Chapman-Enskog Solution Research Articles

Non-linear Burnett order thermal stress coefficient

The formal theory of non-equilibrium statistical is used to obtain the irreversible Burnett order thermal stresses, and time correlation function expressions for both linear and nonlinear transport coefficients are identified. Evaluation of these expressions in the Boltzmann limit shows agreement with the Chapman-Enskog solution of kinetic theory.

Nonequilibrium velocity distributions and recombination rates in electron–ion and ion–ion recombinations in weakly ionized gases

The nonequilibrium velocity distributions and recombination rates in electron–ion and ion–ion recombinations in a heat bath of neutral molecules are investigated by solving the Boltzmann equation with the Monte Carlo simulation. The explicit time-dependent velocity distributions and recombination rates from the initial equilibrium to quasisteady states are obtained for the typical electron–ion dissociative recombination e+N+2→N+N in the heat bath of N2 and for the typical ion–ion recombination H++H− →H+H in the heat bath of H2. The velocity distributions and the recombination rates indicate significant deviations from equilibrium. The decreases in the recombination rates from the equilibrium rates are 30% for e+N+2→N+N and 15% for H++H−→H+H at quasisteady states for the heat bath temperature 300 °K and the charge concentrations X∼10−4 and 10−2, respectively. The Monte Carlo result is compared with the Chapman–Enskog solution by Shizgal and Karplus for H++H−→H+H. The Chapman–Enskog solution indicates a too large nonequilibrium for high charge concentration X≳10−4.

A collisional approach to the calculation of time correlation functions. Transport coefficients of gases

A method of successive approximations is proposed for the evaluation of the time-correlation functions such as those that give the thermal transport coefficients of gases. The method is based on a calculation of the changes in correlations of appropriate functions of the molecular velocity which are a result of collisions in the gas. The decaying rates of the correlations are expressed as integrals of the differential collision cross section. When the first approximation is introduced in the expressions for thermal transport coefficients, results are obtained for the coefficient of binary diffusion and the viscosity and thermal conductivity of single-component systems which are identical with those of the first Chapman-Enskog solutions of the Boltzmann and Enskog equations. For the coefficients of viscosity and thermal conductivity in multicomponent systems, it is shown that the first approximation leads to expressions of the form of the Sutherland and Wassiljewa relations, respectively.

Stefan-Maxwell relations for multicomponent diffusion and the Chapman-Enskog solution of the Boltzmann equations

It is established that corrections factors for higher order Chapman-Enskog approximations to multicomponent diffusion coefficients are equal for a given N-component gas mixture. This enables us to invert the mass flux equations for the diffusion forces and obtain the Stefan-Maxwell equations in any Chapman-Enskog approximation. The impedance matrix for these equations is symmetric regardless of whether or not the multicomponent diffusion coefficients are symmetric. Further, the impedance matrix is the same whether one chooses to invert mass flux equations involving symmetric or asymmetric diffusion coefficients.

Transport properties of two-temperature partially ionized argon

Calculations are presented for the transport properties of a two-temperature argon plasma based on the Chapman-Enskog and perturbed-Lorentzian solutions of the Boltzmann equations. The convergence problems known to exist in the Chapman-Enskog solution for the properties of a one-temperature argon plasma also occur here; this difficulty is removed by using the perturbed Lorentzian solution at low and moderate degrees of ionization. Also, it is found that the lowest order perturbed Lorentzian and Chapman-Enskog approximate solutions provide upper and lower bounds to the actual solution at low and moderate degrees of ionization. Finally, owing to the Ramsauer minimum in the electron-atom cross section, the electrical conductivity of the two-temperature argon plasma is found to decrease with increasing electron temperature (at fixed heavy particle temperature and electron density) at low degrees of ionization, and increase with increasing electron temperature at higher degrees of ionization.

On the Motion of Small Spheres in Gases. II. Thermo-phoresis, Diffusio-phoresis and Related Phenomena

Abstract Using the general theory developed by the author in the previous paper, the problems of thermo-phoresis and diffusio-phoresis have been examined with a view to elucidating the effect of a general gas-surface scattering law on the associated forces. It is found that, in thermo-phoresis, there exists a close coupling between the first anisotropic moment of the gas-surface scattering law and the solution of the Chapman-Enskog conductivity equation. Explicit calculations indicate that for an arbitrary combination of any type of purely elastic scattering, e. g. specular, Lambert or backward, there is no change in the creep velocity of the particle. On the other hand, diffuse scattering with redistribution, leads to a marked decrease in the creep velocity. These conclusions are independent of the force law between gas atoms and depend only on the gas-surface interaction. In diffusio-phoresis, using only the simplest Chapman-Enskog solution for a binary mixture, it is found that unless the gas-surface laws differ greatly for the two species, there is virtually no influence of gas-surface scattering on the particle motion: the effect is due almost entirely to the mass difference and concentration ratio. We also investigate the way in which particles move in gas streams close to boundaries; slip flow is used as an example and it is shown that for diffuse reflection a particle at the wall travels with a speed which is about 60% of the gas speed at the wall.

Dynamic shielding effects in partially ionized gases

The effects of dynamic shielding of charged-particle interactions on the predicted values of hte transport properties of partially ionized argon are considered. It is found that, at low degreesof ionization, dynamic shielding effects are of little importance, because charged-particle encounters are infrequent, so that the perturbed Lorentzian method may be used to obtain a rapidly convergent solution. At higher degrees of ionization, where the Chapman–Enskog solution converges rapidly, dynamic shielding effects can be taken into account using the expressions presented in the appendix to this paper. It is also found that the static shielding model, assuming complete shielding by electrons only, yields results that are quite close to those obtained when dynamic shielding is considered.

The kinetic theory of diffusion and thermal diffusion in adsorbed gases

The derivation of general expressions for the surface diffusion and thermal diffusion coefficient is presented on the basis of the Chapman-Enskog solution of the kinetic equation for an adsorbed gas. The numerical calculations of these coefficients for 36A-40A mixtures adsorbed on the graphite surface are described for intermolecular interaction potentials proposed by Sinagoglu and Pitzer and by Barker and Everett. The method elaborated by Smith and Munn is adapted for numerical calculations on surface binary collisions. The results obtained are compared with those appropriate to transport properties of ordinary three-dimensional gases.

On the initial-value problem in the kinetic theory of gases

The relaxation of an initially non-uniform gas to equilibrium is studied within the framework of the kinetic theory of gases. The macroscopic gas properties are taken to depend on one spatial dimension as well as the time. The amplitude of the non-uniformity is assumed to be small with a length scale large compared with the mean free path, and the Krook model of the Boltzmann collision integral is employed.By applying multi-time scale perturbation methods to this reduced problem, uniformly valid analytical solutions for the macroscopic velocity, density and temperature are obtained. The macroscopic equations appropriate to each stage of the relaxation process are obtained in a straightforward and unambiguous manner. The distribution function obtained is shown to be a re-expansion of the Chapman–Enskog solution of the Krook equation, with additional terms accounting for the relaxation of the initial conditions to a near equilibrium form. The results indicate that the uniformly valid frst approximation to the macroscopic velocity, density and temperature can be obtained from the Navier–Stokes equations, but that no purely macroscopic set of equations will suffice for the determination of higher approximations.

Heat transfer and temperature jump in a polyatomic gas

A variational principle based on the integro-differential form of the linearised Wang Chang-Uhlenbeck equation, with general boundary conditions, is used to evaluate the heat conducted through a polyatomic gas between parallel plates. The result is an accurate, closed form expression for the heat transfer, valid for all degrees of rarefaction, rational in the inverse Knudsen number, and parametrised by the thermal accommodation coefficients, the heat capacity of the internal modes, full range moments of the Chapman-Enskog solution and half range bracket integrals of the free molecular solution. The temperature jump coefficient is obtained from the high density expansion of the heat flux and is dependent on the thermal accommodation coefficients, the internal heat capacity, and moments of the Chapman-Enskog solution. In the limit of vanishing internal specific heat, both the heat transfer and the temperature jump reduce to results previously given for the monatomic gas.

Nonequilibrium Contributions to the Rate of Reaction. IV. Explicit Time-Dependent Solutions

The explicit time dependence of the corrections to the equilibrium rate of reaction and the equilibrium rate of change of the temperature in one-component monatomic systems is obtained by integrating the nonlinear Boltzmann equation with a moment method. The calculations are carried out for the model systems employed in the earlier studies. For sufficiently small values of the ratio of the elastic and reactive relaxation times (τE / τR ≲ 10−2), it was found that in a time τE the time-dependent corrections increased rapidly from zero to an asymptotic value. For times greater than τE, the asymptotic values of the corrections increased or decreased slowly and approximately linearly with time. The maximum values of the corrections as a function of time (in instances where a maximum value was obtained) were compared with the corresponding result obtained employing the Chapman–Enskog solution of the Boltzmann equation. This qualitative comparison indicates that for τE / τR ≈ 10−2 the Chapman–Enskog result differs from the present result by 10%. For τE / τR ≈ 10−4, the deviation between the two results is reduced to 1%–3%. In the limit as τE / τR → 0, the present results coincide exactly with the Chapman–Enskog result.

Existence theory of the linear equations appearing in the Chapman-Enskog solutions to two kinetic equations for liquids

The linear operators appearing in the Chapman-Enskog solutions to Kirkwood's Fokker-Planck kinetic equation and to Rice and Allnatt's kinetic equation are studied in this article. Existence proofs are given for the linearized Chapman-Enskog equations involving either the Fokker-Planck or the Rice-Allnatt operators. It is shown that the Fokker-Planck and Rice-Allnatt operators, defined in the domain appropriate to kinetic theory, are essentially self-adjoint. It is also shown that the spectrum of either of these operators coincides with the spectrum of the self-adjoint extension of the corresponding operator.

Exact and Chapman-Enskog Solutions of the Boltzmann Equation for the Lorentz Model

The Boltzmann equation for the Lorentz model, where noninteracting classical point particles move through a random array of stationary scatterers (hard spheres), can be solved exactly. Also the corresponding Chapman-Enskog solution, and thus the hydrodynamical equation, can be found in closed form, i. e., to all orders in the gradients. A detailed explicit discussion of the relationship between the two solutions is given. The limitations of the Chapman-Enskog development (in the standard meaning of the term) as a method for solving the initial value problem of the Boltzmann equation are discussed, and the resolution of the Hilbert paradox is given.

Nonlinear and nonlocal transport effects in simple fluids

Correlation function formulae for transport coefficients of 2nd order in arbitrarily dense fluids are derived, using a modified Chapman-Enskog solution of the Liouville equation. Some static correlations are neglected. Approximate evaluation for dilute gases gives essentially the same results as the solution of Boltzmann's equation. As an application higher order transport effects in the critical region are estimated. It is conjectured that they are apparent in sound absorption and the line width of Rayleigh scattering if (T−Tc)/Tc≲10−3.

Kinetic Equations for Polyatomic Gases: The 17-Moment Approximation

A set of kinetic equations for polyatomic gases is obained by extending the 13-moment approximation of Grad to solve the inelastic Boltzmann equation of Wang Chang, Uhlenbeck, and de Boer. It is expected that the resulting 17-moment approximation (the four additional moments are the internal temperature and the internal heat flux vector) will be applicable to a wider class of physical problems than the “normal”-type Chapman-Enskog solutions to the inelastic Boltzmann equation. In particular, the kinetic equations should prove quite useful in the study of shock structure and sound propagation in polyatomic gases. As examples, the 17-moment approximation is applied to near equilibrium and relaxation type flows; and all the essential features of the Wang Chang et al. analysis are recovered, such as those involving temperature equilibration, volume viscosity, and the extension of the Navier-Stokes equations to polyatomic gases. In addition, corrections to the Navier-Stokes equations are derived involving terms which are second order in the transport coefficients. Possible generalizations and extensions of the 17-moment approximation are also discussed.

Transport Theory of a Partially Degenerate Plasma

Abstract : The transport coefficients of a high temperature, weakly coupled plasma of non-degenerate ions and partially degenerate electrons (i.e. stellar matter) are calculated by means of a Chapman-Enskog solution of the quantum Lenard-Balescu kinetic equation. Both electron-electron and electron-ion collisions are included. The leading term of the transport coefficients, the Coulomb logarithm, is evaluated exactly and relatively simply for all degrees of electron degeneracy. The method is applicable to any partially degenerate system with weak, long-range particle interactions. (Author)

Combination Rules for Binary Gaseous Mixtures Deduced from Viscosity

In this paper, the results are reported of two separate tests on the validity of three mixing rules that are commonly employed in determining the force parameters for unlike molecular collisions from those for like-molecular collisions where the interaction may be described by means of a two-parameterforcepotential. Both tests utilize the equations for the viscosity of dilute gases based on the Chapman–Enskog solution to the Boltzmann equation, together with highly precise experimental data on the viscosity of pure gases and binary gas mixtures. Both tests indicated that the experimental data are incompatible with the simple combination rules; that is, the arithmetic average for the distance parameter and the geometric mean for the energy parameter. The results of the second test show that the combination rule εijσij6 = (εiiσii6εjjσjj6)1 / 2 is not contradicted by the experimental data. All possible binary combinations of the gases He, Ne, Ar, N2, and CO2 were analyzed for this report.

The Chapman-Enskog solution of the Boltzmann equation: A reformulation in terms of irreducible tensors and matrices: K Kumar,Aus J Phys, 20, June 1967, 205–252

The Chapman-Enskog solution of the Boltzmann equation: A reformulation in terms of irreducible tensors and matrices: K Kumar,Aus J Phys, 20, June 1967, 205–252

Model Dependence of the Slip Coefficient

The slip flow or Kramers problem for rarefied gas flow is studied by the use of a number of models. The models employed include the Bhatnagar-Gross-Krook model, Maxwell molecules, the Gross-Jackson model, Cercignani's variable-collision-frequency model, and a new model which allows variable collision frequency, the correct Chapman-Enskog solution and the correct Prandtl number. The Kramers problem is solved by means of an approximation scheme based on the method of elementary solutions for the constant-collision-frequency models and by a variational method for the variable-collision-frequency model. Numerical results show that the slip coefficient is only slightly model-dependent.

The Chapman-Enskog solution of the generalized Boltzmann equation

The generalization of the Chapman-Enskog method is applied to systems of equations which are obtained when Bogolyubov's functional assumption is introduced into the BBGKY hierarchy. A system of inhomogeneous equations is derived from the solution of which the transport coefficients may be determined. The kernel of these equations is shown to have different left and right eigen functions with zero eigen values. The method, while dependent on the Bogolyubov assumption, is independent of the assumption of a power series expansion for the transport coefficients.