Knudsen Number (Where the Continuum Stops Being True)

This is the page that qualifies all the others. Reynolds, Weber, Euler, Bond, Richardson, Archimedes — every one of them treats a fluid as a continuum with a density, a viscosity and a velocity defined at every point. That is not a property of the fluid. It is a property of the fluid and the scale you are asking about, and the Knudsen number is how you check.

$Kn = \lambda/L$: the average distance a molecule travels between collisions, divided by the size of the thing it is flowing through. When $Kn$ is small a molecule collides with its neighbours many times before it notices a wall, local equilibrium is maintained, and the continuum description is excellent. When $Kn$ approaches 1 the molecule crosses the whole channel between collisions and there is no local anything.

The regimes, with the usual caveat that the boundaries are round numbers chosen for teaching:

$Kn < 0.001$ — continuum. Navier–Stokes with no-slip walls. Everything else on this site applies.
$0.001 < Kn < 0.1$ — slip flow. The bulk gas is still a continuum but the no-slip condition has failed: the gas at the wall moves, and there is a temperature jump there too. Navier–Stokes survives if you replace the boundary conditions, which is what microchannel and MEMS analysis does.
$0.1 < Kn < 10$ — transition. Genuinely hard. Neither limit applies, and there is no patch that saves Navier–Stokes. The honest tools are the Boltzmann equation or direct simulation Monte Carlo. Correlations quoted across this range are interpolations between two limits, not solutions.
$Kn > 10$ — free molecular. Molecules hit walls far more often than each other. There is no viscosity as a transport property and no bulk pressure-driven flow — just ballistic particles and how they accommodate at surfaces.

Those numbers are softer than most conventions on this site. A pressure drop may be tolerably predicted well past $Kn = 0.01$ while a heat transfer coefficient in the same channel is already wrong, because the wall condition that fails first is the one your answer depends on most.

$\lambda$ is not a property of the gas alone. It scales as temperature over pressure, so air's 68 nm at sea level becomes roughly 0.1 mm at 50 km altitude and metres in a vacuum chamber. Three consequences that look unrelated and are the same calculation. A re-entering spacecraft passes through every regime on that list on the way down, which is why hypersonic aerodynamics needs rarefied methods at altitude and continuum CFD below. Aerosol particles smaller than about 1 μm fall faster than Stokes' law predicts, and the Cunningham slip correction that fixes it is a Knudsen number correction. And an ordinary laboratory vacuum line at 1 Pa has a mean free path of centimetres — it is in free molecular flow, and its conductance has nothing to do with viscosity.

One input rule. $L$ must be the smallest dimension that matters — the gap, not the length of the channel — because the walls are what the molecules are failing to equilibrate with. And there is a tidy link to the rest of the shard: $Kn$ is proportional to $Ma/Re$, so rarefaction, compressibility and viscosity are not three independent things. High Mach at low Reynolds number is automatically rarefied, which is exactly the corner of the map a satellite in the upper atmosphere lives in.