Gas Near a Wall: Shortened Mean Free Path, Reduced Viscosity, and the Manifestation of the Knudsen Layer in the Navier–Stokes Solution of a Shear Flow

Abstract

For the gas near a solid planar wall, we propose a scaling formula for the mean free path of a molecule as a function of the distance from the wall, under the assumption of a uniform distribution of the incident directions of the molecular free flight. We subsequently impose the same scaling onto the viscosity of the gas near the wall, and compute the Navier-Stokes solution of the velocity of a shear flow parallel to the wall. This solution exhibits the Knudsen velocity boundary layer in agreement with the corresponding Direct Simulation Monte Carlo computations for argon and nitrogen. We also find that the proposed mean free path and viscosity scaling sets the second derivative of the velocity to infinity at the wall boundary of the flow domain, which suggests that the gas flow is formally turbulent within the Knudsen boundary layer near the wall.