Stokes' Law Calculator (Terminal Settling Velocity)
By Santaji GadepatilinUpdated October 30, 2026
Stokes' law describes how fast a small sphere sinks or rises through a viscous fluid once drag balances its weight and buoyancy. This Stokes' law calculator gives the terminal settling velocity, drag force and particle Reynolds number — useful for sedimentation tanks, dust settling, viscosity measurement with a falling-ball viscometer and emulsions.
How to use the Stokes’ law calculator
Enter the particle diameter, the particle density, the fluid density and the fluid’s dynamic viscosity. The calculator returns the terminal settling velocity (negative means the particle rises), the drag force at that speed and the particle Reynolds number — keep it below about 1 for Stokes’ law to apply. To find viscosity from a falling-ball test, clear the viscosity and enter the measured velocity.
Stokes’ law formula
vt = (ρp − ρf) g d² ÷ (18 μ)
Fd = 3 π μ d v · Rep = ρf v d ÷ μ
Example
A 50 µm quartz sand grain (2,650 kg/m³) settling in water at 20 °C (998 kg/m³, 1.0 mPa·s):
- v = (2650 − 998) × 9.807 × (50 × 10⁻⁶)² ÷ (18 × 0.001) = 0.002250 m/s (2.25 mm/s)
- Re = 998 × 0.002250 × 50 × 10⁻⁶ ÷ 0.001 = 0.112 — below 1, so Stokes’ law applies
- Drag force = 3π × 0.001 × 50 × 10⁻⁶ × 0.002250 = 1.060 × 10⁻⁹ N, equal to the grain’s submerged weight
Settling of quartz particles in water at 20 °C
| Diameter | Settling velocity | Reynolds number | Time to settle 1 m |
|---|---|---|---|
| 1 µm | 0.9 µm/s | 9.0 × 10⁻⁷ | 309 h |
| 5 µm | 22.5 µm/s | 1.1 × 10⁻⁴ | 12.3 h |
| 10 µm | 90 µm/s | 9.0 × 10⁻⁴ | 3.09 h |
| 20 µm | 360 µm/s | 7.2 × 10⁻³ | 46.3 min |
| 50 µm | 2.25 mm/s | 0.11 | 7.41 min |
| 100 µm | 9 mm/s | 0.9 | 1.85 min |
At 100 µm the Reynolds number approaches 1, so Stokes’ law starts to overpredict the velocity of larger grains.
Notes
- Viscosity changes strongly with temperature — water at 5 °C is about 1.5 mPa·s, slowing settling by a third.
- Non-spherical particles settle more slowly than the equivalent sphere.
- For larger objects, use the drag force calculator with a drag coefficient instead.
Frequently asked questions
What is Stokes' law for settling velocity?
v = (ρp − ρf) g d² ÷ (18 μ). Velocity grows with the square of particle diameter, so a particle twice as large settles four times faster.
When is Stokes' law valid?
For creeping flow around a smooth sphere — particle Reynolds number below about 1 (strictly below 0.1 for high accuracy). Larger or faster particles need a drag coefficient correlation instead.
What does a negative velocity mean?
The particle is lighter than the fluid, so it rises instead of sinking — for example oil droplets or air bubbles in water.
How is Stokes' law used to measure viscosity?
In a falling-ball viscometer a ball of known size and density falls a measured distance; rearranging gives μ = (ρp − ρf) g d² ÷ (18 v). Clear the viscosity field and enter the measured velocity.