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 ${10}^{-4}\lesssim \mathcal{I}\lesssim {10}^{-1}$: frictionless, frictional sliding, and 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 nonmonotonic 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 $\mathcal{I}\ge {10}^{-2.5}$ that the growth of the macroscopic friction $\mu \left(\mathcal{I}\right)$ with $\mathcal{I}$ is induced by an increase of collisional dissipation. This implies in that range that $\mu \left(\mathcal{I}\right)-\mu \left(0\right)\sim {\mathcal{I}}^{1-2b}$, 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 $\chi \sim {\mathcal{I}}^b$.

Figure 1

Phase diagram of dense homogeneous inertial frictional flow. In the frictionless and rolling regimes, most energy is dissipated by inelastic collisions, while in the frictional sliding regime energy dissipation is dominated by sliding. Along the phase boundary, grains dissipate equal amounts of energy in collisions and in sliding. For $\mathcal{I}\gtrsim 0.1$, one enters the dilute regime [9]. The dashed line has slope 2.

Figure 2

Ratio of dissipation due to sliding, ${\mathcal{D}}_{\mathrm{slid}}$, to dissipation from collisions, ${\mathcal{D}}_{\mathrm{coll}}$, vs ${\mu }_p$. The triangle has slope 1.

Figure 3

Lever $\mathcal{L}$ vs $\mathcal{I}$, (a) for ${\mu }_p\le 0.3$, and (b) for ${\mu }_p\ge 0.3$. The dashed lines are $\propto {\mathcal{I}}^{-1/2}$, while the dotted line is $\propto {\mathcal{I}}^{-0.22}$.

Figure 4

Autocorrelation of particle velocities, $\tilde{C}\left(\epsilon \right)=\langle V_i^y\left(0\right)V_i^y\left(\epsilon \right)\rangle /\langle V_i^y{\left(0\right)}^2\rangle$, for ${\mu }_p=0.3$ and indicated $\mathcal{I}$. (a) $\tilde{C}\left(\epsilon \right)$ vs $\epsilon$. (b) $\tilde{C}\left(\epsilon \right)$ vs $\epsilon /{\epsilon }_v$. In (b), unfilled symbols correspond to strains larger than $0.01$, not used for fitting, and the solid line shows the fitted form.

Figure 5

(a) Decorrelation strain scale, ${\epsilon }_v$, vs $\mathcal{I}$ for selected ${\mu }_p$. The dotted, dashed, and dot-dashed lines are $\propto {\mathcal{I}}^{1.25},{\mathcal{I}}^{1.1}$, and ${\mathcal{I}}^{0.9}$, respectively. (b) The fraction of sliding contacts, $\chi$, vs $\mathcal{I}$, for various ${\mu }_p$ (symbols as in Fig. 3). Dotted, dashed, and dot-dashed lines have slopes 0.43, 0.41, and 0.27, respectively.

Figure 6

(a) Volume fraction $\varphi$ vs $\mathcal{I}$ for various ${\mu }_p$ (symbols as in Fig. 3). (b) Effective friction $\mu$ vs $\mathcal{I}$ (symbols as in Fig. 3). Inset shows nonmonotonic behavior of $\mu \left(\mathcal{I}\right)-{\mu }_c$, for ${\mu }_p=0,0.3,10$. (c) Decomposition of $\mu$ into collisional and frictional components. Both lines have slope $0.6$.

Figure 7

Illustration of geometrical nonlinearity. If the central grain has an unbalanced force as indicated by the arrow, then the ensuing flow will tend to align the contact normals of the dominant contact forces (thick lines) with the unbalanced force; i.e., the angle $\varphi$ will increase. Geometrically, $d\varphi /dt\propto V$, the velocity of the particle.

Figure 8

$N$ dependence of observables. (a) Dissipation ratio vs $\mathcal{I}$. (b) $\chi$ vs $\mathcal{I}$. (c) $\mathcal{L}$ vs $\mathcal{I}$.

Figure 9

$e$ dependence of observables. (a) Dissipation ratio vs $\mathcal{I}$. (b) $\chi$ vs $\mathcal{I}$. (c) $\mathcal{L}$ vs $\mathcal{I}$.

Figure 10

(a) $\mathrm{\Delta }$ dependence of phase diagram for $\mathrm{\Delta }={10}^{-3.8}$ (squares, solid), ${10}^{-2.8}$ (diamonds, dashed), and ${10}^{-1.8}$ (triangles, dash-dotted). (b) Absence of dependence of $\mathcal{L}$ on time step $\mathrm{\Delta }t$ for ${\mu }_p=0.02$.

Figure 11

(a) $\zeta$ vs $\mathcal{I}$. Dotted and dashed lines have slopes 0.43 and 0.31, respectively. (b) ${\mathcal{L}}_T/\mathcal{L}$ vs $\mathcal{I}$. Dotted and dashed lines have slopes $-0.34$ and $-0.28$, respectively.

Figure 12

Autocorrelation of particle velocities, $C\left(\epsilon \right)=\langle V_i^y\left(0\right)V_i^y\left(\epsilon \right)\rangle$, for indicated ${\mu }_p$ and $\mathcal{I}$ from ${10}^{-4.1}$ (down triangles) to ${10}^{-1.3}$ (circles) [symbols as in Fig. 4 of the main text]. (a)–(d) $C\left(\epsilon \right)/C\left(0\right)$ vs $\epsilon$. (e)–(h) $C\left(\epsilon \right)/C\left(0\right)$ vs $\epsilon /{\epsilon }_v$. In (e)–(h), unfilled symbols correspond to strains larger than $0.01$, not used for fitting, and the solid line shows the fitted form.

Figure 13

Strain scale ${\epsilon }_v$ from collapse of $C\left(\epsilon \right)$. Lines as in Fig. 5 of the main text.