Revisiting dense granular flow dynamics: Compressibility and nonlocality on inclined surfaces

This study presents a compressible nonlocal continuum model for two-dimensional granular flows on inclined surfaces. We derive analytical solutions for the solid volume fraction, ϕ(y*), and the streamwise velocity profile, u*(y*), by employing the method of matched asymptotic expansions. The inner solution for ϕ(y*) is found to be constant at the maximum packing fraction, while the outer solution exhibits a linear decrease, with the matching point determined by mass conservation. Furthermore, we obtain an analytic non-Bagnold u*(y*) that can transit smoothly from the no-slip condition at the rough base to the strain-free condition at the free surface. Not only are the continuity conditions of velocity and shear rate invoked at the matching point, but the continuity of shear work is also introduced as a novel condition. In addition, fitting curves are established for the densest packing volume fraction, ϕmax(θ), and the mean volume fraction, ϕ(θ), based on experimental data, highlighting their dependence on the inclination angle θ. Notably, we reveal that a dimensionless parameter η appears solely in the governing equations of u*(y*). Furthermore, at η ≈ 7.3, a balance is achieved between compressibility and nonlocality.