A random Tetris model for analyzing the flow mechanism in a hopper discharging hard discs through an adjustable obstacle*

We propose a random Tetris model to study hard-disc particles passing through a hopper including an obstacle placed on its centerline. The distance between the obstacle and the exit of the hopper is tunable. Governed by a Gaussian displacement function, each particle can freely move perpendicular to the obstacle-exit direction and towards the exit. Our model rejects a particle’s movement if it creates overlaps between the particle and others, the obstacle, or boundaries of the hopper. Unlike the results in the studies using molecular dynamics simulations, where including an obstacle can increase or decrease the hopper flow rate, our model reveals a monotonically increasing flow rate as we place the obstacle further from the exit of the hopper. We also show that if the variance of the Gaussian displacement function of a particle increases as the particle keeps updating its position successfully, the flow rate always increases as long as the obstacle is close to the exit within a critical range. This can be explained by a free-fly zone forming between the obstacle and the exit. However, if we place the obstacle outside the critical range, the free-fly zone presumably could turn into a jamming-prone zone which causes the flow rate to decrease.