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 hₛ (the thickness at which the flow subsides) as a function of the plane inclination θ 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∕√gh~βh∕hₛ provides reasonable but not perfect collapse of the $u(h)$ curves measured for various θ and mean particle diameter d. Improved collapse is obtained for sand and glass beads by using a recently proposed scaling of the form $u∕√gh={β}h{{0.2em}{0ex}}{tan}²{{0.2em}{0ex}}{θ}∕{h}ₛ{{0.2em}{0ex}}{tan}²{{0.2em}{0ex}}{{θ}}₁$ where ${{θ}}₁$ is the angle at which the ${h}ₛ({θ})$ curves diverge. Measuring the slope ${β}$ for ten different sizes of sand and glass beads, we find a systematic, strong increase of ${β}$ with the divergence angle ${{θ}}₁$ of ${h}ₛ$. Copper materials with different shapes are not well described by either flow rule with $u{~}{h}3∕2$.