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 $\chi\sim {\cal I}^b$.