Jamming transition in two-dimensional hoppers and silos

Jamming of monodisperse metal disks flowing through two-dimensional hoppers and silos is studied experimentally. Repeating the flow experiment $M$ times in a hopper or silo (HS) of exit size $d$, we measure the histograms $h\left(n\right)$ of the number of disks $n$ through the HS before jamming. By treating the states of the HS as a Markov chain, we find that the jamming probability $J\left(d\right)$, which is defined as the probability that jamming occurs in a HS containing $m$ disks, is related to the distribution function $F\left(n\right)\equiv (1∕M){\sum }_{s=n}^{s=\infty }h\left(s\right)$ by $J\left(d\right)=1-F\left(m\right)=1-e^{-\alpha (m-n_o)}$. The decay rate $\alpha$, as a function of $d$, is found to be the same for both hoppers and silos with different widths. The average number of disks $N\equiv 1∕\alpha =⟨n⟩$ passing through the HS can be fitted to $N=A{e}^{B{d}^2}$, $N=A{e}^{B∕(d_c-d)}$, or $N=A{(d_c-d)}^{-\gamma }$. The implications of these three forms for $N$ to the stability of dense flow are discussed.

Figure 1

Schematic diagram of the two-dimensional hopper in the two-dimensional container. (a) The experiment starts with the container in the “up” position when the disks flow toward the hopper $(V_1,H_1,A_1,V_2,H_2,A_2)$. (b) Disks flow back to the top $\left(T\right)$ at the “down” position. The arrows show the paths of the disks.

Figure 2

(a) Schematic diagram of the setup for capturing disk configuration images. (b) Image of the jammed hopper. (c) Image of the fallen disks in the bottom of the container.

Figure 3

(a) Histogram $h\left(n\right)$ of the number of disks $n$ fallen before jamming occurs for the hopper of exit size $d=3.68$. (b) The distribution function $F\left(n\right)$ obtained from $h\left(n\right)$ at this hopper exit size. The inset of (b) is the semilog plots of $F\left(n\right)$ for (i) $d=1.48$, (ii) 3.27, (iii) 3.68, (iv) 3.80, (v) 4.16, (vi) 4.24, (vii) 4.63, (viii) 4.88, (ix) 5.17.

Figure 4

Variation of the decay rate $\alpha$ with hopper or silo exit $d$. The solid line is the fitted curve $\alpha =A{e}^{-B{d}^2}$ with $A=0.846$ and $B=0.275$. The inset shows the same data and the fitted curve with $d^2$ plotted in the $x$ axis.

Figure 5

Jamming probability $J\left(d\right)$ versus hopper or silo exit $d$. The solid line is the fitted curve using Eq. (6). The inset shows the mean number of disks $N$ through the hopper or silo before jamming. The solid line in the inset is from the exponential-square form for $N$. The dashed curve is from the power law and the exponential-reciprocal functional forms which are indistinguishable from each other in this range of $d$.