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 Q and the orifice width w: Q ∼ (w/σ_avg - k), where σ_avg is the average particle diameter, kσ_avg is an offset where Q ∼ 0, the power-law scaling exponent β = d - 1/2, and d 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 β. 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 β varies continuously with the ratio of the viscous drag to the kinetic friction coefficient, λ = ζ/μ. β = d - 1/2 in the λ → 0 limit and d - 3/2 in the λ → ∞ limit, with a midpoint λ_c that depends on the hopper opening angle θ_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 β. The offset k increases with particle stiffness until k ∼ k_max in the hard-particle limit, where k_max ∼ 3.5 is larger for λ → ∞ compared to that for λ → 0. Finally, we show that the simulations of hopper flows of deformable particles in the λ → ∞ limit recapitulate the experimental results for quasi-2D hopper flows of oil droplets in water.