• Version
• 166 Downloads
• 2.72 MB File Size
• 1 File Count
• November 2, 2016 Create Date
• November 2, 2016 Last Updated

National Advisory Committee for Aeronautics, Report - Effect Variable Viscosity and Thermal Conductivity on High Speed Slip Flow Between Concentric Cylinders

The fact that a gas is not a continuum but actually a
collection of molecules in rapid but random motion has
begun to have more and more importance in the aero-
dynamics of high—speed flow. This is due to the expectation
that flow through wind tunnels at low pressure or flight at
extremely high altitudes will not be amenable to analysis
using classical fluid dynamics. When the mean free path
of the molecules l is negligible compared with the macro-
scopic dimension L, which may be wing chord, tunnel
diameter, and so forth, the classical picture should hold as
the molecules are so tightly packed together the gas behaves
just like a mathematical continuum. The ratio l/L is
defined as the Knudsen number Kn, which is a measure of
the degree of gas rarefaction. In terms of the better known
parameters Reynolds number Re and Mach number hf, the
Knudsen number is proportional to M /Re. Hence, although
not a new parameter, it is a convenient one to use when the
degree of rarefaction of the gas is of interest.

Gas dynamics is the continuous-flow regime or Clausius
gas regime for which the N avier-Stokes equations together
with the condition of no slip on the boundaries are valid
and the Knudsen number is extremely small. If the gas
becomes more rarefied and the Knudsen number increases,
the effect of slip along the boundaries becomes noticeable,
although the Navier-Stokes equations remain valid so long
as the Mach number remains small. This phenomenon has
been known for over 75 years and has been the subject of
an extensive study by physicists. Tsien (ref. 1) has sum—
marized this work very well. During this same period of
time the solution of Boltzmann’s integral equation by Enskog
and Chapman, along lines laid down by Hilbert, has led to
the distribution function for a nonuniform gas as an expan-
sion in powers of the Knudsen number. This approach
yields the equations of flow in successive orders of approxi-
mation, the first order being the Navier—Stokes equations,
the second order, the Burnett equations, and so forth.