Rheological response of nonspherical granular flows down an incline

We present an extensive numerical and experimental study, investigating a three-dimensional (3D) granular flow of elongated particles down an inclined plane. Similarly to sheared systems, the average particle orientation is found to enclose a small angle with the flow direction. In the bulk, this behavior is independent of the shear rate. At the surface, however, the particles move in more dilute conditions, and the average orientation strongly depends on the shear rate. A systematic numerical study varying the particle aspect ratio and the plane inclination reveals that the particle size perpendicular to the flow direction, $d_{\mathrm{eff}}$, is an appropriate length scale to define an effective inertial number $I_{\mathrm{eff}}$, which fully captures the impact of the particle shape on the system's rheology. Like in the case of spheres, density and friction result in well-defined functions of the effective inertial number $I_{\mathrm{eff}}$. Thus, we quantify and explain the dependence of the rheological parameters on the aspect ratio, based on the micromechanical details.

Figure 1

Sketch of the experimental setup. Dimensions of the inclined plane: $2\text{m}\times 0.4\text{m}$. Granular material: glass rods with length of $6.6\text{mm}$ and diameter of 1.9 mm. Sample images taken with cameras 1 and 2 and the definitions of the two angles $({\theta }^{\prime }$ and $\varphi )$ are shown.

Figure 2

The surface velocity from laboratory measurements for $\alpha ={26}^{\circ }$ and $\alpha ={33}^{\circ }$ taken at two locations $x=158$ cm (continuous lines) and $x=165$ cm (dashed lines) below the hopper.

Figure 3

Average friction $\overline{\mu }$ as a function of the plane inclination $\alpha$, obtained numerically, in steady-state conditions. Outcomes corresponding to glass particles, several-particle elongation $\xi$, and $g=1m{/}^2$ are shown. The inset illustrates the numerical system.

Figure 4

(a) The average alignment angles ${\theta }_{\text{av}}^{\prime }$ and ${\varphi }_{\text{av}}$ as a function of $\alpha$ at the surface from experiment (empty symbols) and simulation (full symbols). (b) The average surface velocity of the flow as a function of the plane inclination $\alpha$. Experimental data are shown with empty symbols, while the simulation data are indicated with full symbols. All data are obtained with glass particles of aspect ratio $\xi =3.5$.

Figure 5

The average alignment angles ${\theta }_{\text{av}}$ obtained numerically as a function of $\alpha$ for various layers. Red symbols indicate regions at $z<h_m$ and black symbols indicate those at $z>h_m$. The data correspond to glass particles and $g=10\text{m}/{\text{s}}^2$.

Figure 6

Angle of repose ${\alpha }_{c1}$ obtained numerically for glass particles with different elongations $\xi$ and $g=10\text{m}/{\text{s}}^2$. Black markers show the highest angle at which the flow arrests, and red markers show the lowest angle at which steady flow is observed.

Figure 7

Numerical results: the vertical profiles of volume fraction $\varphi \left(z\right)$ (a), macroscopic velocity on the $x$ direction $V_x\left(z\right)$ (b), effective inertial number $I_{\mathrm{eff}}\left(z\right)$ (c), and angular stress eigendirection ${\mathrm{\Phi }}_{\sigma }\left(z\right)$ (d). Results for glass particles and several particle elongations are shown, $\xi \in (1.01,3.0)$; in each figure, different lines correspond to different plane inclinations. Larger inclinations correspond to smaller volume fraction and larger velocity and effective inertial number values.

Figure 8

Numerical results: alignment angle $\theta$, mode, and mean, as functions of particle's aspect ratio $\xi$. The confidential intervals include particles in the bulk region $z\in (2L,h_m-2L)$ and several inclinations. The inset shows the angular shift $\mathrm{\Delta }{\mathrm{\Phi }}_{\sigma \varepsilon }$ between maximal eigendirection of stress and strain, depending on $\xi$. The sketch illustrates the definition of $d_{\text{eff}}$.

Figure 9

Numerical results: (a) Depth-averaged friction $\overline{\mu }$ as a function of depth-averaged inertial number, ${\overline{I}}_{\mathrm{eff}}$. (b) Depth-averaged bulk density from the low-inertia limit $\overline{\rho }-{\rho }_0$, as a function of ${\overline{I}}_{\mathrm{eff}}$. Insets show the dependence of the bulk density at the static limit ${\rho }_0$ and the slope ${\mathrm{\Delta }}_{\rho }$ on the particle aspect ratio $\xi$.

Figure 10

Numerical results, data collapse obtained when plotting the relative friction versus the depth-averaged inertial number ${\overline{I}}_{\mathrm{eff}}$. The continuous lines represent Eq. (7). Results for glass and steel particles with different frictions and $g=1m/s^2$ are shown.