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 $\mathcal{I}$, 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 $10^{-4} \lesssim {\cal I} \lesssim 10^{-1}$: $\mathit{\text{frictionless,}}\mathit{\text{frictional}}\mathit{\text{sliding,}}$ 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 $\mathcal{I}$. 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 ${\cal I}\geq 10^{-2.5}$thatthegrowthofthemacroscopicfriction$\mu({\cal I})$with${\cal I}$ is induced by an increase of collisional dissipation. This implies in that range that $\mu (\mathcal{I})-\mu (0)\sim {\mathcal{I}}^{1-2b}$, where $b\approx 0.2$ is an exponent that characterizes both the dimensionless velocity fluctuations $\mathcal{L}\sim {\mathcal{I}}^{-b}$ and the density of sliding contacts $\chi \sim {\mathcal{I}}^b$.