Hopper flows of deformable particles.

Abstract

Numerous experimental and computational studies show that continuous hopper flows of granular materials obey the Beverloo equation that relates the volume flow rate <i>Q</i> and the orifice width <i>w</i>: <i>Q</i> ∼ (<i>w</i>/<i>σ</i>_avg - <i>k</i>)^<i>β</i>, where <i>σ</i>_avg is the average particle diameter, <i>kσ</i>_avg is an offset where <i>Q</i> ∼ 0, the power-law scaling exponent <i>β</i> = <i>d</i> - 1/2, and <i>d</i> 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 <i>β</i>. 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 <i>β</i> varies continuously with the ratio of the viscous drag to the kinetic friction coefficient, <i>λ</i> = <i>ζ</i>/<i>μ</i>. <i>β</i> = <i>d</i> - 1/2 in the <i>λ</i> → 0 limit and <i>d</i> - 3/2 in the <i>λ</i> → ∞ limit, with a midpoint <i>λ</i>_c that depends on the hopper opening angle <i>θ</i>_w. We also characterize the spatial structure of the flows and associate changes in spatial structure of the hopper flows to changes in the exponent <i>β</i>. The offset <i>k</i> increases with particle stiffness until <i>k</i> ∼ <i>k</i>_max in the hard-particle limit, where <i>k</i>_max ∼ 3.5 is larger for <i>λ</i> → ∞ compared to that for <i>λ</i> → 0. Finally, we show that the simulations of hopper flows of deformable particles in the <i>λ</i> → ∞ limit recapitulate the experimental results for quasi-2D hopper flows of oil droplets in water.