Granular flows in drums of non-uniform widths

Abstract

We study how channel width variations influence the dynamics of free-surface granular flows. For this purpose, we extend a continuum model framework to granular flows passing through channels that narrow or widen. Our theory uses a linearized approximation to an established dense granular flow rheology and a Coulomb friction law to model interaction between flow and sidewalls. We test the theoretical predictions using two novel 40 cm-diameter drums (convex and concave) filled halfway with 2 mm diameter particles rotated at rates in which the shear layer remains shallow and dense. We apply particle tracking velocimetry to enable quantitative comparisons between experimental data and theoretical predictions. We find that our experimental kinematics and energy profiles largely agree with the theoretical predictions. In general, flows through narrowing channels are faster and deeper than flows through widening channels. The influence of width variations grows with increasing flow speed, and the form of the rate dependence changes fundamentally as the regime changes from one in which kinetic energy is dissipated locally to one in which it is advected downstream. For both regimes, theoretical scaling analysis leads us to experimentally validated power laws, in which the exponent depends on the flow regime, and the multiplicative coefficient depends on channel geometry alone. Finally, we discuss how the differences between theoretical predictions and experimental data may be useful for improving our understanding of flows through non-uniform channels.

A continuum model of discrete granular avalanches

“…These include flows over rigid boundaries (Edwards et al 2019;Lin & Yang 2020) and flows over erodible layers. The latter include steady channel flows (Jop, Forterre & Pouliquen 2005;Liu & Henann 2017;Berzi, Jenkins & Richard 2020), transient channel flows (Jop, Forterre & Pouliquen 2007;Capart, Hung & Stark 2015;Parez, Aharonov & Toussaint 2016;Larcher, Prati & Fraccarollo 2018; Capart 2023) and steady, continuous avalanching in rotating drums and draining silos (Hung, Stark & Capart 2016;Hung, Aussillous & Capart 2018;Hung et al 2023). Important flows over erodible beds that have so far resisted this approach, however, are the discrete avalanches produced in very slowly rotated drums (Evesque 1991;Courrech du Pont et al 2005;Liu, Zhou & Specht 2010;Balmforth & McElwaine 2018), in gradually tilted sand boxes (Jaeger, Liu & Nagel 1989;Evesque et al 1993;Aguirre et al 2000) or down continuously supplied sand piles (Lemieux & Durian 2000;Altshuler et al 2008;Arran & Vriend 2018;Alonso-Llanes et al 2022).…”

“…The latter include steady channel flows (Jop, Forterre & Pouliquen 2005; Liu & Henann 2017; Berzi, Jenkins & Richard 2020), transient channel flows (Jop, Forterre & Pouliquen 2007; Capart, Hung & Stark 2015; Parez, Aharonov & Toussaint 2016; Larcher, Prati & Fraccarollo 2018; Capart 2023) and steady, continuous avalanching in rotating drums and draining silos (Hung, Stark & Capart 2016; Hung, Aussillous & Capart 2018; Hung et al. 2023). Important flows over erodible beds that have so far resisted this approach, however, are the discrete avalanches produced in very slowly rotated drums (Evesque 1991; Courrech du Pont et al.…”