Beverloo law for hopper flow derived from self-similar profiles

We use particle simulations to investigate the mass flow in two-dimensional hopper flow and to analyze the dependency of the flow rate with the bottleneck width and the particle diameter. A flow rate law is derived from self-similar velocity and density profiles at the neck. The resulting relation is an enhancement of the Beverloo relation that incorporates the dependency of the density with the neck width. The parameters of the Beverloo relation are interpreted by coupling the hourglass theory with the free-fall arch theory using non-zero arch velocity as accounted by the hourglass theory.