Chapman-Enskog solution

The Chapman-Enskog solution of the Boltzmaim equation [112] leads to the following expressions for the transport coefficients. The viscosity of a pure, monatomic gas can be written as... 

The Burnett Expansion.—The Chapman-Enskog solution of the Boltzmann equation can be most easily developed through an expansion procedure due to Burnett.15 For the distribution function of a system that is close to equilibrium, we may use as a zeroth approximation a local equilibrium distribution function given by the maxwellian form ... 

Chapman-Enskog Solution.—The solution of the Boltzmann equation obtained by Chapman and Enskog involves the assumption... 

Chapman-Enskog solution, 35 coefficicent equations, 28 derivation from Liouville s equation, 41... 

The expression derived for the heat flux from the Chapman-Enskog solution (correct through first order) is [178]... 

Muckenfuss, C., Stefan-Maxwell Relations for Multicomponent Diffusion and the Chapman Enskog Solution of the Boltzmann Equations, J. Chem. Phys., 59, 1747-1752 (1973). 

Is it possible to use the //-theorem and the Chapman-Enskog solution to suggest a generalization of the laws of irreversible thermodynamics so that the latter would apply even when higher-order gradients or nonlinear terms in the hydrodynamic fluxes are taken into account ... 

Chapman-Enskog Solution to the Boltzmann Transport Equation... 

As mentioned previously, the Chapman-Enskog solution is based on a small Knudsen number expansion of the Boltzmann equation. Under the conditions of a small Knudsen number, we substitute the following expansion... 

The alternative approach is to examine the behavior of a dilute gas, i.e., a gas for which one only has to consider binary molecular collisions. This procedure is clearly far more restricted than the correlation function route but the corresponding transport expressions are standard and practical. For example, the Chapman-Enskog solution O) of the Boltzmann equation gives the dilute gas visco-... 

The first-order Chapman-Enskog solution of the Boltzmann equation for the viscosity and thermal conductivity of monatomic species are given by the expressions in terms of effective collision cross sections outlined in Chapter 4. However, in order to be consistent with the original papers, here the equivalent expressions in terms of collision integrals are adopted. 

Equation 5 represents a good approximation for situations in which momentum relaxation takes place considerably faster than nonthermal reaction. The local equilibrium model becomes increasingly inadequate as these rates approach one another, so that the present form of the steady state theory will be least accurate for systems that involve very rapid reactions. Higher order Chapman-Enskog solutions of the Boltzmann equation, which provide successive degrees of refinement, could be incorporated into the theory. Such modifications would introduce additional mathematical structure in Eq. 5, which is probably not needed except for the description of systems that closely approach true steady state behavior. This does not occur for any of the cases of present Interest (vide infra) or. Indeed, for any known nuclear recoil reaction system. For this fundamental reason and also because of the crude level of approximation Involved in our treatment of nonreactive collisions, the further refinement of Eq, 3 has not yet been considered to be worthwhile. 

The Chapman-Enskog solution method, as discussed in Sect. 2.8 for a dilute mono-atomic gas, can be applied to the Enskog s equation as well. Solving the Enskog s equation by the perturbation method to determine /, we find that the zero-order approximation (i.e., / / ° ) of the pressure tensor and the heat flux vector are... 

For a non-Maxwellian distribution, the (CC) term can be approximated by the Chapman-Enskog solution of the Boltzmann equation as presented in detail in Chap. 16 in Chapman and Cowling [25]. The same approach was adapted for solid particles by Gidaspow [49]. The result is ... 

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