Stokes Settling Velocity

Balance a sphere's submerged weight against the viscous drag on it and you get Stokes' 1851 result: terminal velocity rises with the square of diameter and with the density difference, and falls with viscosity. A 0.1 mm sand grain of density 2650 kg/m³ in 20 °C water settles at 0.0090 m/s — 776 metres a day — which is why grit chambers designed to capture 0.2 mm sand at 0.02 m/s work so easily, and why the 0.01 mm silt beside it, settling a hundred times slower, sails straight through.

The squared diameter is the whole argument for coagulation. Clay colloids at 1 μm settle roughly 0.0008 m per day and would take years to reach the floor of any basin; bind ten thousand of them into a 1 mm alum floc and, even at a floc density barely above water, they reach the sludge blanket in minutes. Two limits matter. Stokes' law only holds while the particle Reynolds number stays below about 1, which for sand in water means diameters under roughly 0.1 mm — coarser grit is in the transition regime and settles slower than this equation predicts. And viscosity is strongly temperature-dependent: water at 4 °C is 1.6 times as viscous as at 25 °C, so the same clarifier settles about 40% slower in winter, which is why cold-weather turbidity breakthrough is a seasonal, not a mysterious, event.