On the applicability of Stokes’ hypothesis to low-Mach-number flows

Abstract

Stokes’ hypothesis states that the bulk viscosity of a Newtonian fluid can be set to zero. Although not valid for many fluids, it is common practice to invoke this hypothesis in the study of low-Mach-number, variable-density flows. Based on scaling arguments, we provide a necessary condition for neglecting the bulk viscous pressure from the governing equations. More specifically, we show that the Reynolds number defined with respect to the bulk viscosity must be very large. We further show that even when this condition is not satisfied, the bulk viscous pressure does not need to be taken explicitly into account in the computation of the velocity field because it can be combined with the hydrodynamic pressure.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    Buresti, G.: A note on Stokes’ hypothesis. Acta Mech. 226, 3555–3559 (2015)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Chapman, S., Cowling, T.G.: Mathematical Theory of Non-Uniform Gases. Cambridge University Press, Cambridge (1970)

    Google Scholar 

  3. 3.

    Cramer, M.S.: Numerical estimates for the bulk viscosity of ideal gases. Phys. Fluids 24, 066102 (2012)

    ADS  Article  Google Scholar 

  4. 4.

    Cramer, M.S., Bahmani, F.: Effect of large bulk viscosity on large-Reynolds-number flows. J. Fluid Mech. 751, 142–163 (2014)

    ADS  MathSciNet  Article  Google Scholar 

  5. 5.

    Emanuel, G.: Effect of bulk viscosity on a hypersonic boundary layer. Phys. Fluids A 4, 491–495 (1992)

    ADS  Article  Google Scholar 

  6. 6.

    Emanuel, G.: Bulk viscosity in the Navier–Stokes equations. Int. J. Eng. Sci. 36, 1313–1323 (1998)

    Article  Google Scholar 

  7. 7.

    Ferziger, J.H., Perić, M.: Computational Methods for Fluid Dynamics. Springer, Berlin (1996)

    Google Scholar 

  8. 8.

    Gad-el-Hak, M.: Stokes’ hypothesis for a Newtonian, isotropic fluid. J. Fluids Eng. 117(1), 3–5 (1995)

    Article  Google Scholar 

  9. 9.

    Georgiou, M., Papalexandris, M.V.: Turbulent mixing in T-junctions: the role of the temperature as an active scalar. Int. J. Heat Mass Transf. 115, 793–809 (2017)

    Article  Google Scholar 

  10. 10.

    Georgiou, M., Papalexandris, M.V.: Direct numerical simulation of turbulent heat transfer in a T-junction. J. Fluid Mech. 845, 581–614 (2018)

    ADS  MathSciNet  Article  Google Scholar 

  11. 11.

    Graves, R.E., Argrow, B.M.: Bulk viscosity: past to present. J. Thermophys. Heat Tr. 13(3), 337–342 (1999)

    Article  Google Scholar 

  12. 12.

    Grmella, M.: Extensions of classical hydrodynamics. J. Stat. Phys. 132, 581–602 (2008)

    ADS  MathSciNet  Article  Google Scholar 

  13. 13.

    Grmella, M.: Mass flux in extended and classical hydrodynamics. Phys. Rev. E 89, 063024 (2014)

    ADS  Article  Google Scholar 

  14. 14.

    de Groot, S.R., Mazur, P.: Non-equilibrium Thermodynamics. North Holland Publishing Co., Amsterdam (1962)

    Google Scholar 

  15. 15.

    Jaeger, F., Matar, O.K., Muller, E.A.: Bulk viscosity of molecular fluids. J. Chem. Phys. 148(17), 174504 (2018)

    ADS  Article  Google Scholar 

  16. 16.

    Lebon, G., Jou, D., Casas-Vázquez, J.: Understanding Non-equilibrium Thermodynamics. Springer, Berlin (2008)

    Google Scholar 

  17. 17.

    Lessani, B., Papalexandris, M.V.: Time-accurate calculation of variable density flows with strong temperature gradients and combustion. J. Comput. Phys. 212, 218–246 (2006)

    ADS  MathSciNet  Article  Google Scholar 

  18. 18.

    Majda, A., Sethian, J.: The derivation and numerical solution of the equations for zero-Mach number combustion. Combust. Sci. Technol. 42, 185–205 (1985)

    Article  Google Scholar 

  19. 19.

    Pan, E., Johnsen, E.: The role of bulk viscosity on the decay of compressible, homogeneous, isotropic turbulence. J. Fluid Mech. 833, 717–744 (2018)

    ADS  MathSciNet  Article  Google Scholar 

  20. 20.

    Sivashinsky, G.I.: Hydrodynamic theory of flame propagation in an enclosed volume. Acta Astronaut 6, 631–645 (1979)

    ADS  Article  Google Scholar 

  21. 21.

    Truesdell, C.A., Noll, W.: The Non-Linear Field Theories of Mechanics, 3rd edn. Springer, Berlin (2004)

    Google Scholar 

  22. 22.

    Ván, P., Pavelka, M., Grmella, M.: Extra mass flux in fluid mechanics. J. Non-Equilib. Thermodyn. 42, 131–151 (2017)

    ADS  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Miltiadis V. Papalexandris.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by Andreas Öchsner.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Papalexandris, M.V. On the applicability of Stokes’ hypothesis to low-Mach-number flows. Continuum Mech. Thermodyn. 32, 1245–1249 (2020). https://doi.org/10.1007/s00161-019-00785-z

Download citation

Keywords

  • Bulk viscosity
  • Bulk viscous pressure
  • Stokes’ hypothesis
  • Variable-density flows