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The science
Why sand flow is constant
In a column of grains the weight is carried by friction against the walls (Janssen effect), so the pressure at the bottom saturates and does not grow with the fill height. The outflow therefore does not depend on how much sand is above the hole. The empirical Beverloo law gives the mass flow rate W = C ρb √g (D - k d)5/2 with orifice diameter D, grain diameter d, bulk density ρb (1500 kg/m3 here), C ≈ 0.58 and k ≈ 1.4. The emptying time of a mass M is simply T = M / W, and the flow scales as the square root of gravity.
Water is different
For a liquid Torricelli's law applies, v = √(2 g h), so the flow slows as the head h falls and the level decays as h(t) = (√h0 - (a/2A)√(2g) t)2. The dashed curve in the second graph shows this for comparison, matched to the sand flow at t = 0.
The lower pile
Sand lands on a cone whose slope is the angle of repose φ (here 34 degrees): the pile of volume V has radius r = (3V / (π tan φ))1/3 and height r tan φ until it touches the walls. The grains are drawn as random arrivals at the mean rate W / mgrain.
Jamming
If the orifice is only a few grain diameters wide, arches form and the flow stops. The page marks D < 5 d as jammed (a typical empirical threshold); real jamming is probabilistic.
Checks
(1) A Poisson grain-by-grain stream is compared with the continuum rate; (2) the stepped emptying time of the animated state is compared with M / W; (3) an RK4 integration of Torricelli draining is compared with its closed form.
Honest simplifications: schematic conical bulbs (the drawn fill is scaled by mass, not exact glass volume), a constant packing density, a constant angle of repose, no air or electrostatic effects, and no flow-rate change at the very end of the run.
What to try
- Open the orifice from 2 mm to 4 mm: the flow rate grows about 8 times, much faster than the area (4 times).
- Press Moon: the same sand runs through in 2.5 times longer, a square-root law.
- Make the grains coarse (1 mm) with a 3 mm hole and watch the Jammed warning appear.
- Compare the sand and water curves in the middle graph: water starts equal but slows down.