Alternate Forms of Burnett and Grad Equations

  • Amit Agrawal
  • Hari Mohan Kushwaha
  • Ravi Sudam Jadhav
Part of the Mechanical Engineering Series book series (MES)


In the previous two chapters, a formal derivation of the Burnett and Grad equations was presented. The derivation involved several novel ideas making the approach and the obtained equations invaluable. However, as reviewed in this chapter, both these equations have several limitations because of which working with alternate forms of these equations becomes necessary. Therefore, several variants of these equations have been proposed in the literature. The basic idea behind fixing the equations and the available variants are introduced in this chapter. Comparison of the results as obtained from the variants of the equations for a few specific cases is also presented.


  1. 2.
    Agarwal RK, Yun KY, Balakrishnan R (2001) Beyond Navier–Stokes: Burnett equations for flows in the continuum-transition regime. Phys Fluids 13(10):3061–3085; see also: Erratum: “Beyond Navier–Stokes: Burnett equations for flows in the continuum-transition regime” [Phys. Fluids 13, 3061 (2001)] Physics of Fluids 14, 1818 (2002)Google Scholar
  2. 17.
    Balakrishnan R, Agarwal RK (1997) Numerical simulation of Bhatnagar-Gross-Krook-Burnett equations for hypersonic flows. J Thermophys Heat Transf 11(3):391–399CrossRefGoogle Scholar
  3. 23.
    Bobylev A (2008) Generalized Burnett hydrodynamics. J Stat Phys 132(3):569–580MathSciNetCrossRefGoogle Scholar
  4. 24.
    Bobylev A, Bisi M, Cassinari M, Spiga G (2011) Shock wave structure for generalized Burnett equations. Phys Fluids 23(3):030607CrossRefGoogle Scholar
  5. 31.
    Chapman S, Cowling TG (1970) The mathematical theory of non-uniform gases: an account of the kinetic theory of viscosity, thermal conduction and diffusion in gases. Cambridge University Press, CambridgezbMATHGoogle Scholar
  6. 36.
    Comeaux KA, Chapman DR, MacCormack RW (1995) An analysis of the Burnett equations based on the second law of thermodynamics. In: 33rd Aerospace sciences meeting and exhibit, p 415Google Scholar
  7. 37.
    Dadzie SK (2013) A thermo-mechanically consistent Burnett regime continuum flow equation without Chapman–Enskog expansion. J Fluid Mech 716:R6MathSciNetCrossRefGoogle Scholar
  8. 39.
    De Groot SR, Mazur P (2013) Non-equilibrium thermodynamics. Courier Dover Publications, New YorkzbMATHGoogle Scholar
  9. 52.
    Eu CB (1980) A modified moment method and irreversible thermodynamics. J Chem Phys 73(6):2958–2969MathSciNetCrossRefGoogle Scholar
  10. 63.
    Grad H (1952) The profile of a steady plane shock wave. Commun Pure Appl Math 5(3):257–300MathSciNetCrossRefGoogle Scholar
  11. 64.
    Grad H (1958) Principles of the kinetic theory of gases. Springer, Berlin, pp 205–294Google Scholar
  12. 68.
    Gu XJ, Emerson DR (2009) A high-order moment approach for capturing non-equilibrium phenomena in the transition regime. J Fluid Mech 636:177–216MathSciNetCrossRefGoogle Scholar
  13. 78.
    Jadhav RS, Singh N, Agrawal A (2017) Force-driven compressible plane Poiseuille flow by Onsager-Burnett equations. Phys Fluids 29(10):102002CrossRefGoogle Scholar
  14. 91.
    Mahendra AK, Singh RK (2013) Onsager reciprocity principle for kinetic models and kinetic schemes. arXiv preprint arXiv:13084119Google Scholar
  15. 95.
    McLennan JA (1974) Onsager’s theorem and higher-order hydrodynamic equations. Phys Rev A 10(4):1272MathSciNetCrossRefGoogle Scholar
  16. 103.
    Myong RS (1999) Thermodynamically consistent hydrodynamic computational models for high-Knudsen-number gas flows. Phys Fluids 11(9):2788–2802CrossRefGoogle Scholar
  17. 104.
    Myong R (2011) A full analytical solution for the force-driven compressible Poiseuille gas flow based on a nonlinear coupled constitutive relation. Phys Fluids 23(1):012002CrossRefGoogle Scholar
  18. 107.
    Onsager L (1931) Reciprocal relations in irreversible processes. I. Phys Rev 37(4):405Google Scholar
  19. 108.
    Onsager L (1931) Reciprocal relations in irreversible processes. II. Phys Rev 38(12):2265Google Scholar
  20. 122.
    Romero M, Velasco R (1995) Onsager’s symmetry in the Burnett regime. Phys A Stat Mech Appl 222(1–4):161–172CrossRefGoogle Scholar
  21. 129.
    Shavaliyev MS (1993) Super-Burnett corrections to the stress tensor and the heat flux in a gas of Maxwellian molecules. J Appl Math Mech 57(3):573–576CrossRefGoogle Scholar
  22. 131.
    Singh N, Agrawal A (2016) Onsager’s-principle-consistent 13-moment transport equations. Phys Rev E 93(6):063111CrossRefGoogle Scholar
  23. 133.
    Singh N, Gavasane A, Agrawal A (2014) Analytical solution of plane Couette flow in the transition regime and comparison with direct simulation Monte Carlo data. Comput Fluids 97:177–187MathSciNetCrossRefGoogle Scholar
  24. 136.
    Singh N, Jadhav RS, Agrawal A (2017) Derivation of stable Burnett equations for rarefied gas flows. Phys Rev E 96(1):013106MathSciNetCrossRefGoogle Scholar
  25. 140.
    Struchtrup H (2004) Stable transport equations for rarefied gases at high orders in the Knudsen number. Phys Fluids 16(11):3921–3934MathSciNetCrossRefGoogle Scholar
  26. 141.
    Struchtrup H (2005) Macroscopic transport equations for rarefied gas flows. Springer, BerlinCrossRefGoogle Scholar
  27. 142.
    Struchtrup H, Torrilhon M (2003) Regularization of Grad’s 13 moment equations: derivation and linear analysis. Phys Fluids 15(9):2668–2680MathSciNetCrossRefGoogle Scholar
  28. 146.
    Timokhin MY, Struchtrup H, Kokhanchik A, Bondar YA (2017) Different variants of R13 moment equations applied to the shock-wave structure. Phys Fluids 29(3):037,105CrossRefGoogle Scholar
  29. 148.
    Torrilhon M, Struchtrup H (2004) Regularized 13-moment equations: shock structure calculations and comparison to Burnett models. J Fluid Mech 513:171–198MathSciNetCrossRefGoogle Scholar
  30. 154.
    Uribe F, Garcia A (1999) Burnett description for plane Poiseuille flow. Phys Rev E 60(4A):4063–4078CrossRefGoogle Scholar
  31. 164.
    Weiss W (1995) Continuous shock structure in extended thermodynamics. Phys Rev E 52:R5760–R5763CrossRefGoogle Scholar
  32. 166.
    Woods L (1979) Transport processes in dilute gases over the whole range of Knudsen numbers. Part 1. General theory. J Fluid Mech 93(3):585–607MathSciNetCrossRefGoogle Scholar
  33. 167.
    Woods L, Troughton H (1980) Transport processes in dilute gases over the whole range of Knudsen numbers. Part 2. Ultrasonic sound waves. J Fluid Mech 100(2):321–331CrossRefGoogle Scholar
  34. 177.
    Zhong X, MacCormack RW, Chapman DR (1993) Stabilization of the Burnett equations and application to hypersonic flows. AIAA J 31(6):1036–1043CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Amit Agrawal
    • 1
  • Hari Mohan Kushwaha
    • 1
  • Ravi Sudam Jadhav
    • 1
  1. 1.Department of Mechanical EngineeringIndian Institute of Technology, BombayMumbaiIndia

Personalised recommendations