Abstract

This work mainly concerns with the recently proposed Onsager-Burnett equations [Singh, Jadhav, and Agrawal, Phys. Rev. E 96, 013106 (2017)] for rarefied gas flows, and the progress achieved so far by solving these equations for some benchmark flow problems. Unlike the conventional Burnett equations, the OBurnett equations form a stable and thermodynamically-consistent set of higher- order continuum transport equations. Further, noticeable absence of higher-order derivatives in the OBurnett constitutive relations for stress tensor and heat flux vector renders the equations needing the same number of boundary conditions as that of the Navier-Stokes equations. These two important aspects: thermodynamic consistency and no need of additional boundary conditions, helps to set the OBurnett equations apart from the rest of the higher-order continuum theories. Available results of OBurnett equations for benchmark problems like force-driven plane Poiseuille flow and normal shocks are promising and helps to establish the validity of the equations. The recently proposed Grad's second problem and its solution within the Burnett hydrodynamics is also discussed at length and some important remarks are made in this context.

No abstract

In this paper, we undertake an analytical study of stresses (augmented and Onsager-Burnett) and entropy generation for the plane Poiseuille flow problem, and their variation with Knudsen number. The gas flow is assumed to be 2D laminar, fully developed, compressible, and isothermal; these assumptions make the problem amenable to analytical treatment. The variation of stresses and entropy generation have been analyzed over a large range of Knudsen number. It is found that the augmented and OBurnett normal stresses are of opposite signs to the corresponding Navier-Stokes stresses, while the magnitude of the net normal stress increases with Knudsen number. The magnitude of the augmented Burnett shear stress is insignificant as compared to the augmented Burnett normal stresses. A minimum in the variation of normalized entropy generation against the Knudsen number (Kn) is observed at Kn close to unity, and is being reported for the first time. The magnitude of net entropy generation from the summation of Navier-Stokes and augmented Burnett stresses is found to be positive, even in the transition regime of gas flow. Further, an appearance of minimum or maximum in normalized net shear stress versus Knudsen number, depending upon the lateral position in the micro-channel has also been observed. Altogether, this analysis supports the validity of the Navier-Stokes equation with modified constitutive expression, even for higher Knudsen numbers. Moreover, the significant terms of Burnett stress are pointed out by the analysis, which can help in developing reduced-order model for these equations.

The derivation of analytical equations of non-continuum macroscopic transport phenomena is underpinned by approximate descriptions of the particle distribution function and is required due to the inability of the Navier–Stokes equations to describe flows at high Knudsen number ( Kn ∼ 1). In this paper, we present a compact representation of the second-order correction to the Maxwellian distribution function and 13-moment transport equations that contain fewer terms compared to available moment-based representations. The intrinsic inviscid and isentropic assumptions of the second-order accurate distribution function are then relaxed to present a third-order accurate representation of the distribution function, using which corresponding third-order accurate moment transport equations are derived. Validation studies performed for Grad’s second problem and the force-driven plane Poiseuille flow problem at non-zero Knudsen numbers for Maxwell molecules highlight an improvement over results obtained by using the Navier–Stokes equations and Grad’s 13-moment (G13) equations. To establish the ability of the proposed equations to accurately capture the bulk behavior of the fluid, the results of Grad’s second problem have been validated against the analytical solution of the Boltzmann equation. For the planar Poiseuille flow problem, validations against the direct simulation Monte Carlo method data reveal that, in contrast to G13 equations, the proposed equations are capable of accurately capturing the Knudsen boundary layer.