Chapman-Enskog approximation

By use of the Chapman-Enskog approximate solution method [11], the kinetic pressure tensor can be given by [22] ... 

The kinetic equations for the distribution functions in the zero-order Chapman-Enskog approximation have the form ... 

Let us consider state dependent rate coefficients for chemical reactions appearing in Eqs. (59)-(61). In the zero-order Chapman-Enskog approximation rate coefficients for... 

This equation is the basis of the Gouy-Chapman model of the diffuse charge cloud adjacent to a charged surface. It was discovered hy the french physicist Gouy from Lyon in 1910 [4] and rediscovered in 1913 by Chapman [5] (not the same Chapman as for Chapman-Enskog approximation). 

The discussion to follow has a close resemblance to the Chapman-Enskog approximation method in the kinetic theory of gases, and it may be helpful to review briefly the basis of this method. The distribution function is expanded as... 

The Chapman-Enskog approximation method leads, at all stages, to hydrod5mamic equations which are first order in the time and can therefore be solved subject to given initial conditions. This procedure by which the boundary-value problem is converted into an initial-value problem is, from the mathematical point of view, somewhat mysterious. It appears likely that the procedure will converge only for processes whose scale of variation is of the order of, or less than, the mean free path (or other characteristic length). The reduction to an initial-value problem would then be impossible for rapidly varying processes. 

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. 

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 ... 

The Chapman-Enskog method has been used to solve for steady state tracer diffusion (. ). According to the method the singlet distribution function for the diffusing species 1, present In a trace amount n nj, 1 1) In an otherwise equilibrium fluid. Is approximated by... 

The Chapman-Enskog theory of flow In a one-component fluid yields the following approximation to the momentum balance equation (Jil). 

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... 

Equation (105) is the basis for the determination of gas-phase diffusion coefficients and ultra low vapor pressures using the methods proposed by Davis and Ray (1977), Ravindran et al. (1979), and Ray et al. (1979). Additional information can be gained by writing the Chapman-Enskog first approximation for the gas-phase diffusivity (Chapman and Cowling, 1970),... 

The Chapman-Enskog theory was developed for dilute, monatomic gases for pure substances and for binary mixtures. The extension to multicomponent gas mixtures was performed by Curtiss and Hirschfelder (C12, Hll), who in addition have shown that the Chapman-Enskog results may also be obtained by means of an alternate variational method. Recently Kihara (K3) has shown how expressions for the higher approximations to the transport coefficients may be obtained, which are considerably simpler than those previously proposed by Chapman and Cowling these simpler formulas are particularly advantageous for calculating the coefficients of diffusion and thermal diffusion (M3, M4). 

In these equations, T is the temperature, p is the pressure, X is the mole fraction of species k, m is the molecular mass, R is the universal gas constant, and / is the pure species viscosity. The T>jk are first order (in the Chapman-Enskog theory) binary diffusion coefficients, given by Eq. 12.113. It is actually inappropriate [103] to use a second-order or higher approximation [265] to the binary diffusion coefficients here. For this reason the Dixon-Lewis paper used the notation to emphasize that the first-order approxima-... 

If not available experimentally, the binary diffusion coefficient, Ay, may also be calculated using elementary kinetic theory (Chapman-Enskog equation proposed by Cussler [61]). A first order approximation for DtJ is given by Yakabe et al., as follows [57] ... 

The Chapman-Enskog procedure to approximate the distribution functions /j, /e by a linear perturbation ansatz, the Landau form of the Coulomb collision integral together with the small mass ratio me/m expansions in the classical work of Braginskii results in the friction term R... 

Chapman, Enskog, and Cowling have developed a general method for dealing with nonequilibrium states in not-too-dense gases which allows the approximate evaluation of the proper distribution functions to be used in place of the equilibrium distribution functions. While we shall not go into the details of the method, one of the results is of direct interest in the problem of diffusion in mixtures. 

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