Hong Life CalculatorsUnderstanding the Stokes' law settling velocity calculator — concept and principle
Stokes' law gives the steady speed at which a small sphere sinks (or rises) through a still fluid once gravity, buoyancy and viscous drag are in balance. Enter the particle diameter, particle and fluid densities and the fluid viscosity, and the calculator returns the settling velocity in mm/s and the particle Reynolds number that checks whether the law applies. It is used for sedimentation tanks, soil particle-size analysis, aerosols and centrifuges.
How it is calculated
- v = (ρ_p − ρ_f) · g · d² ÷ (18μ), with d converted from µm to m, μ from mPa·s to Pa·s and g = 9.80665 m/s². The result is shown in mm/s.
- Particle Reynolds number: Re_p = ρ_f · v · d ÷ μ. The law is reliable only when Re_p is well below 1.
The principle
A falling sphere accelerates until the forces on it cancel. Its weight minus buoyancy is (ρ_p − ρ_f)·g·πd³/6. George Gabriel Stokes showed in 1851 that at very low Reynolds number the drag on a sphere is F = 3πμdv. Setting the two equal and solving for v gives the formula. Because drag grows with d but the driving force grows with d³, settling speed rises with the square of diameter. When Re_p approaches 1 or more, inertia in the surrounding flow adds extra drag and Stokes' law overestimates the speed.
Worked example
- Inputs: d = 50 µm = 5 × 10⁻⁵ m, ρ_p = 2650 kg/m³ (quartz), ρ_f = 998 kg/m³ (water), μ = 1.0 mPa·s.
- Density difference times g: (2650 − 998) × 9.80665 = 16,200.6 N/m³.
- d² = 2.5 × 10⁻⁹ m²; 18μ = 0.018 Pa·s.
- v = 16,200.6 × 2.5 × 10⁻⁹ ÷ 0.018 = 2.250 × 10⁻³ m/s = 2.250 mm/s (shown as 2.250081361).
- Re_p = 998 × 0.00225 × 5 × 10⁻⁵ ÷ 0.001 = 0.112, below 1, so the law is valid.
| Particle diameter (µm) | Settling velocity (mm/s) |
|---|---|
| 12.5 | 0.1406300851 |
| 25 | 0.5625203403 |
| 37.5 | 1.265670766 |
| 50 | 2.250081361 |
| 62.5 | 3.515752127 |
| 75 | 5.062683062 |
| 100 | 9.000325444 |
How to read the result
A 50 µm grain (coarse silt, just below the 63 µm sand limit) falls 1 m of water in about 7.4 minutes. A 10 µm particle settles 25 times slower, about 0.09 mm/s, and clay-sized particles below 2 µm take days, which is why fine suspensions stay cloudy and why centrifuges are used. Warmer water settles particles faster because its viscosity is lower.
Common mistakes
- Entering the radius instead of the diameter (the result is four times too small).
- Using the particle density alone instead of the density difference.
- Ignoring Re_p; large sand grains settle more slowly than Stokes' law predicts.
Frequently asked questions
What happens if the particle is lighter than the fluid?
The velocity comes out negative: the particle rises, as oil droplets or bubbles do.
What should I do when Re_p is above 1?
Use a drag-coefficient correlation (for example Schiller-Naumann) and solve iteratively for the terminal velocity.
How is Stokes' law used in a centrifuge?
Replace g with the centrifugal acceleration ω²r, which can be thousands of times larger.
Sources: G. G. Stokes, Transactions of the Cambridge Philosophical Society 9 (1851) 8–106 · ISO 14688-1, Geotechnical investigation and testing — Identification and classification of soil · F. M. White, Viscous Fluid Flow, McGraw-Hill