Stokes’ Law Terminal Velocity Calculator
Calculate the settling (terminal) velocity of a small sphere in a viscous fluid using Stokes’ law.
How it works
v_t = 2·r²·(ρₚ − ρ_f)·g / (9·μ) · Re = ρ_f·v_t·2r / μ
A small sphere sinking or rising in a fluid quickly reaches a steady terminal velocity where the driving weight (minus buoyancy) is balanced by viscous drag. Stokes’ law captures this for slow, smooth flow: speed grows with the square of the radius and the density difference, and falls as the fluid gets more viscous. It only holds at low Reynolds numbers — larger or faster particles create turbulent wakes that this formula does not model.
Worked example
A 0.1 mm (r = 1×10⁻⁴ m) sand grain (ρ = 2500 kg/m³) in water (ρ = 1000, μ = 0.001 Pa·s): v = 2 × (10⁻⁴)² × 1500 × 9.81 ÷ (9 × 0.001) ≈ 0.0327 m/s, or about 33 mm/s.
Sources: Stokes’ law — Wikipedia
Frequently asked questions
When is Stokes’ law valid?
When flow around the particle is smooth and slow — a Reynolds number below roughly 1. Larger grains or big density differences push you past that, and drag then grows faster than the law predicts, so the real speed is lower.
Why can the velocity come out negative?
A negative result means the particle is less dense than the fluid, so buoyancy wins and it rises instead of sinking. The magnitude is still the terminal speed; only the direction flips.
What viscosity value should I use?
Use the fluid’s dynamic viscosity in pascal-seconds: about 0.001 Pa·s for water at room temperature, 0.0000181 for air, and much higher for oils, syrup or glycerol.
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Educational estimate. Check the formula and units for your exact case.