Knudsen number

Remarks

For a gas of hard spheres of diameter $d$ at pressure $p$ and temperature $T$, the mean free path is $\lambda = \dfrac{kT}{\sqrt{2}\, \pi d^2 p}$, where $k$ is the Boltzmann constant. For an ideal gas with adiabatic exponent $\gamma$, it is related to the Mach number $\mathrm{Ma}$ and the Reynolds number $\mathrm{Re}$ by $\mathrm{Kn} = \dfrac{\mathrm{Ma}}{\mathrm{Re}} \sqrt{\dfrac{\gamma \pi}{2}}$. Commonly quoted regimes are continuum flow for $\mathrm{Kn} < 0.01$, slip flow for $0.01 < \mathrm{Kn} < 0.1$, transitional flow for $0.1 < \mathrm{Kn} < 10$ and free molecular flow for $\mathrm{Kn} > 10$; the limits are conventions. In the scaled Boltzmann equation the Knudsen number multiplies the transport term relative to the collision term.

Example: air at sea level has $\lambda \approx 68$ nm, so a flow past a body of size $L = 1$ m has $\mathrm{Kn} \approx 7 \times 10^{-8}$, deep in the continuum regime.

Non-example: the ratio of the molecular diameter to $L$ is not the Knudsen number. It does not involve the density, while the mean free path grows as the gas becomes thinner.