Flow rule for unsteady flows of spherical and non-spherical grains down rough inclined planes

Based on laboratory experiments, Pouliquen (1999) uncovered a universal scaling law for the average velocity v of homogeneous flows of spherical grains down rough inclines [1]: , where g is the gravitational acceleration, h the flow thickness, and h_s(θ) the thickness below which the flow stops depending on the inclination angle θ. Today, this so-called “flow rule” is well established in the field and has served as a critical test for continuum granular flow models [2]. However, based on more accurate measurements for granular materials composed of either spherical or non-spherical grains, Börzsönyi and Ecke (2007) found and pointed out that this revised flow rule was predicted by a two-dimensional granular kinetic theory [3, 4]. In addition, for non-spherical grains, they noticed deviations from this rule at large h/h_s. Both Pouliquen and Börzsönyi and Ecke assumed that the granular flows in their experiments were steady.

Here, we report on new systematic experiments for granular materials composed of spherical glass beads, different kinds of non-spherical sands, and grain-size-equivalent mixtures of these. Their careful analysis reveals a new grain-shape-dependent flow rule that resolves the above contradictions in the current literature and provides quantitative evidence for the statement that the deviations observed by Börzsönyi and Ecke can be attributed to the flows not having reached the steady state.

[1] Pouliquen O. Scaling laws in granular flows down rough inclined planes[J]. Physics of fluids, 1999, 11(3): 542-548.

[2] Kamrin K, Henann D L. Nonlocal modeling of granular flows down inclines[J]. Soft matter, 2015, 11(1): 179-185.

[3] Börzsönyi T, Ecke R E. Flow rule of dense granular flows down a rough incline[J]. Physical Review E, 2007, 76(3): 031301.

[4] Jenkins J T. Dense shearing flows of inelastic disks[J]. Physics of Fluids, 2006, 18(10).