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Macroscopic descriptions of rarefied gases from the elimination of fast
variables

P. J. Dellar (2007) Macroscopic
descriptions of rarefied gases from the elimination of fast variables
Phys. Fluids ** 19** 107101

Preprint available as a PDF ( ElimFastVar.pdf
236K)

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Abstract

The Boltzmann equation describing a dilute monatomic gas is equivalent
to an infinite hierarchy of evolution equations for successive moments
of the distribution function. The five moments giving the macroscopic
mass, momentum, and energy densities are unaffected by collisions
between atoms, while all other moments naturally evolve on a fast
collisional timescale. We show that the macroscopic equations of Chen,
Rao, and Spiegel (2000, Phys.
Lett. A 271, 87), like the familiar Navier-Stokes-Fourier
equations, emerge from using a systematic procedure to eliminate the
higher moments, leaving closed evolution equations for the five moments
unaffected by collisions. The two equation sets differ through their
treatment of contributions from the temperature to the momentum and
energy fluxes. Using moment equations offers a definitive treatment of
the Prandtl number problem using model collision operators, greatly
reduces the labor of deriving equations for different collision
operators, and clarifies the role of solvability conditions applied
to the distribution function. The original Chen-Rao-Spiegel approach
offers greatly improved agreement with experiments for the phase speed
of ultrasound, but when corrected to match the Navier-Stokes-Fourier
equations at low frequencies it then under-estimates the phase speed at
high frequencies. Our introduction of a translational temperature, as
in the kinetic theory of polyatomic gases, motivates a distinction in
the energy flux between advection of internal energy and the work done
by the pressure. Exploiting this distinction yields macroscopic
equations that offer further improvement in agreement with experimental
data, and arise more naturally as an approximation to the infinite
hierarchy of evolution equations for moments.

@article{Dellar07elim,

Author =
{Dellar, P. J.},

Title = {Macroscopic descriptions of rarefied
gases

from the elimination of fast variables},

Journal =
{Phys. Fluids},

Year = {2007},

Volume =
{19},

Pages = {107101--14},

DOI = {doi:10.1063/1.2772271},

URL = {http://link.aip.org/link/?PHF/19/107101}

}