An Introduction to the Theory of the Boltzmann Equation by Stewart Harris

Author:Stewart Harris
Language: eng
Format: epub
Publisher: Kluwer Law International
Published: 2009-06-15T00:00:00+00:00

which defines an infinite set of determined equations (for n = 1, 2, ...) in each order (given m). In lowest order we set m = 0 and find

This implies that f is a local Maxwellian, so that, for example,

In the next approximation the following results are of particular interest:

where μ and κ are the identical values found by Chapman and Enskog for Maxwell molecules. The thermo-fluid equations in this approximation are the Navier–Stokes equations, so that in second order, the Maxwell moment method is fully compatible with the corresponding Chapman–Enskog approximation. This compatibility is not maintained in higher-order approximations, and the question arises as to which method is to be preferred. First we note that the Chapman–Enskog method is applicable, at the expense of a great deal of computational labor (much of which can now be done by computers), to general interparticle force laws, whereas Maxwell’s method is restricted to Maxwell molecules. On the other hand, we must view this positive aspect of the Chapman–Enskog method as actually being far outweighed by the major negative aspect of the method, namely the restriction of its validity to normal solutions, which limits the physical situations in which the method can be applied.

What would be most desirable, of course, would be a method of solving Boltzmann’s equation which includes the strong points of both the Maxwell and Chapman–Enskog methods, so that, if necessary, non-normal solutions for general interparticle potentials could be obtained. These are just the features enjoyed by Grad’s method of solution for Boltzmann’s equation, and we now turn our attention to its development.

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