How small particles fall
Drop a pebble in a lake and it sinks in seconds; stir up mud and the water stays cloudy for hours. The difference is size. For small particles, the fluid’s viscosity dominates and they quickly reach a slow, steady terminal velocity. In 1851 the Irish physicist George Gabriel Stokes worked out the drag on a small sphere, and with it the speed at which particles settle.
The formula
v = (ρp − ρf) × g × d² ÷ (18 μ)
- ρp and ρf are the densities of the particle and the fluid.
- g is the acceleration of gravity, 9.81 m/s².
- d is the particle diameter.
- μ is the dynamic viscosity of the fluid: about 1.0 mPa·s for water at 68 °F and 0.018 mPa·s for air.
If the particle is lighter than the fluid, v is negative and the particle rises, like cream in milk or a bubble in water.
Worked example
A grain of quartz sand, 50 µm across, with a density of 2,650 kg/m³, in water at 68 °F:
- Settling velocity: 1,652 × 9.81 × (50 × 10⁻⁶)² ÷ (18 × 0.001) = 2.25 mm/s.
- Time to fall 3 ft: about 407 seconds, or 6.8 minutes.
- Reynolds number: 0.11, well under 1, so the formula is valid.
- Drag force at terminal velocity: about 1.06 × 10⁻⁹ N, equal to the grain’s weight in water.
Size is everything
Because velocity grows with the square of the diameter, small changes in size make huge differences:
| Particle in water | Diameter | Approximate settling speed |
|---|---|---|
| Fine sand | 100 µm | 9 mm/s |
| Very fine sand | 50 µm | 2.2 mm/s |
| Silt | 10 µm | 0.09 mm/s |
| Clay | 2 µm | 0.004 mm/s |
A clay particle takes about two days to fall 2 ft, which is why muddy water clears so slowly and why soil labs use timed settling to measure clay and silt content.
When Stokes’ law fails
The formula assumes a smooth sphere, a fluid at rest, no nearby walls and a particle Reynolds number below about 1. Above that, the flow around the particle stops being purely viscous and real settling is slower than predicted. Irregular grains also settle more slowly than spheres of the same volume. Very small particles, below about 1 µm in air, are jostled by molecules (Brownian motion) and corrections are needed.
Applications
- Water treatment: settling basins are sized so that particles reach the bottom before the water leaves.
- Soil science: the hydrometer test sorts sand, silt and clay by settling time.
- Air quality: a 10 µm water droplet falls about 3 mm/s in still air, while a 1 µm particle falls about a hundred times slower and can stay aloft for hours.
- Physics: Millikan’s oil-drop experiment used Stokes’ law to find the size of the droplets and then the charge of the electron.