Hydrodynamic effects in density waves of granular flows

Granular flows in a narrow pipe are studied numerically by the model taking account of hydrodynamic effects of fluid surrounding particles. In the simulations density waves are observed over the wide range of the Stokes number, which represents the inertial effect of particles. The mechanisms of formation of the density waves are considered and two types of density waves, which are observed in the systems with zero and finite Stokes numbers, are presented. Power spectra of density waves obtained from the simulations with non-zero Stokes number show $1/f^\alpha$ power law which is similar to that in experiments.