Evaluation of Grad's Second Problem Using Different Higher Order Continuum Theories

In our earlier work (Jadhav, and Agrawal, 2020, “Grad's second problem and its solution within the framework of Burnett hydrodynamics,” ASME J. Heat Transfer, 142(10), p. 102105), we proposed Grad's second problem (examination of steady-state solution for a gas at rest upon application of a one-dimensional heat flux) as a potential benchmark problem for testing the accuracy of different higher order continuum theories and solved the problem within the framework of Burnett hydrodynamics. In this work, we solve this problem within the moment framework and also examine two variants, Bhatnagar–Gross–Krook (BGK)–Burnett and regularized 13 moment equations, for this problem. It is observed that only the conventional form of Burnett equations which are derived retaining the full nonlinear collision integral are able to capture nonuniform pressure profile observed in case of hard-sphere molecules. On the other hand, BGK–Burnett equations derived using BGK-kinetic model predict uniform pressure profile in both the cases. It seems that the variants based on BGK-kinetic model do not distinguish between hard-sphere and Maxwell molecules at least for the problem considered. With respect to moment equations, Grad 13 and regularized 13 moment equations predict consistent results for Maxwell molecules. However, for hard-sphere molecules, since the exact closed form of moment equations is not known, it is difficult to comment upon the results of moment equations for hard-sphere molecules. The present results for this relatively simple problem provide valuable insights into the nature of the equations and important remarks are made in this context.

On the accuracy of macroscopic equations for linearized rarefied gas flows

Many macroscopic equations are proposed to describe the rarefied gas dynamics beyond the Navier-Stokes level, either from the mesoscopic Boltzmann equation or some physical arguments, including (i) Burnett, Woods, super-Burnett, augmented Burnett equations derived from the Chapman-Enskog expansion of the Boltzmann equation, (ii) Grad 13, regularized 13/26 moment equations, rational extended thermodynamics equations, and generalized hydrodynamic equations, where the velocity distribution function is expressed in terms of low-order moments and Hermite polynomials, and (iii) bi-velocity equations and “thermo-mechanically consistent" Burnett equations based on the argument of “volume diffusion”. This paper is dedicated to assess the accuracy of these macroscopic equations. We first consider the Rayleigh-Brillouin scattering, where light is scattered by the density fluctuation in gas. In this specific problem macroscopic equations can be linearized and solutions can always be obtained, no matter whether they are stable or not. Moreover, the accuracy assessment is not contaminated by the gas-wall boundary condition in this periodic problem. Rayleigh-Brillouin spectra of the scattered light are calculated by solving the linearized macroscopic equations and compared to those from the linearized Boltzmann equation. We find that (i) the accuracy of Chapman-Enskog expansion does not always increase with the order of expansion, (ii) for the moment method, the more moments are included, the more accurate the results are, and (iii) macroscopic equations based on “volume diffusion" do not work well even when the Knudsen number is very small. Therefore, among about a dozen tested equations, the regularized 26 moment equations are the most accurate. However, for moderate and highly rarefied gas flows, huge number of moments should be included, as the convergence to true solutions is rather slow. The same conclusion is drawn from the problem of sound propagation between the transducer and receiver. This slow convergence of moment equations is due to the incapability of Hermite polynomials in the capturing of large discontinuities and rapid variations of the velocity distribution function. This study sheds some light on how to choose/develop macroscopic equations for rarefied gas dynamics.

Projection-operator methods for classical transport in magnetized plasmas. Part 2. Nonlinear response and the Burnett equations

The time-independent projection-operator formalism of Breyet al. (PhysicaA, vol. 109, 1981, pp. 425–444) for the derivation of Burnett equations is extended and considered in the context of multispecies and magnetized plasmas. The procedure provides specific formulas for transport coefficients in terms of two-time correlation functions involving both two and three phase-space points. It is shown how to calculate those correlation functions in the limit of weak coupling. The results are used to demonstrate, with the aid of a particular non-trivial example, that the Chapman–Enskog methodology employed by Catto & Simakov (CS) (Phys. Plasmas, vol. 11, 2004, pp. 90–102) to calculate the contributions to the parallel viscosity driven by temperature gradients is consistent with formulas previously derived from the two-time formalism by Brey (J. Chem. Phys., vol. 79, 1983, pp. 4585–4598). The work serves to unify previous work on plasma kinetic theory with formalism usually applied to turbulence. Additional contributions include discussions of (i) Braginskii-order interspecies momentum exchange from the point of view of two-time correlations; and (ii) a simple stochastic model, unrelated to many-body theory, that exhibits Burnett effects. Insights from that model emphasize the role of non-Gaussian statistics in the evaluation of Burnett transport coefficients, including the effects calculated by CS that stem from the nonlinear collision operator. Together, Parts 1 and 2 of this series provide an introduction to projection-operator methods that should be broadly useful in theoretical plasma physics.

A perturbation-based solution of Burnett equations for gaseous flow in a long microchannel

In this paper, an analytical investigation of two-dimensional conventional Burnett equations has been undertaken for gaseous flow through a long microchannel. The analytical solution is obtained by using perturbation analysis around the classical Navier–Stokes solution with appropriate boundary conditions. The perturbation expansion is employed with the smallness parameter $\unicode[STIX]{x1D716}$, taken as the ratio of height to length of the microchannel. The solution for pressure is obtained by solving the cross-stream momentum equation while the velocity distribution is obtained from the streamwise momentum equation. The resulting ordinary differential equations in pressure and velocity are third-order and second-order, respectively. The required boundary conditions for pressure are obtained from direct simulation Monte Carlo (DSMC) data. The obtained analytical solution matches the available DSMC solution well. This is perhaps the first analytical solution of the Burnett equations using the perturbation approach.