PRAMANA
— journal of
c Indian Academy of Sciences °
physics
Vol. 64, No. 6 June 2005 pp. 963–969
Jamming patterns in a two-dimensional hopper KIWING TO Institute of Physics, Academia Sinica, Nankang, Taipei, Taiwan 115. Republic of China E-mail:
[email protected] Abstract. We report experimental studies of jamming phenomenon of monodisperse metal disks falling through a two-dimensional hopper when the hopper opening is larger than three times the size of the disks. For each jamming event, the configuration of the arch formed at the hopper opening is studied. The cumulative distribution functions fd (X) for hoppers of opening size d are measured. (Here X is the horizontal component of the arch vector, which is defined as the displacement vector from the center of the first disk to the center of the last disk in the arch.) We found that the distribution of fd (X) can be collasped into a master curve G(X) = fd (X)µ(d) that decays exponentially for X > 4. The scaling factor µ(d) is a decreasing function of d and is approximately proportional to the jamming probability. Keywords. Jamming; granular flow; hopper. PACS Nos 45.70.Qj; 45.70.Mg; 45.70.Vn
1. Introduction Recently there has been much research effort aimed at trying to understand the basic physics of granular flow in which dissipative inelastic collisions play an important role [1]. It has been known that gravity-driven granular flow in a vertical pipe may change from an accelerating dilute state to a slow dense state when the opening of the pipe is progressively reduced [2]. Since the transition depends on a global length scale which is the diameter of the pipe, the problem is not scalable. In contrast, an ideal hopper can be considered to have an infinitely large inlet so that the only length scale is the size of the opening with respect to the diameter of the grains that flow through it. While the discharge rate decreases with decreasing opening size, the flow may be completely jammed when the opening size is comparable to the grain diameter [3]. Experiments on monodisperse disks flowing in two-dimensional hoppers showed that the jamming transition is stochastic and that the jamming probability is related to the probability of forming an arch at the hopper opening that blocks the flow [4–6]. Within the transition regime, for the jamming probability to approach unity when the hopper opening is reduced, the distribution of the sizes of the arches found in the jammed hopper is directly related to the jamming probability. In our previous study [5], we found that, for a
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Kiwing To hopper with a small opening when jamming was certain, contributions from arches composed of 2, 3, and 4 disks produced a step-like distribution. In this paper, we present our studies on the distribution of the arches found in hoppers with larger openings. We find that in the transition region, where the jamming probability decreases from unity to zero, the arch size distributions at different hopper openings collapse to a universal curve that decays exponentially at large arch size. Furthermore, the scaling factor is proportional to the jamming probability. 2. Experimental set-up and procedures The experimental set-up is similar to that used in [7]. Figure 1 is a schematic diagram of the two-dimensional hopper in a two-dimensional container constructed between two transparent acrylic boards with a 3.5 mm gap between them. Seven hundred metal disks of D = 5 mm diameter and 3 mm thickness are put into the container. Two aluminum plates (W1 , W2 ) at the top portion (T) of the container form a channel where the disks stay before falling down. In the central portion of the container, a pair of transparent acrylic plates (H1 , H2 ), whose horizontal positions can be adjusted independently, constitute a hopper with angle φ = 52◦ and adjustable opening d. Here d is given in units of the disk diameter. This applies to all other lengths in this paper. The container is mounted on a platform that flips the container to the ‘up’ and ‘down’ positions. Figure 1a shows the schematic of the container in the ‘up’ position when the disks start to fall from the top into the hopper under gravity. Disks can flow through the hopper and fall to the space (B) between a pair of metal plates
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Figure 1. Schematic diagram of the container. In the ‘up’ (a) position, disks fall from the top (T) through the hopper (H1 H2 ) to the bottom (B). In the ‘down’ (b) position, disks return to the top (T). The arrows in the figure indicate possible routes of the disks.
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Jamming patterns in a two-dimensional hopper
Figure 2. Schematic diagram of the experimental set-up.
(W3 and W4 ) at the bottom of the container. When jamming occurs in the hopper, disks can overflow from the hopper and fall on the top of the plates W3 and W4 . After 10 s, the flipping platform rotates the container to the ‘down’ position as shown in figure 1b. The container stays in this ‘down’ position for another 10 s so that all the disks can flow back to the space between W1 and W2 through the paths shown by the arrows. Then the platform flips the container back to the ‘up’ position and the experiment repeats. For each hopper opening, the experiments were repeated 400 times. To capture the jamming configuration at the hopper opening we place a light box (L) at the back of the container and a camera (CCD) at the front for capturing images of the hopper (see figure 2). A contact switch is installed to detect the moment when the hopper arrives at the ‘up’ position. The video signal from the camera is captured by a frame grabber (Data Translation, model DT2851) to a computer (PC) 10 s after the container arrives at the ‘up’ position. Figure 3a shows a typical image taken by the camera when the flow was jammed at d = 3.88. From this image, the positions of the disks in the arch that jams the
Figure 3. Typical image (a) taken in the experiments. (b) Schematic diagram of the arch that blocks the flow in the hopper with opening d. The arch ~ goes from the center of the first disk at the right to the center of the vector A last disk at the left. The horizontal component X of the arch vector cannot be smaller than d − 1.
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Kiwing To flow at the hopper opening can be determined. If no disk is found in the image, the flow is not jammed. Hence from the images obtained for a hopper of opening d, the jamming probability J(d) can be calculated as the ratio of jamming events to the number of repetitions Na of the experiments. Since jamming is the result of an arch that blocks the flow at the hopper opening, the jamming probability should be related to the arch-forming process. Therefore we concentrate our attention on the arch and ignore all other disks within the hopper. To analyse the arch quantitatively, we describe the arch using a series of displacement vectors ~ri between the centers of consecutive disks (from left to right) ~ as a vector from the center of the first within the arch. We define an arch vector A ~ is the sum of disk at the left to the center of the last disk at the right. Hence A ~ is recorded and the ~ri as shown in figure 3b. The horizontal component X of A statistics is obtained for each hopper of d between 3.27 and 6.21. 3. Results and discussions Figure 4 shows the cumulative frequencies Fd (X) which are the numbers of arches such that the horizontal component of the arch vector is smaller than X. The legends in the figure indicate the hopper opening d. The inset shows the histograms hd (X) obtained from Fd (X) for d = 3.27, 5.63 and 5.29 using a bin size of 0.2. These three histograms and other histograms (not shown) have several common features: sharp cut-offs at small X, a peak at intermediate X, and decays with similar trends
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Figure 4. Cumulative frequency Fd (X) for hoppers of different openings d. The legend gives the hopper opening size d given in the unit of disk diameter. The inset shows three of the histograms calculated from Fd (X) using 0.2 as the bin size.
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Cumulative distribution function
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Figure 5. Cumulative distribution function fd (X) obtained by normalizing Fd (X) by the number of jamming events Nj whose d-dependence is given in the inset.
at large X. Clearly, arches with X < d − 1 will flow out of the hopper and a sharp cut-off at small X can be expected. On the other hand, arches with large X require more disks to form. So it is less probable to form an arch with large X. Even when the arch is formed its ability to support its own weight and the disks above decreases with increasing X. Hence the histograms decrease to zero at large X. The behaviors of hd (X) at small and large X naturally lead to a peak at intermediate X. To compare more clearly the cumulative frequency for different d, we normalized Fd (X) by the number of jamming events Nj to get the cumulative distribution function fd (X) ≡ Fd (X)/Nj (see figure 5). For d = 3.27, there is a mild kink near X ≈ 3 in fd (X). This is because when X increases from less than 3 to slightly larger than 3, the number of jamming events with arches composed of four disks drops abruptly to zero. Such step-like behavior in hd (X) for d < 3 has been observed and reported in our previous work [5]. For d > 3.5, f (X) becomes smooth due to the overlapping of the possible value of X for arches with more than four disks. Although the cumulative distribution function fd (X) for different d are distinct from each other, they share similar shapes. In fact one can collapse them to a universal curve by multiplying fd (X) with a d-dependent factor µ(d) such that µ(3.72) = 1. Figure 6 shows the scaled distribution function G(X) ≡ fd (X)µ(d) obtained using the cumulative distribution function of d = 3.72 as reference (i.e., µ(3.72) = 1). It falls off exponentially with a decay length of 0.84 for X > 4. The collapse of fd (X) onto G(X) for different d implies that G(X) is independent of d. Since fd (X) = Fd (X)/Nj , µ(d)/Nj = G(X)/Fd (X) must be a constant. Note that the normalisation factor Nj , when divided by the number of repetitions Na Pramana – J. Phys., Vol. 64, No. 6, June 2005
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Figure 6. Scaled distribution function G(X). The inset shows the semi-log plot of G(X). The dashed line in the inset has a slope of −0.84.
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gives the jamming probability J(d) ≡ Nj /Na . Hence µ(d) ∝ J(d). Since µ(3.72) = J(3.72) = 1, we have µ(d) = J(d). Such a relation is indeed observed as shown in figure 7. 968
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Jamming patterns in a two-dimensional hopper It is not surprising to find that a scaling function exists for the statistics fd (X) of hoppers with different openings d. Consider two hoppers H1 and H2 with openings of d1 and d2 , respectively. Clearly, if d1 > d2 , all the arches found in H1 can be found in hopper H2 . However, the arches with X < d1 + 1 found in H2 are not possible in H1 . On the other hand, arches with large X happen far from the hopper opening. So the formation of these arches should not depend on d and fd (X) should behave similarly at large X. Since the process of arch formation in these two hoppers is the same, fd (X) for H1 and H2 should scale for X > d1 + 1. Applying the above argument to other hoppers leads to the scaling possibility. It should be pointed out that the scaled distribution function G(X) is different from the probability a(X) of forming an arch of size X across the two-dimensional hopper. While a(X) contains the physical process related to the disk dynamics, G(X) is obtained by counting the number of events in which the flow is blocked by an arch at the hopper opening. It is probable that a larger arch is formed above the arch that blocks the flow. In these events, only the flow-blocking arch with a smaller value of X contributes to the statistics. Hence, G(X) is biased towards small arches and the effect is more serious for hoppers of small openings. This explains why no scaling was found in hoppers with small openings [5]. 4. Conclusion In this paper, we report our studies on jamming of granular disks in two-dimensional hoppers. A scaled cumulative distribution of the horizontal size X of the arch at the hopper opening was found to decay exponentially with X.
Acknowledgement This work has been supported by the National Science Council of the Republic of China (NSC-93-2112-M-001-015).
References [1] [2] [3] [4] [5] [6] [7]
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