Jamming pattern in a two-dimensional hopper

We perform granular flow experiments using metal disks falling through a two-dimensional hopper. When the opening of the hopper d is small, jamming occurs due to formation of an arch at the hopper opening. We study the statistical properties of the horizontal component X and the vertical component Y of the arch vector that is defined as the displacement vector from the center of the first disk to the center of the last disk in the arch. As d increases, the distribution function of X changes from a steplike function to a smooth function while that of Y remains symmetrical and peaked at Y=0. When the arch vectors are classified according to the number of disk n in the arch, the mean value 〈n〉 is found to increase with d. In addition, the horizontal component X_n and the absolute value of the vertical component |Y_n| in each class have mean values increasing with n. Regarding the arch as a trajectory of a restricted random walker, we derive an expression for the probability density function a_n(X) of forming an n-disk arch. The statistics (〈n〉,〈X_n〉,〈|Y_n|〉 and the fraction g_d(n) of n-disk arches) of the arches generated by a_n(X) agree with those found in the experiment.