The Stokes Number and Why Some Particles Refuse to Follow the Flow
A big drop falling through a cloud ought to sweep up every small droplet in its path. Mostly it doesn’t. The smallest droplets get pushed aside by the air flowing around the falling drop and slip past without touching it. Whether a particle rides the air like that or ploughs straight through it is the question the Stokes number answers. It’s a single dimensionless figure that predicts whether a particle carried by a moving fluid will follow the flow or go its own way.
Two different things named Stokes
The name causes confusion, so first things first. The stokes (symbol St) is a unit. NIST lists it as the old CGS unit of kinematic viscosity, equal to one square centimetre per second, or 10−4 square metres per second. NIST’s guide to the SI says to use m²/s instead. The American Meteorological Society’s glossary notes that the unit was named in honour of Sir George Gabriel Stokes (1819–1903).
The Stokes number is something else. It describes a particle inside a flow, not the fluid, and it’s a ratio of two times, so it has no units. The connection between the two is drag. Viscosity is, as Odinity’s lab report on Stokes’ law and measuring liquid viscosity puts it, a fluid’s resistance to shear, and the Stokes number is built on the same slow-flow drag law that lab tested by dropping a marble through glycerin.
How the Stokes number works
Picture a small particle in air that suddenly starts moving. Drag pulls the particle along, but it can’t match the air’s speed instantly. Its velocity catches up exponentially, and the time constant of that catch-up is called the particle’s relaxation time, or response time. The University of Waterloo aerosol course notes write that catch-up as 1 − e−t/τ, so after one relaxation time, τ, a particle starting from rest has reached about 63% of its final velocity.
For a small sphere moving slowly enough for Stokes drag to apply, the relaxation time is the particle’s density times its diameter squared, divided by 18 times the fluid’s viscosity. Very small particles in gas need an extra slip correction.
The flow has a timescale too. Usually it’s a characteristic length, such as the width of a nozzle, an airway or an obstacle, divided by a characteristic speed. The Stokes number is simply the particle’s relaxation time divided by the flow’s timescale.
The diameter is squared, which matters. A team at Otto von Guericke University Magdeburg points out that making a particle ten times wider makes its response time, and so its Stokes number, a hundred times larger. The same Waterloo course works out that a 30-micrometre glass bead in still air has a relaxation time of about 0.007 seconds. In another example, a 1-micrometre particle as dense as steel (8,000 kg per cubic metre), carried by air at 1 metre per second past a 10-centimetre cylinder, has a Stokes number of about 0.0003.
Much less than 1, much greater than 1
Two limits bracket the behaviour. Caltech’s Christopher Brennen notes that when a particle’s relaxation time is short compared with the timescale of the fluid motion, the particle essentially follows the fluid, so particles with a small Stokes number behave as tracers. The Waterloo notes sketch what happens at a sharp turn in the flow. Particles with a Stokes number much less than 1 stay on the gas streamlines, those near 1 only partly make the turn, and those much greater than 1 carry straight on and hit the wall.
One caution runs through everything below. The Waterloo notes stress that the characteristic dimension “can be defined differently for different applications.” A Stokes number calculated for an airway, a nozzle or a cloud droplet depends on which length and speed the author picked, so thresholds from one field don’t transfer neatly to another.
Illustration of a cloud with small droplets around a larger falling raindrop (artist’s impression).
Where it does real work
Inhalers and the airways
The airways are a branching maze of sharp turns. In a review for the Journal of Aerosol Medicine and Pulmonary Drug Delivery, Chantal Darquenne of UC San Diego explains that inertial impaction mainly affects particles larger than about 5 micrometres. Each time the airflow changes direction, a particle’s momentum tends to keep it on its old path until it strikes an airway wall. Darquenne writes the probability of that as a function of the Stokes number, using the airway’s diameter as the length. The higher the number, the more likely impaction is. She also cites modelling showing that when the Stokes number in the trachea was below about 0.01, aerosol was distributed through the lungs in proportion to the airflow. Geometry matters too. A study in human mouth-throat replicas found that deposition lined up best with a Stokes number calculated from the mouth inlet velocity and the narrowest point of the mouth-throat passage.
Cyclones and dust separators
Industrial cyclones use the same physics on purpose. According to the U.S. EPA’s fact sheet, a cyclone spins dusty gas in a conical chamber. Larger particles have enough inertia to overcome drag, reach the wall and drop into a hopper, while small particles leave with the gas. Among the factors the EPA lists, efficiency rises with bigger or denser particles and faster inlet gas, and falls with higher gas viscosity and a wider cyclone body. Those five line up with the Stokes number if the body’s width is taken as the length: every change that raises efficiency also raises the Stokes number, and every change that lowers it lowers the number. Size matters most because it’s squared, which fits the EPA’s estimate that conventional cyclones remove 70 to 90 percent of particulate matter but only 0 to 40 percent of fine PM2.5.
Impactor samplers
Cascade impactors measure particle sizes in air by sorting them in stages. Air passes through jets aimed at a plate, and each stage has a cut diameter, the size it catches half the time. A 1976 EPA calibration guide defines an impaction parameter, a Stokes number that uses half the jet’s diameter as the length, and gives a value of about 0.2 at the cut point for round jets. It also warns that published studies disagree, and that individual impactors need to be calibrated, so that 0.2 is a working value rather than a law.
Raindrops and cloud droplets
Back to the cloud. In warm clouds, rain forms when larger drops, which fall faster, collect smaller ones. Physicists Florian Poydenot and Bruno Andreotti note in the Journal of Fluid Mechanics that it takes about a million droplets of 10-micrometre radius to build a millimetre-sized raindrop. Measured collision efficiencies roughly collapse onto one curve when plotted against the Stokes number, and they fall off when the smaller drop lacks the inertia to cross the streamlines around the bigger one. Roland Stull’s meteorology textbook gives numbers: collision efficiency is only 0.02 to 0.1 when the small droplet’s radius is 2 to 5 micrometres, but 0.5 to 1 once both drops are larger.
PIV tracer particles
Particle image velocimetry (PIV) measures a flow by seeding it with tiny particles, lighting them twice in quick succession and tracking how far they moved. That only works if they follow the fluid faithfully. In Measurement Science and Technology, the Magdeburg group calls a Stokes number below 0.1 “the first rule to be observed” and cites earlier work finding slip errors below 2% when it stays under 0.2. In their simulated gas jet, 0.37-micrometre tracers (Stokes number about 0.06) tracked the flow, while 3.7-micrometre ones (about 6) lagged so badly that the measured velocity peak shifted downstream.
What it tells you, and what it doesn’t
What’s well established: comparing a particle’s response time with the flow’s timescale works. Particles with Stokes numbers far below 1 go where the fluid goes, and those far above 1 don’t. Response time grows with diameter squared.
What’s a rule of thumb: the specific cut-offs. That means 0.1 for PIV tracers, about 0.2 for an impactor stage and 0.01 in the trachea. Each depends on how its field defines length and speed, and each comes from experiments or models of particular setups. Passing isn’t a guarantee either. The Magdeburg study found that tracers meeting the Stokes criterion still gave biased velocities when only part of the flow was seeded. The simple formula also assumes slow, Stokes-drag flow around the particle, and Brennen gives a different relaxation time when the particle’s Reynolds number is large.
What’s still debated: clouds. Poydenot and Andreotti call the stability of fogs and non-precipitating clouds a longstanding problem. For droplets of roughly 3 to 30 micrometres, neither inertia nor random molecular jostling does much, and collision efficiencies near the minimum have never been measured experimentally. What bridges that gap is disputed. Turbulence is a widely proposed answer, and a 2024 PNAS study combining NASA aircraft observations with simulations found that, in simulations of a cumulus congestus cloud, turbulent collisions made rain form 25 to 30 minutes earlier. It also noted that commonly used models ignore turbulence in droplet collisions, and said turbulence’s role in calmer cloud types needs more work. Poydenot and Andreotti reach a different conclusion. Their calculations point to electrical attraction between droplets as the main effect in that size range, they argue that the turbulence mechanisms they examined matter little there, and they write that it remains uncertain whether any mechanism beyond the ones they modelled is truly necessary.
The Stokes number describes what particles do. The Reynolds number describes the flow carrying them, which Odinity’s lab on calibrating flow meters and exploring Reynolds number measures. For the stokes unit, see the NIST guide to units outside the SI, and for the PIV criterion, the Magdeburg team’s open-access paper.