Linear stability analysis of rapid granular flow on a slope and density wave formation

TL;DR

This paper investigates the linear stability of rapid granular flows on slopes, revealing conditions under which density waves form due to long-wavelength instabilities, supported by hydrodynamic analysis and previous simulations.

Contribution

It provides a hydrodynamic stability analysis explaining density wave formation in granular flows, contrasting with earlier finite wavelength instability studies.

Findings

Long-wavelength instability leads to density wave formation.

Flow becomes unstable against longitudinal perturbations under certain parameters.

Results align with previous molecular dynamics simulations.

Abstract

The linear stability of rapid granular flow on a slope under gravity against the longitudinal perturbation is analyzed using hydrodynamic equations. It is demonstrated that the steady flow uniform along the flow direction becomes unstable against the long-wavelength perturbations longitudinal to the flow direction for certain parameter ranges to form the density wave, in contrast with the finite wavelength instability against the transverse perturbation (Forterre & Pouliquen 2002). It is shown that the instability can be understood as the the long-wave instability of the kinematic waves in a quasi-one dimensional system. The results are compared with our previous molecular dynamics simulations (Mitarai & Nakanishi 2001), where the spontaneous density wave formation has been found.