Boltzmann equation Chapman-Enskog expansion

In principle, one should solve the Boltzmann equation Eq. (65) in order to arrive at explicit expressions for the pressure tensor p and heat flux q, which proves not possible, not even for the simple BGK equation Eq. (11). However, one can arrive at an approximate expression via the Chapman Enskog expansion, in which the distribution function is expanded about the equilibrium distribution function fseq, where the expansion parameter is a measure of the variation of the hydrodynamic fields in time and space. To second order, one arrives at the familiar expression for p and q... 

Hydrodynamic models are derived from the mesoscale model (e.g. the Boltzmann equation) using a Chapman-Enskog expansion in powers of the Knudsen number (Bardos et al., 1991 Cercignani et al, 1994 Chapman Cowling, 1961 Ferziger Kaper, 1972 Jenkins Mancini, 1989). The basic idea is that the collision term will drive the velocity distribution n towards an equilibrium function eq (i-e. the solution to C( eq) = 0), and thus the deviation from equilibrium can be approximated by n -i- Knui. From the... 

In the limit of small Knudsen number, the Chapman-Enskog expansion (Chapman, 1916 Enksog, 1921) of the elastic Boltzmann equation yields a first-order term for CTp of the form... 

Originated from the lattice gas automata (LGA), the LBM has been widely applied in simulating the rarefied gaseous flow in microchannel. Recently the LBM has also found success in applications to the liquid microflows, particularly the electrokinetic flows. For the continuous liquid, the Navier-Stokes equations can be recovered from the Boltzmann equation by Chapman-Enskog expansion or multiscale analysis, in which the Boltzmann equation is split into different scales for space and time variables. The lattice that an LBM operates is usually designated... 

The following mass transfer (species conservation) equation can be derived from lattice Boltzmann equation after Chapman-Enskog expansion [18] (also see Appendix 3). 

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 ... 

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,... 

The viscous stresses only come into play for systems containing significant velocity gradients within the fluid. Nevertheless, very large gradients are not required as the Navier-Stokes equations can be derived from the Chapman-Enskog perturbation theory. On the other hand, for the non-equilibrium boundary layer and shock wave systems, i.e., systems which deviates considerably from equilibrium, higher order expansions are apparently needed [28]. Actually, at least for shock waves the validity of the Maxwell-Boltzmann equation becomes questionable. 

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). 

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... 

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