Chapman—Enskog solutions of the Boltzmann equation

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 ... 

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

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

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. 

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 ... 

A3.1.3.2 THE CHAPMAN-ENSKOG NORMAL SOLUTIONS OF THE BOLTZMANN EQUATION... 

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

The normal solution of the Boltzmann equation, Eq. (77a), was constructed by Enskog and Chapman in order to provide a microscopic foundation for the Navier-Stokes equations of fluid mechanics. Although these equations were derived from phenomenological arguments for continuum fluids, they accurately describe heat and viscous flows in dilute gases under a variety of circumstances. We will show in this section that the method used in constructing the normal solution leads to a derivation of the Navier-Stokes... 

In nonuniform gases the Maxwell distribution Eq. 68 does not satisfy the Boltzmann equation Eq. 66. However, Chapman and Enskogi assumed the Maxwell distribution Eq, 68 may act as the zeroth approximation to the solution of the Boltzmann equation as far as the system is not so far from equihbrium, and write fy in the form... 

For the first-order solution of the Boltzmann equation using Chapman-Enskog method, we get the constitutive laws of the N-S equation as... 

Various attempts have been made to obtain approximate solutions to the Boltzmann equation. Two of these methods were suggested independently by Chapman [10] [11] and by Enskog [24] giving identical results. In this book emphasis is placed on the Enskog method, rather than the Chapman one, as most modern work follows the Enskog approach since it is less intuitive and more systematic, although still very demanding mathematically. 

The next important advance in the theory, and the one that provided the foundation for all later work in this field, was made by Boltzmann, who in 1872 derived an equation for the time rate of change of the distribution function for a dilute gas that is not in equilibrium—the Boltzmann transport equation. (See Boltzmann and also Klein. " ) Boltzmann s equation gives a microscopic description of nonequilibrium processes in the dilute gas, and of the approach of the gas to an equilibrium state. Using the Boltzmann equation. Chapman and Enskog derived the Navier-Stokes equations and obtained expressions for the transport coefficients for a dilute gas of particles that interact with pairwise, short-range forces. Even now, more than 100 years after the derivation of the Boltzmann equation, the kinetic theory of dilute gases is largely a study of special solutions of that equation for various initial and boundary conditions and various compositions of the gas.t... 

In particular, for what class of intermolecular potentials does this equation have a unique solution for given initial and boundary conditions Is the normal solution of Chapman and Enskog in any sense a good approximation to the actual solution, at least sufficiently far from the boundaries, and for sufficiently long times Can one find explicit solutions to the nonlinear Boltzmann equation with sufficient accuracy so that the specifically nonlinear features of the Boltzmann equation can be tested, in shock wave or sound wave or sound propagation experiments, for example ... 

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-... 

One of the solution approaches of the Boltzmann equation is Chapman Enskog method. Here, the velocity distribution function is expanded into a perturbation series with the Knudsen number being the small parameter. 

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,... 

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... 

The popular problems of kinetics theory is the derivation of hydrodynamic equations, in certain conditions, solution of f (r, v,t) transport equation is similar the form that can relate directly to continuous or hydrodynamic description. In certain conditions the transport process is like hydrodynamic limit. In 1911 David Hilbert was who ptropwsed the existence Boltzmann equations solutions (named normal solutions), and these are determinate by initial values of hydrodynamic variables it return to collision invariant (mass, momentum and kinetics energy), Sydney Chapman and David Enskog in 1917 were whose imroUed a systematic process for derivate the hydrodynamic equations (and their corrections of superior order) for these variables. 

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). 

If one replaces L in Eq. (138b) by Lbgk> the linearized Boltzmann equation can be solved for a number of interesting cases. The simplest case where Eqs. (138b) and (138c) have been solved completely is the so-called Kramers problem. Here one considers the flow of a gas in a semi-infinite space bounded by a plane wall with which the molecules make diffusive collisions. For this problem one can show that there is a kinetic boundary layer near the wall and that the Chapman-Enskog normal solution is correct for points that... 

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