New aspects for friction coefficients of finite granular avalanche down a flat narrow reservoir

Abstract

This work examines the bulk internal friction coefficient, \(\mu \), and effective wall friction coefficient, \(\mu _w\), for finite number of nearly identical dry glass spheres in avalanche down a narrow inclined reservoir of smooth frictional bed using a validated discrete element scheme. Instantaneous deviatoric strain rate tensor \(\dot{\gamma }^d_{ij}\) and stress tensor \(\tau _{ij}\) are computed locally to evaluate a three-dimensional constitutive model developed based on the rheology of steady homogeneous surface flows. On one side, the algebraic \(\mu -I\) relation conforms to conventional relation for glass beads, \(\mu =0.34+0.31/(1+0.15/I)\) (Jop et al. in J. Fluid Mech. 541:167–192, 2005, Midi in Eur. Phys. J. E 14:341–365, 2004, Jop in Comptes Rendus Phys. 16:62–72, 2015), when the inertial number \(I>I_{c}=2\times 10^{-2}\). The assumption of collinear \(\tau _{ij}\) and \(\dot{\gamma }^d_{ij}\), however, does not hold and such misalignment agrees to the findings in non-uniform inhomogeneous flows (Cortet et al. in Europhys. Lett. 88(1):14001, 2009). Below \(I_c\), we observe a decaying \(\mu -I\) as found in slowly deforming rheology tests and a simplified model is developed in view of shear-induced dilatation upon yielding. Non-constant effective wall friction coefficient is measured to grow in time and with I towards the sphere-wall sliding friction coefficient in the contact model while preserving the depth-weakening feature as in confined steady surface flows (Richard et al. in Phys. Rev. Lett. 101:248002, 2008, Brodu et al. in Phys. Rev. E 87:022202, 2013). The fact that rotation at one sphere center can divert surface relative velocity across the contact area to render lower sliding friction is considered to develop a model describing how \(\mu _w\) drops with the ratio between rotation-induced velocity and sliding velocity, \(\varOmega \). The simulation data compares fairly well to the predicted monotonic decay of \(\mu _w\) with \(\varOmega \).