Knudsen number ($\mathrm{Kn}={\ell }_{\mathrm{mfp}}/L$)

The Knudsen number is the dimensionless ratio of an intermolecular mean free path ${\ell }_{\mathrm{mfp}}$ to a specified characteristic length $L$ of the apparatus or flow. The mean free path measures the distance travelled between gas–gas collisions; $L$ measures the distance over which boundaries or macroscopic conditions matter. Wall encounters must not be counted as gas–gas collisions in this comparison: doing so would hide the very wall-dominated behavior we want to detect. The same gas can therefore have a small Knudsen number in a large vessel and a large Knudsen number in a tiny opening.

For a dilute hard-sphere gas of identical molecular diameter $d$ in thermal equilibrium, use the conventionHere $n$ is the number density, $p$ the pressure, $T$ the absolute temperature, and $k_B$ the Boltzmann constant; the second equality uses the ideal gas law. The collision cross-section is $\pi d^2$, and the $\sqrt{2}$ accounts for both partners moving with independent equilibrium velocities. Mixtures and velocity-dependent collision cross-sections require a stated molecular model and mean free path convention; the displayed expression is not an exact formula for every real gas. ^{[ref] [ref]}

As a numerical toy example, choose $T=300\mathrm{K}$ and $d=0.37\mathrm{nm}$, without claiming that this is a measured molecular diameter for a particular gas. The formula gives ${\ell }_{\mathrm{mfp}}\simeq 68.1\mathrm{nm}$ at $p=10^5\mathrm{Pa}$. It yields $\mathrm{Kn}\simeq 6.81\times 10^{-6}$ for a $1\mathrm{cm}$ vessel but $\mathrm{Kn}\simeq 0.681$ for a $100\mathrm{nm}$ gap. At $p=1\mathrm{Pa}$ the same toy gas has ${\ell }_{\mathrm{mfp}}\simeq 6.81\mathrm{mm}$, comparable to the centimetre vessel. Reducing either pressure or apparatus size can thus expose molecular transport.

Choose $L$ for the question being asked: a tube diameter compares the mean free path with wall separation, while tube length describes a different geometric scale. For a positive macroscopic field $q$, a local gradient scale can instead be $L_q=|q|/|\nabla q|$, where its gradient is nonzero. A large vessel can contain a thin region with a large local Knudsen number. Small geometric Knudsen number alone does not guarantee a slowly varying near-equilibrium flow. The time, velocity and space scalings must also be stated when the Knudsen number appears as a coefficient in a Boltzmann equation. ^[ref]

For $\mathrm{Kn}\ll 1$, frequent gas–gas collisions can establish a local Maxwellian before macroscopic conditions change appreciably. Under this additional near-equilibrium assumption, the hydrodynamic limit of the Boltzmann equation motivates a continuum approximation with Navier-Stokes equations and heat conduction. A no-slip boundary condition is a leading wall approximation, not a statement about individual molecular velocities.

With a small but finite Knudsen number, a useful slip approximation retains continuum bulk equations and corrects wall velocity and temperature. A Navier slip boundary condition can represent the leading tangential velocity correction, but its slip length must come from the relevant gas–wall interaction model. A nearby Knudsen layer still requires kinetic treatment if its detailed structure is wanted. Slip corrections are perturbations; extending them into order-one Knudsen numbers has no general accuracy guarantee. At order-one Knudsen number, wall and intermolecular collisions both matter; this transitional situation calls for a kinetic theory model. At $\mathrm{Kn}\gg 1$, free molecular flow becomes a useful limiting approximation. These are scale comparisons, with accuracy depending on geometry, flow conditions and the observable, rather than universal sharp numerical boundaries. ^[ref]

A Knudsen layer is the gas region adjoining a boundary where the molecular distribution departs from the bulk near-equilibrium description over distances of order a few mean free paths. Continuum slip and jump conditions describe an effective boundary for the outer solution; they do not resolve this layer. ^[ref]

In free molecular flow, gas–gas collisions are negligible over the trajectories relevant to the observable. Molecules still encounter walls, whose scattering of momentum and energy must be modeled. In a long narrow tube, ${\ell }_{\mathrm{mfp}}$ exceeding the diameter does not by itself prove that every complete transmission path is collision-free: repeated wall reflections can create much longer paths. The approximation must be checked against those paths as well as the width. ^[ref]