Stokes' Drag (F = 6πμrv)

At the scale of dust, droplets and cells, a fluid behaves like honey. Inertia is irrelevant — a particle that stops pushing stops moving almost instantly — and the drag comes entirely from viscous shear, growing linearly with speed, viscosity and radius: $F = 6\pi\mu r v$. This is a different regime, not an approximation of the familiar $\tfrac{1}{2}\rho v^2 A C_d$ drag, which is quadratic in speed and dominated by inertia. Which one applies is decided by the Reynolds number, and the crossover is what makes small things behave so unlike large ones.

The linearity is what produces a terminal settling velocity. Set the drag equal to the submerged weight, $6\pi\mu r v = (\rho_p - \rho_f)\tfrac{4}{3}\pi r^3 g$, and solve: $v = 2(\rho_p-\rho_f)gr^2/9\mu$. A 20 μm silt grain (ρ = 2650 kg/m³) in 20 °C water settles at $3.6\times10^{-4}$ m/s — about 1.3 m per hour. That single number is why a sedimentation basin is sized for hours of detention rather than minutes. And because velocity goes as $r^2$, a 2 μm particle ten times smaller settles a hundred times slower, roughly 13 mm/h, which in any real basin means never. That is the entire justification for coagulation and flocculation: not to make particles heavier, but to make them bigger, because the square is the only lever that matters.

George Gabriel Stokes derived the result in 1851 by solving the flow equations with the inertial terms deleted — legitimate precisely because they are negligible here. Its most famous use came sixty years later, when Millikan balanced charged oil droplets between electric plates and used this law to infer their size from how fast they fell, thereby weighing them and extracting the charge of the electron. The same equation sets centrifuge protocols in the lab, governs how long fine dust hangs in air, and explains why a bacterium swimming is closer to a person swimming in tar than to a person swimming in water.

The range restriction is severe and routinely ignored: Stokes' law wants a particle Reynolds number below about 1, and preferably below 0.1. Beyond that the real drag exceeds what the equation predicts, so it overestimates settling velocity — the failure is in the optimistic direction, which is the worst way for a design equation to fail. That 20 μm silt grain sits at Re ≈ 0.007 and is fine. A 200 μm sand grain computes to 36 mm/s at Re ≈ 14, well outside the range, and settles noticeably slower than the formula claims. Check the Reynolds number of the particle, not of the pipe.

Then the substitution errors. The driving force is the submerged weight, so the density term is the difference $(\rho_p - \rho_f)$, never the particle density alone — buoyancy has already taken its share, and forgetting it inflates a silt calculation by 60%. The $r$ is a radius, and using a diameter gives an answer four times too large in the drag and four times too small in the settling velocity. μ is dynamic viscosity in Pa·s, not kinematic viscosity in centistokes. And the derivation assumes a single rigid sphere alone in an unbounded fluid, which real basins are not: flocs are neither spherical nor rigid, and at any appreciable solids concentration the particles interfere with one another and settle as a hindered mass, slower than any individual calculation predicts.