Russian Mathematics

, Volume 63, Issue 2, pp 44–50 | Cite as

Axisymmetric Helical Flows of Viscous Fluid

  • G. B. SizykhEmail author


In this paper, helical flows are flows in which the velocity vector is collinear to the vorticity vector. For an ideal fluid, examples of stationary helical flows are known (Gromeka-Beltrami flows, ABC-flows, etc.) and it has long been proven that the existence of unsteady helical flows is impossible (Beltrami, 1889). For a viscous fluid, examples of unsteady helical flows are known (Trkal, 1919). But it is still unknown whether there can exist a stationary helical flow of incompressible fluid. In the present paper this question is investigated using the Navier-Stokes equations in the axisymmetric case. It was assumed that the coefficient of proportionality between the vorticity and velocity may depend on the spatial coordinates. It is shown that in the axisymmetric case, the stationary helical flows of viscous incompressible fluid are impossible.

Key words

helical flow Navier-Stokes equations axisymmetric flow of viscous fluid 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Gromeka, I. S. “Some Cases of Motion of Incompressible Fluid”, Uchenye Zapiski Kazanskogo Universiteta III, 1–104 (1882).Google Scholar
  2. 2.
    Beltrami, E. “Considerazioni idrodinamiche”, Rend. Inst. Lombardo Acad. Sci. Lett. 22, 122–131 (1889).zbMATHGoogle Scholar
  3. 3.
    Arnold, V. I. “Sur la Topologie des Écoulements Stationnaires des Fluids Parfaits”, Comptes Rendus Acad Sci Paris 261, 17–20 (1965).Google Scholar
  4. 4.
    Dombre, T., Fisch, U., Greene, J. ‘M., et. al. “Chaotic Streamlines in the ABC Flows”, J. Fluid Mech. 167, 353–391 (1986).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Vereshchagin, V. P., Subbotin, Y. N., Chernykh, N. I. “On the Mechanics of Helical Flows in an Ideal Incompressible Nonviscous Continuous Medium”, Proc. of the Steklov Institute of Math. 284, No. S1, 159–174 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Kovalev, V. P., Sizykh, G. B. “Axisymmetric Helical Flows of Ideal Fluid”, Trudy MFTI 8, No. 3, 171–179 (2016).Google Scholar
  7. 7.
    Truesdell, C. The Kinematics of Vorticity (Indiana Univ. Press, Bloomington, 1954).zbMATHGoogle Scholar
  8. 8.
    Drazin, P. G., Riley, N. The Navier-Stokes Equations: A Classification of Flows and Exact Solutions (Cambridge Univ. Press, 2006).CrossRefzbMATHGoogle Scholar
  9. 9.
    Trkal, V. “Poznámka k Hydrodynamice Vazkých Tekutin”, Časopis pro pĕstování matematiky a fysiky (Praha) 48, No. 3, 302–311 (1919).Google Scholar
  10. 10.
    Kovalev, V. P., Prosviryakov, E. Yu., Sisykh, G. B. “Obtaining Examples of Exact Solutions to Navier-Stokes Equations for Helical Flows by Method of Summing Velocities”, Trudy MFTI 9, No. 1, 71–88 (2017).Google Scholar
  11. 11.
    Byušgens, S. S. “On Helical Flow”, Nauchnye Zapiski Moskovskogo gidromeliorativnogo instituta (MGMI) 17, 73–90 (1948).Google Scholar
  12. 12.
    Lamb, G. Hydrodynamics (Cambridge University Press, Cambridge, 1932; OGIZ. GITTL, Moscow, 1947).zbMATHGoogle Scholar
  13. 13.
    Loitsyanskii, L. G. Mechanics of Fluid and Gas (Drofa, Moscow, 2003) [in Russian].Google Scholar
  14. 14.
    Batchelor, G. Introduction to Fluid Hydrodynamics (Cambridge Univsrsity Press, 1967; Mir, Moscow, 1973).zbMATHGoogle Scholar

Copyright information

© Allerton Press, Inc. 2019

Authors and Affiliations

  1. 1.Moscow Institute of Physics and TechnologyDolgoprudny, Moscow RegionRussia

Personalised recommendations