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Dual stratification of a sand pile formed by trapped kink
Physics Letters A 366 (2007) 591–595 www.elsevier.com/locate/pla
Dual stratification of a sand pile formed by trapped kink Michiko Shimokawa ∗ , Shonosuke Ohta Department of Earth System Science and Technology, Kyushu University, Kasuga, Fukuoka 816-8580, Japan Received 9 January 2007; received in revised form 16 May 2007; accepted 21 May 2007 Available online 26 May 2007 Communicated by A.R. Bishop
Abstract A dual stratification of a sand pile is found out with the experiment of a large Hele–Shaw cell, which is derived from a new trapped kink generated on the pile. The mechanism for the trapped kink is proposed using a geometric model. The results agree with experimental results within experimental error. © 2007 Elsevier B.V. All rights reserved. PACS: 47.32.Cc; 47.70.Mg; 45.70.Ht Keywords: Pattern formation; Sand pile; Stratification; Avalanche; Kink; Repose angle
Granular materials, such as sand, sugar, cement and medicine, are ubiquitous in our daily lives. Though the individual grains seem to behave in a simple manner, their assembly presents complex and strange phenomena [1,2], such as the Brazil-nut effect [3,4], sand dune dynamics [5], the spatial pattern by vertical oscillation [6] and the stratification or segregation pattern of a sand pile [7]. When a mixture consisting of large and small grains is exposed to external vibrations, size segregation of the mixture occurs [3,4]. This is called the Brazil-nut effect or the reverse Brazil-nut effect. In a desert, crescent-shaped dunes called barchans are observed. These move in the wind direction. In a collision, they can sometimes cross each other like solitary waves [5]. When a granular layer vibrates vertically, spatial patterns of standing waves, such as stripes, squares, hexagons, kinks and a disordered state, appear [6]. These granular phenomena are interesting for physics, but their physical properties are not yet fully understood [1,2]. When a binary granular mixture is poured into a quasi-twodimensional Hele–Shaw cell, we observe a segregation pattern or a stratification pattern depending on the experimental conditions [7,8]. The former pattern occurs if the mixture consists of ‘larger smooth grains’ and ‘smaller rough grains’. The latter
* Corresponding author. Tel.: +81 92 583 8647.
E-mail address: [email protected] (M. Shimokawa). 0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.05.067
pattern occurs if the mixture consists of ‘larger rough grains’ (hereafter abbreviated LRG) and ‘smaller smooth grains’ (hereafter abbreviated SSG). The size and repose angle of each grain determine the pattern. Let us focus on a stratification pattern. A stratification pattern has alternate LRG and SSG stripes. The mechanism responsible for the formation of the stratification pattern was considered as follows [9,10]. Size segregation occurs in an avalanche. The upper and lower layers of the avalanche consist of large and small grains, respectively. The avalanche stops at the bottom of a sand pile, where a congestion wave appears. The congestion wave, termed a kink, backs up towards the top of the sand pile. A pair of stripes is formed by the upslope motion of a kink. This process is repeated and the stratification pattern emerges. We are fascinated not only by the static regular pattern but also by the periodic motion of an avalanche and a kink. The kink plays an important role in spontaneous stratification. It has been stated that a kink is generated only at the bottom of a sand pile, because the slope of the sand pile gradually becomes smaller at the bottom [11]. Therefore, a pair of stripes grows towards the top from the bottom. We experimented with a larger apparatus than that used in previous experiments. We discovered that the dual stratification pattern had two periodic stripes as shown in Fig. 1(b) and a new type of a trapped kink, generated on the sand pile slope, was observed. The kink is necessary for the formation of the dual stratification. It was a very surprising that a kink was generated
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Fig. 2. Samples of (a) glass beads (repose angle: 27 degrees, diameter: 0.1 mm) and (b) brown sugar (repose angle: 38 degrees, diameter: 0.2–0.8 mm).
Fig. 1. (Color online.) (a) Sketch of the experimental apparatus. The width of the funnel d is varied, which leads to changes in the flux. The wooden board is shown as W. (b) Dual stratification pattern with the large sand pile. Wavelengths in lower region a and upper region b of the sand pile are shown as λa = 7.7 mm and λb = 3.8 mm (flux: 0.37 g/s).
on the sand pile slope whose inclination does not decrease. Ordinary theories cannot predict kink generation on the slope [11]. In this Letter, we report on the formation process of the new dual stratification observed in our experiment. Experiments have been performed using a vertical Hele– Shaw cell as illustrated in Fig. 1(a). The cell consists of two acrylic sheets of thickness 0.5 cm. The width is 60.0 cm, and the vertical length is 50.0 cm. This size is about twice the conventional size. The two acrylic sheets are mounted parallel on a horizontal base plate. The space between the two parallel sheets is 0.5 cm. Sometimes during the formation of a large sand pile, the foundation of the pile sinks and the pile collapses. Thus, a triangular wooden board W is built to prevent the sand pile from collapsing. The size of W is 12.5 cm long and 8.0 cm high. The sloop of W is 33 degrees, which is equal to the repose angle of the sand pile on the stratification pattern. The wooden board does not influence dual stratification when the pattern is distant from W. The mixture consists of glass beads and brown sugar which correspond to SSG and LRG, respectively, as shown in Fig. 2. The diameter of SSG is 0.1 mm, while that of LRG ranges from 0.2 to 0.8 mm. The repose an-
gles of SSG and LRG are 27 and 38 degrees, respectively. The two types of grains are uniformly mixed in the same volume ratio. The granular mixture is poured through a funnel at the right edge of the Hele–Shaw cell as shown in Fig. 1(a). The width of the funnel d is varied and takes the values 1.6, 2.1 or 3.3 mm. The flux J of the mixture depends on d. The relationship between J and d is given by J = 0.12(d + 1.98)2 − 1.19. We used a SONY DCR-VX2100 digital video camera to observe the dynamic behavior of the new stratification pattern. The shutter speed was 1/60 s and the resolution was 640 × 480 pixels. The videos obtained with the camera were processed at an interval of 0.13 s. The experimental temperature was 20 ± 2 ◦ C. The experiments were performed at a room humidity of 41 ± 4% since special care had to be taken to prevent cohesion due to moisture. The new dual stratification pattern from the experiment with this apparatus is shown in Fig. 1(b). Although ordinary stratification is reported to have a single wavelength [10,11], this pattern has two different wavelengths in the upper and lower regions of the sand pile. The wavelength λa in the lower region and the wavelength λb in the upper region were analyzed by power spectra obtained from the fast Fourier transform algorithm. For a flux of 0.37 g/s, λa and λb are 7.7 mm and 3.8 mm respectively. λa is about twice that of λb . Let us focus on the formation process of the dual stratification. A new type of kink is observed in the emergence of the dual stratification. In previous experiments with a small sand pile, a kink is only produced at the bottom of the sand pile [7– 13]. However, in our experiments with a large sand pile, a kink is generated not only at the bottom but also on the slope. No work is currently available in the published literature on this new kink. We named the kink generated on the slope ‘a trapped kink’ and it is essential for the formation of dual stratification. We investigated the generation of the trapped kink using a digital video camera. The mixture is poured from the edge of the Hele–Shaw cell. Soon an avalanche begins. The binary mixture separates into two layers of LRG and SSG, while it descends as an avalanche. According to the Brazil-nut effect [3,4], LRG concentrates in the upper layer and SSG in the lower layer of an avalanche. We call this region ‘two layer flow’. LRG in the upper layer is faster than SSG in the lower layer. This leads to LRG gathering at the avalanche head. We call this area, which
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Fig. 3. (a) Time series for the head position of avalanche L. (b) Relationship between the length of avalanche L and the length of head LRG δL. The solid line is the line of best fit δL = 0.63L. (c) Distribution of repose angle of sand pile θST and head angle of the solid head LRG θT . Repose angle of LRG θLR is 38.0 degrees. Average values of θST and θT are 33.0 and 38.4 degrees (flux: 0.86 g/s), respectively.
is composed of only LRG at the head, the ‘head LRG’. The head LRG develops with time. When LRG gathers sufficiently at the head, the head LRG stops and it solidifies on the sand pile slope. This head LRG acts as a barrier, blocking further movement of the two layer flow. The two layer flow continues to collide with the stationary materials and solidifies towards the top of the sand pile. These processes lead to the development of the trapped kink. Then, a stripe of LRG and SSG is formed between the top of the sand pile and the position where the trapped kink is generated. After the trapped kink arrives at the top, the next avalanche occurs. Because the head of the solidified avalanche is too steep, the next avalanche cannot stop on the slope. Therefore, the avalanche stops at the bottom. Then, a normal kink is generated at the bottom and moves upslope. In this time, a stripe is formed toward the top from the bottom of the sand pile. After passing this normal kink on the slope, the surface of the slope becomes nearly flat. The avalanche after the normal kink can stop on the sand pile slope with the flat surface, and the trapped kink is generated again. This process is repeated. The frequency of kink generation in the upper region is about twice that in the lower region. Remember that the upslope motion of the kink forms a pair of stripes with alternating LRG and SSG. The lower wavelength λa is produced only by the normal kink, while the upper wavelength λb is produced by the normal kink and the trapped kink. Therefore, λa is about twice that of λb . The fairly constant repetition of the trapped kink and the normal kink creates this dual stratification. In order to understand the properties of trapped kink formation, we measured quantifiable data as shown in Fig. 3: the time evolution regarding the head position of an avalanche L (Fig. 3(a)), the relationship between L and the length of the head LRG δL (Fig. 3(b)), the head angle of the solidified head LRG θT and the repose angle of the sand pile θST (Figs. 3(c), 4). These data were obtained using a digital video camera. The flux of these data J is 0.86 g/s. The influence of W is negligible in the case of positions 10 cm or more from W. All data in Fig. 3 are taken at 10–15 cm from W.
Fig. 4. (Color online.) Snapshots of head LRG. (a) Snapshot when the avalanche descends onto the sand pile slope. (b) Snapshot when the trapped kink is generated. Length of avalanche and head LRG are shown as L and δL, respectively. The position where the trapped kink is generated is shown as LT . Angles for the sand pile and head of the avalanche are shown as θST and θ(t), respectively. Thickness of avalanche is D (flux: 2.15 g/s).
Fig. 3(a) shows the time series regarding the head position of an avalanche L, which is parallel to the slope. L is zero if the head is at the top. We define t = 0 as the time when the avalanche starts to descend at the top of the sand pile. We track motions of an avalanche until a trapped kink is generated. The data for 12 avalanches is shown in this figure. The gradient of Fig. 3(a) shows the velocity of the avalanche along the slope. Initially, L increases almost linearly with time. This means that the avalanche descends at a constant velocity. Later, the velocity of L changes to 0 cm/s from 4.6 cm/s discontinuously, except for a few avalanches. A few avalanches, such as the lowest data of L, slow down for about 0.5 s and then stop. The balance of forces and the linear resistance cannot predict these abrupt stops. The position LT , where an avalanche stops, ranges from 8.1 to 22.1 cm in Fig. 3(a). The statistical distribution of LT is shown in Fig. 5.
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Applying the TK model, we obtain LT =
Fig. 5. Distribution f (LT ) of position where trapped kink is generated. Result of TK model is shown as filled triangle LT = 11.8 ± 2.6 cm. Peak value of distribution obtained from experimental result is 13.2 ± 0.5 cm, which is almost the same as the result of the TK model (flux: 0.86 g/s).
Fig. 3(b) provides a plot of head LRG length δL against L for one avalanche until a trapped kink is generated. δL is parallel to a sand pile slope as shown in Fig. 4. δL increases in direct proportion to L, that is δL = α · L. In the same flux, α is constant, which indicates that the ratio of L and δL is maintained in the avalanche. The value of α in Fig. 3(b) is estimated as 0.63 ± 0.01. This means that 63% of the length of an avalanche is head LRG and the avalanche descends onto the sand pile while maintaining this ratio. This shows that the velocity difference between LRG in the upper layer and SSG in the lower layer is extremely large. Fig. 3(c) shows the distributions of repose angles of a sand pile θST (open triangles) and the head angles of a solidified head LRG θT (open circles). The sample number for each distribution is about 50. These angles are measured with snapshots as shown in Fig. 4(b). The error bar of θT represents the standard deviation. The average values of θST and θT are 33.0 ± 1.3 degrees and 38.4 ± 1.4 degrees (filled circle), respectively. Comparing the two average values, θT is about 5.4 degrees larger than θST . It is found that θT almost agrees with the repose angle of LRG θLR = 38 degrees (filled square). From the results in Fig. 3, we consider an abrupt generation of the trapped kink. Fig. 4 shows images of the head LRG. The lower drawings are schemata of the images shown above. Fig. 4(a) is taken before trapped kink generation (L < LT ), while Fig. 4(b) is taken after trapped kink generation (L = LT ). As mentioned before, for L < LT , head LRG descends as an avalanche and for L = LT , it does not move and is stationary. Comparing (a) with (b), the head angle θ (t) decreases with increasing time. From the geometric relationship shown in Fig. 4, we find θ (t) = θST
D + arctan , δL
(1)
where D represents the thickness of the two layer flow and is approximately constant in an avalanche. Eq. (1) shows that the increase in δL causes the decrease in θ (t). Here, we conjecture that a trapped kink is generated in the condition of θ (t) = θLR . We name this conjecture the trapped kink model (TK model).
D α · tan(θLR − θST )
.
(2)
Substitution of δL = α · L and θ (t) = θLR in Eq. (1) leads to Eq. (2). For a flux of 0.86 g/s, D = 0.7 cm, θT − θST ≈ θLR − θST = 5.4◦ and α = 0.63. Inserting these values into Eq. (2), we obtain LT = 11.8 ± 2.6 cm. The TK model result of 11.8 cm almost fits the experimental value of 13.2 cm as shown in Fig. 5. This result supports the TK model. Next, we consider the phenomena occurring for θ (t) = θLR . The idea of the TK model is based on the formation process of a sand pile. The phenomena occurring at the repose angle of θLR on the pile is often compared to a phase transition. For slopes such that θ (t) ≤ θLR , no flow of sand can occur and the pile appears to be a solid, whereas for θ (t) > θLR , the surface layers of the pile freely flow on the slope as a liquid. That is, the liquid pile at the surface transfers to the solid state in the condition of θ (t) = θLR . According to the above demonstration, our TK model is natural. Let us note the states of the head LRG. At first, for θ (t) > θLR , the head LRG cannot stop and descends as an avalanche like a liquid. Later, for θ (t) = θLR , the head LRG stops suddenly and behaves like a solid. The head LRG changes from a liquid state to a solid state. As a result, the solid head LRG blocks the avalanche behind and a trapped kink is generated. In our experiments with a large sand pile, the slope length is larger than that of previous experiments. Due to the long slope, a lot of LRG gathers at the head of the avalanche, which results in the trapped kink generation. At present, we do not know why LT yields an asymmetrical distribution as shown in Fig. 5. Such features are also observed in other fluxes. It is strange that these distributions are not symmetrical. We suggest that the asymmetric properties result from the interaction between an avalanche and surface grains in the sand slope. An experiment to understand this process would be interesting. The trapped kink is generated by liquid to solid state transition of the head LRG. The relationship between the jamming transition and trapped kink generation is interesting too, and would lead to a better understanding of the granular physics. In summary, we have discovered a dual stratification pattern in an experiment with a large sand pile and a new type of trapped kink causes the formation of this pattern. The trapped kink is generated as a result of the head LRG, which solidifies at the head of the avalanche. When the angle of the head LRG is equal to the repose angle of the LRG, the head LRG solidifies and the trapped kink is generated. The above result on trapped kink generation has been confirmed by the TK model, and the results obtained with the TK model agree with the experimental results within experimental error. Acknowledgements We would like to thank H. Honjo, H. Sakaguchi, H. Katsuragi, M. Isobe, N. Mitarai, H. Nakanishi, A. Nakahara, Y. Nishio and M. Homma for their helpful suggestions, discussion and comments. This Letter has been supported by a Re-
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search Fellowship from the Japan Society for the Promotion of Science for Young Scientists. References [1] H.M. Jaeger, S.R. Nagel, R.P. Behringer, Phys. Today 49 (4) (1996) 32. [2] H.M. Jaeger, S.R. Nagel, R.P. Behringer, Rev. Mod. Phys. 68 (4) (1996) 1259. [3] A.P.J. Breu, H.-M. Ensner, C.A. Kruelle, I. Rehberg, Phys. Rev. Lett. 90 (2003) 01432. [4] M. Schröter, S. Ulrich, J. Kreft, J.B. Swift, H. Swinny, Phys. Rev. E 74 (2006) 011307.
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[5] H. Elbelrhiti, P. Claudin, B. Andreotti, Nature 437 (2005) 720. [6] F. Melo, P.B. Umbanhowar, H.L. Swinney, Phys. Rev. Lett. 75 (1995) 3838. [7] H.A. Makse, S. Havlin, P.R. King, H.E. Stanley, Nature 386 (1997) 379. [8] Y. Grasselli, H.J. Herrmann, Granular Matter 1 (1998) 43. [9] H.A. Makse, P. Cizeau, H.E. Stanley, Phys. Rev. Lett. 78 (1997) 3298. [10] H.A. Makse, R.C. Ball, H.E. Stanley, S. Warr, Phys. Rev. E 58 (1998) 3357. [11] J.M.N.T. Gray, Y.C. Tai, in: H.J. Herrmann, J.-P. Hovi, S. Luding (Eds.), Physics of Dry Granular Media, Kluwer Academic, Boston/London, 1997, p. 697. [12] J.P. Koeppe, M. Enz, J. Kakalious, Phys. Rev. E 58 (1998) R4104. [13] H.A. Makse, H.J. Herrmann, Europhys. Lett. 43 (1998) 1.
Dual stratification of a sand pile formed by trapped kink Michiko Shimokawa ∗ , Shonosuke Ohta Department of Earth System Science and Technology, Kyushu University, Kasuga, Fukuoka 816-8580, Japan Received 9 January 2007; received in revised form 16 May 2007; accepted 21 May 2007 Available online 26 May 2007 Communicated by A.R. Bishop
Abstract A dual stratification of a sand pile is found out with the experiment of a large Hele–Shaw cell, which is derived from a new trapped kink generated on the pile. The mechanism for the trapped kink is proposed using a geometric model. The results agree with experimental results within experimental error. © 2007 Elsevier B.V. All rights reserved. PACS: 47.32.Cc; 47.70.Mg; 45.70.Ht Keywords: Pattern formation; Sand pile; Stratification; Avalanche; Kink; Repose angle
Granular materials, such as sand, sugar, cement and medicine, are ubiquitous in our daily lives. Though the individual grains seem to behave in a simple manner, their assembly presents complex and strange phenomena [1,2], such as the Brazil-nut effect [3,4], sand dune dynamics [5], the spatial pattern by vertical oscillation [6] and the stratification or segregation pattern of a sand pile [7]. When a mixture consisting of large and small grains is exposed to external vibrations, size segregation of the mixture occurs [3,4]. This is called the Brazil-nut effect or the reverse Brazil-nut effect. In a desert, crescent-shaped dunes called barchans are observed. These move in the wind direction. In a collision, they can sometimes cross each other like solitary waves [5]. When a granular layer vibrates vertically, spatial patterns of standing waves, such as stripes, squares, hexagons, kinks and a disordered state, appear [6]. These granular phenomena are interesting for physics, but their physical properties are not yet fully understood [1,2]. When a binary granular mixture is poured into a quasi-twodimensional Hele–Shaw cell, we observe a segregation pattern or a stratification pattern depending on the experimental conditions [7,8]. The former pattern occurs if the mixture consists of ‘larger smooth grains’ and ‘smaller rough grains’. The latter
* Corresponding author. Tel.: +81 92 583 8647.
E-mail address: [email protected] (M. Shimokawa). 0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.05.067
pattern occurs if the mixture consists of ‘larger rough grains’ (hereafter abbreviated LRG) and ‘smaller smooth grains’ (hereafter abbreviated SSG). The size and repose angle of each grain determine the pattern. Let us focus on a stratification pattern. A stratification pattern has alternate LRG and SSG stripes. The mechanism responsible for the formation of the stratification pattern was considered as follows [9,10]. Size segregation occurs in an avalanche. The upper and lower layers of the avalanche consist of large and small grains, respectively. The avalanche stops at the bottom of a sand pile, where a congestion wave appears. The congestion wave, termed a kink, backs up towards the top of the sand pile. A pair of stripes is formed by the upslope motion of a kink. This process is repeated and the stratification pattern emerges. We are fascinated not only by the static regular pattern but also by the periodic motion of an avalanche and a kink. The kink plays an important role in spontaneous stratification. It has been stated that a kink is generated only at the bottom of a sand pile, because the slope of the sand pile gradually becomes smaller at the bottom [11]. Therefore, a pair of stripes grows towards the top from the bottom. We experimented with a larger apparatus than that used in previous experiments. We discovered that the dual stratification pattern had two periodic stripes as shown in Fig. 1(b) and a new type of a trapped kink, generated on the sand pile slope, was observed. The kink is necessary for the formation of the dual stratification. It was a very surprising that a kink was generated
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Fig. 2. Samples of (a) glass beads (repose angle: 27 degrees, diameter: 0.1 mm) and (b) brown sugar (repose angle: 38 degrees, diameter: 0.2–0.8 mm).
Fig. 1. (Color online.) (a) Sketch of the experimental apparatus. The width of the funnel d is varied, which leads to changes in the flux. The wooden board is shown as W. (b) Dual stratification pattern with the large sand pile. Wavelengths in lower region a and upper region b of the sand pile are shown as λa = 7.7 mm and λb = 3.8 mm (flux: 0.37 g/s).
on the sand pile slope whose inclination does not decrease. Ordinary theories cannot predict kink generation on the slope [11]. In this Letter, we report on the formation process of the new dual stratification observed in our experiment. Experiments have been performed using a vertical Hele– Shaw cell as illustrated in Fig. 1(a). The cell consists of two acrylic sheets of thickness 0.5 cm. The width is 60.0 cm, and the vertical length is 50.0 cm. This size is about twice the conventional size. The two acrylic sheets are mounted parallel on a horizontal base plate. The space between the two parallel sheets is 0.5 cm. Sometimes during the formation of a large sand pile, the foundation of the pile sinks and the pile collapses. Thus, a triangular wooden board W is built to prevent the sand pile from collapsing. The size of W is 12.5 cm long and 8.0 cm high. The sloop of W is 33 degrees, which is equal to the repose angle of the sand pile on the stratification pattern. The wooden board does not influence dual stratification when the pattern is distant from W. The mixture consists of glass beads and brown sugar which correspond to SSG and LRG, respectively, as shown in Fig. 2. The diameter of SSG is 0.1 mm, while that of LRG ranges from 0.2 to 0.8 mm. The repose an-
gles of SSG and LRG are 27 and 38 degrees, respectively. The two types of grains are uniformly mixed in the same volume ratio. The granular mixture is poured through a funnel at the right edge of the Hele–Shaw cell as shown in Fig. 1(a). The width of the funnel d is varied and takes the values 1.6, 2.1 or 3.3 mm. The flux J of the mixture depends on d. The relationship between J and d is given by J = 0.12(d + 1.98)2 − 1.19. We used a SONY DCR-VX2100 digital video camera to observe the dynamic behavior of the new stratification pattern. The shutter speed was 1/60 s and the resolution was 640 × 480 pixels. The videos obtained with the camera were processed at an interval of 0.13 s. The experimental temperature was 20 ± 2 ◦ C. The experiments were performed at a room humidity of 41 ± 4% since special care had to be taken to prevent cohesion due to moisture. The new dual stratification pattern from the experiment with this apparatus is shown in Fig. 1(b). Although ordinary stratification is reported to have a single wavelength [10,11], this pattern has two different wavelengths in the upper and lower regions of the sand pile. The wavelength λa in the lower region and the wavelength λb in the upper region were analyzed by power spectra obtained from the fast Fourier transform algorithm. For a flux of 0.37 g/s, λa and λb are 7.7 mm and 3.8 mm respectively. λa is about twice that of λb . Let us focus on the formation process of the dual stratification. A new type of kink is observed in the emergence of the dual stratification. In previous experiments with a small sand pile, a kink is only produced at the bottom of the sand pile [7– 13]. However, in our experiments with a large sand pile, a kink is generated not only at the bottom but also on the slope. No work is currently available in the published literature on this new kink. We named the kink generated on the slope ‘a trapped kink’ and it is essential for the formation of dual stratification. We investigated the generation of the trapped kink using a digital video camera. The mixture is poured from the edge of the Hele–Shaw cell. Soon an avalanche begins. The binary mixture separates into two layers of LRG and SSG, while it descends as an avalanche. According to the Brazil-nut effect [3,4], LRG concentrates in the upper layer and SSG in the lower layer of an avalanche. We call this region ‘two layer flow’. LRG in the upper layer is faster than SSG in the lower layer. This leads to LRG gathering at the avalanche head. We call this area, which
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Fig. 3. (a) Time series for the head position of avalanche L. (b) Relationship between the length of avalanche L and the length of head LRG δL. The solid line is the line of best fit δL = 0.63L. (c) Distribution of repose angle of sand pile θST and head angle of the solid head LRG θT . Repose angle of LRG θLR is 38.0 degrees. Average values of θST and θT are 33.0 and 38.4 degrees (flux: 0.86 g/s), respectively.
is composed of only LRG at the head, the ‘head LRG’. The head LRG develops with time. When LRG gathers sufficiently at the head, the head LRG stops and it solidifies on the sand pile slope. This head LRG acts as a barrier, blocking further movement of the two layer flow. The two layer flow continues to collide with the stationary materials and solidifies towards the top of the sand pile. These processes lead to the development of the trapped kink. Then, a stripe of LRG and SSG is formed between the top of the sand pile and the position where the trapped kink is generated. After the trapped kink arrives at the top, the next avalanche occurs. Because the head of the solidified avalanche is too steep, the next avalanche cannot stop on the slope. Therefore, the avalanche stops at the bottom. Then, a normal kink is generated at the bottom and moves upslope. In this time, a stripe is formed toward the top from the bottom of the sand pile. After passing this normal kink on the slope, the surface of the slope becomes nearly flat. The avalanche after the normal kink can stop on the sand pile slope with the flat surface, and the trapped kink is generated again. This process is repeated. The frequency of kink generation in the upper region is about twice that in the lower region. Remember that the upslope motion of the kink forms a pair of stripes with alternating LRG and SSG. The lower wavelength λa is produced only by the normal kink, while the upper wavelength λb is produced by the normal kink and the trapped kink. Therefore, λa is about twice that of λb . The fairly constant repetition of the trapped kink and the normal kink creates this dual stratification. In order to understand the properties of trapped kink formation, we measured quantifiable data as shown in Fig. 3: the time evolution regarding the head position of an avalanche L (Fig. 3(a)), the relationship between L and the length of the head LRG δL (Fig. 3(b)), the head angle of the solidified head LRG θT and the repose angle of the sand pile θST (Figs. 3(c), 4). These data were obtained using a digital video camera. The flux of these data J is 0.86 g/s. The influence of W is negligible in the case of positions 10 cm or more from W. All data in Fig. 3 are taken at 10–15 cm from W.
Fig. 4. (Color online.) Snapshots of head LRG. (a) Snapshot when the avalanche descends onto the sand pile slope. (b) Snapshot when the trapped kink is generated. Length of avalanche and head LRG are shown as L and δL, respectively. The position where the trapped kink is generated is shown as LT . Angles for the sand pile and head of the avalanche are shown as θST and θ(t), respectively. Thickness of avalanche is D (flux: 2.15 g/s).
Fig. 3(a) shows the time series regarding the head position of an avalanche L, which is parallel to the slope. L is zero if the head is at the top. We define t = 0 as the time when the avalanche starts to descend at the top of the sand pile. We track motions of an avalanche until a trapped kink is generated. The data for 12 avalanches is shown in this figure. The gradient of Fig. 3(a) shows the velocity of the avalanche along the slope. Initially, L increases almost linearly with time. This means that the avalanche descends at a constant velocity. Later, the velocity of L changes to 0 cm/s from 4.6 cm/s discontinuously, except for a few avalanches. A few avalanches, such as the lowest data of L, slow down for about 0.5 s and then stop. The balance of forces and the linear resistance cannot predict these abrupt stops. The position LT , where an avalanche stops, ranges from 8.1 to 22.1 cm in Fig. 3(a). The statistical distribution of LT is shown in Fig. 5.
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Applying the TK model, we obtain LT =
Fig. 5. Distribution f (LT ) of position where trapped kink is generated. Result of TK model is shown as filled triangle LT = 11.8 ± 2.6 cm. Peak value of distribution obtained from experimental result is 13.2 ± 0.5 cm, which is almost the same as the result of the TK model (flux: 0.86 g/s).
Fig. 3(b) provides a plot of head LRG length δL against L for one avalanche until a trapped kink is generated. δL is parallel to a sand pile slope as shown in Fig. 4. δL increases in direct proportion to L, that is δL = α · L. In the same flux, α is constant, which indicates that the ratio of L and δL is maintained in the avalanche. The value of α in Fig. 3(b) is estimated as 0.63 ± 0.01. This means that 63% of the length of an avalanche is head LRG and the avalanche descends onto the sand pile while maintaining this ratio. This shows that the velocity difference between LRG in the upper layer and SSG in the lower layer is extremely large. Fig. 3(c) shows the distributions of repose angles of a sand pile θST (open triangles) and the head angles of a solidified head LRG θT (open circles). The sample number for each distribution is about 50. These angles are measured with snapshots as shown in Fig. 4(b). The error bar of θT represents the standard deviation. The average values of θST and θT are 33.0 ± 1.3 degrees and 38.4 ± 1.4 degrees (filled circle), respectively. Comparing the two average values, θT is about 5.4 degrees larger than θST . It is found that θT almost agrees with the repose angle of LRG θLR = 38 degrees (filled square). From the results in Fig. 3, we consider an abrupt generation of the trapped kink. Fig. 4 shows images of the head LRG. The lower drawings are schemata of the images shown above. Fig. 4(a) is taken before trapped kink generation (L < LT ), while Fig. 4(b) is taken after trapped kink generation (L = LT ). As mentioned before, for L < LT , head LRG descends as an avalanche and for L = LT , it does not move and is stationary. Comparing (a) with (b), the head angle θ (t) decreases with increasing time. From the geometric relationship shown in Fig. 4, we find θ (t) = θST
D + arctan , δL
(1)
where D represents the thickness of the two layer flow and is approximately constant in an avalanche. Eq. (1) shows that the increase in δL causes the decrease in θ (t). Here, we conjecture that a trapped kink is generated in the condition of θ (t) = θLR . We name this conjecture the trapped kink model (TK model).
D α · tan(θLR − θST )
.
(2)
Substitution of δL = α · L and θ (t) = θLR in Eq. (1) leads to Eq. (2). For a flux of 0.86 g/s, D = 0.7 cm, θT − θST ≈ θLR − θST = 5.4◦ and α = 0.63. Inserting these values into Eq. (2), we obtain LT = 11.8 ± 2.6 cm. The TK model result of 11.8 cm almost fits the experimental value of 13.2 cm as shown in Fig. 5. This result supports the TK model. Next, we consider the phenomena occurring for θ (t) = θLR . The idea of the TK model is based on the formation process of a sand pile. The phenomena occurring at the repose angle of θLR on the pile is often compared to a phase transition. For slopes such that θ (t) ≤ θLR , no flow of sand can occur and the pile appears to be a solid, whereas for θ (t) > θLR , the surface layers of the pile freely flow on the slope as a liquid. That is, the liquid pile at the surface transfers to the solid state in the condition of θ (t) = θLR . According to the above demonstration, our TK model is natural. Let us note the states of the head LRG. At first, for θ (t) > θLR , the head LRG cannot stop and descends as an avalanche like a liquid. Later, for θ (t) = θLR , the head LRG stops suddenly and behaves like a solid. The head LRG changes from a liquid state to a solid state. As a result, the solid head LRG blocks the avalanche behind and a trapped kink is generated. In our experiments with a large sand pile, the slope length is larger than that of previous experiments. Due to the long slope, a lot of LRG gathers at the head of the avalanche, which results in the trapped kink generation. At present, we do not know why LT yields an asymmetrical distribution as shown in Fig. 5. Such features are also observed in other fluxes. It is strange that these distributions are not symmetrical. We suggest that the asymmetric properties result from the interaction between an avalanche and surface grains in the sand slope. An experiment to understand this process would be interesting. The trapped kink is generated by liquid to solid state transition of the head LRG. The relationship between the jamming transition and trapped kink generation is interesting too, and would lead to a better understanding of the granular physics. In summary, we have discovered a dual stratification pattern in an experiment with a large sand pile and a new type of trapped kink causes the formation of this pattern. The trapped kink is generated as a result of the head LRG, which solidifies at the head of the avalanche. When the angle of the head LRG is equal to the repose angle of the LRG, the head LRG solidifies and the trapped kink is generated. The above result on trapped kink generation has been confirmed by the TK model, and the results obtained with the TK model agree with the experimental results within experimental error. Acknowledgements We would like to thank H. Honjo, H. Sakaguchi, H. Katsuragi, M. Isobe, N. Mitarai, H. Nakanishi, A. Nakahara, Y. Nishio and M. Homma for their helpful suggestions, discussion and comments. This Letter has been supported by a Re-
M. Shimokawa, S. Ohta / Physics Letters A 366 (2007) 591–595
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