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Stokes flow driven by a Stokeslet in a cone

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Abstract

We consider an axisymmetric Stokes flow in an infinite right circular cone, which has a source of momentum (a Stokeslet) on its axis. It produces an infinite sequence of eddies in the conical flow region. A boundary problem for a stream function is solved. The picture of the streamlines is obtained. We investigate an eddy structure of the flow. The results can be used for constructing nanoreactors while carrying out chemical reactions in strictly localized nanosized spatial regions.

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References

  1. Li D.: Encyclopedia of Microfluidics and Nanofluidics. Springer, New York (2008)

    Book  Google Scholar 

  2. Rivera J.L., Starr F.W.: Rapid transport of water via carbon nanotube. J.Phys. Chem. C 114, 3737–3742 (2010)

    Article  Google Scholar 

  3. Paul D.R.: Creating new types of carbon-based membranes. Science 335(6067), 411–413 (2012)

    Article  Google Scholar 

  4. Chivilikhin S.A., Gusarov V.V., Popov I.Yu.: Flows in nanostructures: hybrid classical-quantum model. Nanosyst. Phys. Chem. Math. 3(1), 7–26 (2012)

    Google Scholar 

  5. Popov I.Yu., Chivilikhin S.A., Gusarov V.V.: Model of fluid flow in nanotube: classical and quantum features. J. Phys. Conf. Ser. 248, 012006/1-8 (2010)

    Article  Google Scholar 

  6. Popov I.Yu.: Statistical derivation of modified hydrodynamic equations for nanotube flows. Phys. Scr. 83, 045601/1-3 (2011)

    Article  Google Scholar 

  7. Happel J., Brenner H.: Low Reynolds Number Hydrodynamics. Prentice-Hall, Englewood Cliffs (1965)

    Google Scholar 

  8. Ackerberg R.C.: The viscous incompressible flow inside a cone. J. Fluid Mech. 21(part 1), 47–81 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  9. Wakiya S.: Axisymmetric flow of a viscous fluid near the vertex of a body. J. Fluid Mech. 78, 737–747 (1976)

    Article  MATH  Google Scholar 

  10. Kim M.U.: Slow viscous rotation of a sphere on the axis of a circular cone. J. Korean Phys. Soc. 10(2), 54–58 (1977)

    Google Scholar 

  11. Hasimoto H., Sano O.: Stokeslets and eddies in creeping flow. Ann. Rev. Fluid Mech. 12, 335–363 (1980)

    Article  MathSciNet  Google Scholar 

  12. Sano O., Hasimoto H.: Three-dimensional Moffatt-type eddies due to a Stokeslet in a corner. J.Phys. Soc. Jpn. 48, 1763–1768 (1980)

    Article  MathSciNet  Google Scholar 

  13. Liu, C.H., Joseph, D.D.: Stokes flow in conical trenches. SIAM J. Appl. Math. 34, 286–296 (1978)

    Google Scholar 

  14. Lecoq N., Masmoudi K., Anthore R., Feullebois F.: Creeping motion of a sphere along the axis of a closed axisymmetric container. J. Fluid Mech. 585, 127–152 (2007)

    Article  MATH  Google Scholar 

  15. Malyuga V.S.: Viscous eddies in a circular cone. J. Fluid Mech. 522, 101–116 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hall O., Gilbert A.D., Hills C.P.: Converging flow between coaxial cones. Fluid Dyn. Res. 41, 011402 (2009)

    Article  MathSciNet  Google Scholar 

  17. Blinova I.V., Kyzyurova K.N., Popov I.Yu.: Nanocones rolling in hydro-thermal medium and flows in conical domains. J. Phys. Conf. Ser. 248, 012013/1-4 (2010)

    Article  Google Scholar 

  18. Blake J.R.: A note on the image system for a Stokeslet in a no-slip boundary. Proc. Camb. Philos. Soc. 70, 303–310 (1971)

    Article  MATH  Google Scholar 

  19. Usha R., Nigam S.D.: Flow in a spherical cavity due to Stokeslet. Fluid Dyn. Res. 11, 75–78 (1993)

    Article  Google Scholar 

  20. Liron, N., Mochon, S.: Stokes flow for a Stokeslet between two parallel flat plates. J. Eng. Math. 10, 287–303 (1976)

    Google Scholar 

  21. Liron N., Shahar R.: Stokes flow due to a Stokeslet in a pipe. J. Fluid Mech. 86(Part 4), 727–744 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  22. Blinova I.V.: Model of non-axisymmetric flow in nanotube. Nanosyst. Phys. Chem. Math. 4(3), 320–323 (2013)

    Google Scholar 

  23. Pozrikidis C.: Computation of periodic Green’s functions of Stokes flow. J. Eng. Math. 30, 79–96 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  24. Popov I.Yu.: Operator extensions theory and eddies in creeping flow. Phys. Scr. 47, 682–686 (1993)

    Article  Google Scholar 

  25. Popov I.Yu.: Stokeslet and the operator extensions theory. Rev. Mat. Univ. Compl. Madrid 9(1), 235–258 (1996)

    MATH  Google Scholar 

  26. Gugel Yu.V., Popov I.Yu., Popova S.L.: Hydrotron: creep and slip. Fluid Dyn. Res. 18, 199–210 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  27. Korn G.A., Korn T.M.: Mathematical Handbook for Scientists and Engineers. McGraw-Hill, New York (1968)

    Google Scholar 

  28. Batchelor G.K.: An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge (2000)

    Book  Google Scholar 

  29. Lebedev, N.N.: Special Functions and Their Applications, 2nd edn. (M.-L.: GIFML) (1963) (in Russian)

  30. Moffatt H.K.: Viscous eddies near a sharp corner. Arch. Mech. Stosow. 2, 365–372 (1964)

    MathSciNet  Google Scholar 

  31. Hackborn W.W.: Asymmetric Stokes flow between parallel planes due to a rotlet. J. Fluid Mech. 218, 531–546 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  32. Shankar P.N.: Moffatt eddies in the cone. J. Fluid Mech. 539, 113–135 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  33. Kononova S.V., Korytkova E.N., Romashkova K.A., Kuznetsov Yu.P., Gofman I.V., Svetlichnyi V.M., Gusarov V.V.: Nanocomposite on the basis of amide imide resin with hydrosilicate nanoparticles of different morphology. J. Appl. Chem. 80, 2064–2070 (2007)

    Google Scholar 

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Authors and Affiliations

  1. St. Petersburg National Research University of Information Technologies, Mechanics and Optics, 49 Kronverkskiy, 197101, St. Petersburg, Russia

    Irina V. Blinova & Igor Yu. Popov

  2. Department of Statistical Science, Duke University, Box 90251, Durham, NC, 27708-0251, USA

    Ksenia N. Kyz’yurova

Authors
  1. Irina V. Blinova
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  2. Ksenia N. Kyz’yurova
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  3. Igor Yu. Popov
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Corresponding author

Correspondence to Igor Yu. Popov.

Additional information

Grant 074-U01 of the Government of Russian Federation, State contract of the Russian Ministry of Education and Science, Grant of Russian Foundation for Basic Researches and Grants of the President of Russia (state contract 14.124.13.2045-MK and Grant MK-1493.2013.1).

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Blinova, I.V., Kyz’yurova, K.N. & Popov, I.Y. Stokes flow driven by a Stokeslet in a cone. Acta Mech 225, 3115–3121 (2014). https://doi.org/10.1007/s00707-014-1117-1

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  • DOI: https://doi.org/10.1007/s00707-014-1117-1

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