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Summary

We have seen that a fluid can be described in terms of its distribution function tex2html_wrap_inline12899 , a function of position, molecular velocity and time. The evolution of the distribution function is governed by the Boltzmann equation and the macroscopic fluid density, velocity and energy can all be found, at any time, from this distribution function. If we are dealing with a rare fluid where the number of molecular collisions is small, and so the majority of the collisions are binary, an expression can be found for the molecular collision function tex2html_wrap_inline12893 . Using this expression the gas can be shown to satisfy the Navier-Stokes equation and explicit expressions can be found for the transport coefficients. An H-theorem can also be proved, using the binary collision function, which states that the distribution function tends towards an equilibrium state which satisfies a Maxwell-Boltzmann distribution. We have also seen that the complex form of the collision function can be replaced by a BGK function which depends only on a single relaxation time. The H-theorem and the derivation of the Navier-Stokes equation and the transport coefficients are only valid for a rare gas where the number of non-binary collisions is negligible.



James Buick
Tue Mar 17 17:29:36 GMT 1998