Interdisciplinary Applied Mathematics

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15.4.1 Classical Solutions

We review here some of the most popular solution methods for the Boltzmann equation; more details can be found in (Cercignani, 1988; Cercignani, 2000; Cercignani et al., 1994), and in the comprehensive review articles (Sharipov and Sleznev, 1998; Aoki, 2001). The degree of success in deriving semianalytical solutions depends on the Knudsen number and the geometry of the flow. More specifically, there are different methods for

the hydrodynamic limit, where Kn ^ 0;

   the free-molecular limit, where Kn ^ ж, and

the transition limit, which is the most difficult regime.

Hydrodynamic Regime: The Grad and Chapman—Enskog Methods

One of the earliest solution approaches for the Boltzmann equation is the method of moments proposed by (Grad, 1949), where the distribution function is represented by a series

where floc is the local Maxwellian distribution obtained from the equilibrium Maxwellian of equation (15.8) by replacing the equilibrium quantities with the local number density n(x,t) and local temperature T(x,t), and H(N) are orthogonal Hermite polynomials. The coefficients a(N) are expressed in terms of the N moments of the distribution function. Similarly, the boundary conditions are handled using projections with the Hermite polynomials, but a closure condition is also needed, which is based on some physical condition.

This method has been used by C.-M. Ho and his colleagues using N =13 moments to predict the pressure distribution in helium flow in a channel with Kn = 0.15    at    the    exit.    The    results    are    shown    in Figure    15.15. On    the

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