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Space-eigenvalue problems in the kinetic theory of gases

Published online by Cambridge University Press:  29 March 2006

M. M. R. Williams  and
J. Spain
M. M. R. Williams
Affiliation:
Nuclear Engineering Department, Queen Mary College, University of London
J. Spain
Affiliation:
Nuclear Engineering Department, Queen Mary College, University of London
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Abstract

The existence of elementary, exponential solutions of the linear Boltzmann equation for gases is examined. Using the hard-sphere model of scattering, it is found that, in problems involving velocity perturbations, there are no discrete non-zero eigenvalues. Thus the relaxation to the asymptotic distribution is non-exponential and is described by the continuum eigenfunctions. For temperature perturbations, however, we find two non-zero discrete eigenvalues whose values are ±0·975 in units of the minimum scattering cross-section. Relaxation to the asymptotic distribution is therefore exponential, although still very rapid.

The conclusions stated above are based upon a truncation of the scattering kernel and a subsequent numerical solution of the resulting integral equations.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 42 , Issue 1 , 4 June 1970 , pp. 85 - 95
Copyright
© 1970 Cambridge University Press

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