Analytical Approach to Continuous and Intermittent Bottleneck Flows

We propose a many-particle-inspired theory for granular outflows from a hopper and for the escape dynamics through a bottleneck based on a continuity equation in polar coordinates. If the inflow is below the maximum outflow, we find an asymptotic stationary solution. If the inflow is above this value, we observe queue formation, which can be described by a shock wave equation. We also address the experimental observation of intermittent outflows, taking into account the lack of space in the merging zone by a minimum function and coordination problems by a stochastic variable. This results in avalanches of different sizes even if friction, force networks, inelastic collapse, or delay-induced stop-and-go waves are not assumed. Our intermittent flows result from a random alternation between particle propagation and gap propagation. Erratic flows in congested merging zones of vehicle traffic may be explained in a similar way.

Figure 1

Density profiles at different times: (a) when the inflow is low and the initial density profile has a hump, (b) the inflow exceeds the maximum outflow and the initial density profile is a step function (shock wave). The simulation results have been obtained by solving the continuity equation with the Godunov scheme, assuming $\rho (12r_{\mathrm{crit}},t)=0.01$ and floating boundary conditions at $r=r_0=2r_{\mathrm{crit}}≔2r_{\mathrm{crit}}(Q_{\mathrm{in}})$ in case (a), but $\rho (2.8r_{\mathrm{crit}},t)=0.1$ and $Q_{\mathrm{out}}=2r_0{q}_{\max }$ (corresponding to the maximum outflow) with $r_0=0.5r_{\mathrm{crit}}$ in case (b). Note that the asymptotic density profile is ${\rho }_{-}(r,Q_{\mathrm{in}})$ in free flow and ${\rho }_{+}(r,Q_{\mathrm{out}})$ in jammed flow.

Figure 2

(a) In agreement with an experiment for granular outflows from a two-dimensional hopper [5], our simulation model generates exponentially distributed avalanche sizes when frictionless particles with coordination problems are jamming at a bottleneck (i.e., theory and experiment show a straight line in a log-linear plot). (b) The standard deviation of the outflow, divided by the average outflow shows 3 regimes: no outflow for $r_0/\Delta r<1/\gamma =2.5$, smooth outflows for large outlets, and intermittent flow in between. (c) The relative proportion of time steps $\Delta t$ with a stopped outflow confirms this picture. Our results are quite insensitive to the selected parameters. For illustration, we chose $\beta =3$, $\gamma =2/5$, $ϵ=0.01$, $Q_{\mathrm{in}}=4/\Delta t$, $v_0=\Delta r/\Delta t$, ${\rho }_{\max }=1/(\Delta r{)}^2$, and, in (a), $r_0=5\Delta r$ (jammed conditions).