This is an updated version of notes from a seminar delivered in 2004. Although the text has been revised this update is still in note form. References missing from the original have been added, as well as appendices giving the data taken from two undergraduate theses undertaken at the University of Queensland. In high altitude flight the average spacing between the molecules of the flow gas is not negligible compared to a typical dimension of the flow field. In this case, the gas does not behave like a continuum and its discrete particle nature must be considered. The continuum assumption "breaks down" and the Navier-Stokes equations can, in theory, no longer be shown to be valid, particularly for high speed flight. The fundamental equation describing the flow at the particle level is the Boltzmann quation, from which the Euler equations, the Navier-Stokes equations and more accurate Burnett equations may be derived under various assumptions. Various scaling parameters have been suggested to identify the regimes in which these different equations are valid, ranging from the Knudsen number, Tsein's (1946) parameter, Cheng's (1961) rarefaction parameter, Bird's (1970) breakdown parameter, and a form of the viscous interaction parameter derived from shock-boundary layer theory. All these depend primarily on the non-dimensional group (Ut_c/L), which must be small for the Navier-Stokes equations to remain valid. U is flow speed, L is a flow length, t_c is a collision time. We show how all these parameters may be derived from the Boltzmann equation and interpreted as the ratio of typical shear stress to pressure in the flow. Cheng's parameter accounts for collision rate in some characteristic region of the flow. It appears to be the best correlation parameter for high-speed blunt body flow. For slender body flow the viscous interaction parameter based on a reference boundary layer temperature, a "modified Tsein parameter" similar to Cheng's parameter, appears to be best. Comparison of Navier-Stokes calculations and direct simulation solutions of the Boltzmann equation show that the Navier-Stokes equations may be adequate for almost rarefied flow (at least for hypersonic flow round a cyclinder)
