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 $h(n)$ 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 $J(d)$, which is defined as the probability that jamming occurs in a HS containing m disks, is related to the distribution function F(n)≡(1∕M)∑ₛ₌ₙ^s=∞h(s) by J(d)=1-F(m)=1-e^-α(m-nₒ). The decay rate α, 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≡1∕α=⟨n⟩ passing through the HS can be fitted to N=Ae^Bd², N=Ae^B∕(dc-d), or N=A(dc-d)^-γ. The implications of these three forms for N to the stability of dense flow are discussed.