Stokes Law Settling Velocity Calculator

George Stokes derived the drag force on a small sphere moving slowly through a viscous fluid in 1851. The result was elegant: drag is proportional to radius, velocity, and viscosity, and to nothing else at all.

Stokes drag force

F_drag = 6 pi eta r v

Where eta is dynamic viscosity (Pa·s), r is sphere radius (m), and v is velocity (m/s).

Terminal (settling) velocity

At terminal velocity, drag equals buoyant-weight:

v_t = 2 r^2 (rho_p - rho_f) g / (9 eta)

Where rho_p is particle density, rho_f is fluid density, and g = 9.81 m/s^2.

When Stokes law applies

The formula is valid when the Reynolds number Re = rho_f v r / eta is much less than 1. This means small, slow, or in very viscous fluid. For particles above ~100 micrometers settling in air, or above ~1 mm in water, Re exceeds 1 and the quadratic drag law (used in the terminal velocity calculator) is more appropriate.

Practical examples

All three use radius, not diameter, because that is what the formula and the input below both take. Getting that wrong costs you a factor of four.

Dust (1 micrometer radius, density 2500 kg/m^3) settling in air: v_t ≈ 0.30 mm/s. From a 1 metre shelf that is nearly an hour in still air, and rooms are never still, which is why fine dust hangs around for as long as it does. Halve the radius and it takes four hours.

Sand grains (100 micrometer radius, density 2650 kg/m^3) settling in water: Stokes predicts v_t ≈ 36 mm/s. Measured values are closer to 21 mm/s, and the calculator will tell you why: Re comes out around 3.6, well past the Re much-less-than-1 condition, so the real drag is higher than Stokes allows for. This is a good sanity check to run before trusting any sedimentation number.

Red blood cells (4 micrometer radius, density ~1100 kg/m^3) in blood plasma (eta ≈ 0.0027 Pa·s): v_t ≈ 0.97 micrometers/s, which is 3.5 mm per hour. That lands squarely inside the normal erythrocyte sedimentation rate (ESR) of 0 to 20 mm/hr used as a medical test. Inflammation raises the ESR because fibrinogen makes cells stack into rouleaux, and a stack of ten cells behaves like one larger particle. Stokes scales with r squared, so clumping is exactly why the test works.

Millikan oil drop experiment

Robert Millikan used Stokes law in 1909 to measure the charge of individual electrons by balancing gravity against an electric field on tiny oil droplets. The experiment earned him the 1923 Nobel Prize.