Generalized Burnett Hydrodynamics

Equations of hydrodynamics (derived from the Boltzmann equation) beyond the Navier-Stokes level are studied by a method proposed earlier by the author. The main question we consider is the following: What is the most natural replacement for classical (ill-posed) Burnett equations?It is shown that, in some sense, it is a two-parameter set of Generalized Burnett Equations (GBEs) derived in this paper. Some equations of this class are even simpler than original Burnett equations. The region of stability in the space of parameters and other properties of GBEs are discussed.     

On computing the Chapman-Enskog and Burnett functions

An expansion and projection technique based on Legendre polynomials is used to solve, in an efficient and accurate way, the Chapman-Enskog equations for viscosity and heat transfer and to compute the Burnett functions required, for example, for Poiseuille-flow problems based on rigid-sphere collisions and the linearized Boltzmann equation.     

On computing the Chapman-Enskog functions for viscosity and heat transfer and the Burnett functions

Hermite cubic splines and collocation are used to solve, in an efficient and accurate way, the Chapman-Enskog equations for viscosity and heat transfer and to compute the Burnett functions required for Poiseuille-flow problems based on rigid-sphere collisions and the linearized Boltzmann equation.     

Onsager-Casimir symmetry properties of the Burnett equations

We demonstrate the approximate nature of the Onsager-Casimir relations for the example of the linearized Burnett equations for a dilute gas. For any discussion of Onsager relations the choice of a correct set of thermodynamic forces and fluxes is of course crucial. By retracing the Chapman-Enskog procedure, we show that the usual expressions for the thermodynamic forces require modifications at the Burnett level. However, inclusion of these terms does not remedy the violation of Onsager symmetry first noticed by McLennan. A modified version of the Onsager symmetry that involves thermodynamic forces derived from an entropy Lagrangian rather than from the entropy itself does remain valid on the Burnett level. Throughout, we allow for the presence of an external potential; the Burnett equations including potential terms are derived in an appendix for a set of variables particularly suited for our discussion. We stress that in discussing Onsager relations one should use the full thermodynamic fluxes rather than their dissipative parts only, in spite of the fact that only the latter contribute to the entropy production.     

Chapman-Enskog and Burnett solutions for a simple, rigid-sphere gas: Numerical solutions using a subtraction technique

Chapman-Enskog and Burnett solutions are obtained for a simple, rigid-sphere gas through application of a subtraction technique to the linearized Boltzmann equation. The numerical results obtained converge and compare well with results reported previously using other techniques. This method is straightforward and may prove quite useful in addressing several other similar problems in the kinetic theory of gases.     

Generalized relativistic Chapman-Enskog solution of the Boltzmann equation

The Chapman-Enskog method of solution of the relativistic Boltzmann equation is generalized in order to admit a time-derivative term associated to a thermodynamic force in its first order solution. Both existence and uniqueness of such a solution are proved based on the standard theory of integral equations. The mathematical implications of the generalization introduced here are thoroughly discussed regarding the nature of heat as chaotic energy transfer in the context of relativity theory.     

Scaling and Expansion of Moment Equations in Kinetic Theory

The set of generalized 13 moment equations for molecules interacting with power law potentials [Struchtrup, Multiscale Model. Simul. 3:211 (2004)] forms the base for an investigation of expansion methods in the Knudsen number and other scaling parameters. The scaling parameters appear in the equations by introducing dimensionless quantities for all variables and their gradients. Only some of the scaling coefficients can be chosen independently, while others depend on these chosen scales–their size can be deduced from a Chapman–Enskog expansion, or from the principle that a single term in an equation cannot be larger in size by one or several orders of magnitude than all other terms.It is shown that for the least restrictive scaling the new order of magnitude expansion method [Struchtrup, Phys. Fluids 16(11):3921 (2004)] reproduces the original equations after only two expansion steps, while the classical Chapman–Enskog expansion would require an infinite number of steps. Both methods yield the Euler and Navier–Stokes–Fourier equations to zeroth and first order. More restrictive scaling choices, which assume slower time scales, small velocities, or small gradients of temperature, are considered as well.     

Discrete ellipsoidal statistical BGK model and Burnett equations

A new discrete Boltzmann model, the discrete ellipsoidal statistical Bhatnagar–Gross–Krook (ESBGK) model, is proposed to simulate nonequilibrium compressible flows. Compared with the original discrete BGK model, the discrete ES-BGK has a flexible Prandtl number. For the discrete ES-BGK model in the Burnett level, two kinds of discrete velocity model are introduced and the relations between nonequilibrium quantities and the viscous stress and heat flux in the Burnett level are established. The model is verified via four benchmark tests. In addition, a new idea is introduced to recover the actual distribution function through the macroscopic quantities and their space derivatives. The recovery scheme works not only for discrete Boltzmann simulation but also for hydrodynamic ones, for example, those based on the Navier–Stokes or the Burnett equations.     

一维空间Riesz分数阶对流扩散方程的格子Boltzmann方法

建立格子Boltzmann方法（LBM）的D1Q3演化模型,研究一类Riesz空间分数阶对流扩散方程的数值求解问题。对分数阶微积分算子中的积分项离散化处理,得到逼近的标准对流扩散方程。结合Taylor展式和Chapman-Enskog多尺度展开技术得到模型的各个方向上的平衡态分布函数,通过D1Q3演化模型正确恢复所要求解的宏观方程。数值算例验证该方法的有效性。     

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.Sloan Foundation Fellow 1968–70. Guggenheim Fellow 1969–70.     

On the invariance of constitutive equations according to the kinetic theory of gases

Iterative techniques for solving the Boltzmann equation in the kinetic theory of gases yield expressions for the stress tensor and heat flux vector that are analogous to constitutive equations in continuum mechanics. However, these expressions are not generally invariant under the Euclidean group of transformations, whereas constitutive equations in continuum mechanics are usually required to be by the principle of material frame indifference. This disparity in invariance properties has led some previous investigators to argue that Euclidean invariance should be discarded as a contraint on constitutive equations. It is proven mathematically in this paper that the results of the Chapman-Enskog iterative procedure have no direct bearing on this issue. In order to settle this question, it is necessary to examine mathematically the effect of superimposed rigid body rotations on solutions of the Boltzmann equation. A preliminary investigation along these lines is presented which suggests that the kinetic theory is consistent with material frame indifference in at least a strong approximate sense provided that the disparity in the time scales of the microscopic and macroscopic motions is extremely large—a condition which is usually a prerequisite for the existence of constitutive equations.On leave from Stevens Institute of Technology, Hoboken, New Jersey 07030.     

Lattice Boltzmann method for the generalized Kuramoto-Sivashinsky equation

In this paper, a lattice Boltzmann model with an amending function is proposed for the generalized Kuramoto-Sivashinsky equation that has the form u_t+uu_x+αu_xx+βu_xxx+γu_xxxx=0. With the Chapman-Enskog expansion, the governing evolution equation is recovered correctly from the continuous Boltzmann equation. It is found that the numerical results agree well with the analytical solutions.     

Stationary Boltzmann equation for a degenerate gas in a slab: Boundary value problem and hydrodynamics

The boundary value problem for the stationary Boltzmann equation for a model gas in a plane slab is solved in full generality. The asymptotic behavior as the size of the slab goes to infinity is studied via a Chapman-Enskog expansion.     

Inhomogeneous similarity solutions of the Boltzmann equation with confining external forces

The Nikolskii transform makes it possible to construct inhomogeneous solutions of the Boltzmann equation from homogeneous ones. These solutions correspond to a gas in expansion, but if we introduce external forces, they can relax toward absolute Maxwellians. This property holds independently of the assumed intermolecular inverse power force. Consequently, for Maxwell molecules and from energy-dependent homogeneous distributions, we construct effectively a class of inhomogeneous similarity distributions with Maxwellian equilibrium relaxation. We review and investigate again the homogeneous distributions which can be written in closed form, for instance, we show that an elliptic exact solution proposed some years ago violates positivity. For Maxwell interaction with singular cross sections, we numerically construct inhomogeneous distributions having Maxwellian equilibrium states and study the Tjon overshoot effect. We show that both the sign and the time decrease of the external force as well as the microscopic model of the cross section contribute to the asymptotic behavior of the distribution. These inhomogeneous similarity solutions include a class of distributions that asymptotically oscillate between different Maxwellians. Two classes of external forces are considered: linear spatial-dependent forces or linear velocity-dependent forces plus source term.     

New Discrete Model Boltzmann Equations for Arbitrary Partitions of the Velocity Space

Modified discrete Boltzmann equations for arbitrary partitions of the velocity space are established. The new equations can be derived from the continuous Boltzmann equation and are a generalization of previous discrete-velocity models. They preserve mass, momentum, and energy, and an H-theorem holds. The new model equations are tested by comparing their solutions with the analytical ones of the continuous Boltzmann equation for the Krook–Wu and the very hard particle models.     

A kinetic derivation of extended irreversible thermodynamics

The basic postulates of the extended irreversible thermodynamics are derived from the kinetic model for a dilute monoatomic gas. Using the Grad 13-moment method to solve the full nonlinear Boltzmann equation for molecules conceived as soft spheres we obtain the microscopic expressions for the entropy flux, the entropy production, and the generalized Pfaffian for the extended definition of entropy as required by such a theory. Some of the physical implications of these results are discussed.Member, Colegio Nacional.     

On the Domain of Hyperbolicity of the Cumulant Equations

In this article we consider the influence of non-equilibirum values of classical variables on the eigenvalues of the advection part of the cumulant equations. Real and finite eigenvalues are a neccessary condition for the cumulant equations to be hyperbolic which can be used to obtain estimates on admissible deviations from equilibrium for a model of particular order still to be valid. We find that this condition puts no constraints on velocity and shear stress values, but specific energy must be positive, normal stress must be bounded by specific energy and heat flux not be too large.