How to use the stokes settling velocity calculator
- Enter the particle diameter and choose micrometers or millimeters.
- Enter the particle density and the fluid density in kilograms per cubic meter, and the fluid viscosity in millipascal seconds.
- Optionally enter a depth to find how long the particle takes to fall that far.
- Read the Stokes velocity, the Reynolds number, the drag-corrected velocity and the times.
Formula
Stokes law: v = (rho_p - rho_f) g d^2 / (18 mu), with g = 9.80665 m/s2
Reynolds number: Re = rho_f v d / mu; Stokes law holds for Re below about 0.1
Corrected: balance weight less buoyancy with drag, using CD = 24 / Re (1 + 0.15 Re^0.687)
If rho_p is below rho_f the particle rises, and the same formulas give the rising speed
The corrected speed is found by solving the force balance numerically and is reasonable for Re up to about 800.
Sand, silt, a steel ball and fog
A 100 micrometer quartz sand grain (2650 kg/m3) in water (998.2 kg/m3, 1.002 mPa s) sinks at 8.98127 mm/s by Stokes law, but the Reynolds number is 0.8947, so the corrected speed of 7.9626 mm/s is better. A 20 micrometer silt grain sinks at 0.359251 mm/s with a Reynolds number of only 0.007158, so Stokes law holds, and it needs 5,567 s to fall 2 m.
A 2 mm steel ball (7850 kg/m3) in oil (880 kg/m3, 50 mPa s) gets 303.788 mm/s from Stokes law, which is too high because Re is 10.7; the corrected value is 194.448 mm/s. A 20 micrometer water droplet in air falls at 11.9 mm/s.
Where Stokes law applies
Stokes law describes the drag on a small smooth sphere moving slowly through a thick fluid, when the flow around it stays smooth. It is used in soil science to size particles by how fast they settle, in water treatment to design settling tanks, in paint and food science for how pigments and droplets separate, and in measuring viscosity with a falling ball.
The settling speed grows with the square of the diameter, so doubling the size makes a particle fall four times faster. For flow regimes in general, see the Reynolds number calculator, and for objects falling through air at high speed, the terminal velocity calculator.
Checking the assumptions
The model is a single sphere in still fluid, far from walls. Real particles are not round, and irregular grains settle more slowly than spheres of the same volume. A cloud of particles hinders itself and settles more slowly than one grain alone. If the Reynolds number is above 1, use the corrected value; if it is above about 800, the drag law changes again and the page is no longer a good guide.
Viscosity depends strongly on temperature: water at 10 C is about 47% thicker than at 25 C, which slows settling. The defaults are examples for water near 20 C, so use values for your own fluid. For buoyancy forces, see the buoyancy and hydrostatic calculator.
Assumptions and limits
- The particle is a smooth sphere in a still, unbounded Newtonian fluid.
- Densities are in kg/m3 and viscosity in mPa s (equal to centipoise).
- The corrected velocity uses the Schiller-Naumann drag law, valid up to a Reynolds number of about 800.
- The default densities and viscosity are examples for quartz sand in water at about 20 C.
Frequently asked questions
What is Stokes law?
A formula for the drag on a small sphere moving slowly in a viscous fluid. It gives the settling speed as (density difference) x g x d squared / (18 x viscosity).
When does it stop working?
When the Reynolds number is above about 0.1 to 1. Then use the corrected velocity shown on the page.
Why does a bigger particle fall faster?
Weight grows with the cube of the diameter, but drag grows only with the diameter, so the speed grows with its square.
What if the particle is lighter than the fluid?
It rises, and the page reports the rising speed.