Steady Flow and its Instability of Gravitational Granular Flow

Granular matter consists of a large number of macroscopic particles; the thermal noise has no effect on the particle motion and the interactions are dissipative. These features make the granular flow very different from molecular fluids. One of the simplest situations to see the complex behavior of granular flow is the gravitational granular flow on a slope. When the inclination angle is small enough, the material stays at rest. The material begins to flow beyond a critical angle; the interaction between the particles is dominated by sustained contacts in the dense and slow flow for the small inclination angle, while the low-density rapid flow is realized for the large enough inclination, where the interaction is dominated by inelastic collisions. Understanding these flowing behaviors is a challenging problem of non-equilibrium statistical physics, not to mention its technological importance. In this thesis, we investigate the fundamental properties of gravitational granular flow. We study the steady flow and its instability by molecular dynamics simulations, and analyze the steady flow by a hydrodynamic model. Firstly, we clarify the difference between the low-density collisional flow and the dense frictional flow. In the molecular dynamics simulation of the granular material, the soft sphere model with elastic force and dissipation is often used. We examine how the dynamical behavior of steady granular flow changes in the inelastic hard sphere limit of the soft sphere model with keeping the restitution coefficient constant. We find distinctively different limiting behaviors for the two flow regimes, i.e., the collisional flow and the frictional flow. In the collisional flow, the hard sphere limit is straightforward; the number of collisions per particle per unit time converges to a finite value and the total contact time fraction with other particles goes to zero. For the frictional flow, however, we demonstrate that the collision rate diverges as a power of the particle stiffness so that the time fraction of the multiple contacts remains finite even in the hard sphere limit although the contact time fraction for binary collisions tends to zero. The second subject is to test the applicability of a continuum model to the collisional granular flow in which spinning motion of each grain is considered. In granular material, the angular momentum of the spinning motion is not always negligible because of their macroscopic size. It is known that the mean spin often deviates from the vorticity of the mean velocity near the boundary. Such a deviation may affect the flow behavior through the coupling of the particle spin and the velocity field. We apply the micropolar fluid model to the collisional flow, which is a continuum model in which the angular velocity field is considered as well as the density and the velocity fields. We demonstrate that, using a simple estimate for the parameters in the theory, the model equations quantitatively reproduce the velocity and the angular velocity profiles obtained from the numerical simulation of the rapid granular flow on a slope. Finally, we investigate the steady granular flow on a slope and its instability in the low-density regime using the molecular dynamics simulation. We determine the parameter region where the steady collisional flow is realized upon changing the inclination angle and the density of particles. Then we demonstrate that, when the system size is long enough, the collisional granular flow shows clustering instability. It is shown that the uniform flow is less stable for longer system size and/or for lower density.