Model for density waves in gravity-driven granular flow in narrow pipes

A gravity-driven flow of grains through a narrow pipe in vacuum is studied by means of a one-dimensional model with two coefficients of restitution. Numerical simulations show clearly how density waves form when a strikingly simple criterion is fulfilled: that dissipation due to collisions between the grains and the walls of the pipe is greater per collision than that which stems from collisions between particles. Counterintuitively, the highest flow rate is observed when the number of grains per density wave grows large. We find strong indication that the number of grains per density wave always approaches a constant as the particle number tends to infinity, and that collapse to a single wave, which was often observed also in previous simulations, occurs because the number of grains is insufficient for multiple wave formation.

Figure 1

Spatiotemporal plots of grain trajectories. (a) Gaseous regime $\mu =0.8,\nu =0.3$. (b) Split velocity transition point $\mu =\nu =0.55$. (c) and (d): Different multiwave patterns, $\left(\mu ,\nu \right)=\left(0.55,0.65\right)$ and (0.30,0.85), respectively. (e) and (f): Collapse to a single wave; $\left(\mu ,\nu \right)=\left(0.70,0.95\right)$ and (0.15, 0.85), respectively. Further details in text.

Figure 2

(Color online) (a) Number of grains per waves and (b) average grain velocity as functions of particle number $N$ for three sets of parameters (same in both panels). The dashed line in the above panel is $N=N_w$, i.e., a single density wave is present.

Figure 3

(Color online) (a): Plot of $\text{log}{N}_w$ where $N_w$ is number of grains per density wave as a function of $\mu$ and $\nu$ in the asymptotic limit. (b): Plot of average flow rate as a function of $\mu$ and $\nu$ in the asymptotic limit. In both panels, encircled letters refer to parameter pairs in Fig. 1.

Figure 4

Frequency spectrum of the wave pattern of Fig. 1d. The graph is the average over the power spectra of 200 subsamples of a long time series of $\approx {10}^7$ data points.