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.     

On the velocity distributions of granular gases

We present a new approach to determine velocity distributions in granular gases to improve the Sonine polynomial expansion of the velocity distribution function, at higher inelasticities, for the homogeneous cooling regime of inelastic hard spheres. The perturbative consistency is recovered using a new set of dynamical variables based on the characteristic function and we illustrate our approach by computing the first four Sonine coefficients for moderate and high inelasticities. The analytical coefficients are compared with molecular dynamics simulations results and with a previous approach by Huthmann et al.     

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     

Kinetic theory of field effects in nonuniform molecular gases

Effects of magnetic and electric fields on transport phenomena in dilute polyatomic gases are reviewed within the framework of first order Enskog theory. The established technique of approximate operator inversion is used to give first order approximations of the transport coefficients. Instead of the customary expansion of polarization into orthogonal polynomials a more general treatment is chosen here so as to accomodate recent experimental observations. The polarizations produced by macroscopic fluxes are assumed to be eigenfunctions of the collision operator within the subspace of functions anisotropic in angular momentum. The formalism is extended to mixtures in a way to let the final expressions assume the same form as for pure gases. The obtained transport coefficients obey several symmetry relations and inequalities. Additional inequalities are now also derived for the matrix describing the saturated field effects.     

SOME PROGRESS IN THE LATTICE BOLTZMANN MODEL

A lattice Boltzmann equation model has been developed by using the equilibrium distribution function of the Maxwell－Boltzmann-like form, which is third order in fluid velocity u_α. The criteria of energy conservation between the macroscopic physical quantities and the microscopic particles are introduced into the model, thus the thermal hydrodynamic equations containing the effect of buoyancy force can be recovered in terms of the Taylor and Chapman－Enskog asymptotic expansion methods. The two-dimensional thermal convection phenomena in a square cavity and between two concentric cylinders have been calculated by implementing a heat flux boundary condition. Both numerical results are in good agreement with the conventional numerical results.     

Calculation of the collision integral linear kernel for Maxwellian molecules in the isotropic case

Application of the method of nonlinear moments to solve the Boltzmann equation generates the need to sum a series that is the expansion of the distribution function in basis functions. This series converged only if the Grad test is fulfilled. Such a limitation can be removed if the expansion of the distribution function is summed over the index related to only the expansion in velocity magnitude. In this case, the distribution function and the collision integral become expanded in only spherical harmonics and the expansion coefficients satisfy integro-differential equations. The kernels of these equations are the sums of the Sonine polynomials in the velocities of colliding and outgoing particles multiplied by matrix elements of the collision integral. For a number of arguments, the direct calculation of the kernels requires that a very large number of terms in the sum be taken into consideration. In this respect, an approach seems to be promising in which the asymptotics of the matrix elements and Sonine polynomials at large indices are used and summation over index is replaced by integration. In this paper, we apply this approach to calculate the linear kernel in the isotropic case, assuming that interaction between particles is described by a pseudopower law. With this approach, the collision integral kernel can be calculated with a high accuracy using as little as a few tens of series terms and the asymptotic estimate of the residue.     

Ionenbewegung unter dem Einfluß von Umladungsstößen

The motion of ions under the influence of an electric field and a density gradient is investigated by taking into account only charge-transfer collisions between the ions and a neutral gas background. The two model cases of constant mean free path and constant mean free time are considered. For weak electric fields the Boltzmann equation can be solved using a special expansion of the velocity distribution. The expansion in Sonine polynomials as given by Chapman and Cowling does not converge. For medium and high electric fields the velocity distribution is calculated by collecting the contributions of the different paths of the ions. The inhomogeneous drift in high electric fields is investigated for a small perturbation by gradients of the electric field, the ion density and of the mean free path.     

Granular gas of viscoelastic particles in a homogeneous cooling state

Kinetic properties of a granular gas of viscoelastic particles in a homogeneous cooling state are studied analytically and numerically. We employ the most recent expression for the velocity-dependent restitution coefficient for colliding viscoelastic particles, which allows us to describe systems with large inelasticity. In contrast to previous studies, the third coefficient a₃ of the Sonine polynomials expansion of the velocity distribution function is taken into account. We observe a complicated evolution of this coefficient. Moreover, we find that a₃ is always of the same order of magnitude as the leading second Sonine coefficient a₂; this contradicts the existing hypothesis that the subsequent Sonine coefficients a₂,a₃…, are of an ascending order of a small parameter, characterizing particles inelasticity. We analyze evolution of the high-energy tail of the velocity distribution function. In particular, we study the time dependence of the tail amplitude and of the threshold velocity, which demarcates the main part of the velocity distribution and the high-energy part. We also study evolution of the self-diffusion coefficient D and explore the impact of the third Sonine coefficient on the self-diffusion. Our analytical predictions for the third Sonine coefficient, threshold velocity and the self-diffusion coefficient are in a good agreement with the numerical finding.     

具有小Knudsen数的Boltzmann方程的奇异扰动解法

本文提出一种解小Knudsen数的Boltzmann方程的方法,它可以消去Hilbert展开和Enskog展开中的久期项改进正规解,消去Grad展开中的久期项改进初始层解,并把求边界层解归结为求解线性代数方程组。 关键词：     

Navier–Stokes Transport Coefficients of <Emphasis Type="Italic">d</Emphasis>-Dimensional Granular Binary Mixtures at Low Density

The Navier–Stokes transport coefficients for binary mixtures of smooth inelastic hard disks or spheres under gravity are determined from the Boltzmann kinetic theory by application of the Chapman–Enskog method for states near the local homogeneous cooling state. It is shown that the Navier–Stokes transport coefficients are not affected by the presence of gravity. As in the elastic case, the transport coefficients of the mixture verify a set of coupled linear integral equations that are approximately solved by using the leading terms in a Sonine polynomial expansion. The results reported here extend previous calculations (Garzó, V., Dufty, J.W. in Phys. Fluids 14:1476–1490, 2002) to an arbitrary number of dimensions and provide explicit expressions for the seven Navier–Stokes transport coefficients in terms of the coefficients of restitution and the masses, composition, and sizes of the constituents of the mixture. In addition, to check the accuracy of our theory, the inelastic Boltzmann equation is also numerically solved by means of the direct simulation Monte Carlo method to evaluate the diffusion and shear viscosity coefficients for hard disks. The comparison shows a good agreement over a wide range of values of the coefficients of restitution and the parameters of the mixture (masses and sizes).     

正则长波方程的格子Boltzmann模型

提出了用于正则长波方程的5-Bit格子Boltzmann模型。应用Chapman-Enskog展开和多重尺度技术,通过选择平衡态分布函数的高阶矩,得到了时间尺度t₀上的守恒律,从而给出三阶精度的算法。模型中的参数通过稳定性分析给出。     

Slip Boundary Conditions for the Compressible Navier–Stokes Equations

The slip boundary conditions for the compressible Navier–Stokes equations are derived systematically from the Boltzmann equation on the basis of the Chapman–Enskog solution of the Boltzmann equation and the analysis of the Knudsen layer adjacent to the boundary. The resulting formulas of the slip boundary conditions are summarized with explicit values of the slip coefficients for hard-sphere molecules as well as the Bhatnagar–Gross–Krook model. These formulas, which can be applied to specific problems immediately, help to prevent the use of often used slip boundary conditions that are either incorrect or without theoretical basis.     

Na-Ne混合气体扩散系数的计算

用面积分方法计算钠原子与氖原子间的相互作用势,根据计算扩散系数的Chapman-Enskog理论,文中计算了Na-Ne混合气体在温度200～700K范围内的扩散系数.计算结果与已有的实验结果符合得相当好.     

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.     

Nonlinear Transport in Inelastic Maxwell Mixtures Under Simple Shear Flow

The Boltzmann equation for inelastic Maxwell models is used to analyze nonlinear transport in a granular binary mixture in the steady simple shear flow. Two different transport processes are studied. First, the rheological properties (shear and normal stresses) are obtained by solving exactly the velocity moment equations. Second, the diffusion tensor of impurities immersed in a sheared inelastic Maxwell gas is explicitly determined from a perturbation solution through first order in the concentration gradient. The corresponding reference state of this expansion corresponds to the solution derived in the (pure) shear flow problem. All these transport coefficients are given in terms of the restitution coefficients and the parameters of the mixture (ratios of masses, concentration, and sizes). The results are compared with those obtained analytically for inelastic hard spheres in the first Sonine approximation and by means of Monte Carlo simulations. The comparison between the results obtained for both interaction models shows a good agreement over a wide range values of the parameter space.     

Perturbation analysis of a stationary nonequilibrium flow generated by an external force

The stationary flow of a gas in a slab under the action of a constant external force parallel to the walls is analyzed in the context of the Bhatnagar-Gross-Krook model kinetic equation. The force produces spatial gradients along the coordinate normal to the walls. By performing a perturbation expansion in powers of the force, we obtain the hydrodynamic fields up to fifth order in the force. Then the velocity distribution function and all its moments are evaluated to third order. The expansion coefficients are polynomials in the space variable of a degree increasing linearly with the expansion order. Although the series expansion is only asymptotic, it shows how the state of the system is modified by a variation of the external force beyond the linear regime.     

A multi-energy-level lattice Boltzmann model for Maxwell’s equations without sources

A lattice Boltzmann model for the Maxwell’s equations without sources is proposed by taking separate sets of distribution functions for the electric and magnetic fields and using the higher-order moment method. The higher-order moment method is based on the solution of a series of partial differential equations obtained by multi-energy-level techniques, multi-scale techniques, and Chapman–Enskog expansion, As numerical examples, some classical electromagnetic phenomena, such as the electric field and equipotential lines around an electrostatic dipole, the electric and magnetic fields around oscillating dipoles are given. These numerical results agree well with classical ones.     

Zernike aberration coefficients transformed to and from Fourier series coefficients for wavefront representation

The set of Fourier series is discussed following some discussion of Zernike polynomials. Fourier transforms of Zernike polynomials are derived that allow for relating Fourier series expansion coefficients to Zernike polynomial expansion coefficients. With iterative Fourier reconstruction, Zernike representations of wavefront aberrations can easily be obtained from wavefront derivative measurements.     

Variational principles for relativistic transport coefficients

Variational expressions for the transport coefficients of a one-component, relativistic gas are derived from the linearized relativistic Boltzmann equation for both quantum and classical gases. These expressions depend on functions _χ of the energy of the particles comprising the gas in such a way that: a) if _χ differs from a solution of the linearized Boltzmann equation by ε, then the value of the variational expression calculated with this _χ differs from the true value of the corresponding transport coefficient by ε²; and b) the value of the variational expression is always less than this true value. It is shown that values of the transport coefficients obtained by expanding _χ in a particular set of orthogonal polynomials and keeping only the first nontrivial term in the expansion are equivalent to those obtained using the Grad method of moments. It follows therefore that values obtained using this later method represent lower bounds on the true values. We also show that one can obtain simple, closed-form expressions for the various transport coefficients corresponding to an arbitrary number of terms in an expansion of the trial function _χ in the above-mentioned set of orthogonal polynomials. Finally we point out that all of our results can be carried over to the nonrelativistic case by taking the limit c → ∞.