Gravity-driven flux of particles through apertures

TL;DR

This work addresses gravity-driven granular discharge through circular apertures and the observed deviations from simple free-fall behavior due to confinement and packing. By integrating 3D experiments with DEM simulations, the authors decompose the flux into velocity and packing contributions and introduce a dimensionless flux ratio $F$ that measures confinement relative to a free-fall limit, leading to a minimal, physically grounded description. They show the free-fall flux $Q_{ff} = A \rho_g \sqrt{g D} \; \phi_{ff}$ with $\phi_{ff} \approx \phi_{RLP} \approx 0.555$, and that $\langle u_z \rangle_A \approx \sqrt{g D}$, while boundary layers near the aperture cause an exponential relaxation over a confinement length $\lambda = n d$ with $n \approx 10$–$15$, yielding $F \approx 1 - e^{-D/(n d)}$. This framework provides a granular-analog to a Froude-like description of discharge, clarifying how boundary effects and packing structure govern flux and outlining a route beyond empirical Beverloo scaling to a universal, physics-based understanding of granular bottlenecks.

Abstract

The gravity-driven discharge of granular material through an aperture is a fundamental problem in granular physics and is classically described by empirical laws with different fitting parameters. In this Letter, we disentangle the mass flux into distinct velocity and packing contributions by combining three-dimensional experiments and simulations. We define a dimensionless flux ratio that captures confinement-driven deviations from a free-fall limit, which is recovered when the aperture is large compared to the grain size. For spherical cohesionless grains, the deviations from the free-fall limit are captured by a single exponential correction factor over a characteristic length scale of $\sim$ 10-15 grain diameters. This is shown to be the scale over which the packing structure is modified due to the boundary. Building on the $\sqrt{gD}$ exit-velocity scaling, we propose a kinematic framework that explains the universality of granular discharge beyond empirical descriptions.

Figures (4)

• Figure 1: Examples of granular discharge for particles of different diameters $d$ through circular apertures of diameter $D$: (a) $5$ mm, (b) $10$ mm, (c) $20$ mm, (d) $42$ mm. Rows correspond to $d=4.25, \, 2.10, \, 1.15$ mm from top to bottom. Scale bars are 1 cm. For larger $D/d$, denser flows are observed. No flows are observed for $D/d \lesssim 3$.
• Figure 2: Measured mass flux $Q$ from discharge experiments for a range of $D$ and $d$. The solid line shows the model free-fall flux given by Eq. \ref{['eq:eq_freeFall_flux']}, $Q_{\rm ff} = A \, \rho_g \sqrt{g\, D} \, \phi_{\rm ff}$, independent of $d$. Inset: Mass–time data $M(t)$ for $D=20$ mm [Fig. \ref{['fig:Fig1_Imaging']}(c)] across all grain sizes in the region of interest. The slope of each curve yields $Q$. For $D \gg d, \, Q_{\rm ff}$ predicts the flux.
• Figure 3: (a) DEM simulations of the mean vertical velocity at the aperture, $\langle u_z \rangle_A / \sqrt{gD}$, for various $d$. Values remain between 0.8 and 1, approaching 1 for $D \gg d$. (b) Packing fraction from all simulations collapses in wall-normal coordinates when rescaled by particle size ($R = D/2$). Far from walls (simulations): $\phi_{\rm ff} = 0.56$. (c) Area-averaged packing fraction, $\langle \phi \rangle_A = Q / A\rho_g\sqrt{gD}$, from experiments with $D \geq 50d$. Mean value of (experiments) $\phi_{\rm ff} \simeq 0.54$ (dashed line).
• Figure 4: Normalized flux $F = Q / Q_{\rm ff}$ vs relative aperture size $D/d$ for all experiments. Data collapse onto a single curve, described by $F = 1 - {\rm exp}(-D/nd)$, with $n=13$ (solid line) [RMSE: 0.04]. Beverloo’s relation \ref{['eq:eq_Beverloo']}, with $C =0.58$, $k = 1.5$, nedderman1982flow and $\phi_b = 0.6$, is shown normalized by $Q_{\rm ff}$ (dashed line). Inset: Evolution of $F$ with a Knudsen number, $\mathrm{Kn} = (D/d)^{-1}$.