Stokes Law Settling Velocity Calculator
Find how fast a small particle sinks through a fluid, from the particle size, the two densities and the viscosity.
What it does
Put a grain of sand in a glass of water and watch it sink. It comes to rest on the bottom in a couple of minutes, which feels slow until you remember that the grain is being pushed down by gravity and held back by the friction of the water moving around it. The balance between those two forces is what Stokes law describes.
The sediment survey work behind this page puts the law in plain words: Stokes law says the viscous resistance to the fall of a particle equals the net downward force on it, which is the weight of the particle less the buoyant force of the fluid. The result is a fall velocity that depends on the diameter of the particle, the two densities and the viscosity of the fluid, and nothing else.
The same source notes where the law is safe to use. It holds for quartz sediment particles finer than 0.062 mm, and even there the assumptions are listed: the particles must be big enough that Brownian movement is not important, smooth, rigid and near spherical, and they must be falling at a steady speed in a fluid of unlimited extent with no other particles close by.
This page runs that formula. Give it the diameter of the particle, its density, the density and the viscosity of the fluid, and it returns the settling velocity, the Reynolds number of the flow so you can see whether the law applies, the specific gravity of the particle and the time needed to settle a given depth.
The limits are honest. The formula ignores the inertia of the fluid and the shape of the particle, and it is a single particle in clear fluid. In a suspension the neighbours change the fall speed, and for anything larger than about 0.1 mm the true speed sits below the calculated one.
How to use
- Choose metric or imperial for the particle size, the densities and the answer.
- Enter the density of the particle. Use 2650 kg per m3 for quartz and 2710 for calcite.
- Enter the density of the fluid. Water at 20 degrees C is 998 kg per m3.
- Enter the diameter of the particle in the unit you chose. The answer grows with the square of this number, so small errors in the diameter matter a lot.
- Enter the dynamic viscosity of the fluid, which you can take from a table for the temperature you are working at.
- Read the settling velocity, then check the Reynolds number. Below 1 the law holds, above 1 the real speed is lower, and above about 1000 the law does not apply at all.
Formula
- v
- Terminal settling velocity of the particle, in metres per second.
- d
- Diameter of the particle, in metres.
- rho_p
- Density of the particle, in kilograms per cubic metre.
- rho_f
- Density of the fluid, in kilograms per cubic metre.
- mu
- Dynamic viscosity of the fluid, in pascal seconds.
The Reynolds number of the falling particle is the density of the fluid times the settling velocity times the diameter, divided by the viscosity. It is the test of whether the law is honest: below 1 the flow around the particle is creeping and the law is good, between 1 and 1000 the inertial effects of the fluid reduce the real speed below the calculated one, and above 1000 the law no longer describes the fall at all.
Worked example
Example: a 0.1 mm quartz grain settling in water at 20 degrees C
Defaults: metric units, a particle density of 2650 kg per m3, a fluid density of 998 kg per m3, a diameter of 0.1 mm, a viscosity of 1.002 mPa s and a depth of 1 m.
- The diameter in metres is 0.1 / 1000 = 0.0001 m, so the square of it is 1e-8 m2.
- The density difference is 2650 - 998 = 1652 kg per m3.
- The numerator is 9.80665 x 1e-8 x 1652 = 1.62007e-4.
- The denominator is 18 x 0.001002 = 0.018036.
- Settling velocity: 1.62007e-4 / 0.018036 = 8.9824e-3 m per s, which is 8.9824 mm per s.
- The Reynolds number is 998 x 8.9824e-3 x 0.0001 / 0.001002 = 0.8947, so the creeping flow assumption holds.
- To fall 1 m takes 1 / 8.9824e-3 = 111.3 s, and the tool shows 8.9824 mm per s, 0.8947, 2.655, 111.3 s and the creeping flow range.
Reference table
| Diameter | Settling velocity | Reynolds number | Time to fall 1 m |
|---|---|---|---|
| 0.01 mm | 0.0898 mm per s | 0.0009 | 11,132.9 s |
| 0.05 mm | 2.2456 mm per s | 0.1118 | 445.3 s |
| 0.1 mm | 8.9824 mm per s | 0.8947 | 111.3 s |
| 0.2 mm | 35.9294 mm per s | 7.1572 | 27.8 s |
| 0.5 mm | 224.5590 mm per s | 111.8313 | 4.5 s |
Limits
- The law is written for a single smooth spherical particle falling in a fluid of unlimited extent, and real particles are neither.
- It ignores the inertia of the fluid, so above a Reynolds number of about 1 the true fall speed is lower than the calculated one.
- It says nothing about a suspension, where neighbouring particles slow each other down through hindered settling.
- It is a settled fall speed, so it does not describe the start of the motion or any flocculation that changes the effective size of the particle.
Common mistakes
- Using the diameter in millimetres where the formula wants metres. The answer comes out a million times too small.
- Leaving the fluid density out of the density difference. In water the buoyant force removes a good part of the weight, and a particle that is barely denser than the fluid falls very slowly.
- Using the kinematic viscosity where the dynamic viscosity belongs. The two differ by the density of the fluid.
- Ignoring the Reynolds number. Above 1 the calculated speed is too generous, and above 1000 the formula is meaningless.
- Expecting the fall speed to scale with the size of the particle. It scales with the square of the size, so fine particles settle far more slowly than the eye expects.
Try it yourself
Quick checks. Answers sit tucked under each question.
How fast does a 0.2 mm quartz grain fall in the same water?
35.9294 mm per s, which is 27.8 s to fall a metre. The speed is four times the 8.9824 mm per s of the 0.1 mm grain because the diameter has doubled, and the Reynolds number rises to 7.1572, so the real fall speed will be a little below the calculated one.
What if the fluid is twice as viscous?
The fall speed halves to 4.5002 mm per s, and the time to fall a metre doubles to 222.2 s. The viscosity sits in the denominator of the formula, so the speed is inversely proportional to it.
How long does a 0.01 mm particle take to settle one metre?
11,132.9 s, which is just over three hours. The speed is only 0.0898 mm per s, and the Reynolds number of 0.0009 shows the flow is deep in the creeping range where the law is at its best.
Questions
What is Stokes law?
The rule that says the viscous drag on a small particle falling through a fluid equals the net downward force on it, which gives a fall speed proportional to the square of the diameter and to the difference between the particle density and the fluid density.
How small does a particle have to be for the law to work?
Small enough for the Reynolds number to stay below 1. For quartz in water that means roughly 0.1 mm or less, and the sediment survey text behind this page puts the safe range at under 0.062 mm.
What does the Reynolds number tell me?
It says whether the creeping flow assumption behind the law holds. Below 1 the calculation is good, from 1 to 1000 the real fall speed is below the calculated one, and above 1000 the law does not apply.
Does the temperature matter?
Yes, through the viscosity. Water at 20 degrees C has a dynamic viscosity of 1.002 mPa s, and colder water is thicker, so the same particle settles more slowly in winter than in summer.
Why does a heavier particle settle faster?
Because the driving force is the difference between the particle density and the fluid density, while the drag depends on the size and the speed. A particle twice as dense as another of the same size falls twice as fast, which is why gold sinks through water much faster than a plastic bead.
What is hindered settling?
The slowing of particles that fall in a crowd. When the particles are close together their wakes interact and the whole suspension settles more slowly than the formula for a single particle predicts.
Sources
- US Geological Survey, techniques and methods of water resources investigations 5-C1: laboratory methods, with the Stokes law for settling
- US Army Corps of Engineers hydrologic engineering center: particle settling velocity and fall velocity models
- US Geological Survey, some fundamentals of particle size analysis, fall velocity of a particle