Dynamic compressibility of dense granular shear flows

Abstract

It has been conjectured by Bagnold [1] that an assembly of hard non-deformable spheres could behave as a compressible medium when slowly sheared, as the average density of such a system effectively depends on the confining pressure. Here we use discrete element simulations to show the existence of transverse and sagittal waves associated to this dynamical compressibility. For this purpose, we study the resonance of these waves in a linear Couette cell and compare the results with those predicted from a continuum local constitutive relation.Acoustics in granular media is a rapidly developing field that is attractive both from the fundamental point of view and for its applications in material science, soil mechanics and geophysics [2,3]. It provides a unique tool to probe the mechanical response of such materials, to detect heterogeneities, aging or avalanche precursors. In the static case, grain elasticity is the only restoring force; elastic waves present most of the known behaviors of mechanical waves: velocity dispersion, non-linearity associated to contact geometry, scattering associated to disorder, anisotropy, aging, mode coupling, wave-guiding in the presence of macroscopic heterogeneities, fragility, etc [3][4][5][6][7][8]. Recent studies [9][10][11] have focused on the failure of the mean field description of elasticity when approaching the jamming transition separating the solid-like from the liquid-like regime. The importance of non-affine displacements in this limit was revealed by the structure of the vibration spectrum, which exhibits an anomalous excess of low frequency 'soft' modes [12][13][14]. In the following we wish instead to investigate wave propagation on the liquid side of the jamming transition.Wave propagation in granular shear flows have never been studied for itself. It has been revealed indirectly by instabilities, in situations where acoustic waves are spontaneously emitted during a dense granular flow (see [2] for a review): when sufficiently dry, avalanches flowing at the surface of a dune produce a loud sound known as the 'song of dunes ' [15-18]; vibrations are also produced during the discharge of a granular flow from a silo or any elongated tube [19]; soft mono-disperse particles flowing on an inclined plane near the angle of repose exhibit spontaneous oscillations at the resonant frequency of elastic waves [20,21]. These phenomena have mostly been associated to the elastic deformations of the grains. However, Bagnold [1] and followers [2,16,18] have hypothesized the existence of vibrations driven by the coupling between normal stress and shear rate.In this Letter we theoretically demonstrate the existence of such waves in dense slow shear flows of hard non-deformable grains. We show that these waves are due to a compressibility of dynamical origin, whose scaling with the volume fraction φ is related to the non-affine cooperative motion of the grains. The dispersion relation presents three branches, two of which becoming nondissipative when approaching the jamming point. Our a...

Partial regularisation of the incompressible 𝜇(I)-rheology for granular flow

“…In particular, Trulsson et al (2013) showed that acoustic waves could be transmitted through shear flows when compressibility was taken into account and Pouliquen & Forterre (2009) and Kamrin & Henann (2015) were able to model the point of transition between static and moving material on a frictional inclined plane. However, these effects introduce additional complexity into the governing equations and require new numerical schemes to be developed in order to solve them.…”

Effective friction of granular flows made of non-spherical particles

“…More generally, this approach has been successfully applied in different geometries [2,3], e.g. silo discharge [11], granular chute flows [12][13][14][15] and granular column collapse [16,17], as well as to describe dynamic compressibility effects associated with spontaneous oscillatory motion [18,19]. Interestingly, experiments and numerical simulations have recently provided evidence for limie-mail: somfai.ellak@wigner.mta.hu tations of the μ(I)-rheology, when some non-local effects come into play (see e.g.…”

Compressibility regularizes the 𝜇(I)-rheology for dense granular flows

“…Forterre & Pouliquen (2002) showed that kinetic theory taken in the compressible limit could reproduce the instability leading to the longitudinal vortices observed in their experiments. Dense granular flows in the inertial regime might also be strongly affected by acoustic waves, resonances and instabilities due to dynamic density fluctuations as recently observed in numerical and experimental studies (Melosh 1979;Börzsönyi et al 2009;Brodu et al 2013;Trulsson et al 2013;Krishnaraj & Nott 2016). A classic example of such instability is also found in the pulsating flows frequently observed out of silos (Muite et al 2004).…”

“…At this point, it is interesting to estimate the importance of the dynamical pressure developing while shearing a granular media and its possible coupling with the main flow (Trulsson et al 2013). Based on the experimental scaling (3.2) and the definition of the inertial number (3.1), we may express the friction coefficient µ and the pressure at equilibrium p eq as a function of the volume fraction φ:…”