Granular packing is often characterized by a linear relationship between packing fraction $\varphi$ and dimensionless shear rate. We examine this relationship in simple and complex heterogeneous flow geometries. At high inertial number $\left(I\right)$, we observe that the packing fraction depends approximately linearly on $I$ and is seemingly independent of the geometry. However, at low $I$ the packing fraction varies nonlinearly with $I$ and this variation is dependent on the specific geometry and conditions. This response is analogous to the behavior of the stress ratio $\left(\mu \right)$, where, at low $I$, $\mu$ depends on the geometry of the system, an effect that is often attributed to nonlocal effects of the granular rheology. We demonstrate that, in steady isochoric flow geometries, the $\varphi$ and $\mu$ responses are equivalent and $\varphi$ may be determined locally from $\mu$, even at low $I$. However, in a more complex, nonisochoric geometry, such as a pseudo-two-dimensional hopper, this $\varphi \left(\mu \right)$ relationship is not recovered in dense regions. We show that during transient startup in a shear cell $\varphi$ and $\mu$ follow a similar response to that seen in these nonisochoric flows. Hence we conclude that the observed discrepancy arises because the hopper is transient in a Lagrangian sense. The simple $\varphi \left(\mu \right)$ relationship seen in isochoric flows is insufficient in these more complex systems because of differences in the temporal evolution of the stress and packing fraction.

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• Granular packing
• Dry granular materials
• Granular flows
• Molecular dynamics