Chapman—Enskog solutions of the

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 ... 

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 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 ... 

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. 

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

Chapman-Enskog Expansion As we have seen above, the momentum flux density tensor depends on the one-particle distribution function /g, which is itself a solution of the discrete Boltzman s equation (9.80). As in the continuous case, finding the full solution is in general an intractable problem. Nonetheless, we can still obtain a useful approximation through a perturbative Chapman-Enskog expansion. 

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

Chapman-Enskog theory provides the basis for the multicomponent transport properties laid out by Hirschfelder, Curtiss, and Bird [178] and by Dixon-Lewis [103]. The multi-component diffusion coefficients, thermal conductivities, and thermal diffusion coefficients are computed from the solution of a system of equations defined by the L matrix [103], seen below. It is convenient to refer to the L matrix in terms of its nine block submatrices, and in this form the system is given by... 

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... 

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 ... 

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-... 

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... 

Thus it is seen that the solution of the first approximation equation of Chapman and Enskog, Eq. 75, is given by a solution of the variational principle... 

This is the proof of the variational principle that Eq. 93 gives the solution of the Chapman-Enskog equation. The variational principle given in the form of Eq. 93 is much more convenient for practical purposes, because we need not consider restrictions other than simple ones such as the auxiliary conditions, Eqs. 70, 71, and 72. In the case of thermal conduction in a simple gas, Eq. 93 reduces to... 

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. 

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

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... 

Example 5.1-2 Comparing two estimates of gas diffusion Use the Chapman-Enskog theory and the empirical correlation in Equation 5.1-9 to estimate the diffusion of hydrogen in nitrogen at 21 °C and 2 atmospheres. The experimental value is 0.38 cm /sec. Solution For the Chapman-Enskog theory, the key parameters are... 

Enskog obtained a solution by expanding A, (Wj) and 6, (W,) in a finite series of Sonine polynomials [60,114,178]. This solution is usually found to converge in only one or two terms. More discussion of the solution itself can be found in Chapman and Cowling [60] and in Hirschfelder, Curtiss, and Bird [178]. We are concerned here with using the Enskog result to obtain transport coefficients. 

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