Revealing a hidden compressible nonlocal parameter for granular flows down an incline via asymptotic analysis

Asymptotic analysis is applied to solve steady granular flows down an incline using a nonlocal gradient expansion model (Bouzid et. al) with dilatancy effect (Dsouza & Nott). The y-momentum equation is solved with both shearing and non-shearing boundary conditions for the solid volume fraction φ(y). The inner solution at the base is matched to the outer solution at the surface. The x-momentum equation is handled similarly with a strain-free surface and a non-solid yet no-slip base. Special care was made to ensure continuity of velocity u(y), shear rate, and shear work at the matching point. Both solutions are validated numerically. Our solution for φ(y) replicates the literature trend: a nearly constant layer is decayed rapidly near the free surface. The predicted u(y) also captures a diversion from the Bagnold velocity profile with a magnitude decay to show the nonlocal effects. Intriguingly, the degree of nonlocality and compressibility is characterized by a newly discovered dimensionless nonlocal parameter, η, that compares the effects of nonlocal spatial correlation, effective gravity, and compressibility modification. Flow behaviors with vanishing or nearly-infinite η are analyzed to demonstrate the solution robustness while gaining insights.