Generalized Chapman-Enskog Method Research Articles

Continuum Models for Bulk Viscosity and Relaxation in Polyatomic Gases

Bulk viscosity and acoustic wave propagation in polyatomic gases and their mixtures are studied in the frame of one-temperature and multi-temperature continuum models developed using the generalized Chapman–Enskog method. Governing equations and constitutive relations for both models are written, and the dispersion equations are derived. In the vibrationally nonequilibrium multi-component gas mixture, wave attenuation mechanisms include viscosity, thermal conductivity, bulk viscosity, diffusion, thermal diffusion, and vibrational relaxation; in the proposed approach these mechanisms are fully coupled contrarily to commonly used models based on the separation of classical Stokes–Kirchhoff attenuation and relaxation. Contributions of rotational and vibrational modes to the bulk viscosity coefficient are evaluated. In the one-temperature approach, artificial separation of rotational and vibrational modes causes great overestimation of bulk viscosity whereas using the effective internal energy relaxation time yields good agreement with experimental data and molecular-dynamic simulations. In the multi-temperature approach, the bulk viscosity is specified only by rotational modes. The developed two-temperature model provides excellent agreement of theoretical and experimental attenuation coefficients in polyatomic gases; both the location and the value of its maximum are predicted correctly. One-temperature dispersion relations do not reproduce the non-monotonic behavior of the attenuation coefficient; large bulk viscosity improves its accuracy only in the very limited frequency range. It is emphasized that implementing large bulk viscosity in the one-temperature Navier–Stokes–Fourier equations may lead to unphysical results.

Extended continuum models for shock waves in CO2

Three continuum models extending the conventional Navier–Stokes–Fourier approach for modeling the shock wave structure in carbon dioxide are developed using the generalized Chapman–Enskog method. Multi-temperature models are based on splitting multiple vibrational relaxation mechanisms into fast and slow processes and introducing vibrational temperatures of various CO2 modes. The one-temperature model takes into account relaxation processes through bulk viscosity and internal thermal conductivity. All developed models are free of limitations introduced by the assumptions of a calorically perfect gas and constant Prandtl number; thermodynamic properties and all transport coefficients are calculated rigorously in each cell of the grid. Simulations are carried out for Mach numbers 3–7; the results are compared with solutions obtained in the frame of other approaches: multi-temperature Euler equations, model kinetic equations, and models with constant Prandtl numbers. The influence of bulk viscosity and Prandtl number on the fluid-dynamic variables, viscous stress, heat flux, and total enthalpy is studied. Bulk viscosity plays an important role in sufficiently rarefied gases under weak deviations from equilibrium; in multi-temperature models, non-equilibrium effects are associated with slow relaxation processes rather than with bulk viscosity. Using a constant Prandtl number yields over-predicted values of the heat flux. Contributions of various energy modes to the total heat flux are evaluated, with emphasis on the compensation of translational–rotational and vibrational energy fluxes.

Hydrodynamic states of electron plasma of semiconductors in the generalized Chapman–Enskog method

The hydrodynamic states of polar semiconductors are investigated. Such states are described by the temperature, mass velocity, and density of the number of particles of the electron subsystem, which is considered as rarefied one. The phonon subsystem of a semiconductor is considered to be equilibrium. The electron-phonon interaction in the system is described by the Fröhlich Hamiltonian. The theory is based on Bogolyubov’s linear kinetic equation for the electron distribution function in the presence of a weak electric field. The reduced description of the system is investigated by the generalized Chapman–Enskog method, which leads to a nonlinear equation for the electron distribution function in the reduced description. The zone structure of the system is neglected, focusing on the study of fundamental issues of relaxation of temperature and velocity. Gradients of hydrodynamic parameters are considered small; the same smallparameters determine the smallness of the electric field. In the basic order of the perturbation theory, the distribution function of the system satisfies a nonlinear equation. The method of spectral theory of the collision integral operator shows that the study of the basic approximation is reduced to an analysis of a first-order quasilinear equation, which, in particular, has a solution that we found earlier.

Multi-temperature vibrational energy relaxation rates in CO2

Rates of vibrational energy relaxation in carbon dioxide are studied in the framework of the three-temperature kinetic-theory approach. Vibrational–translational transitions in the bending mode and inter-mode exchange of vibrational quanta are considered. In the zero-order approximation of the generalized Chapman–Enskog method, the energy relaxation rates in the coupled symmetric–bending and asymmetric modes are expressed in terms of thermodynamic forces similar to chemical reaction affinities, and a compact representation for the vibrational energy production rates is proposed. Linearized theory is developed, and analytical ratios of linearized relaxation rates to those defined by the original Landau–Teller (LT) theory are obtained. The relaxation rates are calculated using the Schwartz–Slawsky–Herzfeld (SSH) and forced harmonic oscillator models for the vibrational energy transition probabilities in the temperature range 200 K–10 000 K. For inter-mode exchanges, using the SSH theory yields significantly underpredicted relaxation rates. The ranges of applicability for the LT formula and linearized theory are estimated; the original LT formula for inter-mode vibrational energy exchanges is not capable of accounting for the excitation of both vibrational modes; linearized models yield better results. Possible steps for improving the numerically efficient LT model are proposed.

Influence of angular momentum on transport coefficients in rarefied gases

In the present work, the influence of intrinsic angular momentum on transport coefficients in a single-component gas flow is studied on the basis of the generalized Chapman–Enskog method. A self-consistent set of flow equations is derived, both for an inviscid and viscous flow approximations. It is shown that when angular momentum is considered, the shear viscosity coefficient is a tensor of rank 4, the thermal conductivity coefficient is a tensor of rank 2. The presence of cross-coupling between the heat flux and flux of angular momentum is also demonstrated. Numerical results for the thermal conductivity coefficient, coefficient of coupling between angular momentum and heat fluxes, as well as the bulk viscosity are presented for the case of 2-dimensional flow of atomic hydrogen.

Corrections to the Landau kinetic equation for a weakly dissipative randomly driven system and the fluctuation-dissipation theorem

This paper is devoted to the derivation of a kinetic equation for many-body one-component dissipative systems in an external random field. The system under consideration was discussed in the recent paper by Sliusarenko et al. [J. Math. Phys. 56, 043302 (2015)], where a generalization of the Vlasov kinetic equation was obtained. The potential interaction is assumed to be small (the corresponding small parameter λ), the dissipation interactions and the correlation functions of the external random field are considered as small quantities estimated by one small parameter μ for the simplicity. The kinetic equation is obtained up to the terms of the orders λ2μ0, λ1μ1, λ0μ2 inclusive. In weakly spatially nonuniform states of the system, this gives corrections to the Landau–Vlasov kinetic equation caused by the dissipation and external random field. In the case λ ≫ μ after the mean free time, the system reaches a state approximately described by the Maxwell distribution. The dissipation and random field will lead to the evolution of the system temperature. A time evolution equation for the temperature in spatially uniform states is derived on the basis of the obtained kinetic equation by a generalized Chapman–Enskog method which takes into account that the kinetic equation is an approximate one. This temperature time evolution equation is investigated up to the terms of the order μ3 inclusive. It is shown that under some conditions the final stage of the system evolution is a steady state. A fluctuation-dissipation theorem for this state is discussed. In this steady state, the system is described by a distribution function that contains corrections to the Maxwell distribution.

Kinetic theory of two-temperature polyatomic plasmas

We investigate the kinetic theory of two-temperature plasmas for reactive polyatomic gas mixtures. The Knudsen number is taken proportional to the square root of the mass ratio between electrons and heavy-species, and thermal non-equilibrium between electrons and heavy species is allowed. The kinetic non-equilibrium framework also requires a weak coupling between electrons and internal energy modes of heavy species. The zeroth-order and first-order fluid equations are derived by using a generalized Chapman–Enskog method. Expressions for transport fluxes are obtained in terms of macroscopic variable gradients and the corresponding transport coefficients are expressed as bracket products of species perturbed distribution functions. The theory derived in this paper provides a consistent fluid model for non-thermal multicomponent plasmas.

Plasma kinetic coefficients with account for relaxation processes

The hydrodynamics of a completely ionized two-component electron–ion plasma is investigated on the basis of the Landau kinetic equation with account for component temperature and velocity relaxation. The term relaxation is understood here in a narrow sense as a process which can take place in a spatially homogeneous system. The investigation is made on the basis of a generalized Chapman–Enskog method. The relaxation is investigated in the vicinity of the standard hydrodynamic state of plasma: the deviations of the component temperatures and velocities from their standard hydrodynamic values are considered to be small. The integral equations for the component distribution functions are investigated in a perturbation theory in small gradients and small relaxation parameters. The component distribution functions of the first-order both in gradients and in relaxation parameters are calculated. Relaxation corrections to the energy and momentum fluxes and to the kinetic coefficients of the standard hydrodynamic state of the system are obtained.

Transport coefficients in nonequilibrium gas-mixture flows with electronic excitation

In the present paper, a one-temperature model of transport properties in chemically nonequilibrium neutral gas-mixture flows with electronic excitation is developed. The closed set of governing equations for the macroscopic parameters taking into account electronic degrees of freedom of both molecules and atoms is derived using the generalized Chapman-Enskog method. The transport algorithms for the calculation of the thermal-conductivity, diffusion, and viscosity coefficients are proposed. The developed theoretical model is applied for the calculation of the transport coefficients in the electronically excited N/N(2) mixture. The specific heats and transport coefficients are calculated in the temperature range 50-50,000 K. Two sets of data for the collision integrals are applied for the calculations. An important contribution of the excited electronic states to the heat transfer is shown. The Prandtl number of atomic species is found to be substantially nonconstant.

Numerical and theoretical study of the shock stand-off distance in non-equilibrium flows

A theoretical model based on a quasi-one-dimensional formulation is developed which allows determination of the shock stand-off distance at the stagnation point of blunt bodies in hypersonic non-equilibrium flows. Despite the simple ideal dissociating gas model implemented in the theoretical approach, it gives insight into the main physics governing the shock stand-off problem. More detailed and precise data are obtained by a numerical simulation where vibrational and chemical relaxation processes as well as their interactions are taken into account. The physical modelling of these processes is based on a kinetic approach and on a generalized Chapman–Enskog method of solving the Boltzmann equation. Explicit formulae for rate constants and vibrational energy consumption are derived and incorporated into the general conservation equations. Good agreement between theoretical, numerical and experimental results is achieved which ensures a reliable and mutual validation of the different methods.

On the theory of bulk viscosity and relaxation pressure

A general theory of bulk viscosity and relaxation pressure, based on the generalized Chapman-Enskog method, is given, and also simple qualitative theories of them. Problems associated with the definition of temperature are discussed. Formulae are given for the bulk viscosity coefficient and the thermal conductivity of a gas possessing rotational degrees of freedom. The influence of bulk viscosity on the absorption and dispersion of sound and on the structure of a shock wave in nitrogen is investigated.

Transport properties of nonequilibrium polyatomic gas mixtures

It is shown that the well-known Hirschfelder-Euken correction to the thermal conductivity of a polyatomic gas mixture given by the first approximation in Sonine polynomials can be less than the corresponding exact value (for a Lorentz mixture of light and heavy molecules interacting in accordance with Coulomb's law) by a factor of 3.4. Fairly high accuracy is achieved in the second approximation in Sonine polynomials. Within the framework of the latter, similar corrections to the nonequilibrium heat and diffusion fluxes are found. On the basis of the generalized Chapman-Enskog method a more general case is studied. In this case some of the nonelastic collision integrals is also taken into account in calculating the transport coefficients. The transport coefficients are either represented in terms of the well-known formulas for fast and retarded internal molecular energy exchange or convenient approximate expressions are obtained.

Generalized Chapman–Enskog method in the kinetic theory of reacting ionized gases

In this paper the problem of kinetic coefficient calculations for a nonequilibrium mixture of chemically reacting ionized gases is investigated. The study is based on the generalized Chapman–Enskog method (GCEM) proposed by Alexeev [Theor. Exp. Chem. 5, 541 (1969)]. This work should be considered as a direct continuation of the paper [Phys. Rev. E 49, 2809 (1994)]. An effective method for the solution of the basic equation with non-self-adjoint operator (A equation) of the GCEM is developed for the case of reacting ionized gases and the calculation of the bracket expressions for elastic collisions of charged particles is given. Several concrete results of the kinetic coefficient calculations are discussed.

Application of the generalized Chapman-Enskog method to the transport-coefficient calculation in a reacting gas mixture.

In this paper we investigate the calculation of the kinetic coefficients for a nonequilibrium mixture of chemically reacting gases. This study is based on the generalized Chapman-Enskog method (GCEM) proposed by Alexeev. An alternative method for the resolution of the basic linearized integral equations of the GCEM is developed in this paper. The algorithm of this method is adjusted to include triple inelastic collisions and the resolution method of the integral equation with non-self-adjoint operator was found to be the most effective for practical use. Some issues and examples of calculation are discussed.

Kinetic derivation of the gas-dynamic equation for multicomponent mixtures of light and heavy particles

The gas-dynamic equations for multicomponent mixtures of particles with large mass disparity and frozen internal degrees of freedom are derived by means of the methods of the kinetic theory. It is found that the macroscopic description may be one-velocity and two-temperature. A special feature is the need to use a multivelocity solution of the Boltzmann equation for the heavy components and also the method of introducing their effective temperature. The corresponding modification of the generalized Chapman-Enskog method is given. The Stefan-Maxwell relations are shown to be meaningful as local relations for the diffusion velocities of the components in going over from the multivelocity solution of the kinetic equations to the one-velocity macroscopic description. The transport properties, the exchange terms, and the Stefan-Maxwell relations in the system of gas-dynamic equations written in the Navier-Stokes approximation are calculated in an arbitrary approximation in Sonine polynomials. A criticism of the generalized Chapman-Enskog method is considered and the necessary explanations are given.

Transport properties of nonequilibrium gas mixtures.

Transport properties of binary mixtures of diatomic gases in vibrational nonequilibrium are analyzed and the corresponding terms are established. In order to solve the Boltzmann equation, a generalized Chapman-Enskog method is used, valid whatever the degree of nonequilibrium, contrarily to previous analyses in which only the extreme regimes were considered, at weak or strong nonequilibrium, and generally for pure gases only. The transport coefficients, first written in terms of collision integrals, are finally expressed as functions of macroscopic quantities, known or experimentally attainable, following Mason and Monchick type approximations and with the harmonic-oscillator model. Finally, examples of calculations are given for ${\mathrm{N}}_{2\mathrm{\ensuremath{-}}}$${\mathrm{H}}_{2}$ and ${\mathrm{O}}_{2\mathrm{\ensuremath{-}}}$${\mathrm{N}}_{2}$ mixtures. Results are in good agreement with available experimental data. Comparisons with approximate formulas, extensively used in the literature, show also that, at least for the investigated mixtures, the use of these formulas seems roughly justified.

Region of applicability and the main features of the generalized Chapman-Enskog method

The present paper is made necessary by the publication of the foregoing paper in this issue by Kolesnichenko [1]. It considers the basic propositions of the generalized Chapman-Enskog method and analyzes the arguments put forward by Kolesnichenko [1] and the validity of the method. The position of the results obtained by Kolesnichenko [14–17] is indicated. Nonequilibrium flows of multiatomic gases in which there occur processes of exchange of internal energy of the molecules in collisions between them and chemical reactions (such processes are called inelastic) are encountered frequently in nature and technology. It is therefore naturally of interest to derive gas-dynamic equations for such flows. The methods of the kinetic theory of gases were first used to obtain equations describing the limiting cases of very fast inelastic processes that take place in times of the order of the molecule-molecule collision times (equilibrium case) and very slow inelastic processes that take place over times of the order of the characteristic flow time (relaxation case). In [2–5], an algorithm was proposed for deriving gas-dynamic equations valid for arbitrary ratios of the rates of the elastic and inelastic processes and reducing to the well-known equations for the limiting cases already mentioned. The algorithm is called the generalized Chapman-Enskog method (abbreviated to the generalized method). The development, modification, and analysis of its properties can be found in [4, 6–13]. In [1], Kolesnichenko has questioned the validity of this algorithm.

Kinetic theory of dense fluids. II. The generalized Chapman-Enskog solution

In the second paper of this series we solve the kinetic equation proposed in the previous paper by a method following the spirit of Chapman and Enskog (generalized Chapman-Enskog method). The zeroth-order solution to the kinetic equation leads to the Euler equations in hydrodynamics for real fluids, and the first-order solution to the Navier-Stokes equations for real fluids. General formulas for transport coefficients such as viscosity and heat-conductivity coefficients are obtained for dense fluids, which are given in terms of time-correlation functions of fluxes conjugate to the thermodynamic forces. The results have the same formal structures as the time-correlation functions in linear response theory except for the collision operator appearing in place of the Liouville operator in the evolution operator for the system.