus on the surface velocities in the transitory regime,which can be measured much more accurately than bulkflow. Vertical gradients of the velocity are small near thesurface, so that what is observed at the surface is notsubstantially different from what happens in the top layers[6–8]. For shallow layers, we established in Ref. [6] that!r is well fitted by!r  nerfr RcW; (1)where Rc andW parameterize the location and width of theshear zones, and nerfx denotes the normalized errorfunction 1=2 1=2 errorfx. Figure 4(a) illustrates thatthis relation breaks down for deeper layers—mainly be-cause of precession. Moreover, before precession sets in, acareful analysis reveals that the left-right symmetry of!ris broken: the right (large r) tail of !r is significantlysteeper than its left (small r) tail for H * 0:45.FIG. 3. 3D flow profiles around transition point. Left: Fiveslices for H=Rs ’ 0:6 and heights Hb in the material as indi-cated—here !p ’ 0. The kink in the patterns corresponds thelocation of the shear zone. Right: Five slices at identical heightsHb for larger H=Rs ’ 0:75, where !p ’ 0:8. The upper layersshow no deformation of the pattern indicating the absence ofshear. Below: Sketches of the shear zones within the bulk(black)—the dashed curves indicate the heights Hb where thepatterns were created.1-2PRL 96, 118001 (2006) P H Y S I C A L R E V I E W L E T T E R Sweek ending24 MARCH 2006These changes in the surface profiles reflect thetopological change in the 3D flow structure. For shallowlayers, two flow profiles obtained for the same height H,but for two different slip radii (Rs1 and Rs2), are simplyrelated by a translation of the radial coordinate via!r Rs1jRs1;H  !r Rs2jRs2;H—the shear zone is alocal phenomenon, insensitive to the location of the centerof the shear cell [6]. For deep layers, when r becomes anonlinear function, this symmetry is absent, since the flowis sensitive to where the center is—the shear mode isglobal. The precession and symmetry breaking of !rboth reflect the crossover from a local to a global shearmode—consistent with the change in the 3D structure ofthe shear zones.To capture the symmetry breaking and precession, !rcould be fitted to a four parameter fit: !r  p 1pnerf s2, where   r Rc=W. Such fits con-firm the independence of !p to details and the onset ofsymmetry breaking before precession. More insight can begained when postulating that!r  nerfr; (2)and calculating r : nerf1!r from our experimen-tally obtained velocity profiles. The resulting r, shownin Fig. 4(b), clearly bring out the deviations from thesimple form, Eq. (1). For shallow layers r is a linearfunction since !r obeys Eq. (1): r  a0  a1r r Rc=W. For deep layers, r becomes a nonlinearfunction that saturates at a precession-dependent value forr # 0. To fit r for deep layers, the most obvious choice isto add a symmetry breaking term r2 —the resulting fit isFIG. 4. Surface velocity profiles !r for Rs  95 mmand increasing layer depth H. Thick curves: H  10;20; . . . ; 80 mm; Thin curves H  15; 25; . . . ; 75 mm; Dashedcurves H  56; 57; . . . ; 69 mm. (a) Precession gradually sets infor H * 60 mm. (b) Corresponding profiles of r (dots),compared to cubic fits given by Eq. (3) (curves). Similar topanel (a), H  10; 15; 20; . . . ; 55; 56; . . . ; 70 mm.11800rather poor. However, r can be fitted well by a cubicpolynomial [see Fig. 4(b)]:r  a0  a1r a3r3: (3)We do not see strong a priori reasons why this fit wouldrepresent the true analytical form of the flow profiles.However, the surprise is that a1 vanishes for large H (sincethe slope near the intercept becomes zero), while a3 is zerofor small H. Hence, by the addition of one fit parameter tothe ‘‘pure’’ error-function form [Eq. (1)], !r for both theshallow and deep regimes can be expressed in a simpleform. Moreover, both precession and symmetry breakingare captured by this single extra parameter.Location of the shear zone.—For shallow layers, where!r is given by Eq. (1), the center location Rc is indepen-dent of the particle properties and was found to follow thesimple scaling law [6]:1 Rc=Rs  H=Rs5=2: (4)The recent theory of Unger et al. [7] predicts the location ofthe shear zones at the surface, Rmodel, for all layer depthsand deviates from the scaling relation Eq. (4). In Fig. 5 ourdata for the center of the shear zones is compared to boththe power law and Unger’s numerical result. For generalshear zones, there is no unique choice for the center and wehave tested the following three definitions:R1 : where !r  0:5; (5)R2 : where @r!r is maximal; (6)R3 : where @rr!r strain rate is maximal: (7)For shallow layers, both R1 and R2 are better described byEq. (4) than by Rmodel, while R3 appears to be betterdescribed by Rmodel. For deep layers, the situation ismore complicated, with neither model describing any ofthe three measured curves in detail.Mesoscopic Scale.—While the onset of precession andlocation of the shear zones are set by the system scales HandRs, the width of the shear zones are set by a mesoscopic0 20 H (mm) 60 8050 100Ri (mm)FIG. 5. Shear zone positions versus layer depth H for Rs 95 mm, where circles, pluses, and diamonds correspond toR1, R2, and R3 [Eqs. (5)–(7)]; the solid curve is the scalingform given by Eq. (4) and the dashed curve is the result by Ungeret al. [7].1-30 H (mm) 900 0.5a 1 (mm-1)a 3 (mm-3)010-5(c)0.2 H/Rs (mm) 0.804(d)a1 a30.000.05a 1 RS2/3 d1/3a 3 RS2 d0 H (mm) 80040 (a)W’ (mm)0.1 1.0 0.22.0 (b)H/RSW’/(RS2/3d1/3)FIG. 6. (a) Width of shear zones W0 as a function of H forRs  45, 65, and 95 mm. The saturation of W0 for large H isrelated to the transition to precession—the eventual vanishing ofW0 is a fitting artefact caused by the flattening of the velocityprofile. (b) The nondimensionalized width W0R2=3s d1=3 follows auniversal curve as a function of H=Rs, which for small heights isapproximated well by a power law with exponent 2=3 (straightline). (c) Fit parameters a1 for Rs  65 mm (diamonds) and95 mm (+) and a3 for Rs  65 mm (triangles) and 95 mm (stars)as a function of H. (d) Nondimensionalized fit parameters as afunction of H=Rs (see text).PRL 96, 118001 (2006) P H Y S I C A L R E V I E W L E T T E R Sweek ending24 MARCH 2006length scale governed by both H and grain size d. Earlierwork for shallow layers was inconclusive about the func-tional dependence of the width on layer height [6]. To beable to discuss deep layers also, we define a widthW0 as theinterval where ! grows from !p  0:1	 1!p to0:9	 1!p—for shallow layers, W0  1:812W [seeEq. (1)]. We will now show strong evidence for a meso-scopic scale (Fig. 6), which grows as H2=3 and whichdetermines the width of the shear zones.The first evidence comes directly from W0. For shallowlayers,W0 is independent of Rs, but for deep layers both themicroscale (d) and macroscale (Rs) play a role [Fig. 6(a)].When we postulate that the ratio of the relevant lengthscale to the grain size d (which we take as 0.7 mm) growsas H=d, this suggests to plot the dimensionless widthW0=Rs d1 as a function of the dimensionless heightH=Rs and adjust  to get optimal collapse. As shown inFig. 6(b), this leads to a good data collapse for   2=3,and moreover, for shallow layers, the rescaled width itselfgrows as H=Rs2=3. Secondly, the fit parameters a1 and a3,plotted in Fig. 6(c) for two values of Rs, have dimensionsof inverse length and inverse length cubed. It follows thata1R2=3s d1=3 and a3R2sd should collapse when plotted as afunction of H=Rs, as is indeed the case [Fig. 6(d)].Outlook and open questions.—Detailed studies of thegranular flows in split bottom geometries [6–8] show awide range of smooth and robust flows. Three regimes canbe distinguished by fitting the surface flows to Eq. (1)–(3).11800Shallow layers occur for H=Rs & 0:45 where !p is zero,a3 is zero and Eq. (1) fits the data well—here the shearzones completely reach the free surface [6]. For 0:45 &H=Rs & 0:65, a crossover regime is reached where the left-right symmetry of !r is broken, both a1 and a3 arenonzero but !p is essentially zero. Deep layers occur forH=Rs * 0:65, where the reflection symmetry of the shearzones is strongly broken, precession sets in, a1 tends tozero and where there is increasing shear in the verticaldirection [8]. This robust behavior of grain flows in splitbottom geometries will provide important testing groundfor the development of (continuum) theories [7,9,10].There are three questions that we think deserve particu-lar attention. (i) What mechanism selects the functionalform of !r? In particular, the tails of these profilesdeserve special attention, since they can be exponential,Gaussian, or more complex as shown in Fig. 4 [3,6,11].(ii) What sets the nontrivial exponent 2=3 for the scaling ofthe width of the shear zones? (iii) Should the transition toprecession be conceived as a smooth crossover or as asharply defined transition? The smooth growth of !pwithH=Rs suggests a crossover (Fig. 2), while the (critical)vanishing of a1 with (H=Rs) [Figs. 6(c) and 6(d)] suggestsa sharp transition.We gratefully acknowledge discussions with X. Cheng,H. Jaeger, S. Nagel, T. Unger, J. Kertesz, M. Depken, andW. van Saarloos, and financial support from ‘‘NederlandseOrganisatie voor Wetenschappelijk Onderzoek (NWO)’’and ‘‘Stichting Fundamenteel Onderzoek der Materie(FOM).’’1-4[1] GDR MiDi, Eur. Phys. J. E 14, 341 (2004).[2] R. Nedderman, Statics and Kinematics of GranularMaterials (Cambridge University Press, Cambridge,England, 1992).[3] D. M. Mueth et al., Nature (London) 406, 385 (2000);D. M. Mueth, Phys. Rev. E 67, 011304 (2003).[4] D. W. Howell, R. P. Behringer, and C. T. Veje, Phys. Rev.Lett. 82, 5241 (1999).[5] W. Losert, L. Bocquet, T. C. Lubensky, and J. P. Gollub,Phys. Rev. Lett. 85, 1428 (2000); W. Losert and G. Kwon,Advances in Complex Systems 4, 369 (2001); M. Toiya,J. Stambaugh, and W. Losert, Phys. Rev. Lett. 93, 088001(2004).[6] D. Fenistein and M. van Hecke, Nature (London) 425, 256(2003); D. Fenistein, J.-W. van de Meent, andM. van Hecke, Phys. Rev. Lett. 92, 094301 (2004).[7] T. Unger, J. Török, J. Kertész, and D. E. Wolf, Phys. Rev.Lett. 92, 214301 (2004).[8] X. Cheng et al., Phys. Rev. Lett. 96, 038001 (2006).[9] J. Kertész (private communications).[10] M. Depken, W. van Saarloos, and M. van Hecke, Phys.Rev. E 73, 031302 (2006).[11] T. S. Komatsu, S. Inagaki, N. Nakagawa, and S. Nasuno,Phys. Rev. Lett. 86, 1757 (2001).

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