Stokes Number (particle inertia)

Put a particle in a moving gas and ask it to turn a corner. Whether it manages to is the whole of this number. The gas turns because it must — it cannot pass through the obstacle — and the particle turns only to the extent that drag has time to make it. The Stokes number is the ratio of the time the particle needs to be redirected against the time the gas gives it.

The numerator is the relaxation time $\tau_p = \rho_p d^2 / 18\mu$, which comes straight out of balancing a particle's momentum against Stokes drag: a particle released into still air loses its velocity exponentially with this time constant. It is a genuinely physical quantity — multiply it by the velocity and you get the stopping distance, the length a particle would coast if the gas vanished. The denominator, $L/v$, is how long the gas takes to negotiate the obstacle. Divide one by the other and you have $\mathrm{Stk}$.

Below about 0.1 the particle follows the streamlines and is not collected by inertia at all. Above about 1 it goes more or less straight on and hits. Between those two the collection efficiency climbs steeply, and every inertial collection device on earth — the cyclone, the cascade impactor, the fibrous filter, the scrubber, the nose — is engineered to put the particles it wants to catch above that band and the ones it wants to pass below it.

The squared diameter is why this works so well as a size classifier. A particle four times bigger has sixteen times the Stokes number, so a device that is nearly transparent to 1 µm can be nearly opaque to 4 µm. It is also why the cut of a cascade impactor is sharp: each stage's efficiency curve is steep in diameter even though it is gentle in Stokes number.

The relaxation time assumes Stokes drag, and that assumption has a range. Stokes solved the creeping-flow problem in 1851 for a sphere at vanishing Reynolds number, and $\tau_p = \rho_p d^2/18\mu$ is his solution rearranged. Past a particle Reynolds number $\mathrm{Re}_p = \rho_f d v / \mu$ of about 1, drag rises faster than linearly with velocity, the exponential decay of velocity is no longer exponential, and $\tau_p$ is simply not that expression any more. For a particle in air at 10 m/s, $\mathrm{Re}_p = 1$ arrives at a diameter around 1.5 µm — which is uncomfortably small, and means that a great many practical impaction calculations are being done a little outside the law they invoke. The error is in a known direction: real drag exceeds Stokes drag, so the true relaxation time is shorter, and this formula overstates the Stokes number. Use a drag correlation with a Reynolds correction when $\mathrm{Re}_p$ gets past 1, and be aware that the published Stokes-number thresholds were established in the range where the simple law holds.

There is a correction at the other end too. Below about a micrometre a particle is comparable in size with the mean free path of the gas molecules, and it slips between them rather than seeing a continuum. The Cunningham slip correction multiplies the relaxation time by a factor of about 1.16 at 1 µm and 2.9 at 0.1 µm, making small particles more mobile than this equation says. It is why sub-micrometre aerosol is so hard to collect by inertia, and why filters catch it by diffusion and interception instead.

The characteristic length is a choice, and it is the choice that most often makes two people's Stokes numbers disagree. A cyclone can be characterized by its body diameter, its inlet width, or its vortex-finder diameter, and these differ by factors of two to five. An impactor is characterized by its nozzle diameter, which at least is unambiguous; a fibrous filter by the fibre diameter, which is a distribution rather than a number. When a paper says collection begins near $\mathrm{Stk} = 0.5$, it means near 0.5 with its own length. Carrying that threshold onto a different length is the commonest error on this page.

One last point that saves a great deal of confusion. Density and diameter appear only as $\rho_p d^2$, which is why aerosol science works in aerodynamic diameter — the diameter of a unit-density sphere with the same inertia. Any size that came out of an impactor or a cyclone is already an aerodynamic diameter, and it should be paired with a density of 1000 kg/m³, not with the material's own. Pairing a true density with an aerodynamic diameter counts the same correction twice.