- Email: [email protected]
Dynamical behaviour of rotated granular mixtures
Physica A 270 (1999) 115–124
www.elsevier.com/locate/physa
Dynamical behaviour of rotated granular mixtures a Graduate
Sanjay Puria;b; ∗ , Hisao Hayakawaa
School of Human and Environmental Studies, Kyoto University, Sakyo-ku, Kyoto 606-01, Japan b School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110 067, India
Abstract We present a phenomenological model for the dynamics of granular mixtures in a rotating drum. We demonstrate that our model successfully captures many of the experimental features known in the context of this problem. The results presented here are supplementary to those c 1999 Elsevier presented in a detailed paper (S. Puri, H. Hayakawa, cond-mat/9901260.). Science B.V. All rights reserved. PACS: 83.70.F; 64.75 Keywords: Granular mixtures; Rotating drum; Phenomenological modelling; Radial and axial segregation
1. Introduction There has been much recent interest in the dynamical behaviour of granular materials [1–3], which are assemblies of mesoscopic particles with typical sizes ranging from 10 m–1 cm. These particles rapidly dissipate energy on collision. Therefore, interesting physical phenomena are observed in granular materials only under the e ects of external driving forces, e.g., vibration, rotation, pouring, etc. In this paper, we focus upon a particularly interesting experimental problem [4–13], viz., the dynamical behaviour of granular mixtures in a rotating drum. In a recent work [14], we have formulated and studied a phenomenological model for this problem. Here, we brie y discuss the physical basis of our modelling and present results which are supplementary to those presented in Ref. [14]. ∗
Corresponding author. E-mail address: [email protected] (S. Puri)
c 1999 Elsevier Science B.V. All rights reserved. 0378-4371/99/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 9 9 ) 0 0 1 5 3 - 3
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This paper is organized as follows. Section 2 brie y summarizes the experimental results available for this problem. In Section 3, we discuss the phenomenological modelling of granular mixtures in a rotating drum. Sections 4 and 5 present some analytical and numerical results obtained from our phenomenological model. Finally, Section 6 concludes this paper with a summary and discussion of our results.
2. Overview of experimental results There have been many experimental studies [4–13] of the technologically relevant problem of granular mixtures in a rotating drum. Typically, these are in the context of mixtures of rough (say, sand or S) and smooth (say, glass or G) granular materials. The general consensus is that the di erence in Coulombic friction between the two species plays an important role in the dynamical behaviour. It should be kept in mind that granular assemblies made from the same material (e.g., glass) but with di erent grain sizes will, in general, have di erent Coulombic friction coecients. Therefore, the characterization of granular mixtures as a combination of rough and smooth grains is of wide applicability. Let us brie y summarize the experimental results available in this context. For small drum-rotation frequencies, the granular material sloshes around in the bottom of the drum. This regime has been of limited experimental interest. At very high rotation frequencies, the granular material is thrown about all over the drum and this regime has also received little attention. Experimental interest has primarily focused upon an intermediate frequency regime, where the granular material exhibits a steady-state behaviour, with a characteristic S-shaped surface pro le. In this range of frequencies, the rotated granular material is thought to consist of two distinct regions [15]. There is a bulk region, which rotates as a solid and accretes particles to (or depletes particles from) a surface layer, which is characterized by laminar
ow. The S-shaped surface pro le results from a balance between ow in the laminar layer and accretion from (or depletion to) the bulk. We will focus upon this intermediate frequency regime here. The relevant experimental results can be summarized as follows: (a) There is a rapid radial segregation with the smooth material (G) accumulating at the drum walls; and the rough material (S) accumulating in the central region. The time-scale of radial segregation is only a few rotation time-periods. (b) The radially-segregated state may be destabilized on longer time-scales – resulting in an axially segregated state, which consists of alternating bands rich in S and G, respectively. These bands coarsen extremely slowly in time. (c) The axial bands in (b) may be radially non-uniform, with di erent widths at the centre and edges of the drum. Before we proceed, we should stress that every experiment does not necessarily show all the features in (a) – (c) above. For example, some experiments directly exhibit axial segregation whereas others only show radial segregation. Furthermore, the degree of
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radial non-uniformity of bands in the axially segregated state varies from one experiment to another. Clearly, a reasonable phenomenological model should account for all these di erent possibilities.
3. Phenomenological modelling In recent work [14], we have proposed a phenomenological model, which successfully replicates the behaviours described in (a) – (c) of Section 2. We have presented a detailed discussion of our modelling in Ref. [14] and refer the interested reader to that paper. In the present exposition, we con ne ourselves to elucidating the physical ingredients of our modelling. Our phenomenological model is based on a time-dependent order parameter eld (r; t), which is the local density di erence of S and G. The spatial coordinate r ≡ (x; y) is de ned in a 2-dimensional reference frame, which corresponds to the average surface pro le of the rotated homogeneous mixture. The x- and y-coordinates refer to the radial and axial directions, respectively. Essentially, the reference frame corresponds to the laminar surface layer in which granular particles ow, collide and scatter. We consider the bulk of the granular material to be a solid – in accordance with conventional experimental wisdom in this regard. The evolution of the order parameter eld results from stochastic particle exchanges and is determined by the following factors: (i) There is a current of G particles driven by the slope gradient because any mixture of S and G has a repose angle which is steeper than that for pure G. Thus, local
uctuations in the density are reinforced. We make the further reasonable assumption that the S-shaped surface pro le is in local equilibrium with the order parameter (composition) eld, because the ows leading to formation of the surface pro le occur on faster time-scales than the stochastic exchanges leading to evolution of (r; t). (ii) There is a di usive process due to random collisions of particles in the laminar layer. (iii) We penalize interfaces between domains which are rich in S and G due to stresses resulting from mismatches in shapes and sizes of the di erent types of grains. In our earlier paper [14], we have combined the factors in (i) – (iii) to obtain an evolution equation for (r; t). Without going into further details, we present the dimensionless form of our model here: @ (r; t) = −∇ · (1 − (r; t)2 )∇ a(R2 − x2 ) + [b + a(R2 − x2 )] (r; t) @t − tanh−1 [ (r; t)] + ∇2 (r; t) ≡ −∇ · J (r; t) ;
(1)
where J (r; t) is the current of (r; t). Eq. (1) is reminiscent of the Cahn–Hilliard (CH) equation for phase separation in binary alloys but has some crucial di erences. Moreover, the mechanism for segregation of granular mixtures is completely di erent
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from that for binary alloys, where phase separation results from a competition between energetic and entropic considerations. The dimensionless scaling which leads to Eq. (1) is obtained from the various parameters (e.g., mobility, di usion constant, etc.) associated with the factors in (i) – (iii). We should stress that all these parameters are related to the drum-rotation frequency ! [14]. As we have stressed in our introductory remarks, granular matter is strongly dissipative and exhibits interesting dynamical behaviour only in the presence of external driving (rotation, in the present case). Our dimensionless model in Eq. (1) is characterized by two parameters a and b; and the dimensionless drum radius R. We assume that the drum is of in nite extent in the axial direction. In the radial direction, we impose at boundary conditions on the order parameter at x = ± R, viz., @ (r; t) =0: (2) @x x = ± R Furthermore, we impose the physical requirement that these should be no radial current at the drum edge, viz., @ 2 2 a(R −x )+[b+a(R2 −x2 )] (r; t)−tanh−1 [ (r; t)]+∇2 (r; t) x = ± R = 0 : @x (3) Before we present detailed results, it is relevant to intuitively understand how our model in Eq. (1) replicates the various behaviours discussed in (a) – (c) of Section 2. The term a(R2 − x2 ) in the argument of the current rapidly drives G to the drum edges and S to the drum centre (if a ¿ 0). The linear term in (in the argument of the current) generates segregation instabilities if [b + a(R2 − x2 ) − 1] ¿ 0. However, these are characterized by a much slower time-scale than the radial-segregation mechanism. For an appropriate choice of parameter values, these uctuations may grow and lead to a crossover from radial to axial segregation. We will discuss this crossover in the next section. Finally, we should remark that, with appropriate adjustment of parameter values in our model, we can eliminate any or all of the features speci ed in (a) – (c) of Section 2. 4. Crossover from radial to axial segregation To understand the mechanism of the crossover from radial to axial segregation, it is relevant to examine the linear stability of the radially-segregated static solution s (x). This is obtained as the zero-current solution of the 1-dimensional version of Eqs. (1) – (3) [14]. We will show plots of s (x) vs. x later but, typically, the radiallysegregated pro le is G-rich when |x| ' R and S-rich when |x| ' 0. For arbitrary parameter values, the linear stability analysis of s (x) is analytically intractable and has to be performed numerically. However, we can obtain analytical results in a simple limit where a → 0 and R → ∞ with aR2 = O(1). In this simplifying
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limit, let us examine the stability of the interface between G-rich and S-rich regions for the case of a granular mixture consisting of equal amounts of S and G. We introduce the Green’s function G(r; r0 ) satisfying ∇ · {(1 − (r; t)2 )∇G(r; r0 )} = (r − r0 ). Then, we can rewrite Eq. (1) as an integro-di erential equation: Z @ (r0 ; t) = (r; t) − ; (4) dr 0 G(r; r0 ) @t R where the Lagrange multiplier accounts for the conservation law dr (r; t) = const: In Eq. (4), we have introduced the “chemical potential” (r; t) in analogy with the CH equation. This is de ned as (r; t) = −a(R2 − x2 ) − [b + a(R2 − x2 )] (r; t) + tanh−1 [ (r; t)] + ∇2 (r; t) : (5) Let us consider a small deformation of the interface for the radially-segregated static solution s (x). In the vicinity of the interface, the order parameter eld can be approximated as (r) ' s (nˆa · (r − ra )) ≡ s (), where nˆa and ra refer to the normal unit vector and position of the interface, respectively. Assuming a small deviation from the interface solution, e.g., (r; t) = s () + (r; t), we can compute the chemical potential in the linear approximation as (r; t) ' − Ä s0 () + L (r; t) ; L≡−
1 @2 + 2 @ 1−
2 s
− [b + a(R2 − 2 )] :
(6)
In Eq. (6), Ä is the local curvature of the interface; s0 () = d s ()=d; and L = L† . Furthermore, s0 () satis es L s0 () + 2a(1 + s ()) = 0, where the nal correction term is negligible in the limit a → 0, R → ∞ and aR2 = O(1). Thus, s0 () is a zero eigenfunction of L (i.e., L s0 () = 0) and this enables us to use results known for the CH equation. We know that @ =@t ' − vn s0 , where vn is the interface velocity normal to the interface. Assuming that s0 () ' s () with s = s (0) − s (R), we can use Eq. (4) to obtain the dynamical equation for interface motion as follows: Z (7) ( s )2 dra0 0 G0 (ra ; ra0 0 )vn (ra0 0 ) = Ä(ra ) − 0 ; R where = d s0 ()2 , and 0 = s . Notice that we have simpli ed the de ning equation for the Green’s function as ∇2 G0 (r; r0 ) = (r − r0 ), which is reasonable in the weak-segregation limit. Eq. (7) is equivalent to the dynamical equation for an interface for the CH equation [16]. It is known in the context of the CH equation that a at interface is unstable due to a Mullins–Sekerka instability [17]. A similar argument applies in the present context also – at least in the simple limit considered.
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Fig. 1. Static radially-segregated pro les s (x) vs. x obtained from a numerical solution of the 1-dimensional version of our model in Eqs. (1) – (3). The parameter values were b = 0:9 and a = 0:03; and the radius was R = 4 dimensionless units. The initial conditions consisted of small-amplitude random uctuations about background values 0 = ± 0:3. The resultant static pro les are denoted by the indicated symbols. The solid lines correspond to an analytic approximation described in Ref. [14].
5. Numerical results We implemented a simple Euler-discretized version of Eqs. (1) – (3) on a 2-dimensional lattice of size [ − R; R] × [0; L], where R is the drum radius and L is the length of the drum – both in dimensionless units. Periodic boundary conditions were applied in the axial direction, and the boundary conditions in Eqs. (2) and (3) were applied at x = ± R. The discretization mesh sizes were x = 0:5 and t = 0:01. The parameter values for results shown here are b = 0:9 and a = 0:03. It is physically appropriate to set b ¡ 1, as this corresponds to a situation where there is no segregation when a = 0 – in accordance with our expectation that the homogeneous mixture will not segregate unless it is rotated. The initial conditions on (r; t) for our simulations consisted of uniformly-distributed random uctuations about a background value 0 . The numerical results shown here are primarily for the case 0 = −0:3, corresponding to a mixture of 35% S and 65% G. In our earlier paper [14], we have presented detailed results for the case 0 = 0. Fig. 1 shows radially-segregated pro les obtained from a 1-dimensional version of the model in Eqs. (1) – (3). The initial conditions had background values of 0 = ±0:3;
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Fig. 2. Evolution of a homogeneous initial condition, obtained from a 2-dimensional simulation of our model in Eqs. (1) – (3). The parameter values were b = 0:9 and a = 0:03. Other simulation details are provided in the text. The initial condition consisted of small uctuations about a background value 0 = −0:3. Regions rich in S( ¿ 0) are marked in black and regions rich in G( ¡ 0) are not marked in the frames.
and the drum radius was R = 4. The initially homogeneous state is rapidly driven to the static pro le s (x) on a time-scale t ' 10 dimensionless units. Recall that the time-scale of radial segregation is approximately 1 drum rotation. Thus, we make the approximate identi cation that 1 drum rotation ' 10 dimensionless time units for these parameter values. In Fig. 1, the solid lines denote analytic approximations to the radially-segregated pro les [14]. The analytic pro les are seen to be in reasonable agreement with the numerically obtained static solutions for the weak-segregation limit depicted in Fig. 1. In the strong-segregation limit (corresponding to large values of a), our analytic approximation is not reasonable [14]. Fig. 2 shows the temporal evolution of a homogeneous initial condition with 0 = −0:3 in the complete 2-dimensional version of our model. The drum radius was R = 4 (as before) and we set L = 128. In Fig. 2, the S-rich regions ( ¿ 0) are marked in black and the G-rich regions ( ¡ 0) are not marked. The rst frame (t = 20) corresponds to a radially-segregated state, but with ¡ 0 everywhere (see Fig. 1). The radially-segregated state is established by the rapid ow of G to the drum edges,
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Fig. 3. Axial variation of the order parameter pro les for the evolution depicted in Fig. 2. We plot (x; y; t) vs. y for dimensionless times t = 20; 600; 800 and 20 000. The values of x are (a) x = 0, at the drum centre; (b) x = R=2; and (c) x = R, at the drum edge.
corresponding to the regime described under (a) in Section 2. As the drum continues to be rotated, uctuations grow on a slower time-scale – as seen for t = 50; 200; 500. These form S-rich domains in the central region (e.g., t = 600; 700; 800), which grow out until they make contact with the drum edges (e.g., t =4000). The axially-segregated state coarsens extremely slowly, corresponding to the regime described under (b) in Section 2. The segregated bands are characterized by a radial non-uniformity (e.g., t = 20 000), in accordance with experimental results described under (c) in Section 2.
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We have investigated the time-dependence of the width of these bands for the case 0 =0, corresponding to a mixture of 50% S and 50% G. We nd that the width w(t) ∼ ln t – both at the centre of the drum and at the edges [14]. This is in accordance with recent experimental results of Kakalios et al. [18]. Finally, Fig. 3 shows the variation of the order parameter along the axial coordinate (y) for some of the frames in Fig. 2. The uppermost frame (Fig. 3(a)) plots (x; y; t) vs. y at x = 0, corresponding to the centre of the drum. The t = 20 state is depicted by a dotted line, which is just below = 0, and corresponds to the radially-segregated state shown in Fig. 1. The density uctuations at the drum centre grow with the passage of time but continue to have an average value which is richer in S than the initial homogeneous state – in agreement with experimental results [8]. The middle and lower frames of Fig. 3 show the temporal evolution of (x; y; t) vs. y for x = R=2 and x = R, respectively.
6. Summary and discussion Let us end this paper with a brief summary and discussion of the results presented here. We have formulated a phenomenological model for the dynamical behaviour of granular mixtures in a rotating drum. Our model is 2-dimensional (in the surface plane of the rotated mixture) and based on a few simple physical inputs. We nd that our model can replicate a range of experimental results available for this problem. This is gratifying and suggests that coarse-grained models will prove useful in modelling various universal features of the dynamics of granular materials. In the context of the rotating-drum problem, there are a number of directions for further study. For example, we would like to undertake a complete examination of the dynamical behaviours contained in our model and simple generalizations thereof. Apart from the main experimental observations outlined in Section 2, there are also a number of subsidiary phenomena reported in some experiments, e.g., travelling waves, etc. We would like to investigate the ability of our model to replicate all of these experimental results. More importantly, we believe that it is relevant to consider the complete 3-dimensional version of our model, accounting for particle ows in the bulk. The conventional experimental belief that the bulk undergoes rotation as a solid is being questioned by recent magnetic resonance imaging (MRI) experiments [19,20]. An appropriate extension of our model would include an evolution equation for the velocity eld of the bulk granular material. This is a direction of particular interest to us.
Acknowledgements SP is grateful for the warm hospitality of Kyoto University, where this work was done. His stay at Kyoto was supported by the Kyoto University Foundation. Both
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authors are grateful to R. Kobayashi, M. Nakagawa and Y. Shiwa for many useful discussions. SP is also grateful to H.J. Herrmann and S. Lipson for various critical inputs on this work. This work is, in part, supported by a grant from the Ministry of Education, Science, Sports and Culture of Japan (No. 09740314). References [1] H.M. Jaeger, S.R. Nagel, R.P. Behringer, Rev. Mod. Phys. 68 (1996) 1259. [2] A. Mehta (Ed.), Granular Matter: An Interdisciplinary Approach, Springer, New York, 1994. [3] R.P. Behringer J. Jenkins (Eds.), Powders and Grains 97: Proc. 3rd Int. Conf. on Powders and Grains, Balkema, Rotterdam, 1997. [4] Y. Oyama, Bull. Inst. Phys. Chem. Res. Jpn. Rep. 18 (1939) 600. [5] M.B. Donald, B. Roseman, British Chem. Eng. 7 (1962) 749, 823. [6] S. Das Gupta, D.V. Khakhar, S.K. Bhatia, Chem. Eng. Sci. 46 (1991) 1531. [7] S. Das Gupta, D.V. Khakhar, S.K. Bhatia, Powder Technol. 67 (1991) 145. [8] M. Nakagawa, Chem. Eng. Sci. 49 (1994) 2544. [9] G.H. Ristow, M. Nakagawa, cond-mat/9806067. [10] O. Zik, D. Levine, S.G. Lipson, S. Shtrikman, J. Stavans, Phys. Rev. Lett. 73 (1994) 644. [11] S. Thomae, in: S. Puri, S. Dattagupta (Eds.), Nonlinearities in Complex Systems, Narosa, Delhi, 1997. [12] K. Ueda, R. Kobayashi, T. Yanagita, in: Statistical Physics: Proc. 2nd Tohwa University International Meeting, World Scienti c, Singapore, 1998, 141 p. [13] K. Ueda, R. Kobayashi, Technical Report of IEICE, NLP 96-141, NC 96-95 (1997-02), 1, 1997. [14] S. Puri, H. Hayakawa, cond-mat/9901260. [15] J. Rajchenbach, Phys. Rev. Lett. 65 (1990) 2221. [16] K. Kawasaki, T. Ohta, Physica A 118 (1983) 175. [17] H. Hayakawa, T. Koga, J. Phys. Soc. Japan 59 (1990) 3542; and references therein. [18] J. Kakalios et al., unpublished. [19] K.M. Hill, A. Caprihan, J. Kakalios, Phys. Rev. Lett. 78 (1997) 50. [20] K.M. Hill, A. Caprihan, J. Kakalios, Phys. Rev. E 56 (1997) 4386.
www.elsevier.com/locate/physa
Dynamical behaviour of rotated granular mixtures a Graduate
Sanjay Puria;b; ∗ , Hisao Hayakawaa
School of Human and Environmental Studies, Kyoto University, Sakyo-ku, Kyoto 606-01, Japan b School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110 067, India
Abstract We present a phenomenological model for the dynamics of granular mixtures in a rotating drum. We demonstrate that our model successfully captures many of the experimental features known in the context of this problem. The results presented here are supplementary to those c 1999 Elsevier presented in a detailed paper (S. Puri, H. Hayakawa, cond-mat/9901260.). Science B.V. All rights reserved. PACS: 83.70.F; 64.75 Keywords: Granular mixtures; Rotating drum; Phenomenological modelling; Radial and axial segregation
1. Introduction There has been much recent interest in the dynamical behaviour of granular materials [1–3], which are assemblies of mesoscopic particles with typical sizes ranging from 10 m–1 cm. These particles rapidly dissipate energy on collision. Therefore, interesting physical phenomena are observed in granular materials only under the e ects of external driving forces, e.g., vibration, rotation, pouring, etc. In this paper, we focus upon a particularly interesting experimental problem [4–13], viz., the dynamical behaviour of granular mixtures in a rotating drum. In a recent work [14], we have formulated and studied a phenomenological model for this problem. Here, we brie y discuss the physical basis of our modelling and present results which are supplementary to those presented in Ref. [14]. ∗
Corresponding author. E-mail address: [email protected] (S. Puri)
c 1999 Elsevier Science B.V. All rights reserved. 0378-4371/99/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 9 9 ) 0 0 1 5 3 - 3
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This paper is organized as follows. Section 2 brie y summarizes the experimental results available for this problem. In Section 3, we discuss the phenomenological modelling of granular mixtures in a rotating drum. Sections 4 and 5 present some analytical and numerical results obtained from our phenomenological model. Finally, Section 6 concludes this paper with a summary and discussion of our results.
2. Overview of experimental results There have been many experimental studies [4–13] of the technologically relevant problem of granular mixtures in a rotating drum. Typically, these are in the context of mixtures of rough (say, sand or S) and smooth (say, glass or G) granular materials. The general consensus is that the di erence in Coulombic friction between the two species plays an important role in the dynamical behaviour. It should be kept in mind that granular assemblies made from the same material (e.g., glass) but with di erent grain sizes will, in general, have di erent Coulombic friction coecients. Therefore, the characterization of granular mixtures as a combination of rough and smooth grains is of wide applicability. Let us brie y summarize the experimental results available in this context. For small drum-rotation frequencies, the granular material sloshes around in the bottom of the drum. This regime has been of limited experimental interest. At very high rotation frequencies, the granular material is thrown about all over the drum and this regime has also received little attention. Experimental interest has primarily focused upon an intermediate frequency regime, where the granular material exhibits a steady-state behaviour, with a characteristic S-shaped surface pro le. In this range of frequencies, the rotated granular material is thought to consist of two distinct regions [15]. There is a bulk region, which rotates as a solid and accretes particles to (or depletes particles from) a surface layer, which is characterized by laminar
ow. The S-shaped surface pro le results from a balance between ow in the laminar layer and accretion from (or depletion to) the bulk. We will focus upon this intermediate frequency regime here. The relevant experimental results can be summarized as follows: (a) There is a rapid radial segregation with the smooth material (G) accumulating at the drum walls; and the rough material (S) accumulating in the central region. The time-scale of radial segregation is only a few rotation time-periods. (b) The radially-segregated state may be destabilized on longer time-scales – resulting in an axially segregated state, which consists of alternating bands rich in S and G, respectively. These bands coarsen extremely slowly in time. (c) The axial bands in (b) may be radially non-uniform, with di erent widths at the centre and edges of the drum. Before we proceed, we should stress that every experiment does not necessarily show all the features in (a) – (c) above. For example, some experiments directly exhibit axial segregation whereas others only show radial segregation. Furthermore, the degree of
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radial non-uniformity of bands in the axially segregated state varies from one experiment to another. Clearly, a reasonable phenomenological model should account for all these di erent possibilities.
3. Phenomenological modelling In recent work [14], we have proposed a phenomenological model, which successfully replicates the behaviours described in (a) – (c) of Section 2. We have presented a detailed discussion of our modelling in Ref. [14] and refer the interested reader to that paper. In the present exposition, we con ne ourselves to elucidating the physical ingredients of our modelling. Our phenomenological model is based on a time-dependent order parameter eld (r; t), which is the local density di erence of S and G. The spatial coordinate r ≡ (x; y) is de ned in a 2-dimensional reference frame, which corresponds to the average surface pro le of the rotated homogeneous mixture. The x- and y-coordinates refer to the radial and axial directions, respectively. Essentially, the reference frame corresponds to the laminar surface layer in which granular particles ow, collide and scatter. We consider the bulk of the granular material to be a solid – in accordance with conventional experimental wisdom in this regard. The evolution of the order parameter eld results from stochastic particle exchanges and is determined by the following factors: (i) There is a current of G particles driven by the slope gradient because any mixture of S and G has a repose angle which is steeper than that for pure G. Thus, local
uctuations in the density are reinforced. We make the further reasonable assumption that the S-shaped surface pro le is in local equilibrium with the order parameter (composition) eld, because the ows leading to formation of the surface pro le occur on faster time-scales than the stochastic exchanges leading to evolution of (r; t). (ii) There is a di usive process due to random collisions of particles in the laminar layer. (iii) We penalize interfaces between domains which are rich in S and G due to stresses resulting from mismatches in shapes and sizes of the di erent types of grains. In our earlier paper [14], we have combined the factors in (i) – (iii) to obtain an evolution equation for (r; t). Without going into further details, we present the dimensionless form of our model here: @ (r; t) = −∇ · (1 − (r; t)2 )∇ a(R2 − x2 ) + [b + a(R2 − x2 )] (r; t) @t − tanh−1 [ (r; t)] + ∇2 (r; t) ≡ −∇ · J (r; t) ;
(1)
where J (r; t) is the current of (r; t). Eq. (1) is reminiscent of the Cahn–Hilliard (CH) equation for phase separation in binary alloys but has some crucial di erences. Moreover, the mechanism for segregation of granular mixtures is completely di erent
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from that for binary alloys, where phase separation results from a competition between energetic and entropic considerations. The dimensionless scaling which leads to Eq. (1) is obtained from the various parameters (e.g., mobility, di usion constant, etc.) associated with the factors in (i) – (iii). We should stress that all these parameters are related to the drum-rotation frequency ! [14]. As we have stressed in our introductory remarks, granular matter is strongly dissipative and exhibits interesting dynamical behaviour only in the presence of external driving (rotation, in the present case). Our dimensionless model in Eq. (1) is characterized by two parameters a and b; and the dimensionless drum radius R. We assume that the drum is of in nite extent in the axial direction. In the radial direction, we impose at boundary conditions on the order parameter at x = ± R, viz., @ (r; t) =0: (2) @x x = ± R Furthermore, we impose the physical requirement that these should be no radial current at the drum edge, viz., @ 2 2 a(R −x )+[b+a(R2 −x2 )] (r; t)−tanh−1 [ (r; t)]+∇2 (r; t) x = ± R = 0 : @x (3) Before we present detailed results, it is relevant to intuitively understand how our model in Eq. (1) replicates the various behaviours discussed in (a) – (c) of Section 2. The term a(R2 − x2 ) in the argument of the current rapidly drives G to the drum edges and S to the drum centre (if a ¿ 0). The linear term in (in the argument of the current) generates segregation instabilities if [b + a(R2 − x2 ) − 1] ¿ 0. However, these are characterized by a much slower time-scale than the radial-segregation mechanism. For an appropriate choice of parameter values, these uctuations may grow and lead to a crossover from radial to axial segregation. We will discuss this crossover in the next section. Finally, we should remark that, with appropriate adjustment of parameter values in our model, we can eliminate any or all of the features speci ed in (a) – (c) of Section 2. 4. Crossover from radial to axial segregation To understand the mechanism of the crossover from radial to axial segregation, it is relevant to examine the linear stability of the radially-segregated static solution s (x). This is obtained as the zero-current solution of the 1-dimensional version of Eqs. (1) – (3) [14]. We will show plots of s (x) vs. x later but, typically, the radiallysegregated pro le is G-rich when |x| ' R and S-rich when |x| ' 0. For arbitrary parameter values, the linear stability analysis of s (x) is analytically intractable and has to be performed numerically. However, we can obtain analytical results in a simple limit where a → 0 and R → ∞ with aR2 = O(1). In this simplifying
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limit, let us examine the stability of the interface between G-rich and S-rich regions for the case of a granular mixture consisting of equal amounts of S and G. We introduce the Green’s function G(r; r0 ) satisfying ∇ · {(1 − (r; t)2 )∇G(r; r0 )} = (r − r0 ). Then, we can rewrite Eq. (1) as an integro-di erential equation: Z @ (r0 ; t) = (r; t) − ; (4) dr 0 G(r; r0 ) @t R where the Lagrange multiplier accounts for the conservation law dr (r; t) = const: In Eq. (4), we have introduced the “chemical potential” (r; t) in analogy with the CH equation. This is de ned as (r; t) = −a(R2 − x2 ) − [b + a(R2 − x2 )] (r; t) + tanh−1 [ (r; t)] + ∇2 (r; t) : (5) Let us consider a small deformation of the interface for the radially-segregated static solution s (x). In the vicinity of the interface, the order parameter eld can be approximated as (r) ' s (nˆa · (r − ra )) ≡ s (), where nˆa and ra refer to the normal unit vector and position of the interface, respectively. Assuming a small deviation from the interface solution, e.g., (r; t) = s () + (r; t), we can compute the chemical potential in the linear approximation as (r; t) ' − Ä s0 () + L (r; t) ; L≡−
1 @2 + 2 @ 1−
2 s
− [b + a(R2 − 2 )] :
(6)
In Eq. (6), Ä is the local curvature of the interface; s0 () = d s ()=d; and L = L† . Furthermore, s0 () satis es L s0 () + 2a(1 + s ()) = 0, where the nal correction term is negligible in the limit a → 0, R → ∞ and aR2 = O(1). Thus, s0 () is a zero eigenfunction of L (i.e., L s0 () = 0) and this enables us to use results known for the CH equation. We know that @ =@t ' − vn s0 , where vn is the interface velocity normal to the interface. Assuming that s0 () ' s () with s = s (0) − s (R), we can use Eq. (4) to obtain the dynamical equation for interface motion as follows: Z (7) ( s )2 dra0 0 G0 (ra ; ra0 0 )vn (ra0 0 ) = Ä(ra ) − 0 ; R where = d s0 ()2 , and 0 = s . Notice that we have simpli ed the de ning equation for the Green’s function as ∇2 G0 (r; r0 ) = (r − r0 ), which is reasonable in the weak-segregation limit. Eq. (7) is equivalent to the dynamical equation for an interface for the CH equation [16]. It is known in the context of the CH equation that a at interface is unstable due to a Mullins–Sekerka instability [17]. A similar argument applies in the present context also – at least in the simple limit considered.
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Fig. 1. Static radially-segregated pro les s (x) vs. x obtained from a numerical solution of the 1-dimensional version of our model in Eqs. (1) – (3). The parameter values were b = 0:9 and a = 0:03; and the radius was R = 4 dimensionless units. The initial conditions consisted of small-amplitude random uctuations about background values 0 = ± 0:3. The resultant static pro les are denoted by the indicated symbols. The solid lines correspond to an analytic approximation described in Ref. [14].
5. Numerical results We implemented a simple Euler-discretized version of Eqs. (1) – (3) on a 2-dimensional lattice of size [ − R; R] × [0; L], where R is the drum radius and L is the length of the drum – both in dimensionless units. Periodic boundary conditions were applied in the axial direction, and the boundary conditions in Eqs. (2) and (3) were applied at x = ± R. The discretization mesh sizes were x = 0:5 and t = 0:01. The parameter values for results shown here are b = 0:9 and a = 0:03. It is physically appropriate to set b ¡ 1, as this corresponds to a situation where there is no segregation when a = 0 – in accordance with our expectation that the homogeneous mixture will not segregate unless it is rotated. The initial conditions on (r; t) for our simulations consisted of uniformly-distributed random uctuations about a background value 0 . The numerical results shown here are primarily for the case 0 = −0:3, corresponding to a mixture of 35% S and 65% G. In our earlier paper [14], we have presented detailed results for the case 0 = 0. Fig. 1 shows radially-segregated pro les obtained from a 1-dimensional version of the model in Eqs. (1) – (3). The initial conditions had background values of 0 = ±0:3;
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Fig. 2. Evolution of a homogeneous initial condition, obtained from a 2-dimensional simulation of our model in Eqs. (1) – (3). The parameter values were b = 0:9 and a = 0:03. Other simulation details are provided in the text. The initial condition consisted of small uctuations about a background value 0 = −0:3. Regions rich in S( ¿ 0) are marked in black and regions rich in G( ¡ 0) are not marked in the frames.
and the drum radius was R = 4. The initially homogeneous state is rapidly driven to the static pro le s (x) on a time-scale t ' 10 dimensionless units. Recall that the time-scale of radial segregation is approximately 1 drum rotation. Thus, we make the approximate identi cation that 1 drum rotation ' 10 dimensionless time units for these parameter values. In Fig. 1, the solid lines denote analytic approximations to the radially-segregated pro les [14]. The analytic pro les are seen to be in reasonable agreement with the numerically obtained static solutions for the weak-segregation limit depicted in Fig. 1. In the strong-segregation limit (corresponding to large values of a), our analytic approximation is not reasonable [14]. Fig. 2 shows the temporal evolution of a homogeneous initial condition with 0 = −0:3 in the complete 2-dimensional version of our model. The drum radius was R = 4 (as before) and we set L = 128. In Fig. 2, the S-rich regions ( ¿ 0) are marked in black and the G-rich regions ( ¡ 0) are not marked. The rst frame (t = 20) corresponds to a radially-segregated state, but with ¡ 0 everywhere (see Fig. 1). The radially-segregated state is established by the rapid ow of G to the drum edges,
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Fig. 3. Axial variation of the order parameter pro les for the evolution depicted in Fig. 2. We plot (x; y; t) vs. y for dimensionless times t = 20; 600; 800 and 20 000. The values of x are (a) x = 0, at the drum centre; (b) x = R=2; and (c) x = R, at the drum edge.
corresponding to the regime described under (a) in Section 2. As the drum continues to be rotated, uctuations grow on a slower time-scale – as seen for t = 50; 200; 500. These form S-rich domains in the central region (e.g., t = 600; 700; 800), which grow out until they make contact with the drum edges (e.g., t =4000). The axially-segregated state coarsens extremely slowly, corresponding to the regime described under (b) in Section 2. The segregated bands are characterized by a radial non-uniformity (e.g., t = 20 000), in accordance with experimental results described under (c) in Section 2.
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We have investigated the time-dependence of the width of these bands for the case 0 =0, corresponding to a mixture of 50% S and 50% G. We nd that the width w(t) ∼ ln t – both at the centre of the drum and at the edges [14]. This is in accordance with recent experimental results of Kakalios et al. [18]. Finally, Fig. 3 shows the variation of the order parameter along the axial coordinate (y) for some of the frames in Fig. 2. The uppermost frame (Fig. 3(a)) plots (x; y; t) vs. y at x = 0, corresponding to the centre of the drum. The t = 20 state is depicted by a dotted line, which is just below = 0, and corresponds to the radially-segregated state shown in Fig. 1. The density uctuations at the drum centre grow with the passage of time but continue to have an average value which is richer in S than the initial homogeneous state – in agreement with experimental results [8]. The middle and lower frames of Fig. 3 show the temporal evolution of (x; y; t) vs. y for x = R=2 and x = R, respectively.
6. Summary and discussion Let us end this paper with a brief summary and discussion of the results presented here. We have formulated a phenomenological model for the dynamical behaviour of granular mixtures in a rotating drum. Our model is 2-dimensional (in the surface plane of the rotated mixture) and based on a few simple physical inputs. We nd that our model can replicate a range of experimental results available for this problem. This is gratifying and suggests that coarse-grained models will prove useful in modelling various universal features of the dynamics of granular materials. In the context of the rotating-drum problem, there are a number of directions for further study. For example, we would like to undertake a complete examination of the dynamical behaviours contained in our model and simple generalizations thereof. Apart from the main experimental observations outlined in Section 2, there are also a number of subsidiary phenomena reported in some experiments, e.g., travelling waves, etc. We would like to investigate the ability of our model to replicate all of these experimental results. More importantly, we believe that it is relevant to consider the complete 3-dimensional version of our model, accounting for particle ows in the bulk. The conventional experimental belief that the bulk undergoes rotation as a solid is being questioned by recent magnetic resonance imaging (MRI) experiments [19,20]. An appropriate extension of our model would include an evolution equation for the velocity eld of the bulk granular material. This is a direction of particular interest to us.
Acknowledgements SP is grateful for the warm hospitality of Kyoto University, where this work was done. His stay at Kyoto was supported by the Kyoto University Foundation. Both
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authors are grateful to R. Kobayashi, M. Nakagawa and Y. Shiwa for many useful discussions. SP is also grateful to H.J. Herrmann and S. Lipson for various critical inputs on this work. This work is, in part, supported by a grant from the Ministry of Education, Science, Sports and Culture of Japan (No. 09740314). References [1] H.M. Jaeger, S.R. Nagel, R.P. Behringer, Rev. Mod. Phys. 68 (1996) 1259. [2] A. Mehta (Ed.), Granular Matter: An Interdisciplinary Approach, Springer, New York, 1994. [3] R.P. Behringer J. Jenkins (Eds.), Powders and Grains 97: Proc. 3rd Int. Conf. on Powders and Grains, Balkema, Rotterdam, 1997. [4] Y. Oyama, Bull. Inst. Phys. Chem. Res. Jpn. Rep. 18 (1939) 600. [5] M.B. Donald, B. Roseman, British Chem. Eng. 7 (1962) 749, 823. [6] S. Das Gupta, D.V. Khakhar, S.K. Bhatia, Chem. Eng. Sci. 46 (1991) 1531. [7] S. Das Gupta, D.V. Khakhar, S.K. Bhatia, Powder Technol. 67 (1991) 145. [8] M. Nakagawa, Chem. Eng. Sci. 49 (1994) 2544. [9] G.H. Ristow, M. Nakagawa, cond-mat/9806067. [10] O. Zik, D. Levine, S.G. Lipson, S. Shtrikman, J. Stavans, Phys. Rev. Lett. 73 (1994) 644. [11] S. Thomae, in: S. Puri, S. Dattagupta (Eds.), Nonlinearities in Complex Systems, Narosa, Delhi, 1997. [12] K. Ueda, R. Kobayashi, T. Yanagita, in: Statistical Physics: Proc. 2nd Tohwa University International Meeting, World Scienti c, Singapore, 1998, 141 p. [13] K. Ueda, R. Kobayashi, Technical Report of IEICE, NLP 96-141, NC 96-95 (1997-02), 1, 1997. [14] S. Puri, H. Hayakawa, cond-mat/9901260. [15] J. Rajchenbach, Phys. Rev. Lett. 65 (1990) 2221. [16] K. Kawasaki, T. Ohta, Physica A 118 (1983) 175. [17] H. Hayakawa, T. Koga, J. Phys. Soc. Japan 59 (1990) 3542; and references therein. [18] J. Kakalios et al., unpublished. [19] K.M. Hill, A. Caprihan, J. Kakalios, Phys. Rev. Lett. 78 (1997) 50. [20] K.M. Hill, A. Caprihan, J. Kakalios, Phys. Rev. E 56 (1997) 4386.