Why sand drains like a clock, not a liquid.
-Beverloo mass flow W (g/s)
-DEM flux (grains per time unit)
-Hydrostatic rho g H (kPa)
-Janssen saturation pressure (kPa)
-Emptying time of the silo (s)
-Orifice state
The science
Granular matter does not drain like water. Friction against the walls carries most of the weight of the bed, so the pressure at the bottom saturates (Janssen, 1895): p(h) = rho g lambda (1 - exp(-h/lambda)) with lambda = D_c / (4 mu K), where K is the lateral stress ratio (about 0.4). Near the top it matches hydrostatics, p ~ rho g h.
Because the pressure at the outlet does not depend on the height, the mass flow does not either. Beverloo's correlation for a 3D hopper is W = C rho sqrt(g) (D - k d)^(5/2) with C ~ 0.58, k ~ 1.4: grains of diameter d effectively see an orifice smaller by about 1.4 diameters, and the flow scales with the 5/2 power (3/2 in 2D).
The animated grains are a real discrete-element simulation: 2D discs with a spring-dashpot normal force F_n = k delta - c v_n and Coulomb friction limited to mu F_n, in units where the grain diameter and gravity are 1. Narrow orifices (D/d below about 3) form stable arches and clog.
Honest simplifications: the disc simulation is 2D and in dimensionless units, so its flux is compared in trend, not in kg/s; the tabulated W uses the standard 3D Beverloo constants and ignores fines, humidity and air drag.