Rarefied gas flow approximations

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]