Discrete relaxation modes for a hard-sphere gas

1968 ◽  
Vol 46 (21) ◽  
pp. 2463-2469 ◽  
Author(s):  
M. Rahman ◽  
M. K. Sundaresan
Keyword(s):  
Boltzmann Equation ◽  
Discrete Spectrum ◽  
Hard Sphere ◽  
Spherically Symmetric ◽  
Symmetric State ◽  
Elementary Method ◽  
Linearized Boltzmann Equation ◽  
Relaxation Modes

The discrete spectrum of a linearized Boltzmann equation for a hard-sphere gas is studied, and the results of Kuščer and Williams (1967) for a spherically symmetric state are rederived by a more elementary method. Extension is made to nonisotropic states, and it is shown that the discrete spectrum is empty for [Formula: see text].

Discrete versus continuum relaxation modes of a hard sphere gas

1984 ◽  
Vol 62 (2) ◽  
pp. 97-103 ◽  
Author(s):  
Bernard Shizgal
Keyword(s):  
Discrete Spectrum ◽  
Cross Section ◽  
Cross Sections ◽  
Hard Sphere ◽  
Expansion Method ◽  
Gas Mixture ◽  
Discrete Ordinate Method ◽  
Discrete Ordinate ◽  
Relaxation Modes ◽  
The Continuum

The nature of the discrete spectrum of the linear Boltzmann collision operators for a simple gas and for a gas mixture is studied numerically with a discrete ordinate method. The discrete ordinate method is found to give a large number of discrete eigenvalues whereas the expansion method with Burnett functions yields only a few converged eigenvalues. The hard sphere cross section is used in the present paper although the methods employed are readily applicable to other cross sections. The approach of the eigenvalues to the continuum boundary is studied in detail and a comparison with a previous asymptotic Wentzell–Kramers–Brillouin (WKB) analysis yields excellent agreement.

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