Stokes’ Law Settling Velocity Calculator
Find how fast a small sphere sinks or rises through a still fluid. Stokes’ law gives the settling velocity, the drag force and the particle Reynolds number, and runs backwards as a falling-ball viscometer or to find a particle’s size or density.
How to Use
- Pick what to find: the settling velocity, the fluid’s viscosity (falling-ball viscometer), the particle diameter or the particle density.
- Choose a fluid to fill in its density and viscosity, or type your own values in any unit.
- Enter the particle’s diameter and density, and the measured speed if you are solving backwards. Speeds can be in µm/s, mm/s, cm/s, m/h and more.
- Read the answer, the drag force, the particle Reynolds number and the time to fall a set distance. Check the Reynolds number: Stokes’ law only holds below about 1.
- Press a preset for a ready-made example, and open Show Work for each step and conversion.
Worked Example
A grain of fine sand in water. A quartz grain 0.1 mm across (r = 0.00005 m, density 2,650 kg/m³, typical) in water at 20 °C (998.2 kg/m³, 0.0010016 Pa·s). The density difference is 1,651.8 kg/m³, so v = 2 × 1,651.8 × 9.80665 × 0.00005² ÷ (9 × 0.0010016) = 8.985 mm/s, about 11.1 s to fall 10 cm. The check: Re = 998.2 × 0.008985 × 0.0001 ÷ 0.0010016 = 0.895, just inside the Stokes range.
A falling-ball viscometer. A 2 mm steel ball (7,850 kg/m³) falls through glycerol (1,261 kg/m³) at a steady 10.2 mm/s. Solving for viscosity: μ = 2 × (7,850 − 1,261) × 9.80665 × 0.001² ÷ (9 × 0.0102) = 1.408 Pa·s. Re = 1,261 × 0.0102 × 0.002 ÷ 1.408 = 0.018, comfortably in the Stokes range.
The common mistake: putting the diameter in for the radius. Stokes’ law uses the radius, squared. Entering the sand grain’s 0.1 mm diameter as r gives 35.94 mm/s, four times the real 8.985 mm/s. The other trap is skipping the Reynolds check: the same formula gives a 0.5 mm grain 224.6 mm/s, but at Re = 112 the law no longer applies.
Show Work
Formulas
Stokes, Millikan and the Settling Tank
George Gabriel Stokes, Lucasian Professor of Mathematics at Cambridge, derived the drag on a slowly moving sphere in 1851, in a paper on the effect of air friction on pendulums. He had already written down the equations of viscous flow that now carry his name with Claude-Louis Navier’s. Stokes noted that the result explains why clouds float: their droplets are so small that they fall only very slowly.
The law’s most famous use came in 1909–1913, when Robert Millikan and Harvey Fletcher timed charged oil droplets falling between two plates. The fall speed, through Stokes’ law, gave each droplet’s size and weight, and the field needed to hold it still gave its charge, which always came in whole multiples of one elementary charge. Millikan had to add a slip correction for droplets so small that air no longer behaves as a smooth fluid around them, the effect Ebenezer Cunningham had described in 1910.
Property values used here: water and air at 20 °C as on the Fluid Flow Calculator, glycerol at 20 °C from the CRC Handbook of Chemistry and Physics, and typical figures for quartz sand (2,650 kg/m³), steel (7,850 kg/m³), mineral dust, red blood cells and blood plasma, which vary from sample to sample.
About This Tool
This calculator applies Stokes’ law to a smooth, solid sphere falling or rising at its steady speed through a still fluid far from any wall. It solves for the settling velocity, the viscosity, the particle diameter or the particle density, and always reports the drag force, the particle Reynolds number and the time to fall a distance you choose. It warns when Re passes 1, where the law overestimates the speed, and when sub-micrometre particles in a gas need a slip correction. Real particles are rarely perfect spheres, and a wall close to the ball (a narrow viscometer tube) slows it, so treat results as estimates for those cases.
Everything runs in your browser; nothing you enter is sent anywhere.
Related tools: Terminal Velocity Calculator, Fluid Flow Rate & Reynolds Number Calculator, and Capillary Rise & Surface Tension Calculator.
Frequently Asked Questions
What is Stokes’ law?
For a small sphere moving slowly through a fluid the drag is F = 6πμrv. Setting it equal to the weight minus the buoyancy gives the steady settling speed v = 2(ρp − ρf)gr² ÷ (9μ). A 0.1 mm quartz sand grain (2,650 kg/m³) in water at 20 °C settles at 2 × 1,651.8 × 9.80665 × 0.00005² ÷ (9 × 0.0010016) = 8.985 mm/s.
When does Stokes’ law stop working?
When the particle Reynolds number Re = ρf v d ÷ μ rises past about 1, a wake forms and the drag grows faster than v. The 0.1 mm sand grain is at Re = 0.895, near the edge; Oseen’s correction puts its drag about 17% higher. A 0.5 mm grain would give 224.6 mm/s at Re = 112, far outside the law, so its real speed is much lower.
How does a falling-ball viscometer work?
Drop a ball of known size and density into the liquid, time it over a measured distance once it has stopped speeding up, and solve Stokes’ law for μ. A 2 mm steel ball (7,850 kg/m³) falling at 10.2 mm/s through glycerol (1,261 kg/m³) gives μ = 2 × 6,589 × 9.80665 × 0.001² ÷ (9 × 0.0102) = 1.408 Pa·s, within 0.3% of the 1.412 Pa·s listed for glycerol at 20 °C.
Why does fine dust hang in the air so long?
Settling speed falls with the square of the size. A 10 µm mineral dust speck (2,500 kg/m³, typical) falls through air at 7.52 mm/s and takes about 3.3 minutes to drop 1.5 m; a 2.5 µm particle falls at 0.47 mm/s, sixteen times slower, so any draught keeps it aloft.
Why does mud take days to clear?
Clay particles are tiny. A 2 µm clay grain of 2,650 kg/m³ settles through still water at only 0.0036 mm/s, taking about 3.2 days to fall 1 m, while 20 µm silt settles 100 times faster at 0.36 mm/s. That is why settling tanks and sieve-free grain-size tests (the hydrometer method) work by timing the fall.
How do I use the Stokes’ Law Settling Velocity Calculator?
Just type your numbers. The answer shows up right away — there is no button to press. Change anything and it updates by itself.
Is it free? Does it work without internet?
Yes to both. It is free with no sign-up, and once the page has loaded it keeps working even with no internet.
Where does my data go?
Nowhere — every calculation runs on your own device. Nothing you enter is uploaded, logged, or stored.
Common Use Cases
Sediment and water treatment
A grain that settles 1 mm/s through water at 20 °C is a 33.36 µm quartz particle; anything finer needs a longer tank or a flocculant.
Measuring viscosity
A 2 mm steel ball timed at 9.804 s over 10 cm of glycerol gives 1.408 Pa·s.
Air quality
10 µm dust falls through still air at 7.52 mm/s, 199 s from head height of 1.5 m; 2.5 µm (PM2.5) particles fall sixteen times slower.
Blood and lab centrifuges
A lone red blood cell (8 µm, 1,100 kg/m³ in plasma at 1,025 kg/m³, typical) settles at only 2.179 µm/s under normal gravity, which is why samples are spun in a centrifuge.
Physics homework
Mistaking the diameter for the radius makes the 0.1 mm sand grain fall at 35.94 mm/s instead of 8.985 mm/s, four times too fast.
Last updated: