Fluid Dynamics

, Volume 53, Issue 1, pp 152–168 | Cite as

Status of the Navier–Stokes Equations in Gas Dynamics. A Review

  • V. S. Galkin
  • S. V. Rusakov


The existing ideas on the status of the Navier–Stokes equations are changed in taking into account the following facts: generally speaking, the terms of these equations neglected in the boundary layer equations are of the order of certain Burnett terms in the conservation equations; the Navier–Stokes equations cannot be used to describe slow nonisothermal gas flows since in this case it is necessary to take the Burnett temperature stresses into account; and in the transport relations the Burnett terms determine certain effects (for example, the mechanocaloric effect).

Key words

Navier–Stokes equations Burnett equations Chapman–Enskog method boundary layer slow nonisothermal flows Burnett effects bulk viscosity 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases (Cambridge University Press, Cambridge, 1952; Izd-vo Inost. Lit., Moscow, 1960).zbMATHGoogle Scholar
  2. 2.
    M. N. Kogan, Rarefied Gas Dynamics (Plenum Press, New York, 1969; Nauka, Moscow, 1967).Google Scholar
  3. 3.
    J. H. Ferziger and H. G. Kaper, Mathematical Theory of Transport Processes in Gases (North-Holland, Amsterdam, 1972; Mir, Moscow, 1976).Google Scholar
  4. 4.
    H. S. Tsien, “Superaerodynamics: Mechanics of Rarefied Gases,” J. Aeronaut. Sci. 13 (12), 653–664 (1946).CrossRefGoogle Scholar
  5. 5.
    M. Z. Krzywoblocki, “On the Two-Dimensional Laminar Boundary Layer Equations for Hypersonic Flow in Continuum and Rarefied Gases,” J. Aeronaut. Soc. India (1), 1–10 (1953).Google Scholar
  6. 6.
    Yu. P. Lun’kin, “Boundary Layer Equations and the Boundary Conditions for Them in the Case of Weakly Rarefied Gas Flow with Supersonic Velocities,” Prikl. Mat. Mekh. 21 (5), 597–605 (1957).MathSciNetGoogle Scholar
  7. 7.
    V. S. Galkin, “Slip Effects in Hypersonic Weakly Rarefied Gas Flows past Bodies,” Inzh. Zh. 3 (1), 27–36 (1963).Google Scholar
  8. 8.
    V. N. Zhigulev, “Equation of Motion of a Nonequilibrium Mediumwith Regard to Radiation,” Inzh. Zh. 4 (2), 231–241 (1964).Google Scholar
  9. 9.
    Y. Sone, Molecular Gas Dynamics. Theory, Techniques, and Applications (Birkhauser, Boston-Basel-Berlin, 2007).CrossRefzbMATHGoogle Scholar
  10. 10.
    Yu. P. Golovachev, Numerical Simulation of Viscous Gas Flows in the Shock Layer (Nauka, Fizmatlit, Moscow, 1996) [in Russian].zbMATHGoogle Scholar
  11. 11.
    M. N. Kogan, V. S. Galkin, and O. G. Friedlender, “Stresses Arising in Gases As a Result of Temperature and Concentration Inhomogeneities,” Usp. Fiz. Nauk 119 (1), 111–125 (1976).ADSCrossRefGoogle Scholar
  12. 12.
    M. S. Cramer, “Numerical Estimates for the Bulk Viscosity of Gases,” Phys. Fluids 24, 066102, 1–23 (2012).Google Scholar
  13. 13.
    R. E. Graves and B. M. Argrow, “Bulk Viscosity: Past to Present,” J. Thermophys. Heat Transfer 13 (3), 337–342 (1999).CrossRefGoogle Scholar
  14. 14.
    E. A. Nagnibeda and E. V. Kustova, Kinetic Theory of Transport Processes and Relaxation in Nonequilibrium Reacting Gas Flows (University Press, Saint-Petersburg, 2003) [in Russian].zbMATHGoogle Scholar
  15. 15.
    M. A. Rydalevskaya, Statistic and Kinetic Models in Physicochemical Gas Dynamics (University Press, St.-Petersburg, 2003) [in Russian].Google Scholar
  16. 16.
    A. Ern and V. Giovangigli, Multicomponent Transport Algorithms (Springer-Verlag, New-York-Berlin-Heidelberg, 1994).zbMATHGoogle Scholar
  17. 17.
    V. Giovangigli, Multicomponent FlowModeling (Birkhauser, Boston-Basel-Berlin, 1999).CrossRefzbMATHGoogle Scholar
  18. 18.
    V. M. Zhdanov, V. S. Galkin, O. A. Gordeev, and I. A. Sokolova, Physicochemical Processes in Gas Dynamics. Handbook, Vol. 3, in: S. A. Losev (Ed.) Models of the Molecular Transfer Process in Physicochemical Gas Dynamics (Fizmatlit, Moscow, 2013) [in Russian].Google Scholar
  19. 19.
    V. S. Galkin, M. N. Kogan, and N. K. Makashev, “Generalized Chapman–Enskog Method,” Dokl. Akad. Nauk SSSR 220 (2), 304–307 (1975).ADSzbMATHGoogle Scholar
  20. 20.
    M. N. Kogan, V. S. Galkin, and N. K. Makashev, “Generalized Chapman–Enskog Method: Derivation of the Nonequilibrium Gasdynamic Equations,” in: Rarefied Gas Dynamics. 11th Int. Symp. Cannes, 1978, Vol. 2 (Paris, 1979), pp. 693–734.Google Scholar
  21. 21.
    V. S. Galkin, M. N. Kogan, and N. K. Makashev, “Region of Applicability and the Main Features of the Generalized Chapman–Enskog Method,” Fluid Dynamics 19 (3), 449–458 (1984).ADSCrossRefzbMATHGoogle Scholar
  22. 22.
    B. V. Alekseev and I. T. Grushin, Transfer Processes in Reacting Gases and Plasma (Energoatomizdat, Moscow, 1994) [in Russian].Google Scholar
  23. 23.
    R. Brun “Transport Properties of NonequilibriumGas Flows,” in: M. Capitelli (Ed.), Molecular Physics and Hypersonic Flows (Kluwer Acad. Publ., Netherlanders, 1996), pp. 361–382 (1996).CrossRefGoogle Scholar
  24. 24.
    V. A. Matsuk and V. A. Rykov, “The Chapman–Enskog Method for a Mixture of Gases,” Dokl. Akad. Nauk SSSR 233 (1), 49–51 (1977).ADSGoogle Scholar
  25. 25.
    V. A. Matsuk, “The Chapman–Enskog Method for a Chemically Reacting Gas Mixture with Allowance for the Internal Degrees of Freedom,” Zh. Vychisl.Mat. Mat. Fiz. 18 (4), 1043–1048 (1978).MathSciNetGoogle Scholar
  26. 26.
    T. C. Lin and R. E. Street, “Effect of Variable Viscosity and Thermal Conductivity on High-Speed Slip Flow between Concentric Cylinders,” NACA Rep., No. 1175 (1954).Google Scholar
  27. 27.
    V. S. Galkin and M. Sh. Shavaliev, “Gasdynamic Equations of Higher Approximations of the Chapman–Enskog Method. Review,” Fluid Dynamics 33 (4), 469–487 (1998).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    H. K. Cheng, “The Blunt Body Problem in Hypersonic Flow at Low Reynolds Number,” IAS Paper, No. 63–92 (1963).Google Scholar
  29. 29.
    G. A. Tirskii, “Continuum Models in the Problems of Hypersonic Rarefied Gas Flow past Bodies,” Prikl. Mat. Mekh. 61 (6), 903–930 (1997).MathSciNetGoogle Scholar
  30. 30.
    M. M. Kuznetsov and V. S. Nikol’skii, “Kinetic Analysis of Hypersonic Polyatomic Gas Flow in the Thin Three-Dimensional Shock Layer,” Uch. Zap. TsAGI 16 (3), 38–49 (1985).Google Scholar
  31. 31.
    H. K. Cheng, C. J. Lee, E. Y. Wong, and H. T. Yang, “Hypersonic Slip Flows and Issues on Extending Continuum Model Beyond the Navier–Stokes Level,” AIAA Paper, No. 89-1663 (1989).CrossRefGoogle Scholar
  32. 32.
    V. Ya. Neiland, V. V. Bogolepov, G. N. Dudin, and I. I. Lipatov, Asymptotic Theory of Supersonic Viscous Flows (Fizmatlit, Moscow, 2003; Elsevier, Amsterdam, 2007).Google Scholar
  33. 33.
    L. D. Landau and E. M. Lifshitz, Fluid Mechanics (2nd Ed.) (Pergamon Press, 1987; Nauka, Moscow, 1986).zbMATHGoogle Scholar
  34. 34.
    M. N. Kogan, “Kinetic Theory in Aerothermodynamics,” Prog. Aerospace Sci. 29 (4), 271–354 (1992).ADSCrossRefGoogle Scholar
  35. 35.
    V. Yu. Aleksandrov and O. G. Fridlender, “Slow Gas Motions and the Negative Drag of a Strongly Heated Spherical Particle,” Fluid Dynamics 43 (3), 485–492 (2008).ADSCrossRefzbMATHGoogle Scholar
  36. 36.
    V. Yu. Aleksandrov, “Drag of a Strongly Heated Sphere at Small Reynolds Numbers,” Fluid Dynamics 46 (5), 794–808 (2011).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    L. D. Landau and E. M. Lifshitz, Theoretical Physics. Vol. 10, E. M. Lifshitz and L. P. Pitaevskii, Physical Kinetics (Fizmatlit, Moscow, 2001) [in Russian].Google Scholar
  38. 38.
    V. S. Galkin and S. V. Rusakov, “Gas Dynamics Equations of Slow Flows of Polyatomic Gas Mixtures with Inhomogeneous Temperature and Concentrations,” Prikl. Mat. Mekh. 79 (2), 218–235 (2015).zbMATHGoogle Scholar
  39. 39.
    V. S. Galkin and S. V. Rusakov, “Transformations of the Burnett Components of Transport Relations in Gases,” Fluid Dynamics 49 (1), 131–136 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    H. Primakoff, “The Translational Dispersion of Sound in Gases,” J. Acoust. Soc. America 14 (1), 14–18 (1942).ADSCrossRefGoogle Scholar
  41. 41.
    M. Greenspan, “Transmission of Sound Waves in Gases at Very Low Pressures,” in: Physical Acoustics, Vol. 2, Pt. A, Properties of Gases, Liquids and Solutions (Acad. Press, N.Y., 1965), pp. 1–45.Google Scholar
  42. 42.
    J. D. Foch and G. W. Ford, “The Dispersion of Sound in Monatomic Gases,” in: J. De Bour and G. E. Ulenbeck (Eds.), Studies in Statistical Mechanics, Vol. 5 (North-Holland, Amsterdam, 1970), pp. 101–231.Google Scholar
  43. 43.
    J. D. Foch, G. E. Ulenbeck, and M. F. Losa, “Theory of Sound Propagation in Mixtures of Monatomic Gases,” Phys. Fluids 15 (7), 1224–1232 (1972).ADSCrossRefzbMATHGoogle Scholar
  44. 44.
    H. Honma, D. Q. Xu, and H. Oguchi, “KineticModel Approach to the Shock Structure Problem: A Detailed Aspect, in: Rarefied Gas Dynamics. Proc. 17th Int. Symp. VCH. Weinheim, 1991, pp. 161–166.Google Scholar
  45. 45.
    V. S. Galkin and S. V. Rusakov, “On Asymptotic Theory of Dispersion of Sound in a Binary Gas Mixture,” Prikl. Mat. Mekh. 78 (2), 194–200 (2014).Google Scholar
  46. 46.
    V. S. Galkin and S.V. Rusakov, “Asymptotic Theory of the Parameters ofAsymmetry of aWeak ShockWave,” Prikl. Mat. Mekh. 77 (4), 573–584 (2013).Google Scholar
  47. 47.
    M. Sh. Shavaliev, “Investigation of theWeak ShockWave Structure and Propagation of Small Perturbations in GasMixtures Using the Burnett Equations,” Prikl. Mat. Mekh. 63 (3), 444–456 (1999).MathSciNetzbMATHGoogle Scholar
  48. 48.
    V. S. Galkin and V. A. Zharov, “Solution of Problems of Sound Propagation andWeak ShockWave Structure in a Polyatomic Gas Using the Burnett Equations,” Prikl. Mat. Mekh. 65 (3), 467–476 (2001).zbMATHGoogle Scholar
  49. 49.
    F. E. Lumpkin and D. R. Chapman, “Accuracy of the Burnett Equations for Hypersonic Real Gas Flows,” J. Thermophys. and Heat Transfer 6 (3), 419–425 (1992).ADSCrossRefGoogle Scholar
  50. 50.
    V. I. Roldughin and V. M. Zhdanov, “Non-Equilibrium Thermodynamics and Kinetic Theory of GasMixtures in the Presence of Interfaces,” Advances Colloid Interface Science 98, 121–215 (2002).CrossRefGoogle Scholar
  51. 51.
    V. M. Zhdanov and V. I. Roldugin, “NonequilibriumThermodynamics and Kinetic Theory of Rarefied Gases,” Usp. Fiz. Nauk 168 (4), 407–438 (1998).CrossRefGoogle Scholar
  52. 52.
    V. S. Galkin, “Degeneracy of the Chapman–Enskog Series for Transport Properties in the Case of Slow Steady-StateWeakly-Rarefied Gas Flows,” Fluid Dynamics 23 (4), 614–620 (1988).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    S. M. Dikman and L. P. Pitaevskii, “Convection of a New Type in the Magnetoactive Plasma,” Zh. Eksp. Teor. Fiz. 78 (5), 1750–1759 (1980).ADSGoogle Scholar
  54. 54.
    A. B. Mikhailovskii, Theory of Plasma Instabilities, Vol. 2, Instability of Inhomogeneous Plasma (Atomizdat, Moscow, 1977) [in Russian].Google Scholar
  55. 55.
    A. B. Mikhailovskii, “Hydrodynamic Theory of Plasma Rotation in a Tokamak,” Fiz. Plasmy 9 (3), 594–603 (1983).Google Scholar
  56. 56.
    V. S. Tsypin, “Rotation and Transfer of Collisionless Plasma in a Tokamak,” Fiz. Plasmy 11 (10), 1163–1166 (1985).Google Scholar
  57. 57.
    F. R. W. McCourt, J. J. M. Beenakker, W. E. Kohler, and I. Kuscer, Nonequilibrium Phenomena in Polyatomic Gases, Vol. 2 (Clarendon Press, Oxford, 1991).Google Scholar
  58. 58.
    G. A. Pavlov and Yu. V. Troshchiev, “Investigation of Thermal Regimes inMedia with Volume Heat Release,” Zh. Tekhn. Fiz. 83 (1), 34–39 (2013).Google Scholar
  59. 59.
    G. A. Pavlov, “Burnett Kinetic Coefficients in Dense Charged and Neutral Media,” Zh. Tekhn. Fiz. 80 (4), 152–155 (2010).Google Scholar
  60. 60.
    V. S. Galkin and S. V. Rusakov, “On Theory of the Volume Viscosity and Relaxational Pressure,” Prikl. Mat. Mekh. 69 (6), 1051–1064 (2005).MathSciNetzbMATHGoogle Scholar
  61. 61.
    M. H. Ernst, “Transport Coefficients and Temperature Definition,” Physica 32 (2), 252–272 (1966).ADSMathSciNetCrossRefGoogle Scholar
  62. 62.
    M. Sh. Shavaliev, “Transport Phenomena in the Burnett Approximation for MulticomponentGas Mixtures,” Fluid Dynamics 9 (1), 96–104 (1974).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  63. 63.
    V. S. Galkin, “Burnett Equations forMulticomponentMixtures of Polyatomic Gases,” Prikl. Mat. Mekh. 64 (4), 590–609 (2000).zbMATHGoogle Scholar
  64. 64.
    V. S. Galkin and V. A. Zharov, “Transport Properties of Gases in the Burnett Approximation,” Prikl. Mat. Mekh. 66 (3), 434–447 (2002).MathSciNetzbMATHGoogle Scholar
  65. 65.
    V. S. Galkin and S. V. Rusakov, “Requirements to the Accuracy of the Burnett Transport Coefficients,” Fluid Dynamics 47 (6), 802–805 (2012).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  66. 66.
    S. Takata, S. Yasuda, K. Aoki, and T. Shibata, “Various Transport Coefficients Occurring in Binary Gas Mixtures and their Database,” in Rarefied Gas Dynamics. 23rd Int. Symp. N.-Y. Amer. Inst. Phys., 2003, pp. 106–113.Google Scholar
  67. 67.
    M. Sh. Shavaliev, “Super-Burnett Corrections to the Stress Tensor and Heat Flux in a Gas of Maxwellian Molecules,” Prikl. Mat. Mekh. 57 (3), 168–171 (1993).zbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Zhukovsky Central Aerohydrodynamic Institute (TsAGI)Zhukovsky, Moscow oblastRussia

Personalised recommendations