Collision model Boltzmann-Enskog

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... 

Data representation can be considered truly satisfactory only when it has a molecular basis. The first such successful approach was that of Enskog (Enskog 1922) for a system of hard spheres in which he made empirical modifications to the Boltzmann theory to account for the finite size of the molecules. Use of the Boltzmann equation, which considers only binary collisions, is valid for this model, since multiple collisions have a low probability. Enskog obtained expressions relating the diffusion, viscosity and thermal conductivity for the dense system, subscript E, to the dilute-gas values, superscript (0),... 

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. 

For convenience, the Maxwell-Boltzmann equation (2.67) with the collision operator definition (2.69) is rewritten here defining the point of departure in Enskog s model development ... 

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