Chapman Enskog Approach Research Articles

Kinetic stability of Chapman–Enskog plasmas

In this paper, we investigate the kinetic stability of classical, collisional plasma – that is, plasma in which the mean-free-path $\lambda$ of constituent particles is short compared with the length scale $L$ over which fields and bulk motions in the plasma vary macroscopically, and the collision time is short compared with the evolution time. Fluid equations are typically used to describe such plasmas, since their distribution functions are close to being Maxwellian. The small deviations from the Maxwellian distribution are calculated via the Chapman–Enskog (CE) expansion in $\lambda /L \ll 1$ , and determine macroscopic momentum and heat fluxes in the plasma. Such a calculation is only valid if the underlying CE distribution function is stable at collisionless length scales and/or time scales. We find that at sufficiently high plasma $\beta$ , the CE distribution function can be subject to numerous microinstabilities across a wide range of scales. For a particular form of the CE distribution function arising in strongly magnetised plasma (viz. plasma in which the Larmor periods of particles are much smaller than collision times), we provide a detailed analytic characterisation of all significant microinstabilities, including peak growth rates and their associated wavenumbers. Of specific note is the discovery of several new microinstabilities, including one at sub-electron-Larmor scales (the ‘whisper instability’) whose growth rate in certain parameter regimes is large compared with other instabilities. Our approach enables us to construct the kinetic stability maps of classical, two-species collisional plasma in terms of $\lambda$ , the electron inertial scale $d_e$ and the plasma $\beta$ . This work is of general consequence in emphasising the fact that high- $\beta$ collisional plasmas can be kinetically unstable; for strongly magnetised CE plasmas, the condition for instability is $\beta \gtrsim L/\lambda$ . In this situation, the determination of transport coefficients via the standard CE approach is not valid.

Mesoscopic Kinetic Approach of Nonequilibrium Effects for Shock Waves.

A shock wave is a flow phenomenon that needs to be considered in the development of high-speed aircraft and engines. The traditional computational fluid dynamics (CFD) method describes it from the perspective of macroscopic variables, such as the Mach number, pressure, density, and temperature. The thickness of the shock wave is close to the level of the molecular free path, and molecular motion has a strong influence on the shock wave. According to the analysis of the Chapman-Enskog approach, the nonequilibrium effect is the source term that causes the fluid system to deviate from the equilibrium state. The nonequilibrium effect can be used to obtain a description of the physical characteristics of shock waves that are different from the macroscopic variables. The basic idea of the nonequilibrium effect approach is to obtain the nonequilibrium moment of the molecular velocity distribution function by solving the Boltzmann-Bhatnagar-Gross-Krook (Boltzmann BGK) equations or multiple relaxation times Boltzmann (MRT-Boltzmann) equations and to explore the nonequilibrium effect near the shock wave from the molecular motion level. This article introduces the theory and understanding of the nonequilibrium effect approach and reviews the research progress of nonequilibrium behavior in shock-related flow phenomena. The role of nonequilibrium moments played on the macroscopic governing equations of fluids is discussed, the physical meaning of nonequilibrium moments is given from the perspective of molecular motion, and the relationship between nonequilibrium moments and equilibrium moments is analyzed. Studies on the nonequilibrium effects of shock problems, such as the Riemann problem, shock reflection, shock wave/boundary layer interaction, and detonation wave, are introduced. It reveals the nonequilibrium behavior of the shock wave from the mesoscopic level, which is different from the traditional macro perspective and shows the application potential of the mesoscopic kinetic approach of the nonequilibrium effect in the shock problem.

Separation of ion component from solid hydrocarbon materials by laser ablation

Plasma usually consists of multiple ion component. Ion-component separation occurs in various conditions, and profoundly affects the plasma dynamic evolution. In this work, ion-component separation in two-ion-component plasma is investigated in the hydrodynamic condition. Starting from the Landau-Fokker-Planck equations of two-ion-component plasma, the ion transport equations are reduced through the Chapman-Enskog approach. The transport equations are then transformed into a set of linear algebraic equations and solved by expanding the perturbed ion distribution functions into the series of Sonine polynomials. The diffusive ion mass flows with inclusion of baro-diffusion, thermo-diffusion and electro-diffusion are thus obtained. With these efforts, the complete ion fluid equations are presented, which can be used to describe the processes of ion-component separation. We evaluate ion-component separation in the case of a solid CH plate target ablated with a laser pulse, by solving the ion diffusion equation with the hydro states output from the one-dimensional radiative hydro code Multi-1D. The simulation results show that ion-component separation mainly occurs around ablation front and under-dense region, and that the effect of ion-species separation on plasma hydrodynamic evolution is minor and can be neglected. For those physical processes sensitive to ion concentration such as Thomson scattering, however, the effect of ion-component separation is significant, which means that ion-component separation should be included in the study of laser plasma interaction.

Order-by-order anisotropic transport coefficients of a magnetised fluid: a Chapman-Enskog approach

We derive the first and second-order expressions for the shear, the bulk viscosity, and the thermal conductivity of a relativistic hot boson gas in a magnetic field using the relativistic kinetic theory within the Chapman-Enskog method. The order-by-order off-equilibrium distribution function is obtained in terms of the associate Laguerre polynomial with magnetic field-dependent coefficients using the relativistic Boltzmann-Uehling-Uhlenbeck transport equation. The order-by-order anisotropic transport coefficients are evaluated in powers of the dimensionless ratio of kinetic energy to the fluid temperature for finite magnetic fields. In a magnetic field, the shear viscosity (in all order) splits into five different coefficients. Four of them show a magnetic field dependence as seen in a previous study [1] using the relaxation time approximation for the collision kernel. On the other hand, bulk viscosity, which splits into three components (in all order), is independent of the magnetic field. The thermal conductivity shows a similar splitting but is field-dependent. The difference in the first and second-order results are prominent for the thermal conductivities than the shear viscosity; moreover, the difference in the two results is most evident at low temperatures. The first and second-order results seem to converge rapidly for high temperatures.

Probing bulk viscosity in relativistic flows.

We derive an analytical connection between kinetic relaxation rate and bulk viscosity of a relativistic fluid in d spatial dimensions, all the way from the ultra-relativistic down to the near non-relativistic regime. Our derivation is based on both Chapman-Enskog asymptotic expansion and Grad's method of moments. We validate our theoretical results against a benchmark flow, providing further evidence of the correctness of the Chapman-Enskog approach; we define the range of validity of this approach and provide evidence of mounting departures at increasing Knudsen number. Finally, we present numerical simulations of transport processes in quark-gluon plasmas, with special focus on the effects of bulk viscosity which might prove amenable to future experimental verification. This article is part of the theme issue 'Fluid dynamics, soft matter and complex systems: recent results and new methods'.

Second-order dissipative hydrodynamics for plasma with chiral asymmetry and vorticity

By making use of the chiral kinetic theory in the relaxation-time approximation, we derive an Israel-Stewart type formulation of the hydrodynamic equations for a chiral relativistic plasma made of neutral particles (e.g., neutrinos). The effects of chiral asymmetry are captured by including an additional continuity equation for the axial charge, as well as the leading-order quantum corrections due to the spin of particles. In a formulation of the chiral kinetic theory used, we introduce a symmetric form of the energy-momentum tensor that is suitable for the description of a weakly nonuniform chiral plasma. By construction, the energy and momentum are conserved to the same leading order in the Planck constant as the kinetic equation itself. By making use of such a chiral kinetic theory and the Chapman-Enskog approach, we obtain a set of second-order dissipative hydrodynamic equations. The effects of the fluid vorticity and velocity fluctuations on the dispersion relations of chiral vortical waves are analyzed.

High‐order discontinuous Galerkin nonlocal transport and energy equations scheme for radiation hydrodynamics

SummaryThe nonlocal theory of the radiative energy transport in laser‐heated plasmas of arbitrary ratio of the characteristic inhomogeneity scale length to the photon mean free paths is applied to define the closure relations of a hydrodynamic system. The corresponding transport phenomena cannot be described accurately using the Chapman–Enskog approach, that is, with the usual fluid approach dealing only with local values and derivatives. Thus, we directly solve the photon transport equation allowing one to take into account the effect of long‐range photon transport. The proposed approach is based on the Bhatnagar–Gross–Krook collision operator using the photon mean free path as a unique parameter. Such an approach delivers a calculation efficiency and an inherent coupling of radiation to the fluid plasma parameters in an implicit way and directly incorporates nonequilibrium physics present under the condition of intense laser energy deposition due to inverse bremsstrahlung. In combination with a higher order discontinuous Galerkin scheme of the transport equation, the solution obeys both limiting cases, that is, the local diffusion asymptotic usually present in radiation hydrodynamics models and the collisionless transport asymptotic of free‐streaming photons. In other words, we can analyze the radiation transport closure for radiation hydrodynamics and how it behaves when deviating from the conditions of validity of Chapman–Enskog method, which is demonstrated in the case of exact steady transport and approximate multigroup diffusion numerical tests. As an application, we present simulation results of intense laser‐target interaction, where the radiative energy transport is controlled by the mean free path of photons. Copyright © 2016 John Wiley & Sons, Ltd.

Chapman–Enskog expansion for the Vicsek model of self-propelled particles

Using the standard Vicsek model, I show how the macroscopic transport equations can be systematically derived from microscopic collision rules. The approach starts with the exact evolution equation for the N-particle probability distribution and, after making the mean-field assumption of molecular chaos, leads to a multi-particle Enskog-type equation. This equation is treated by a non-standard Chapman–Enskog expansion to extract the macroscopic behavior. The expansion includes terms up to third order in a formal expansion parameter ϵ, and involves a fast time scale. A self-consistent closure of the moment equations is presented that leads to a continuity equation for the particle density and a Navier–Stokes-like equation for the momentum density. Expressions for all transport coefficients in these macroscopic equations are given explicitly in terms of microscopic parameters of the model. The transport coefficients depend on specific angular integrals which are evaluated asymptotically in the limit of infinitely many collision partners, using an analogy to a random walk. The consistency of the Chapman–Enskog approach is checked by an independent calculation of the shear viscosity using a Green–Kubo relation.

Arc plasma devices: Evolving mechanical design from numerical simulation

Wide ranges of technological applications involve arc plasma devices as the primary plasma source for processing work. Recent findings exhibit the existence of appreciable thermal non-equilibrium in these so-called thermal plasma devices. Commercially available magnetohydrodynamic codes are not capable of handling such systems due to unavailability of non-equilibrium thermodynamic and transport property data and self-consistent models. A recipe for obtaining mechanical design of arc plasma devices from numerical simulation incorporating two-temperature thermal non-equilibrium model is presented in this article with reference to the plasma of the mixture of molecular gases like nitrogen and oxygen. Such systems are technologically important as they correspond to the plasma devices operating with air, oxygen plasma torches in cutting industries and plasma devices using nitrogen as shielding gas. Temperature field, associated fluid dynamics and electrical characteristics of a plasma torch are computed in a systematic manner to evaluate the performance of a conceived design using a two-fluid CFD model coupled with a two-temperature thermodynamic and transport property code. Important effects of different nozzle designs and plasma gases obtained from the formalism are discussed. Non-equilibrium thermodynamic properties are computed using modified two-temperature Saha equations and transport properties are computed using standard Chapman–Enskog approach.

Models of thermal conductivity of multilayer wear resistant coatings

The thermal conductivity of non-metallic nano- and microstructured materials is a key parameter when considering wear resistant coatings. Here, different models of phonon transport and estimation of thermal conductivity are analyzed. The hopping mechanism of phonon transport was found to be applicable for small-grain-size materials in understanding thermal conductivity, whereas large grain size materials can be studied using the correlation function approach. The Chapman–Enskog approach to the Boltzmann transport equation assuming that the transport of phonons is controlled by scattering on the grain boundaries has been analyzed. The Fourier law of thermal conductivity is obtained with the thermal conductivity inversely proportional to the specific surface of the boundaries.

Electron transport calculations for plasma in high Chapman-Enskog approximations

The semi-quantum model of electronic transport in strongly coupled plasma is the most adequate model for experimental data. The model is improved here: first, a higher Chapman-Enskog approach is found, and second, a more accurate expression for the Debye radius is found. So, the applicability of the model is expanded to laege densities. The accuracy of the calculation of electrical and thermal conductivity is improved by 0.5–0.1%.

Self-diffusion in granular gases: Green–Kubo versus Chapman–Enskog

We study the diffusion of tracers (self-diffusion) in a homogeneously cooling gas of dissipative particles, using the Green-Kubo relation and the Chapman-Enskog approach. The dissipative particle collisions are described by the coefficient of restitution epsilon which for realistic material properties depends on the impact velocity. First, we consider self-diffusion using a constant coefficient of restitution, epsilon=const, as frequently used to simplify the analysis. Second, self-diffusion is studied for a simplified (stepwise) dependence of epsilon on the impact velocity. Finally, diffusion is considered for gases of realistic viscoelastic particles. We find that for epsilon=const both methods lead to the same result for the self-diffusion coefficient. For the case of impact-velocity dependent coefficients of restitution, the Green-Kubo method is, however, either restrictive or too complicated for practical application, therefore we compute the diffusion coefficient using the Chapman-Enskog method. We conclude that in application to granular gases, the Chapman-Enskog approach is preferable for deriving kinetic coefficients.

Higher-order continuum approximation for rarefied gases

The Hilbert–Chapman–Enskog expansion of the kinetic equations in mean flight times is believed to be asymptotic rather than convergent. It is therefore inadvisable to use lower order results to simplify the current approximation as is done in the traditional Chapman–Enskog procedure, since that is an iterative method. By avoiding such recycling of lower order results, one obtains macroscopic equations that are asymptotically equivalent to the ones found in the Chapman–Enskog approach. The new equations contain higher order terms that are discarded in the Chapman–Enskog method. These make a significant impact on the results for such problems as ultrasound propagation. In this paper, it is shown that these results turn out well with relatively little complication when the expansions are carried to second order in the mean free time, for the example of the relaxation or Bhatnagar–Gross–Krook model of kinetic theory.

Hydrodynamics and transport coefficients for dilute granular gases

The hydrodynamics of granular gases of viscoelastic particles, whose collision is described by an impact-velocity dependent coefficient of restitution, which is developed using the Chapman-Enskog approach. We derive the hydrodynamic equations and the according transport coefficients with the assumption that the shape of the velocity distribution function follows adiabatically the decaying temperature. We show numerically that this approximation is justified up to intermediate dissipation. The transport coefficients and the coefficient of cooling are expressed in terms of the elastic and dissipative parameters of the particle material and by the gas parameters. The dependence of these coefficients on temperature differs qualitatively from that obtained with the simplifying assumption of a constant coefficient of restitution which was used in previous studies. The approach formulated for gases of viscoelastic particles may be applied also for other impact-velocity dependencies of the restitution coefficient.

Chapman–Enskog closure approximation in the kinetic theory of dilute turbulent gas-particulate suspensions

The Chapman–Enskog approach is applied to a generalised Fokker–Planck equation for the ensemble-averaged phase-space number density of particles to find a closure approximation for the third-order fluctuating velocity correlations in the particle phase of a turbulent dilute gas-particulate suspension. The resulting closed set of continuum equations is shown to be free of empirical parameters, provided the particles are sufficiently large and the turbulence in the both phases is locally isotropic. Special attention is paid to the case where the particle phase is near equilibrium state. In this case the transport equations for diagonal components of the particle Reynolds stresses reduce to a conservation equation for the turbulent kinetic energy. The effective energy transfer coefficient is calculated and the boundary conditions at the rigid wall are formulated. The resulting system of continuum equations and boundary conditions is analysed for fully developed flow between the vertical plane walls of a dilute but densely loaded gas-particulate suspension. Such a flow models a number of practical applications, e.g. the flow in the riser section of fast (circulating) fluidised bed.

Mass‐ and momentum transfer in hydrogen/air and methane/air mixtures

AbstractIn this article we present results for the viscosity and the mass transfer rates of hydrogen/air, hydrogen/oxygen, methane/air and methane/oxygen mixtures in the temperature range from 1000 to 7000 K and a pressure range from 103 to 106 Pa. In addition, the combustion ratio is varied from 0 to ∞. The transport properties are calculated from the first order solution of the Chapman Enskog approach to the Boltzmann equation, assuming chemical equilibrium composition. An extensive literature study was performed to derive experimental and/or theoretically based data for the respective binary interaction potentials. The values of the collision integrals, as derived from a complex numerical integration procedure, are correlated to an approximation formula. In addition, the exact solutions of the kinetic theory are compared to frequently used empirical mixture rules. For the mixture viscosity an easy approximation formula is deduced from the gas‐kinetic theory.

Nonequilibrium effects in model reactive systems: The role of species temperatures.

The nonequilibrium effects for a model reactive system, A+B\ensuremath{\rightarrow} products, that arise from the perturbation of the distribution function from the Maxwellian are studied. The main objective is the calculation of the fractional decrease of the nonequilibrium rate coefficient from the equilibrium value. This effect is examined with the Chapman-Enskog method of solution of the Boltzmann equation which treats the reactive processes as a weak perturbation. The approach is referred to as weak nonequilibrium. The reactive process causes the temperatures of the two species to differ from the system temperature and this effect can play an important role in the determination of the departure of the rate coefficient from the equilibrium value. A second method is an extension of the Chapman-Enskog approach and involves the expansion of the distribution functions about Maxwellians at different temperatures and is referred to as strong nonequilibrium (SNE). A third approach is a modification of SNE and is referred to as modified strong nonequilibrium. The three methods are described and departures of the rate coefficients from their equilibrium values are computed for each case and compared, along with an explicitly time-dependent solution of the Boltzmann equation.

Light-induced viscous flow of a one-component gas.

A new phenomenon, light-induced viscous flow in a one-component gas, is observed in ${\mathrm{CH}}_{3}$F upon velocity-selective vibrational excitation. The effect originates from a state-dependent kinetic collision rate in combination with nonuniform illumination. The theory for this effect is derived using a Chapman-Enskog approach. Good agreement between theory and experiments in the near-hydrodynamic regime is found.

Transport coefficients for a dense fluid of square-well, rough spheres

Transport coefficients are derived for a dense fluid of square-well, rough spheres. This molecular model may be described as a rough spherical core surrounded by a square-well potential. The Chapman–Enskog approach is used to obtain the transport coefficients. The expressions include the ’’first’’ order contributions arising from the deviation of the singlet distribution function from local equilibrium in addition to the ’’zeroth’’ order terms involving the local equilibrium function. The kinetic equation is also derived.

Transport Properties of a Nonequilibrium Partially Ionized Gas

A modified Chapman-Enskog approach for an ionized gas is presented. The method allows the electron and heavy particle temperatures to differ while requiring all species to have the same macroscopic velocity to zero order. Expressions are obtained for the transport properties of such a gas and a complete set of self-contained hydrodynamic equations is given. As an example, the transport properties of a fully ionized gas containing a single ion species are evaluated using the cutoff Coulomb potential. In the limit of equal electron and ion temperatures, the results are in agreement with previous results.