1 Introduction

Granular matter shows solid-like or fluid-like behaviors. It is intrinsically different from a classic solid or fluid. Granular system is far from equilibrium. It shows complicated nonlinear phenomena such as force chain and force network ^{[1, 2]} ,surface wave ^{[3, 4]} ,mix and segregation ^{[5, 6, 7]} , jamming ^{[8, 9, 10, 11]} ,granular flow ^{[12, 13, 14, 15, 16]} ,and granular clock ^{[17, 18, 19, 20]} .

Granular flow is ubiquitous in industry,agriculture,and natural phenomena. It is closely related to typical geological disasters such as avalanches,mudslides,and debris flow ^{[21, 22, 23, 24]} . Therefore,the phenomena of granular flow has attracted much attention from physical,mechanical, and engineering societies ^{[25, 26, 27]} . The flow states of granular flow are normally classified into three classes: dilute flow,dense flow,and jammed. The particles in the three states behave like gas,liquid,and solid. In the dilute flow state,the system is dominated by the two-particle collisions,and the flow rate is controlled by the channel width and the exit width. In the dense flow state,the system is dominated by multi-particle interactions,and the flow rate is controlled only by the exit width ^[14] . The transitions among the three states have attracted much attention. For example,To et al. ^[9] studied the jamming transition in a hopper. Chen et al. ^[12] and Hou et al. ^[13] studied the dilute-to-dense transition of granular flow under the influence of electric field. Hou et al. ^[14] provided the scaling behavior of a dilute-to-dense transition of granular flow. Many experiment and simulation works have discovered the complex dynamical phenomena of granular flow. Rericha et al. ^[28] showed that granular flow showed supersonic shock when it passed a wedge. Drozd et al. ^[29] found three velocity fluctuations in dense granular flow. Tripathi and Khakhar ^[30] showed that steady flow could be achieved at a narrow range of angle by experiments on a bumpy inclined plane. Kumaran and Maheshwari ^[31] found a transition from disordered to ordered motion in granular chute flow with rough base.

In the research of granular packing,the structural characteristics are usually detected by the pair correlation function g(r) ^{[10, 32]} . Recently,the tools from the complex network theory are introduced into the field of granular packing ^{[33, 34, 35, 36, 37]} . Ben-Nun et al. ^[34] studied the complex network structure in the comminution of granular materials. Tordesillas et al. ^[36] studied the force cycles and force chains in a sheared granular material. Hu et al. ^[37] studied the force network characteristics of marginally jammed and deeply jammed solids.

In this paper,we study the relation between the packing configuration and the granular flow in a channel with two successive bottlenecks. Different from our previous experiment of granular flow with turnings ^[38] ,here we study a straight chute with channel width shrinks and analyze the problem from a complex network perspective. In industry and agriculture,there are often successive bottlenecks in granular chutes or pipes. However,a comprehensive study of the bottleneck effects on granular flow is missing. Bottlenecks can have different forms such as exiting width shrinking,turning,and merging flow. Hajra et al. ^[39] studied the mixing and segregation of the granular flow in a zigzag channel. Alonso-Marroquin et al. ^[40] found that the hopper flow rate could be enhanced by placing an obstacle before the exit. These works have revealed the intriguing effect of bottlenecks,but neglect the consecutive effect of bottlenecks on granular phase transition.

Here,we focus on the effect of two bottlenecks on the phase transition of granular flow. When changing the channel width between the two bottlenecks,the granular flow shows a dilute-to-dense transition. In particular,a bistable phenomenon is observed. The occurrence of a bistable phenomenon is induced by the initial packing configuration in the hopper. Then, we adopt statistical tools from the complex network theory to analyze the fabric networks of the granular packing configuration. We define a two-dimensional (2D) packing clustering coefficient,which can better characterize the fabric networks’ mechanical stability. 2 Experimental setup

The granular chute flow system is on a reclined plane with the angle 30° to the horizon (see Fig. 1). The channel is constructed by specially shaped glass spacers between a metal base and a glass cover plate. The gap between the base plate and the cover plate is 3 mm so as to maintain a quasi-2D granular flow of stainless steel beads with the diameter (2.5 ± 0.001) mm. There is a reservoir of particles at the top of the channel. The first section of the channel has a width of 50 mm. The final section has a width of 20 mm and a length of 200 mm. The channel between the two sections is the main section. It has a length D of 500 mm,and the adjustable width D is in the range from 20 mm to 50 mm. Therefore,the channel width first shrinks from 50 mm to D (upper bottleneck),and then shrinks from D to 20 mm (lower bottleneck).

At the exit of the top hopper,there is a thin plate controlling the flow of particles. At the beginning,the thin plate is totally closed. Steel beads are poured into the reservoir,and a granular packing is formed. Then,the thin plate is withdrawn very quickly,and thus the granular flow is initiated by gravity. At the final exit,an electronic balance is used to record the total mass of beads falling out of the channel. The balance has a precision of 0.1 g and the measuring frequency is 5 Hz. The mass of a bead is about 0.06 g. The flow rate can be obtained afterwards by the mass time series.

3 Experiment results

We first report the experiment result of the flow rate. The main section width D is the only adjustable parameter. In our experiment,the final exit width is fixed. Therefore,the dilute flow can be defined as the state where the flow rate increases with D,while the dense flow is the state where the flow rate will not change with D.

Figure 2 shows the variations of the mean flow rate versus D. From the figure,we can see that the flow state shows a dilute-to-dense transition in the main section. When D 26 mm,the flow is dilute,and the interaction of the particles is dominated by two-particle collisions. When D is large,the channel is dense,and the interaction of particles is dominated by multi-particle collisions. The flow rate is almost the same for the dense flow. Interestingly,the transition shows a bistable style at 24 mm D 26 mm. In this region,the final flow state can be either dilute or dense. Once established,the flow state will not switch between the dilute flow and the dense flow. It is verified that the dilute flow in this region is steady because the flow remains dilute after 30 minutes of continuous experiment.

Figure 3 shows the time series of the flow rate when D = 25 mm. At this width,the system is bistable. Close observations show that the initial flow rate is different. When the initial flow rate is relatively high (F ≈ 50 g·s ⁻¹ ),the flow rate will maintain a high value and the flow will remain dilute. When the initial flow rate is relatively low (F ≈ 40 g·s ⁻¹ ),the flow rate will stay at a low value and the flow will be dense. Further observation shows that the two bottlenecks show different interactions with high and low initial flow rates.

Figure 4 shows snapshots of the channel. It is seen from Fig. 4(a) that when the initial flow rate is high,the upper bottleneck becomes dense first. Because the outflow from the upper bottleneck is lower than the transition condition of the lower bottleneck,the lower bottleneck is dilute. Thus,the channel is in the dilute flow. From Fig. 4(b),one can see that when the initial flow rate is relatively low,the upper bottleneck is dilute,and produces a higher outflow rate. In this case,the lower bottleneck become dense first. Then,the dense flow propagates upwards to the upper bottleneck. Thus,the channel is finally dense. This dependence on the initial flow rate only happens when 24 mm D 26 mm. When D < 24 mm,the upper bottleneck is always dense first,and the channel is dilute. When D > 26 mm,the lower bottleneck is always dense first,and the channel is dense.

4 Analysis of packing fabric network

As shown above,the difference in the initial flow rate will change the interaction style of the bottlenecks and thus induce a bistable behavior. To find the reason for the initial flow rate difference,we analyze the initial packing configuration. Although the particles are poured into the hopper randomly,there are subtle differences between the packing configurations. We first capture snapshots of the beads in the hopper. Then,the beads and their contacts are recognized with a particle tracking program.

Figure 5 shows the two typical packing configurations in the hopper and their fabric network. The pictures show only a small portion of the packing. In the fabric network,the red dots represent the particles,and the blue lines represent the contacts among the particles. The initial packing can be either loose or dense,producing quite different fabric networks.

To quantify the differences between the two packing,we adopt the tools from the complex network theory to analyze their fabric networks. We first show the degree distribution. In network language,the degree is the number of links of a node,the same as the coordination number of particles in the packing. The degree distribution P (K) is defined as the probability of finding a node with the degree K. Figure 6 shows the degree distribution of the fabric network. For the loose packing,the peak is shown at K = 4. For the dense packing,58% of the particles are with the maximum coordination number of a two-dimensional packing,i.e.,K = 6. The particles with K = 6 are more stable. As a result,the dense packing is more stable.

Moreover,3-particle cycles (triangular structure) dominate in the fabric network of dense packing. No 5-particle cycle or 6-particle cycle shows up in the dense packing. Many 5-particle cycles and 6-particle cycles can be found in the loose packing. Triangular structure is mechanically stable,and it will benefit the stability of the system. In the complex network theory,one can use the clustering coefficient to represent the stability of a fabric network. If the neighbor set of a node i is denoted as G_i ,because node i has K_i neighbors,the maximal possible link number in G_i is K_i (K_i − 1)/2,while the actual number of links in G_i is e _i . Therefore,the classic local clustering coefficient at the node i can be defined as

where K_i is the degree of the node i. However,in a quasi-two-dimensional bead packing,the maximal possible number in a particle’s neighbor set is only K_i . We define a 2D packing local clustering coefficient at the particle i as Obviously,the 2D packing local clustering coefficient (see E_q. (2)) will be one if the fabric network is in the form of triangles. Then,the clustering coefficient of the whole network is defined as the average of the local clustering coefficient of all nodes in the network where N is the total number of nodes in the network. By definition,the value of C is in the range between zero and one. For the two typical packing configurations,using E_qs. (1) and (3),we can find that the clustering coefficient C is 0.243 for loose packing while 0.373 for dense packing. Using E_qs. (2) and(3),we can get C = 0.436 for loose packing and C = 0.860 for dense packing. Therefore,the dense packing configuration is more stable than the loose packing because its fabric network has a much bigger clustering coefficient. When the exit of the hopper is open,the loose packing will induce a higher initial flow rate,while the dense packing will induce a lower initial flow rate. In this way,we have explained the reason for the difference of the initial flow rate. 5 Conclusions

In summary,we have studied the effect of initial packing configuration on the phase transition of granular chute flow with two bottlenecks. The granular flow shows a dilute-to-dense transition with a nontrivial bistable phenomenon. The bistable phenomenon is closely related to the initial packing in the reservoir. The degree distribution and clustering coefficient show different characteristics for the loose packing and the dense packing. Under gravity,the dense packing is relatively more stable and can produce lower initial flow rate. The interaction of bottlenecks with the initial flow rate results in the bistable phenomenon. In industrial and agricultural processes,a higher flow rate is often desirable. However,the granular matter is often densely packed before flowing. This might lead to a lower flow rate. In some geological disasters,the granular packing often becomes loose first due to the external effect. There can be a larger flow rate and thus a massive loss and casualty. This situation must be detected and avoided.

Finally,we note that the rectangular shapes of the bottlenecks will not affect the granular behavior. In fact,at the two sidewalls near the exit,the beads pile up like two “soft wedges”. Therefore,the granular flow behavior will not be different if the sidewalls are changed to wedges. Nevertheless,if the lower exit is too wide,the lower bottleneck will not affect the granular flow. Therefore,the lower bottleneck must be small enough to produce the bistable phenomenon.

This work connects the fabric network structure with the granular flow state. The results are valuable for the optimization of transportation and processing of granular materials in industry, agriculture,and mining. The work can also shed some light on the researches of related complex flow problems such as avalanche,slides,debris flow,vehicular traffic,and pedestrian traffic.