Beyond the Navier–Stokes equations: Burnett hydrodynamics

García‐Colín, L. S.; Velasco, R. M.; Uribe, F. J.

doi:10.1016/j.physrep.2008.04.010

Cited by 118 publications

(84 citation statements)

References 245 publications

(364 reference statements)

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“…It is widely known that, when conducting a Chapman-Enskog type of expansion in K n on the Boltzmann kinetic equation without any spatial diffusion modification, the second order hydrodynamic model beyond that of Navier-Stokes-Fourier, namely the Burnett equations, violates many conventional mechanical and thermodynamical properties [14][15][16][17]. We note that the volume-mass diffusion contributions in our hydrodynamic set of equations (for example, in the momentum equation 34b) are also of Burnett order.…”

Section: Discussionmentioning

confidence: 99%

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Analysis of the thermomechanical inconsistency of some extended hydrodynamic models at high Knudsen number

Dadzie

Reese

2012

Phys. Rev. E

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This version is available at https://strathprints.strath.ac.uk/39217/ Strathprints is designed to allow users to access the research output of the University of Strathclyde. Unless otherwise explicitly stated on the manuscript, Copyright © and Moral Rights for the papers on this site are retained by the individual authors and/or other copyright owners. Please check the manuscript for details of any other licences that may have been applied. You may not engage in further distribution of the material for any profitmaking activities or any commercial gain. You may freely distribute both the url (https://strathprints.strath.ac.uk/) and the content of this paper for research or private study, educational, or not-for-profit purposes without prior permission or charge.Any correspondence concerning this service should be sent to the The collision integral dominates in the small local Knudsen number regime, which is associated with the exact traditional continuum limit. We find a sub-domain of the continuum range which the conventional Knudsen number classification does not account for appropriately. In this sub-domain, it is possible to obtain a fully mechanically-consistent volume (or mass) diffusion model that satisfies the second law of thermodynamics on the grounds of extended non-local-equilibrium thermodynamics.

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Section: Discussionmentioning

confidence: 99%

“…Typical examples include the class of high order hydrodynamics models obtained as terms beyond Navier-Stokes-Fourier order when approximating the original Boltzmann kinetic equation. [14][15][16][17];…”

Section: Introductionmentioning

confidence: 99%

Analysis of the thermomechanical inconsistency of some extended hydrodynamic models at high Knudsen number

Dadzie

Reese

2012

Phys. Rev. E

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“…In the attempt to resolve inconsistencies related to Burnett's hydrodynamic-regime equations, various set of equations, consisting essentially of modifying or adjusting the originals have been proposed (see, for example, Garcia-Colin et al 2008). Among these is the class of models proposed to remedy particularly the problem of instability under small perturbations in time or space.…”

Section: Introductionmentioning

confidence: 99%

“…The comprehensive review by Garcia-Colin et al (2008) describes the successes and failures of Burnett equations. In that review, standing problems related to Burnett's equations are listed as follows: (i) the boundary conditions are in general unknown; (ii) the convergence properties of the Chapman-Enskog series are not known; (iii) there is no proof about the Burnett equations exhibiting a positive definite entropy production; (iv) the Burnett equations are frame dependent in rotating coordinates; (v) the equilibrium state is linearly unstable under longitudinal perturbations.…”

Section: Introductionmentioning

confidence: 99%

A thermo-mechanically consistent Burnett regime continuum flow equation without Chapman–Enskog expansion

Dadzie

2013

J. Fluid Mech.

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Chapman-Enskog expansion is the orthodox approach to derive continuum flow models from Boltzmann's kinetic equation for dilute gases. Beyond the Navier-Stokes-Fourier order, these models known as Burnett hydrodynamic-regime equations violate a number of fundamental mechanical and thermodynamic principles in their original forms. This has generated a widely investigated problem in the kinetic theory of gases. In this short article, we derive a Burnett hydrodynamic-regime continuum model that is systematically consistent with all known mechanical and thermodynamic principles without using any series' expansion. Close comparison with the conventional Burnett hydrodynamic set of equations is considered and their linear stabilities around an equilibrium point under small perturbations are presented.

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“…A kinetic model for non-vibrating polyatomic gases 25 the molecular mean free path is much smaller than the characteristic flow length) macroscopic equations, such as the Navier-Stokes, Burnett, and Grad equations, can be used in some instances (Galkin & Rusakov 2005;Greenshields & Reese 2007;Garcia-Colin, Velasco & Uribe 2008;Gu & Emerson 2009;Rana, Torrilhon & Struchtrup 2013;Rahimi & Struchtrup 2014), in the transition and free-molecular regimes the Boltzmann equation itself should be solved. However, its intricate collision operator makes a solution difficult to obtain by deterministic numerical methods.…”

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confidence: 99%

A kinetic model of the Boltzmann equation for non-vibrating polyatomic gases

White

Scanlon

et al. 2014

J. Fluid Mech.

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A kinetic model of the Boltzmann equation for non-vibrating polyatomic gases is proposed, based on the Rykov model for diatomic gases. We adopt two velocity distribution functions (VDFs) to describe the system state; inelastic collisions are the same as in the Rykov model, but elastic collisions are modelled by the Boltzmann collision operator (BCO) for monatomic gases, so that the overall kinetic model equation reduces to the Boltzmann equation for monatomic gases in the limit of no translational-rotational energy exchange. The free parameters in the model are determined by comparing the transport coefficients, obtained by a Chapman-Enskog expansion, to values from experiment and kinetic theory. The kinetic model equations are solved numerically using the fast spectral method for elastic collision operators and the discrete velocity method for inelastic ones. The numerical results for normal shock waves and planar Fourier/Couette flows are in good agreement with both conventional direct simulation Monte Carlo (DSMC) results and experimental data. Poiseuille and thermal creep flows of polyatomic gases between two parallel plates are also investigated. Finally, we find that the spectra of both spontaneous and coherent Rayleigh-Brillouin scattering (RBS) compare well with DSMC results, and the computational speed of our model is approximately 300 times faster. Compared to the Rykov model, our model greatly improves prediction accuracy, and reveals the significant influence of molecular models. For coherent RBS, we find that the Rykov model could overpredict the bulk viscosity by a factor of two.

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Beyond the Navier–Stokes equations: Burnett hydrodynamics

Cited by 118 publications

References 245 publications

Analysis of the thermomechanical inconsistency of some extended hydrodynamic models at high Knudsen number

Analysis of the thermomechanical inconsistency of some extended hydrodynamic models at high Knudsen number

A thermo-mechanically consistent Burnett regime continuum flow equation without Chapman–Enskog expansion

A kinetic model of the Boltzmann equation for non-vibrating polyatomic gases

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