Power-Law Scaling in Granular Rheology across Flow Geometries

Based on discrete element method simulations, we propose a new form of the constitutive equation for granular flows independent of packing fraction. Rescaling the stress ratio $\mu$ by a power of dimensionless temperature $\mathrm{\Theta }$ makes the data from a wide set of flow geometries collapse to a master curve depending only on the inertial number $I$. The basic power-law structure appears robust to varying particle properties (e.g., surface friction) in both 2D and 3D systems. We show how this rheology fits and extends frameworks such as kinetic theory and the nonlocal granular fluidity model.

Figure 1

Planar shear geometries tested: (a) simple shear, (b) shear with gravity, (c) chute flows ($\theta =60°$ and 90°), and (d) concave flows. Dashed lines indicate schematic velocity profiles. Red particles’ velocity is imposed to form rigid walls.

Figure 2

Relations between $\mu$, $I$, and $\mathrm{\Theta }$ in various planar shear configurations. Noncollapse of $\mu$ vs $I$ in 3D (a) and 2D (e), and similar noncollapse of $\mathrm{\Theta }$ vs $I$ (b),(f). Dashed lines in (a) and (b) are ${\mu }_{\mathrm{loc}}(I)$ and ${\mathrm{\Theta }}_{\mathrm{loc}}(I)$ in Eq. (2), respectively. Multiplying $\mu$ by ${\mathrm{\Theta }}^p$ makes the scattered points collapse onto a master curve $f(I)$ (dashed trend lines for ${\mu }_p=0.4$ and solid trend lines for ${\mu }_p=0.1$). In 3D, $p=1/6$ (c),(d) and, in 2D, $p=1/8$ (g),(h). The surface friction does not appear to change the exponent but changes the master curve (d),(h).

Figure 3

(a) Inclined chute geometry with no-slip sides and a rough floor; DEM velocity pictured. Red particles fixed. (b) The distribution of $\log \mathrm{\Theta }$. (c) Comparing the $\mu (I,\mathrm{\Theta })$ trend line previously obtained in planar shear tests to this geometry. (d) Comparing velocity from continuum model solutions to DEM data, viewed down the chute’s center-plane (fixed $y$). (e) Comparing DEM velocity contours to those obtained from solving the $\mu (I)$ relation, and (f) the $\mu (I,\mathrm{\Theta })$ relation. The unit of velocity is $(|\overrightarrow{G}|H{)}^{1/2}$ where $H=20d$. All figures are for $\tan \theta =0.5$ except (c). See Supplemental Material for $\tan \theta =0.6$ case.

Figure 4

DEM data in various planar shear flows of 3D spheres with ${\mu }_p=0.4$. (a) $\varphi$ decreases linearly with $\mu$ for $\varphi \gtrsim 0.5$: $\varphi \approx 0.69-0.27\mu$ (dashed line). (b) Our findings predict $\tilde{g}=gd/\delta v\sim (0.69-\varphi {)}^2$ (dashed parabola) for ${10}^{-2.5}\lesssim I\lesssim {10}^{-1}$ and $\tilde{g}\sim \text{constant}$ for ${10}^{-1}\lesssim I\lesssim 1$ (horizontal dashed line).