Flow rule of dense granular flows down a rough incline

We present experimental findings on the flow rule for granular flows on a rough inclined plane using various materials including sand and glass beads of various sizes and four types of copper particles with different shapes. We characterize the materials by measuring $h_s$ (the thickness at which the flow subsides) as a function of the plane inclination $\theta$ on various surfaces. Measuring the surface velocity $u$ of the flow as a function of flow thickness $h$, we find that for sand and glass beads the Pouliquen flow rule $u/\sqrt{gh} \sim \beta h/h_s$ provides reasonable but not perfect collapse of the $u(h)$ curves measured for various $\theta$ and mean particle diameter $d$. Improved collapse is obtained for sand and glass beads by using a recently proposed scaling of the form $u/\sqrt{gh} =\beta \cdot h \tan^2\theta /h_s\ \tan^2\theta_1$ where $\theta_1$ is the angle at which the $h_s(\theta)$ curves diverge. Measuring the slope $\beta$ for ten different sizes of sand and glass beads, we find a systematic, strong increase of $\beta$ with the divergence angle $\theta_1$ of $h_s$. The copper materials with different shapes are not well described by either flow rule with $u \sim h^{3/2}$.