What this calculator does
Stokes' law gives the terminal velocity of a small sphere settling through a fluid, where viscous drag rather than turbulence dominates. It applies to silt settling in water, droplets in air and particles in industrial separation.
The key feature is the square of the diameter. A particle twice as wide settles four times as fast, which is why fine sediment stays suspended for so long while coarse grains drop out almost immediately.
The formula
Multiply gravity by the square of the diameter and by the difference between particle and fluid density, then divide by eighteen times the dynamic viscosity. The result is valid only while the Reynolds number stays low.
| Term | Meaning |
|---|---|
| Stokes' law | The viscous drag relationship for slow flow around a sphere. |
| Density difference | Particle density minus fluid density. A negative value means the particle rises rather than settles. |
| Low Reynolds number | The condition for validity, generally below about 1, where viscous forces dominate over inertia. |
The inputs explained
| Field | What to enter |
|---|---|
| Particle diameter (m) | The particle diameter in metres. 0.0001 m is 100 micrometres, about the size of fine sand. |
| Particle density (kg/m³) | The particle density in kilograms per cubic metre. Quartz sand is about 2,650. |
| Fluid density (water ≈ 1000, air ≈ 1.225) (kg/m³) | The fluid density. Water is about 1,000; air about 1.225. |
| Fluid dynamic viscosity (water ≈ 0.001) (Pa·s) | A number, measured in Pa·s. Starts at 0.001. |
When to use it
Predicting sedimentation
How quickly suspended particles settle determines clarification times in water treatment and in natural water bodies.
Sizing a separation process
Particle size grading by settling speed relies directly on this relationship.
Understanding why dust lingers
Very fine particles settle so slowly that air movement keeps them suspended almost indefinitely.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does particle size change the settling speed?
Particles of increasing diameter in water.
| Particle diameter | Terminal velocity (mm/s) | Reynolds number at this speed |
|---|---|---|
| 50 µm | 2.247 mm/s | 0.1124 (Stokes flow: valid) |
| 100 µm | 8.989 mm/s | 0.8989 (Stokes flow: valid) |
| 200 µm | 35.958 mm/s | 7.192 (Re ≥ 1: Stokes’ law may not hold) |
Questions
Why does size matter so much?
Because weight grows with the cube of diameter while viscous drag grows only with the diameter itself. The ratio between them therefore scales with the square, which is why fine particles settle so disproportionately slowly.
When does Stokes' law stop applying?
Once the Reynolds number rises much above 1, meaning the particle is large or fast enough for inertia to matter. Beyond that the drag relationship changes and the standard drag equation is needed instead.
What if the particle is less dense than the fluid?
The density difference goes negative and the particle rises rather than settles, at the same rate. This is exactly how bubbles and oil droplets behave in water.
Does particle shape matter?
Considerably. The law assumes a sphere, and irregular or flat particles experience more drag and settle more slowly than their equivalent volume sphere would.
For terminal velocity dominated by turbulent drag instead, see the terminal velocity calculator. For the flow regime, see the Reynolds number calculator.