Chapman–Enskog solutions to arbitrary order in Sonine polynomials I: Simple,rigid-sphere gas

The Chapman–Enskog solutions of the Boltzmann equations provide a basis for the computation of important transport coefficients for both simple gases and gas mixtures. These coefficients include the viscosities, the thermal conductivities, and diffusion coefficients. The Chapman–Enskog solutions are also useful for computation of the associated slip and jump coefficients near surfaces. Generally, these solutions are expressed in terms of Sonine polynomial expansions. While it has been found that relatively, low-order expansions (of order 4) can provide reasonable precision in the computation of the transport coefficients (to about 1 part in 1000), the adequacy of the low-order expansions for computation of the slip and jump coefficients still needs to be explored. Also of importance is the fact that such low-order expansions do not provide good convergence (in velocity space) for the actual Chapman–Enskog solutions even though the transport coefficients derived from these solutions appear to be reasonable. Thus, it is of some interest to explore Sonine polynomial expansions to higher orders. It is our purpose in this paper to report the results of our investigation of high-order, standard, Sonine polynomial expansions for the viscosity and the thermal conductivity related Chapman–Enskog solutions for a simple, rigid-sphere gas where we have carried out our calculations using expansions to order 150 and where our reported values for the transport coefficients have been demonstrated to converge to at least 25 significant digits. We note that, for a rigid-sphere gas, all of the relevant integrals needed for these solutions are evaluated analytically as pure fractions and, thus, results to any desired precision may be obtained. This work also indicates how results may be obtained in a similar fashion for realistic intermolecular potential models, and how gas-mixture problems may also be addressed with some additional effort.     

Transport Coefficients for Inelastic Maxwell Mixtures

The Boltzmann equation for inelastic Maxwell models (IMM) is used to determine the Navier–Stokes transport coefficients of a granular binary mixture in d-dimensions. The Chapman–Enskog method is applied to solve the Boltzmann equation for states near the (local) homogeneous cooling state. The mass, heat, and momentum fluxes are obtained to first order in the spatial gradients of the hydrodynamic fields, and the corresponding transport coefficients are identified. There are seven relevant transport coefficients: the mutual diffusion, the pressure diffusion, the thermal diffusion, the shear viscosity, the Dufour coefficient, the pressure energy coefficient, and the thermal conductivity. All these coefficients are exactly obtained in terms of the coefficients of restitution and the ratios of mass, concentration, and particle sizes. The results are compared with known transport coefficients of inelastic hard spheres (IHS) obtained analytically in the leading Sonine approximation and by means of Monte Carlo simulations. The comparison shows a reasonably good agreement between both interaction models for not too strong dissipation, especially in the case of the transport coefficients associated with the mass flux     

惰性气体原子间相互作用势比较研究

原子间相互作用势是预测惰性气体输运性质的必要输入条件. 文章对描述惰性气体原子间相互作用的Lennard-Jones势、指数排斥势、Hartree-Fock-Dispersion-B (HFD-B)势和唯象势的形式和特点进行了分析. 基于Chapman-Enskog方法, 计算得到了惰性气体在300–5000 K温度区间内基于四种原子相互作用势的黏性和热导率, 并与文献报道的实验和理论计算结果进行了比较. 研究结果表明, 基于Hartree-Fock排斥理论与色散理论发展起来的HFD-B势能够合理反映惰性气体原子相互作用的趋势与特征, 因而可以较好地预测惰性气体的宏观输运性质.     

First-principle calculation for the high-temperature diffusion coefficients of small pairs: the H–Ar Case

Argon has been widely used as a diluent for high-temperature reacting flow experiments. Numerical simulation of such process requires accurate diffusion coefficients for the H–Ar pair. All available potential energy functions for the H–Ar system have been empirically extrapolated in the repulsive region and are probably not reliable for prediction of H–Ar binary diffusion coefficient at high temperatures. We perform calculations using the restricted coupled cluster theory with single and double excitation (plus triple corrections) [RCCSD(T)] and suitable basis sets to obtain accurate potential energies. The calculated potential energy function is corrected for basis set superposition error and is validated against molecular beam scattering data. Comparisons with previous literature potential functions are also made. Using the Chapman–Enskog theory we carried out first-principle calculation of high-temperature diffusion coefficients by direct numerical integration of the collision integrals using the RCCSD(T) potential function. The computed diffusion coefficients are validated against available experimental data. Comparisons are also made with results obtained from transport compilations and packages commonly used in combustion simulation.     

A Monte Carlo model for determination of binary diffusion coefficients in gases

A Monte Carlo method has been developed for the calculation of binary diffusion coefficients in gas mixtures. The method is based on the stochastic solution of the linear Boltzmann equation obtained for the transport of one component in a thermal bath of the second one. Anisotropic scattering is included by calculating the classical deflection angle in binary collisions under isotropic potential. Model results are compared to accurate solutions of the Chapman–Enskog equation in the first and higher orders. We have selected two different cases, H₂ in H₂ and O in O₂, assuming rigid spheres or using a model phenomenological potential. Diffusion coefficients, calculated in the proposed approach, are found in close agreement with Chapman–Enskog results in all the cases considered, the deviations being reduced using higher order approximations.     

A kinetic theory of dense fluids

A new kinetic equation is developed which incorporates the desirable features of the Enskog, the Rice-Allnatt, and the Prigogine-Nicolis-Misguich kinetic theories of dense fluids. Advantages of the present theory over the latter three theories are (1) it yields the correct local equilibrium hydrodynamic equations, (2) unlike the Rice-Allnatt theory, it determines the singlet and doublet distribution functions from the same equation, and (3) unlike the Prigogine-Nicolis-Misguich theory, it predicts the kinetic and kinetic-potential transport coefficients. The kinetic equation is solved by the Chapman-Enskog method and the coefficients of shear viscosity, bulk viscosity, thermal conductivity, and self-diffusion are obtained. The predicted bulk viscosity and thermal conductivity coefficients are singular at the critical point, while the shear viscosity and self-diffusion coefficients are not.     

New properties of a class of generalized kinetic equations

We give some properties of a new class of hard-sphere kinetic equations of great generality, introduced earlier by Polewczak. The assumptions used to obtain the general class are very weak, and the equations include not only the standard and revised Enskog equations, but also generalizations thereof that can be expected to yield essentially exact transport coefficients. In particular, there is a natural two-particle realization that is obtained from maximizing the information entropy subject to prescribed two-particle and one-particle probability distribution functions;k-particle analogs fork > 2 also naturally follow. We obtain Liapunov functionals for the whole class of equations under consideration and discuss the question of which of these functionals can be expected to play the role ofH-functions. We also obtain several more special results that include new lower bounds on the potential part of theH-function for the revised Enskog equation. The bounds are instrumental in obtaining global existence theorems and also imply that the necessary condition for invertibility of the nonequilibrium extension of local activity as a functional of local density is satisfied.     

Molecular dynamics simulations of one-dimensional Lennard-Jones systems

One-dimensional Lennard-Jones systems are investigated by molecular dynamics simulations. The full Lennard-Jones potential is compared to the repulsive Lennard-Jones potential. It is found that the pair correlation function and the normalized velocity autocorrelation function agree at high densities and high temperature. However, the diffusion coefficient indicates that the attractive potential introduces additional correlations into particle dynamics which are not reflected in the statics. These results are in agreement with three-dimensional studies.     

Kinetic Models for Granular Flow

The generalization of the Boltzmann and Enskog kinetic equations to allow inelastic collisions provides a basis for studies of granular media at a fundamental level. For elastic collisions the significant technical challenges presented in solving these equations have been circumvented by the use of corresponding model kinetic equations. The objective here is to discuss the formulation of model kinetic equations for the case of inelastic collisions. To illustrate the qualitative changes resulting from inelastic collisions the dynamics of a heavy particle in a gas of much lighter particles is considered first. The Boltzmann–Lorentz equation is reduced to a Fokker–Planck equation and its exact solution is obtained. Qualitative differences from the elastic case arise primarily from the cooling of the surrounding gas. The excitations, or physical spectrum, are no longer determined simply from the Fokker–Planck operator, but rather from a related operator incorporating the cooling effects. Nevertheless, it is shown that a diffusion mode dominates for long times just as in the elastic case. From the spectral analysis of the Fokker–Planck equation an associated kinetic model is obtained. In appropriate dimensionless variables it has the same form as the BGK kinetic model for elastic collisions, known to be an accurate representation of the Fokker–Planck equation. On the basis of these considerations, a kinetic model for the Boltzmann equation is derived. The exact solution for states near the homogeneous cooling state is obtained and the transport properties are discussed, including the relaxation toward hydrodynamics. As a second application of this model, it is shown that the exact solution for uniform shear flow arbitrarily far from equilibrium can be obtained from the corresponding known solution for elastic collisions. Finally, the kinetic model for the dense fluid Enskog equation is described.     

Shock compression of simple molecules

A simple approach is developed to calculate shock compression of simple molecules. This approach is based upon an accurate analytic representation of the Lennard-Jones fluids in conjunction with the Enskog theory, which is used to calculate the molecular diameter as a function of temperature along the Hugoniot. The model permits rapid, yet reliable calculations. It is applied to N₂, O₂, H₂, D₂, CH₄, CO, and CO₂. The results are tested by the comparison with experimental data and with other calculations. The computed Hugoniots agree reasonably with experimental results for many (but not all) simple molecules and are comparable to those of more complicated models.     

Dynamic properties of Lennard-Jones fluids and liquid metals

The dependence of the dynamic properties of liquid metals and Lennard-Jones fluids on the characteristics of the interaction potentials is analyzed. Molecular-dynamics simulations of liquids in analogous conditions but assuming that their particles interact either through a Lennard-Jones or a liquid-metal potential were carried out. The Lennard-Jones potentials were chosen so that both the effective size of the particles and the depth of the potential well were very close to those of the liquid-metal potentials. In order to investigate the extent to which the dynamic properties of liquids depend on the short-range attractive interactions as well as on the softness of the potential cores, molecular-dynamics simulations of the same systems but assuming purely repulsive interactions with the same potential cores were also performed. The study includes both single-particle dynamic properties, such as the velocity autocorrelation functions, and collective dynamic properties, such as the intermediate scattering functions, the dynamic structure factors, the longitudinal and transverse current correlations, and the transport coefficients.     

RESEARCH NOTE WCA repulsive and attractive contributions to the thermodynamic properties at the vapour-liquid equilibrium

The Lennard-Jones attractive and repulsive contributions of intermolecular forces (as separated in the Weeks-Chandler-Andersen (WCA) theory) to the pressure and chemical potential of coexisting vapour and liquid phases are obtained by using an equation of state recently proposed by us. Some comments are given about the computer simulation results obtained by Plackov and Sadus (1997, Fluid Phase Equilib., 134, 77) using the McQuarrie-Katz separation of the intermolecular potential.     

Vapour pressure of non-polar fluids and their repulsive and attractive contributions

Using a simple equation of state, based on the Weeks-Chandler-Andersen separation of the intermolecular potential, we have obtained the contributions of repulsive and attractive intermolecular forces to the thermodynamic properties of coexisting vapour and liquid phases of a Lennard-Jones (LJ) fluid.

In order to obtain the vapour pressure of real non-polar fluids, we take the LJ fluid as a reference model, and propose a new perturbative contribution, which is dependant on the temperature and on the acentric factor of the substance. Using the complete perturbed equation, we determine the corresponding repulsive and attractive contributions to the vapour pressure of non-polar fluids. The results show that the attractive vapour pressure of non-polar fluids increases with increasing acentric factor, i.e., larger deviation of the molecular shape from spherical symmetry.

This procedure could be extended to separate the repulsive and attractive contributions of the intermolecular forces to other thermodynamic properties of non-polar fluids as well as of polar fluids and fluid mixtures.     

Thermodynamic properties and self-diffusion coefficients of simple liquids

Pair distribution functions and thermodynamic properties for a system of 108 particles interacting with a hard-sphere modified Lennard-Jones (12, 6) potential have been calculated by the Monte Carlo technique for a range of reduced liquid densities and two reduced temperatures. It is shown that the introduction of a hard core significantly modifies the Lennard-Jones parameters for the noble gases. The friction coefficients proposed in the transport theories of Rice and Allnatt and of Palyvos and Davis have been calculated using the exact pair distribution functions. In the normal liquid range, contributions to the frictional force on a particle arising from hard sphere collisions and quasi-Brownian motion are shown always to be comparable. This result is contrary to the conditions under which the principal assumption of Brownian theory is applicable.     

On the generalization of the Enskog equation to the case of steep repulsive potentials

For a classical homogeneous system of particles interacting via steeply repulsive potentials a generalization of the Enskog equation is proposed. This kinetic equation has the properties that it reduces to the usual Enskog equation in the limit of hard-sphere potentials and that the total instead of the kinetic energy is conserved in the system. The expression for the potential energy obtained is correct at arbitrary densities in equilibrium.     

A Fokker–Planck Model of the Boltzmann Equation with Correct Prandtl Number for Polyatomic Gases

We propose an extension of the Fokker–Planck model of the Boltzmann equation to get a correct Prandtl number in the Compressible Navier–Stokes asymptotics for polyatomic gases. This is obtained by replacing the diffusion coefficient (which is the equilibrium temperature) by a non diagonal temperature tensor, like the Ellipsoidal-Statistical model is obtained from the Bathnagar–Gross–Krook model of the Boltzmann equation, and by adding a diffusion term for the internal energy. Our model is proved to satisfy the properties of conservation and a H-theorem. A Chapman–Enskog analysis shows how to compute the transport coefficients of our model. Some numerical tests are performed to illustrate that a correct Prandtl number can be obtained.     

On the Enskog-Thorne theory for a binary mixture of dissimilar rigid spheres

Thorne's method for obtaining transport coefficients in a binary rigid-sphere mixture is reexamined. First, a close look is taken at the way in which the point where the Enskog functionsx _ij are evaluated is introduced. Second, the calculation of the fluxes in the system and the transport coefficients is given. Thorne's results are found to be correct and independent of the choice of the point where the transfer plane is located. This does not hold true for the diffusion flux. It is shown that a different diffusion force is obtained for each selection and that only those diffusional effects which are of first order in the density are consistent with irreversible thermodynamics.     

Equation of state for solids under high pressure,with application to alkali halides

A new equation of state has been derived from an analysis of the relations between energy, pressure and volume combined with a Padé approximation for the pressure. This modified equation of state is first tested by comparison with the exact expressions that are obtained for four-potential models, the Lennard-Jones, the Morse, the Born-Mayer, and the Varshni-Shukla potentials, with excellent results. It is also compared with experimental high-pressure data for four alkali halide crystals with quite good results.     

Collective modes of one-dimensional lennard-jones systems

The density fluctuations of one-dimensional Lennard-Jones systems are investigated by molecular dynamics simulation. The full Lennard-Jones potential is compared to the repulsive Lennard-Jones potential. It is found that the behavior of the density fluctuations at small wave vectors is determined by the repulsive portion of the potential. The variation of the fluctuations with density is explained. It is shown that these systems do not display hydrodynamics.