Enskog expansion method

Alternative estimates of the transport coefficients can be obtained from the rigorous Chapman-Enskog expansion method of mono-atomic gases at low densities (e.g., [24] [25] [12] [61] (p 202) [28]). The transport coefficients deduced from the Chapman-Enskog kinetic theory with the rigid elastic spheres interaction model yield (e.g., [39] sect 8.2 [61], p 202) ... 

The Enskog [24] expansion method for the solution of the Boltzmann equation provides a series approximation to the distribution function. In the zero order approximation the distribution function is locally Maxwellian giving rise to the Euler equations of change. The first order perturbation results in the Navier-Stokes equations, while the second order expansion gives the so-called Burnett equations. The higher order approximations provide corrections for the larger gradients in the physical properties like p, T and v. 

The Chapman-Enskog method may be applied directly to Eq. (223) to obtain/g as an expansion in powers of the operator V, and in this manner one can verify that the solution obtained is in fact the usual normal solution, in the linear approximation. On the other hand, the closed form (223) may be useful for rapidly varying processes to which the Chapman-Enskog expansion is not applica.ble. 

We shall now solve the Kramers equation (7.4) approximately for large y by means of a systematic expansion in powers of y-1. Straightforward perturbation theory is not possible because the time derivative occurs among the small terms. This makes it a problem of singular perturbation theory, but the way to handle it can be learned from the solution method invented by Hilbert and by Chapman and Enskog for the Boltzmann equation.To simplify the writing I eliminate the coefficient kT/M by rescaling the variables,... 

Enskog s procedure for solution of an equation such as Eq. (226) involves the introduction of a perturbation parameter X with an expansion about a condition of local equilibrium, which arises if the LHS of Eq. (226) equals zero. The method entails putting Eq. (226) in the form... 

In spite of have been proposed many approximated solutions to Boltzmann equation (including the Grad s method of 13 moments, expansions of generalized polynomial, bimodal distributions functions), however the Chapman-Enskog is the most popular outline for generalize hydrodynamic equations starting from kinetics equations kind Boltzmann (James William, 1979 Cercignani, 1988). 

Equations (6.85) and (6.86) refiect the local equilibrium nature of the solution to Eq. (6.83). It is, therefore, seen that the so-called Chapman-Enskog method of the solution is based on an expansion about local equilibrium conditions. Pitfalls of this approach have been previously noted. Writing, without loss of generality. 

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