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) is identical with (1/M) sigma(s=n) to (s=infinity) h(s) by J(d) = 1 - F(m) = 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 is identical with 1/alpha = [n] passing through the HS can be fitted to N = A e(Bd2), 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.