By Mikhail N. Kogan (auth.)
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Extra info for Rarefied Gas Dynamics
A. Frieman, in: "Rarefied Gas Dynamics," Third SympoSium, Academic Press, New York, 1963, and C. H. Su, Phys. Fluids, Vol. 8 (1964). See also V. N. Zhigulev, Doklady Akad. Nauk SSSR, Vol. 5 (1965); M. N. 6 (1966). 48 EQUA TIONS OF THE KINETIC THEORY OF GASES [CH. II Let the gas be located in a vessel of volume V. (t. 9) where S is the surface of the vessel, and 11 is the normal to it directed toward the gas. It is easy to see that the last integral determines the probability for the i-th particle to pass through the vessel wall while s particles are located in the states zi' ...
When b < d, all the molecules collide with the same impact parameter, and, therefore, the molecules of each group again possess the same velocity after the collision. If we examine these same groups of molecules within the framework of the assumption used in deriving the Boltzmann equation, then each molecule of one group could collide with a molecule of the other group with any impact parameter. The result would be that the molecules would possess a whole range of velocities after collision. Clearly, the behavior of the system is quite different in these two cases.
Fig. 8 2. The Boltzmann equation may be transformed somewhat for hard-sphere molecules of diameter d. Let I/J be the angle between the relative velocity vector g and the line of the molecules at the instant of collision (Fig. 8). It is easy to see that b=dsinljJ and db=dcosljJdljJ. Then Eq. 10) where k is the unit vector along the line of centers and dQ = sin 1/1 dl/l de. is a surface element on the unit sphere. The integration is carried out over the whole sphere. Therefore, the integral must be divided by two to avoid counting the same collisions twice.
Rarefied Gas Dynamics by Mikhail N. Kogan (auth.)