Velocity profile of granular flows inside silos and hoppers

摘要

We measure the flow of granular materials inside a quasi-two dimensional siloas it drains and compare the data with some existing models. The particlesinside the silo are imaged and tracked with unprecedented resolution in bothspace and time to obtain their velocity and diffusion properties. The dataobtained by varying the orifice width and the hopper angle allows us tothoroughly test models of gravity driven flows inside these geometries. All ofour measured velocity profiles are smooth and free of the shock-likediscontinuities ("rupture zones") predicted by critical state soil mechanics.On the other hand, we find that the simple Kinematic Model accurately capturesthe mean velocity profile near the orifice, although it fails to describe therapid transition to plug flow far away from the orifice. The measured diffusionlength $b$, the only free parameter in the model, is not constant as usuallyassumed, but increases with both the height above the orifice and the angle ofthe hopper. We discuss improvements to the model to account for thedifferences. From our data, we also directly measure the diffusion of theparticles and find it to be significantly less than predicted by the VoidModel, which provides the classical microscopic derivation of the KinematicModel in terms of diffusing voids in the packing. However, the experimentaldata is consistent with the recently proposed Spot Model, based on a simplemechanism for cooperative diffusion. Finally, we discuss the flow rate as afunction of the orifice width and hopper angles. We find that the flow ratescales with the orifice size to the power of 1.5, consistent with dimensionalanalysis. Interestingly, the flow rate increases when the funnel angle isincreased.