Bottlenecks in granular flow: When does an obstacle increase the flow rate in an hourglass?

Bottlenecks occur in a wide range of situations from pedestrians, ants, cattle, and traffic flow to the transport of granular materials. We examine granular flow across a bottleneck using simulations of monodisperse disks. Contrary to expectations but consistent with previous work, we find that the flow rate across a bottleneck actually increases if an obstacle is optimally placed before it. Using the hourglass theory and a velocity-density relation, we show that the peak flow rate corresponds to a transition from free flow to congested flow, similar to the phase transition in traffic flow.

Figure 1

Snapshot of the simulation of gravity-driven granular flow with no obstacle (a), and with an obstacle of $6.3d$ diameter (b). The color encodes particle speed (darker is faster). The parameters governing the flow are hopper angle $\theta$, bottleneck width $D_0$, angle of convergence $\alpha$, and aperture $A$ created by the obstacle.

Figure 2

Flow rate without obstacle vs neck width in (a), and vs hopper angle using $C^{*}=C(D_0/d-c_0)$ in (b); the best fit is in agreement with the Beverloo relation, Eq. (1).

Figure 3

Flow rate vs obstacle diameter in (a), and vs obstacle height in (b), for different values of hopper angle $\theta$. In (a) the center of the obstacle is $9.46d$ above the center of the neck. In (b) the obstacle diameter is $9.46d$. Differences in initial conditions cause statistical fluctuations, so for each parameter suite the simulation was repeated 32 times with the particles in different initial positions. The error bars indicate standard errors. In (c) we show the ratio between the peak flow rate and the flow rate without an obstacle vs hopper angle $\theta$ for both cases. The peak flow rate vs angle of convergence is fitted in (d) using the Beverloo relation with effective gravity $g^{*}=g\cos \theta$.

Figure 4

Mean velocity (a) and mean density (d) at the bottleneck vs aperture. The lines show the multilinear fit of the data. The squares in (b) show the flow rate $J$ vs aperture calculated as the number of refilling particles per second. The circles in (b) show the microscopic flow rate $J_{\text{mic}}$ calculated from Eq. (2). The lines in (b) correspond to the model using the fitting in (a) and (c). Each dot in (d) shows the mean velocity vs mean density in a snapshot of the simulation. The circles show the time average of both density and velocity at different apertures. The bilinear fitting is performed using the slopes obtained from zones I and II in (a) and (c). Zone III corresponds to the hysteresis loop shown in the inset.