Hopper flows of deformable particles.

Numerous experimental and computational studies show that continuous hopper flows of granular materials obey the Beverloo equation that relates the volume flow rate and the orifice width : ∼ (/ - ), where is the average particle diameter, is an offset where ∼ 0, the power-law scaling exponent = - 1/2, and is the spatial dimension. Recent studies of hopper flows of deformable particles in different background fluids suggest that the particle stiffness and dissipation mechanism can also strongly affect the power-law scaling exponent . We carry out computational studies of hopper flows of deformable particles with both kinetic friction and background fluid dissipation in two and three dimensions. We show that the exponent varies continuously with the ratio of the viscous drag to the kinetic friction coefficient, = /. = - 1/2 in the → 0 limit and - 3/2 in the → ∞ limit, with a midpoint that depends on the hopper opening angle . We also characterize the spatial structure of the flows and associate changes in spatial structure of the hopper flows to changes in the exponent . The offset increases with particle stiffness until ∼ in the hard-particle limit, where ∼ 3.5 is larger for → ∞ compared to that for → 0. Finally, we show that the simulations of hopper flows of deformable particles in the → ∞ limit recapitulate the experimental results for quasi-2D hopper flows of oil droplets in water.