Beyond the Navier-Stokes equations: Burnett hydrodynamics

Abstract This work is mainly concerned with the extension of hydrodynamics beyond the Navier–Stokes equations, a regime known as Burnett hydrodynamics. The derivation of the Burnett equations is considered from several theoretical approaches. In particular we discuss the Chapman–Enskog, Grad’s method, and Truesdell’s approach for solving the Boltzmann equation. Also, their derivation using the macroscopic approach given by extended thermodynamics is mentioned. The problems and successes of these equations are discussed and some alternatives proposed to improve them are mentioned. Comparisons of the predictions coming from the Burnett equations with experiments and/or simulations are given in order to have the necessary elements to give a critical assessment of their validity and usefulness.

[1]  R. Temam.  Navier-Stokes Equations , 1977 .

[2]  Russel E. Caflisch,et al.  Shock profile solutions of the Boltzmann equation , 1982 .

[3]  G. Uhlenbeck,et al.  Studies in statistical mechanics , 1962 .

[4]  Continuum description of rarefied gas dynamics. III. The structures of shock waves. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[5]  Angelo Marcello Anile,et al.  Relativistic fluids and magneto-fluids , 2005 .

[6]  C. G. Hoover,et al.  Links between microscopic and macroscopic fluid mechanics , 2003 .

[7]  Ramesh K. Agarwal,et al.  Beyond Navier–Stokes: Burnett equations for flows in the continuum–transition regime , 2001 .

[8]  Sauro Succi,et al.  Colloquium: Role of the H theorem in lattice Boltzmann hydrodynamic simulations , 2002 .

[9]  W. Wakeham,et al.  Status and future developments in the study of transport properties , 1992 .

[10]  A. Bobylev,et al.  The Chapman-Enskog and Grad methods for solving the Boltzmann equation , 1982 .

[11]  Alexander N. Gorban,et al.  Hydrodynamics from Grad's equations: What can we learn from exact solutions? , 2002, Annalen der Physik.

[12]  Henning Struchtrup,et al.  Failures of the Burnett and super-Burnett equations in steady state processes , 2005 .

[13]  J. Kestin,et al.  Equilibrium and transport properties of the noble gases and their mixtures at low density , 1984 .

[14]  Alejandro L. Garcia.  Numerical methods for physics , 1994 .

[15]  D. Morgenstern.  Analytical Studies Related to the Maxwell-Boltzmann Equation , 1955 .

[16]  Exact summation of the Chapman-Enskog expansion from moment equations , 2000 .

[17]  R. G. Lord.  Direct simulation Monte Carlo calculations of rarefied flows with incomplete surface accommodation , 1992 .

[18]  B. Z. Cybyk,et al.  Direct Simulation Monte Carlo: Recent Advances and Applications , 1998 .

[19]  C. G. Hoover,et al.  Temperature maxima in stable two-dimensional shock waves , 1997 .

[20]  Alejandro L. Garcia,et al.  Comparison of Kinetic Theory and Hydrodynamics for Poiseuille Flow , 2002 .

[21]  P. Resibois,et al.  Approximate kinetic theory of hard-sphere fluids near equilibrium: II. A quasihydrodynamic approximation for the velocity autocorrelation function , 1975 .

[22]  L. García-Colín,et al.  On the validity of the Onsager relations in an inert multiple dilute gas mixture , 2005 .

[23]  L. H. Söderholm.  Hybrid Burnett Equations: A New Method of Stabilizing , 2005 .

[24]  A. Avoird.  OVERVIEW ON INTERMOLECULAR POTENTIALS , 1992 .

[25]  Velasco,et al.  Kinetic approach to generalized hydrodynamics. , 1991, Physical Review A. Atomic, Molecular, and Optical Physics.

[26]  D. Ronis.  Statistical mechanics of systems nonlinearly displaced from equilibrium I , 1979 .

[27]  L. Mestel.  Stellar magnetism , 1999 .

[28]  Duncan A. Lockerby,et al.  High-resolution Burnett simulations of micro Couette flow and heat transfer , 2003 .

[29]  L. C. Woods.  An Introduction to the Kinetic Theory of Gases and magnetoplasmas. By L. C. WOODS. Oxford University Press, 1993. 300 pp. £27.50. , 1995, Journal of Fluid Mechanics.

[30]  L. García-Colín.  On the Burnett and higher order equations of hydrodynamics , 1983 .

[31]  G. Galdi.  An Introduction to the Mathematical Theory of the Navier-Stokes Equations : Volume I: Linearised Steady Problems , 1994 .

[32]  M. Sabbane,et al.  Analyse du flux de Poiseuille bidimensionnel via l'équation de Boltzmann , 2004 .

[33]  T. R. Kirkpatrick,et al.  Generic Long-Range Correlations in Molecular Fluids , 1994 .

[34]  Boundary layer variational principles: a case study. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[35]  Daniel A. Erwin,et al.  Testing continuum descriptions of low-Mach-number shock structures , 1991, Journal of Fluid Mechanics.

[36]  E. P. Muntz,et al.  Molecular velocity distribution functions in an argon normal shock wave at Mach number 7 , 1983 .

[37]  Manuel Torrilhon,et al.  Regularized 13-moment equations: shock structure calculations and comparison to Burnett models , 2004, Journal of Fluid Mechanics.

[38]  Tommaso Ruggeri.  On the shock structure problem in non-equilibrium Thermodynamics of gases , 1996 .

[39]  Ruggeri.  Breakdown of shock-wave-structure solutions. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[40]  Charles Muckenfuss.  Some Aspects of Shock Structure According to the Bimodal Model , 1962 .

[41]  S. Chapman,et al.  On the Law of Distribution of Molecular Velocities, and on the Theory of Viscosity and Thermal Conduction, in a Non-Uniform Simple Monatomic Gas , 1916 .

[42]  M. Alaoui,et al.  Poiseuille flow driven by an external force , 1992 .

[43]  Harold Grad,et al.  The profile of a steady plane shock wave , 1952 .

[44]  Geoffrey Ingram Taylor,et al.  The conditions necessary for discontinuous motion in gases , 1910 .

[45]  U. Weinert.  Multi-temperature generalized moment method in Boltzmann transport theory , 1982 .

[46]  Shi Jin,et al.  Regularization of the Burnett Equations via Relaxation , 2001 .

[47]  L. García-Colín.  EXTENDED IRREVERSIBLE THERMODYNAMICS : AN UNFINISHED TASK , 1995 .

[48]  G. Pham-Van-diep,et al.  Nonequilibrium Molecular Motion in a Hypersonic Shock Wave , 1989, Science.

[49]  L. C. Woods,et al.  The thermodynamics of fluid systems , by L. C. Woods. Pp 359. £12·50. 1985. ISBN 0-19-856180-6 (Oxford University Press) , 1987, The Mathematical Gazette.

[50]  L. García-Colín.  Some Thoughts About Hydrodynamics , 2003 .

[51]  Alejandro L. Garcia,et al.  Molecular simulations of sound wave propagation in simple gases , 2001 .

[52]  M. Greenspan.  Attenuation of Sound in Rarefied Helium , 1949 .

[53]  Weiss.  Continuous shock structure in extended thermodynamics. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[54]  Louis Rosenhead,et al.  The Second Coefficient of Viscosity of Liquids and Gases , 1952 .

[55]  D. Prialnik.  An Introduction to the Theory of Stellar Structure and Evolution , 2000 .

[56]  Santos,et al.  Hilbert-class or "normal" solutions for stationary heat flow. , 1989, Physical review. A, General physics.

[57]  B. Somov,et al.  Magnetohydrodynamic discontinuities in space plasmas: Interrelation between stability and structure , 1996 .

[58]  A. N. Gorban,et al.  Constructive methods of invariant manifolds for kinetic problems , 2003 .

[59]  T. G. Cowling,et al.  The mathematical theory of non-uniform gases , 1939 .

[60]  C. Ott,et al.  Multi-Dimensional Explorations in Supernova Theory , 2006, astro-ph/0612460.

[61]  L. Rayleigh.  Aerial Plane Waves of Finite Amplitude , 1910 .

[62]  Renée Gatignol,et al.  Kinetic theory for a discrete velocity gas and application to the shock structure , 1975 .

[63]  I. G. Currie.  Fundamental mechanics of fluids , 1974 .

[64]  M. Greenspan.  Propagation of Sound in Rarefied Helium , 1950 .

[65]  On the gravitational instability of a dissipative medium , 2001, astro-ph/0110295.

[66]  Kun Xu,et al.  Super-Burnett solutions for Poiseuille flow , 2003 .

[67]  O. A. Ladyzhenskaia.  Solution “in the large” of the nonstationary boundary value problem for the Navier‐Stokes system with two space variables , 1959 .

[68]  A. Chorin,et al.  Computational Fluid Mechanics , 1989 .

[69]  D. E. Rosner,et al.  Comparison of Burnett and DSMC predictions of pressure distributions and normal stress in one-dimensional, strongly nonisothermal gases , 1999 .

[70]  A. Majda,et al.  Vorticity and incompressible flow , 2001 .

[71]  J. Ball,et al.  The Scientific Life and Influence of¶Clifford Ambrose Truesdell III , 2002 .

[72]  G. Martínez-Pinedo,et al.  Theory of core-collapse supernovae , 2006, astro-ph/0612072.

[73]  Salomons,et al.  Modeling shock waves in an ideal gas: Going beyond the Navier-Stokes level. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[74]  Rosenau,et al.  Extending hydrodynamics via the regularization of the Chapman-Enskog expansion. , 1989, Physical review. A, General physics.

[75]  Gary S. Settles,et al.  High-speed imaging of shock waves, explosions and gunshots , 2006 .

[76]  Mikhail S. Ivanov,et al.  COMPUTATIONAL HYPERSONIC RAREFIED FLOWS , 1998 .

[77]  J. Kestin,et al.  EQUILIBRIUM AND TRANSPORT PROPERTIES OF GAS MIXTURES AT LOW DENSITY : ELEVEN POLYATOMIC GASES AND FIVE NOBLE GASES , 1990 .

[78]  Yoshikazu Giga,et al.  Nonlinear Partial Differential Equations , 2004 .

[79]  On the theory of the thickness of weak shock waves , 1948 .

[80]  William G. Hoover,et al.  Time Reversibility, Computer Simulation, And Chaos , 1999 .

[81]  Kazuo Aoki,et al.  Numerical analysis of the Poiseuille and thermal transpiration flows between two parallel plates on the basis of the Boltzmann equation for hard‐sphere molecules , 1989 .

[82]  Alejandro L. Garcia,et al.  On the validity of hydrodynamics in plane Poiseuille flows , 1997 .

[83]  D. Burnett,et al.  The Distribution of Velocities in a Slightly Non‐Uniform Gas , 1935 .

[84]  H. Alsmeyer,et al.  Density profiles in argon and nitrogen shock waves measured by the absorption of an electron beam , 1976, Journal of Fluid Mechanics.

[85]  L. Loeb,et al.  Kinetic Theory of Gases , 2018, Foundations of Plasma Physics for Physicists and Mathematicians.

[86]  In the Chapman-Enskog Expansion¶the Burnett Coefficients Satisfy¶the Universal Relation ω3+ω4+θ3= 0 , 2002 .

[87]  J. Brey.  Long time behavior of the Burnett transport coefficients , 1983 .

[88]  D. Burnett.  The Distribution of Molecular Velocities and the Mean Motion in a Non-Uniform Gas , 1936 .

[89]  M. Lampis.  New approach to the Mott-Smith method for shock waves , 1977 .

[90]  Pierre Resibois,et al.  Classical kinetic theory of fluids , 1977 .

[91]  Duncan A. Lockerby,et al.  Capturing the Knudsen layer in continuum-fluid models of non-equilibrium gas flows , 2005 .

[92]  Asymptotic theory of the Boltzmann system, for a steady flow of a slightly rarefied gas with a finite Mach number: General theory , 2000 .

[93]  William A. Wakeham,et al.  Intermolecular Forces: Their Origin and Determination , 1983 .

[94]  R. M. Velasco,et al.  Slip boundary conditions in Couette flow , 1999 .

[95]  L. Holway.  Existence of Kinetic Theory Solutions to the Shock Structure Problem , 1964 .

[96]  J. Peixoto,et al.  Physics of climate , 1992 .

[97]  C. Cercignani.  The Boltzmann equation and its applications , 1988 .

[98]  J. Smoller.  Shock Waves and Reaction-Diffusion Equations , 1983 .

[99]  Dispersion and absorption of sound in monatomic gases: An extended kinetic description , 1999 .

[100]  Rho-Shin Myong,et al.  Velocity slip in microscale cylindrical Couette flow: the Langmuir model , 2005 .

[101]  Ramesh K. Agarwal,et al.  COMPUTATIONAL FLUID DYNAMICS OF WHOLE-BODY AIRCRAFT , 1999 .

[102]  Santos,et al.  Heat and momentum transport far from equilibrium. , 1987, Physical review. A, General physics.

[103]  G. R. Eisler,et al.  Numerical computation of the hypersonic leading edge problem using the Burnett equations , 1976 .

[104]  Dean R. Chapman,et al.  Comparison of Burnett, super-Burnett and Monte Carlo solutions for hypersonic shock structure , 1989 .

[105]  Carlo Cercignani,et al.  The structure of an infinitely strong shock wave , 1999 .

[106]  Andrés Santos,et al.  Perturbation analysis of a stationary nonequilibrium flow generated by an external force , 1994 .

[107]  E. A. Mason,et al.  Transport Properties of Fluids: The Corresponding–States Principle: Dilute Gases , 1996 .

[108]  P. Holmes,et al.  Turbulence, Coherent Structures, Dynamical Systems and Symmetry , 1996 .

[109]  Kazuo Aoki,et al.  Shock-wave structure for a binary gas mixture: finite-difference analysis of the Boltzmann equation for hard-sphere molecules , 2001 .

[110]  On the stability of the Jeffery–Hamel flow , 1997 .

[111]  S M Yen,et al.  NUMERICAL SOLUTION OF THE NONLINEAR BOLTZMANN EQUATION FOR NONEQUILIBRIUM GAS FLOW PROBLEMS , 1984 .

[112]  S. Silich,et al.  Shock-wave propagation in the nonuniform interstellar medium , 1995 .

[113]  Duncan A. Lockerby,et al.  New directions in fluid dynamics: non-equilibrium aerodynamic and microsystem flows , 2003, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[114]  L. García-Colín,et al.  Non-linear constitutive and diffusion equations in the Burnett regime , 1981 .

[115]  Xiaolin Zhong,et al.  Stabilization of the Burnett Equations and Application to Hypersonic Flows , 1993 .

[116]  Jonathan Robert Dorfman,et al.  Nonlinear transport equations in statistical mechanics , 1981 .

[117]  R. B. Lindsay,et al.  Absorption of Sound in Fluids , 1951 .

[118]  J. McLennan.  Convergence of the Chapman‐Enskog Expansion for the Linearized Boltzmann Equation , 1965 .

[119]  A. Bobylev,et al.  Instabilities in the Chapman-Enskog Expansion and Hyperbolic Burnett Equations , 2006 .

[120]  Mikhail Naumovich Kogan,et al.  Rarefied Gas Dynamics , 1969 .

[121]  Pavón,et al.  Nonlocal and nonlinear effects in shock waves. , 1991, Physical review. A, Atomic, molecular, and optical physics.

[122]  F. Uribe,et al.  Burnett description for plane Poiseuille flow. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[123]  P. Gnoffo.  Planetary-Entry Gas Dynamics , 1999 .

[124]  F. Uribe,et al.  The Burnett equations for a Lorentzian mixture , 1993 .

[125]  Uribe,et al.  Shock wave profiles in the burnett approximation , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[126]  H. Brenner.  Navier–Stokes revisited , 2005 .

[127]  Kirk Tr.  Short-wavelength collective modes and generalized hydrodynamic equations for hard-sphere particles. , 1985 .

[128]  Andrés Santos,et al.  Nonlinear Poiseuille flow in a gas , 1998 .

[129]  D. Risso,et al.  GENERALIZED HYDRODYNAMICS FOR A POISEUILLE FLOW : THEORY AND SIMULATIONS , 1998 .

[130]  Salomons,et al.  Usefulness of the Burnett description of strong shock waves. , 1992, Physical review letters.

[131]  Denis J. Evans,et al.  Temperature profile for Poiseuille flow , 1997 .

[132]  M. De.  Handbuch der Physik , 1957 .

[133]  Alexander N. Gorban,et al.  Method of invariant manifolds and regularization of acoustic spectra , 1994 .

[134]  Denis Serre,et al.  Handbook of mathematical fluid dynamics , 2002 .

[135]  Ching-Jen Chen,et al.  Fundamentals of turbulence modeling , 1998 .

[136]  J. Foch.  On Higher Order Hydrodynamic Theories of Shock Structure , 1973 .

[137]  I. Prigogine.  Le domaine de validit? de la thermodynamique des ph?nom?nes irr?versibles , 1949 .

[138]  I. Karlin,et al.  Stabilization of the lattice boltzmann method by the H theorem: A numerical test , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[139]  S. Loyalka,et al.  Plane Poiseuille flow: Rigid sphere gas , 1990 .

[140]  Eu,et al.  Generalized hydrodynamics, normal-stress effects, and velocity slips in the cylindrical Couette flow of Lennard-Jones fluids. , 1989, Physical review. A, General physics.

[141]  M. Lindenfeld,et al.  Nonlinear transport in the Boltzmann limit , 1979 .

[142]  J. Meixner,et al.  Entropy and Entropy Production , 1973 .

[143]  H. Grad.  Principles of the Kinetic Theory of Gases , 1958 .

[144]  Roddam Narasimha,et al.  Structure of a plane shock layer , 1962 .

[145]  Y. Ohr.  Improvement of the Grad 13 moment method for strong shock waves , 2001 .

[146]  Kun Xu,et al.  Microchannel flow in the slip regime: gas-kinetic BGK–Burnett solutions , 2004, Journal of Fluid Mechanics.

[147]  T. D. Creighton.  An explosion of sound , 2006 .

[148]  Nonlinear viscosity and Grad's method. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[149]  Eu,et al.  Nonlinear transport processes and fluid dynamics: Cylindrical Couette flow of Lennard-Jones fluids. , 1988, Physical review. A, General physics.

[150]  W. G. Hoover.  Computational Statistical Mechanics , 1991 .

[151]  Mario Pulvirenti,et al.  Mathematical Theory of Incompressible Nonviscous Fluids , 1993 .

[152]  Forrest E. Lumpkin,et al.  Accuracy of the Burnett equations for hypersonic real gas flows , 1991 .

[153]  Renormalization of the Chapman–Enskog Expansion: Isothermal Fluid Flow and Rosenau Saturation , 1998 .

[154]  R. Caflisch.  The half-space problem for the boltzmann equation at zero temperature , 1985 .

[155]  C. J. Greenshields,et al.  The structure of shock waves as a test of Brenner's modifications to the Navier–Stokes equations , 2006, Journal of Fluid Mechanics.

[156]  L. Garrido,et al.  Systems far from equilibrium , 1980 .

[157]  L. C. Woods.  Transport processes in dilute gases over the whole range of Knudsen numbers. Part 1. General theory , 1979, Journal of Fluid Mechanics.

[158]  G. Lebon,et al.  Extended irreversible thermodynamics , 1993 .

[159]  R. M. Velasco,et al.  Inconsistency in the Moment’s method for solving the Boltzmann equation , 2004 .

[160]  Toshiyuki Nakanishi,et al.  Poiseuille-type flow of a rarefied gas between two parallel plates driven by a uniform external force. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[161]  P. Terry,et al.  Suppression of turbulence and transport by sheared flow , 2000 .

[162]  Harold Grad,et al.  Asymptotic Theory of the Boltzmann Equation , 1963 .

[163]  S. Grossmann.  The onset of shear flow turbulence , 2000 .

[164]  R. Fernández-Feria,et al.  Kinetic theory of binary gas mixtures with large mass disparity , 1987 .

[165]  R. M. Velasco,et al.  BURNETT DESCRIPTION OF STRONG SHOCK WAVES , 1998 .

[166]  L. Rayleigh,et al.  The theory of sound , 1894 .

[167]  S. Woosley,et al.  EVOLUTION AND EXPLOSION OF MASSIVE STARS * , 1978, Reviews of Modern Physics.

[168]  J. McLennan.  Burnett Coefficients and Correlation Functions , 1973 .

[169]  Louis Rosenhead,et al.  The steady two-dimensional radial flow of viscous fluid between two inclined plane walls , 1940, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[170]  E. W. McDaniel,et al.  Transport Properties of Ions in Gases , 1988 .

[171]  H. Struchtrup.  Macroscopic transport equations for rarefied gas flows , 2005 .

[172]  R. Illner,et al.  The mathematical theory of dilute gases , 1994 .

[173]  Sébastien Candel,et al.  A second-order description of shock structure , 1995 .

[174]  Inconsistencies in moment methods. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[175]  J. Lebowitz,et al.  Approximate kinetic theory of hard-sphere fluids near equilibrium. I. Formal theory , 1975 .

[176]  S. Chapman.  On the Kinetic Theory of a Gas. Part II: A Composite Monatomic Gas: Diffusion, Viscosity, and Thermal Conduction , 1918 .

[177]  Denis J. Evans,et al.  The heat flux vector for highly inhomogeneous nonequilibrium fluids in very narrow pores , 1995 .

[178]  Clifford Ambrose Truesdell,et al.  Fundamentals of Maxwell's kinetic theory of a simple monatomic gas , 1980 .

[179]  W. Mccomb,et al.  The physics of fluid turbulence. , 1990 .

[180]  M. Ernst,et al.  Nonanalytic dispersion relations for classical fluids , 1975 .

[181]  I. Kuščer,et al.  Slip Coefficients for General Gas‐Surface Interaction , 1972 .

[182]  H. Mott-Smith,et al.  The Solution of the Boltzmann Equation for a Shock Wave , 1951 .

[183]  M. Greenspan.  Propagation of Sound in Five Monatomic Gases , 1956 .

[184]  Martin Kröger,et al.  From hyperbolic regularization to exact hydrodynamics for linearized Grad's equations. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[185]  R. Menikoff,et al.  The Riemann problem for fluid flow of real materials , 1989 .

[186]  S. Chandrasekhar.  On stars, their evolution and their stability. , 1984, Science.

[187]  F. John.  Partial differential equations , 1967 .

[188]  C. K. Wong,et al.  Theory of nonlinear transport in Burnett order , 1978 .

[189]  Evans,et al.  Isothermal shear-induced heat flow. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[190]  R. Lewis,et al.  A Unifying Principle in Statistical Mechanics , 1967 .

[191]  Á. R. Vasconcellos,et al.  Microscopic Approach to Irreversible Thermodynamics III: Generalized Constitutive Equations , 1995 .

[192]  Tai-Ping Liu,et al.  Nonlinear Stability of Shock Waves for Viscous Conservation Laws , 1985 .

[193]  T. Carleman,et al.  Problèmes mathématiques dans la théorie cinétique des gaz , 1957 .

[194]  Gérard A. Maugin,et al.  Infernal Variables and Dissipative Structures , 1990 .

[195]  H. Primakoff.  The Translational Dispersion of Sound in Gases , 1942 .

[196]  I. Gyarmati.  On the Wave Approach of Thermodynamics and some Problems of Non-Linear Theories , 1977 .

[198]  S. P. Gill,et al.  Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena , 2002 .

[199]  A. Orlov,et al.  Weak shock structure on the basis of modified hydrodynamical equations , 1993 .

[200]  G. Bird.  Molecular Gas Dynamics and the Direct Simulation of Gas Flows , 1994 .

[201]  R. Peierls,et al.  More Surprises in Theoretical Physics , 1992 .

[202]  E. Jaynes,et al.  E. T. Jaynes: Papers on Probability, Statistics and Statistical Physics , 1983 .

[203]  Kinetic Theory of a Dilute Gas System under Steady Heat Conduction , 2002, cond-mat/0202003.

[204]  J. Potter.  Shock Waves: Numerical Integration of the Burnett Equations for Shock Structure in a Maxwell Gas , 1977 .

[205]  Daniel A. Erwin,et al.  Nonequilibrium gas flows. I: A detailed validation of Monte Carlo direct simulation for monatomic gases , 1991 .

[206]  L. García-Colín,et al.  Beyond the navier-stokes regime in hydrodynamics , 1981 .

[207]  C. Ott,et al.  A new mechanism for gravitational-wave emission in core-collapse supernovae. , 2006, Physical review letters.

[208]  Iliya V. Karlin,et al.  Structure and approximations of the chapman-enskog expansion for the linearized grad equations , 1992 .

[209]  D. Gilbarg,et al.  The Structure of Shock Waves in the Continuum Theory of Fluids , 1953 .

[210]  García-Colín,et al.  Bobylev's instability , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[211]  Y. Sone.  Flows induced by temperature fields in a rarefied gas and their ghost effect on the behavior of a gas in the continuum limit , 2000 .

[212]  M. Hirsch,et al.  Differential Equations, Dynamical Systems, and Linear Algebra , 1974 .

[213]  A. Orlov.  Transport phenomena in a weak shock wave , 1992 .

[214]  R. M. Velasco,et al.  Generalized Hydrodynamics in Gases , 1995 .

[215]  Santos,et al.  Velocity distribution for a gas with steady heat flow. , 1989, Physical review. A, General physics.