Force analysis of jamming with disks of different sizes in a two-dimensional hopper

We calculate using force analysis the jamming probability of mixed disks that move downwards under gravity in a two-dimensional hopper. Jamming happens when disks form an arc across the hopper opening and we determine the stability of the forces acting at different arc points. To test the accuracy of our technique, we performed hopper experiments (side-wall separation: 0.2cm) using disks (average thickness: 0.198 0.02cm) at various d/R values where d is the hopper opening dimension and R is the disk radius. For N identical disks, the jamming probability is a sigmoidal function of N. We measured the jamming probability at d/R = 2.97, 3.48, 4.0, 4.3 and 4.97, and verified the presence of three possible flow regimes: (1)jam-free continuous flow for N less than the onset N_d, (2)transitory flow where the probability increases with N for N_d≤N≤N_u and (3)an N-independent jammed state for N greater than the saturation threshold N_u. Good agreement is found between theory and experiments. The existence of three flow regimes is confirmed for all d/R values with the N_u value increasing with d/R - the 2D hopper is more robust against jamming when relatively small homogeneous disks are used. We also experimented with mixtures of N large and n small disks and confirmed that jamming becomes more likely with increasing N for a constant n (and vice versa) until the maximum jamming probability is attained.