A ONE DIMENSIONAL MATHEMATICAL MODEL FOR THE BOLTZMANN EQUATION
Technical note no. 10
CENTRE NATIONAL D'ETUDES ET DE RECHERCHES AERONAUTIQUES BRUSSELS (BELGIUM)
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A one dimensional model for the Boltzmann equation, based on a Fokker Planck-like differential collision operator is proposed, to study the validity of the usual approximation methods as well as the behavior of solutions near the free molecule limit. The corresponding fluid dynamic equations are derived and the existence and structure of shocks are studied. The distribution function is obtained for the unsteady, spatially uniform problem and the existence of an infinite number of relaxation times and modes is proved. Linearized versions of the model equation are derived. The near free molecule limit for solutions of the full model equation is briefly investigated for steady state, and a boundary layer type of behavior is found to exist. The leading terms in the deviation from free molecule data are found to be of the order of the cube root of an interaction parameter playing a role analogous to the Knudsen number.
- Fluid Mechanics