Phase Diagram for Inertial Granular Flows

Flows of hard granular materials depend strongly on the interparticle friction coefficient \(\mu_p\) and on the inertial number \({\cal I}\), which characterizes proximity to the jamming transition where flow stops. Guided by numerical simulations, we derive the phase diagram of dense inertial flow of spherical particles, finding three regimes for \(10^{-4} \lesssim {\cal I} \lesssim 10^{-1}\): \textit{ frictionless, frictional sliding, } and {\it rolling}. These are distinguished by the dominant means of energy dissipation, changing from collisional to sliding friction, and back to collisional, as \(\mu_p\) increases from zero at constant \({\cal I}\). The three regimes differ in their kinetics and rheology; in particular, the velocity fluctuations and the stress ratio both display non-monotonic behavior with \(\mu_p\), corresponding to transitions between the three regimes of flow. We rationalize { the phase boundaries between these regimes}, show that energy balance yields scaling relations { between microscopic properties} in each of them, and { derive the strain scale at which particles lose memory of their velocity. For the frictional sliding regime most relevant experimentally, we find for \({\cal I}\geq 10^{-2.5}\) that the growth of the macroscopic friction \(\mu({\cal I})\) with \({\cal I}\) is induced by an increase of collisional dissipation. This implies in that range that \(\mu({\cal I})-\mu(0)\sim {\cal I}^{1-2b}\), where \(b\approx 0.2\) is an exponent that characterizes both the dimensionless velocity fluctuations \({\cal L}\sim {\cal I}^{-b}\) and the density of sliding contacts \(\chi\sim {\cal I}^b\).