Stokes' law

Stokes' law is the exact expression for the drag force on a small, slow-moving sphere in a viscous fluid. Its form is startlingly clean — F = 6πηrv, where η is the dynamic viscosity of the fluid, r is the radius of the sphere, and v is the speed. There are no fudge factors, no dimensionless drag coefficients, and no shape parameters: a small enough sphere in a thick enough fluid feels exactly this much drag and no more.

The law was derived by George Gabriel Stokes in 1851 as an exact solution of the Navier-Stokes equations in the low-Reynolds-number limit, where the inertia of the fluid can be ignored relative to its viscosity. It applies when Re = ρ·v·L / η is much less than 1 — which means slow motion, small objects, viscous fluids, or some combination. A grain of pollen falling in still air, a bacterium swimming in water, and a ball-bearing sinking in glycerine are all firmly in the Stokes regime.

Two famous applications followed. First, Robert Millikan's oil-drop experiment (1909) used Stokes' law to measure the terminal velocity of charged oil droplets in air and from it inferred the charge on a single electron — one of the foundational experimental results of atomic physics. Second, Jean Perrin (1908) used Stokes' law to analyse the Brownian motion of suspended particles and produced the first definitive measurement of Avogadro's number, clinching the atomic hypothesis.

Outside its regime of validity the law breaks down badly. By Re ≈ 1 the inertial contribution to drag is comparable to the viscous one, and by Re ≈ 1000 the drag is almost entirely quadratic in velocity and Stokes is off by orders of magnitude. Within its regime, though, it is one of the cleanest exact results in continuum mechanics.