Supplementary material for ’ High - Speed Tracking of Rupture and Clustering in Freely Falling Granular Streams ’

The discharge of grains from a hopper or silo has received considerable attention, both because it is relevant to numerous industries, and a model system for examining general features of granular flow [1, 2]. These studies have generally focused on the motion of grains inside the hopper and the flow rate of material out of the nozzle. It is known that these systems can exhibit intermittent jamming and density waves which could potentially effect clustering. To check that the details of the nozzle flow do not govern the cluster formation, we performed experiments with both cylindrical and conical nozzles, nozzles with smooth and rough walls, and externally vibrated the nozzle. We observed nearly identical clusters in all cases. The nozzle and reservoir are the same as described in [3]. Briefly, a 9 cm diameter resevior of grains feeds a nozzle which consists of a porous, 13 cm long, 16 mm diameter cylinder with flat disk at the base containing a circular aperture. Due to the Janssen effect, these dimensions guarantee a steady flow rate provided the reservoir is sufficiently full [4, 5]. The mean velocity of grains leaving the nozzle u0 varies with the diameter of the opening D0, from 0.15 m/s for D0 = 1 mm to 0.25 m/s for D0 = 6 mm, roughly independent of grain diameter or material. After leaving the nozzle, the grains accelerate under gravity so the downward velocity grows with depth z as u(z) =  u0 + 2gz, causing the stream to elongate and stretch as it falls. While the stream is stretching due to gravity it simultaneously thins, which is evident in the images of the stream near the nozzle. The average diameter of the stream decays with depth as D(z) ∼ D0(1 + 2gz/u0) so that u(z)D(z) ∼ constant. This implies that the packing density of the stream remains roughly constant as the stream falls.