Asymptotic analysis for the propagation and arresting process of a finite dry granular mass down a rough incline

This work presents an asymptotic analysis for the propagation and arresting process of a two-dimensional finite granular mass down a rough incline in a shallow configuration. Bulk shear stress and arresting mechanism are formulated according to the coherence length model that considers momentum transport at a length scale over which grains are spatially correlated. A Bagnold-like streamwise velocity and a non-zero transverse velocity are solved and integrated into a surface kinematic condition to give an advection–diffusion equation for the bulk surface profile, $h(x,t)$, that is solved using the matched asymptotic method. These flow solutions are further employed to determine composite solutions for a flow-front trajectory and a local coherence length, $l(x,t)$, which reveals smooth growth of $h(x,t)$ and $l(x,t)$ from zero at the propagating front with $l(x,t)\ll h(x,t)$. At the rear, $h(x,t)$ vanishes but $l(x,t)$ asymptotes to a constant that depends on inclination angle. According to the arresting mechanism, the location where $l(x,t)\sim h(x,t)$ is solved to the leading order to locate the deposition front so that its propagation dynamics can be derived. A finite flow arrest time, $T_{d}$, and the corresponding finite run-out distance, $L_{d}$, are evaluated when all the flowing mass has passed the deposition front and are employed to construct a modified front trajectory with the deposition effect. The predicted run-out distance and front trajectory profile compare reasonably well with experimental data in the literature on inclinations at angles higher than the material repose angle.