Scaling laws in granular continuous avalanches in a rotating drum

Transcription

1 Physica A 356 (25) Scaling laws in granular continuous avalanches in a rotating drum N. Seu lveda, G. Krstulovic, S. Rica Deartamento de Física, Facultad de Ciencias Físicas y Matemáticas de la Universidad de Chile, Blanco Encalada 28, Santiago, Chile Available online 9 June 25 Abstract We study the front shae of avalanches roduced in a rotating two-dimensional drum artially filled with small glass beads. We focus our work on the study of the length and shae of granular fronts, in articular how they do scale in terms of the hysical arameters involved. A single scaling law for the length is found. This scaling law is also relevant for the behavior of the full shae of the fronts. More than 3 different fronts shae, for different values of the arameters collase into an universal curve. r 25 Elsevier B.V. All rights reserved. Keywords: Granular matter; Granular flows; Avalanches Since ioneering works by Bagnold [1] granular flows under gravity have been considered in many different contexts and geometries. The relevance of this henomenon is great because of the industrial rocess of various materials, and natural catastrohes such as landslides, clay and snow avalanches [2]. In site of a long history of exeriments on granular flows, the henomenon encounters several fundamental difficulties in its theory and alication. More than a decade ago, Jaeger et al. [3] and Rajchenbach [4] have recognized that a continuous avalanche regime could be realized in a rotating drum geometry, as one increases the rotation seed of the drum. Indeed for low angular frequencies an Corresonding author. address: rica@dfi.uchile.cl (S. Rica) /$ - see front matter r 25 Elsevier B.V. All rights reserved. doi:1.116/j.hysa

2 N. Seúlveda et al. / Physica A 356 (25) intermittent regime is observed [3], meanwhile as one increases the rotating seed, a continuous avalanche regime is established [4]. As soon as the drumstarts to rotate, the bulk of beads climb-u with the drum until the uer beads loose their stability falling down, roducing an avalanche. Ultimately, those rolling beads arrive at the bottom of the cell loosing its motion, then they are carried u again, creating a continuous cycle. Inside the cell grains exeriment two different regimes: while grains are ascending the inter-article distance remains fixed, like in a solid, meanwhile falling grains, that is the avalanche, behave like a fluid. During the whole rocess there is a continuous exchange of articles between the solid and fluid art. A kinetic model describing these rocesses was roosed in Ref. [5]. However, the model redicts a arabolic shae of the rolling grains at the interface [6] instead of the characteristic S shae rofile observed in exeriments [4]. Later, Zik et al. [7] studied segregation in a rotating drum describing the observed S shae of the interface for a half-filled rotating drum. More recently, Ottino and collaborators [8] reorted measurements of the shear seed in the rolling layer and Rajchenbach [9] modifies Bagnold stress to exlain constant shear rate for dense and raid flows. In this article we reort a rotating drumexeriment artially filled with glass beads. We study the stationary avalanches finding scaling laws for the length and the full rofile of the fronts when the angular velocity, the drumradius, the volume filled by the beads and the size of the articles vary. Exeriment. The exerimental setu consists of a two-dimensional Hele Shaw rotating drumartially filled with small glass beads (Fig. 1a). The two faces of the drumwere built in Plexiglas, each one with a furrow inside where a rubber o-ring is laced making uniform cell sacing about b ¼ 1:8 mm. We have worked with three different cell radii (a): 4.5, 12.5 and 16.5 cm. The cell can rotate around an axis erendicular to gravity driven by a motor with constant ower and a system of a set of ulleys allowing us to exlore a large range of angular velocity between.3 and 6 rad/s. We will focus our study on the length and shae of the front, in articular how do fronts scale in terms of the hysical arameters involved? Those are: the angular velocity o, the radius a of the drum, the size D of the articles and the initial filling Fig. 1. (a) Exerimental setu. (b) Scheme of the exeriment, arameters and variables. (c) Grains flying arabolically like in a ball mill.

3 18 N. Seúlveda et al. / Physica A 356 (25) fraction. Although the distance b between both faces of the drumremains constant in the full set of exeriments, we could anticiate a deendence on this arameter because of dimensional analysis. This deendence is strange because the lexiglas being smoother than the rubber o-ring, one may exect that friction among the cell sides would not be relevant, therefore, one should not exect a deendence of scaling laws on the thickness b. We summarize in the following table, the range of variations of the arameters keeing in mind that the angular frequency o and the filling fraction (determined by the initial angular arc f ) extended at rest (Fig. 1b) are varied continuously. For all the grains, the reose angle haens to be almost constant and y r 32. o (rad/s) f (deg) a (cm) D ðmmþ b ðmmþ o a=g a=b b=d # # # In the exeriment it is not ossible to use the comlete range of o given by our setu, because there are two natural bounds. For large o, ifo 2 4g=a the systemacts as a centrifuge, moreover, before that a non-steady behavior arises when grains laced in the uer art are thrown arabolically fromthe border to the interior (Fig. 1c). This behavior is the basic rincile of a ball mill in the mining industry: because of a large effective friction between the rubber at the bottomand the granular bed, the material could be sticked inside the cylinder for large angles. The material does not sli downhill as in an avalanche, but they do fly inside the cell. Finally, for tiny grains air-drag is also imortant at large angular seeds. On the other hand, for very small o, only discrete avalanches can be seen [3,4], those avalanches resent an intermittent behavior [3]. This last rotating frequency is about o I :2 rad=seg. Measurements were made using a digital video camera working at a standard seed of 3 frame/s, from the movies one gets the full color rofile (see Fig. 2, left), then each icture is treated in grayscale with Photosho. Next, the figures are transformed into a binary scale using Matlab and, finally, the data for the rofile interface are obtained and rocessed (see Fig. 2, right). Scaling law for the front length. Using dimensionless arameters the only ossible general law for the rofile angular length is rffiffi a f max ¼ f 1 o ; b g D ; a b ; f. (1) The function f 1 ðþ should be rather comlex, thus we try to exlore a articular limit, mainly a=bb1, and o ffiffiffiffiffiffi a=g 51 but keeing oa4 gd and finite, that is an intermediate asymtotic. It will be desirable to have the limit of a small filling

4 N. Seúlveda et al. / Physica A 356 (25) fraction f 51 to recover the essential asects of an avalanche over a flat surface, however, that limit is difficult to reach, e.g. for f ¼ 4 the rofile is almost undistinguishable. On the other hand, a large electrostatic charge er article aears in such a small amount of grains. Finally, the limit b=db1 is only artially reached because we are limited by electrostatic effects aearing in very small articles and the increase of b needs a different cell that will change dramatically the rough boundary. In order to settle relation (1) we use the fact that there is only one dimensionless arameter involving o, so first, the measurements were made for fixed article diameter D ¼ mm, initial filling fraction f ¼ 65 and cell radius a ¼ 16:5cm while the angular frequency, o, varying between the range considered. Over a large number of exerimental oints (see Fig. 3, left ), a linear regression of their Fig. 2. Left: a digital hotograh of a front for o a=g ¼ :25, a ¼ 16:5cm, D ¼ 7 15 mm. Right: a olar lot r=a of different fronts obtained at (end front increases with o) o ¼, o ¼ :8 s 1, o ¼ :13 s 1, o ¼ :18 s 1, o ¼ :47 s 1 and o ¼ :59 s 1. Note that the osition of front of the avalanche remains almost at the same lace: 27 5, indeendently of the rotating frequency max 1.4 max ω[rad/s] ω gd ab Fig. 3. Left: summary for the rofile length f max =f for different initial filling fractions, cell radiis, article diameters as function of o. : a ¼ 16:5cm; D ¼ mm; E: a ¼ 16:5cm; D ¼ mm;.: a ¼ 16:5cm; D ¼ mm; %: a ¼ 12:5cm; D ¼ mm; a ¼ 4:5cm, D ¼ mm; Right: full collase for all data the front length f max =f vs. o ffiffiffiffiffiffi ab=gd.

5 182 N. Seúlveda et al. / Physica A 356 (25) logarithms with 95% confidence bounds leads the law f max o 1=6:12 with an excellent correlation coefficient of.995. Naturally, this ower law has no sense in the limit o!, which is clearly omitted in our study because oa4 ffiffiffiffiffiffi gd. Next, we study the deendance of f 1 ðþ varying the other arameters a and f, but keeing constant the article diameter D. We have noticed that if one lots f max =f as a function of o a=g for different values of a and f all different oints collase in a single curve. Writing f max =f ¼ f ~ 1 ðo a=g ; b=d; a=b; f Þ we can deduce that there is no (or very weak) deendence of a=b and f in the function f ~ 1 ðþ. Finally, we have done exeriments under the same exerimental rocedure with glass articles of different sizes as in the table. We see that the scaling f max =f ðo a=g Þ 1=6 fails as we change D (see Fig. 3, left). Thus, f ~ 1 deends weakly on the arameter b=d for the range considered. The best fit of all data is the following (see Fig. 3, right) f max b 1=12:8 rffiffiffi a 1=6:12 ¼ð1:77 :7Þ o f D g. (2) That indicates that o ffiffiffiffiffiffi ab=gd should be the relevant dimensionless arameter. No existing theory or model [1,9] reveals such a arameter u-to-date. Scaling law for the shae of the fronts. In this section, we will study the full shae of the rofile of the avalanche. Dimensional analysis indicates that the rofile deendence should be (in olar coordinates, see Fig. 1b) rffiffi rðfþ a ¼ f a 2 o ; b g D ; a b ; f ; f (3) being f 2 ðþ a new universal function. However, because of the mass conservation we could guess a ossible deendence of f 2 ðþ on the similarity law (1) or (2) in the region of arameters considered stated in the revious section. If the area of the rofile is conserved, something to be checked a osteriori, one has that the area at rest and that for arbitrary o are the same. Let be h ¼ a r, thus we have a 2 2 ðf sin f Þ¼ 1 Z fmax Z fmax ða 2 ða hðfþþ 2 ÞÞ df a hðfþ df, (4) 2 f min where the integration is realized over the angles of the rofile. The last aroximation considers h5a and we have set f min after Fig. 2, right. Making the change of variable f ¼ f ~ max f one transforms f into the interval ½; 1Š. Redefining the rofile as (hereafter f ~ 1 stands by f ~ 1 ðo a=g ; b=d; a=b; f Þ) rðfþ a ¼ f ~ 1 sin f ~hð f 1 f ~ ¼ f=f f ~ 1 Þ. (5) Here hð ~ fþ ~ is a new universal function that will describe the shae of any front after the roer scaling. Conservation of mass imlies that R 1 ~ hð fþ ~ d f ~ ¼ 1: In Fig. 4, left, we see many different fronts: h=a ¼ 1 r=a vs. olar angle f for different arameters o, a, L, D. On the right-hand side of Fig. 4 we have the lot of the front scaled using relation (5), there we can see a good collase into the universal

6 N. Seúlveda et al. / Physica A 356 (25) a h [rad] h Fig. 4. Left: different front rofiles for different arameters value. Right: full collase of the universal front ~ hð ~ fþ. curve hð ~ fþ ~ within a range of confidence of 1%. The collase is good only for o a=g :7. That means that for large rotation frequencies the area is no longer conserved somewhat reasonably because of the dilatance of the grain layers at high rotation seed. Conclusion. We reort a large number of measurements of the size and shae of fronts of avalanches roduced by a rotating two-dimensional drum. We have found a scaling law (2) for the size of the front. Moreover, because of mass conservation this scaling law determines the full shae of fronts characterized by an universal curve lotted in Fig. 4, right. Hundred different fronts, for different values of the arameters, collase into this universal curve. Finally, the authors acknowledge the grant Fondecyt and Fonda (Chile), they would like to thank S. Contreras and M.A. Soto for their hositality at the Biohysics laboratory, and they are grateful to N. Fraysse (Nice) and I. Aranson (Argonne) for gentle grains rovision and R. Walker for a careful reading of the manuscrit. References [1] R.A. Bagnold, Proc. Roy. Soc. (London) A 225 (1954) 49, rerinted in The Physics of Sediment Transort by Wind and Water, American Society of Civil Engineers, New York, 1988, [2] E.J. Hofinger, Ann. Rev. Fluid Mech. 15 (1983) 47. [3] H.M. Jaeger, C. Liu, S. Nagel, Phys. Rev. Lett. 62 (1989) 4. [4] J. Rajchenbach, Phys. Rev. Lett. 65 (199) [5] J.P. Bouchaud, M. Cates, Ravi Prakash, S.F. Edwards, J. Phys. I 4 (1994) [6] P.-G. de Gennes, C.R. Acad. Sci., Ser. IIb 321 (1995) 51. [7] O. Zik, et al., Phys. Rev. Lett. 73 (1994) 644. [8] N. Jain, J.M. Ottino, R.M. Luetow, Phys. Fluids 14 (22) 572. [9] J. Rajchenbach, Phys. Rev. Lett. 9 (23)