Advertisement

Computational Mathematics and Mathematical Physics

, Volume 58, Issue 11, pp 1804–1816 | Cite as

Mathematical Modelling of Flagellated Microswimmers

  • M. A. Zaitsev
  • S. A. Karabasov
Article
  • 21 Downloads

Abstract

The motion of a flagellated microorganism in a free space based on specifying the space-time shape of its centreline is studied. To solve the governing Stokes equations subject to non-slip boundary conditions on the microorganism body, a computational algorithm based on the finite element method is proposed. Results of computations on meshes of various density, domains of various sizes, and the solution of the benchmark problem of flow around the Stokes sphere used for verification are presented. Using the Lighthill–Gueron–Liron theory, a semi-analytical solution of the same problem of motion of a flagellated microorganism in which the corresponding coefficients of viscous drag are found by additional test computations is obtained. It is shown that the theory and the results of direct numerical simulation are in good agreement.

Keywords:

microswimmers Stokes equations finite element method simulation 

Notes

ACKNOWLEDGMENTS

The work by S.A. Karabasov was supported the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska-Curie project no. 703526.

We are grateful to Prof. V. Kantsler of Warwick University for the illustration of flagellated microswimmer motion shown in Fig. 1.

REFERENCES

  1. 1.
    J. S. Guasto, R. Rusconi, and R. Stocker, “Fluid mechanics of planktonic microorganisms,” Ann. Rev. Fluid Mech. 44, 373 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    A. P. Berke, L. Turner, H. C. Berg, and E. Lauga, “Hydrodynamic attraction of swimming microorganisms by surfaces,” Phys. Rev. Lett. 101, 038102 (2008).CrossRefGoogle Scholar
  3. 3.
    G. Li and J. X. Tang, “Accumulation of microswimmers near a surface mediated by collision and rotational brownian motion,” Phys. Rev. Lett. 103, 078101 (2009)CrossRefGoogle Scholar
  4. 4.
    P. Denissenko, V. Kantsler, D. J. Smith, and J. Kirkman-Brown, “Human spermatozoa migration in microchannels reveals boundary-following navigation,” Proc. Natl. Acad. Sci. U.S.A. 109, 8007 (2012).CrossRefGoogle Scholar
  5. 5.
    V. Kantsler, J. Dunkel, M. Polin, and R. E. Goldstein, “Ciliary contact interactions dominate surface scattering of swimming eukaryotes,” Proc. Natl. Acad. Sci. U.S.A. 110, 187 (2013).CrossRefGoogle Scholar
  6. 6.
    N. Liron, “The LGL (Lighthill–Gueron–Liron) theorem—historical perspective and critique,”. Math. Meth. App. Sci. 24, 17-18, (2001).MathSciNetzbMATHGoogle Scholar
  7. 7.
    L. D. Landau and E. M. Lifshitz, Fluid Mechanics, 3rd ed. (Nauka, Moscow, 1986; Pergamon Press, Oxford, 1987).Google Scholar
  8. 8.
    T. D. Montenegro-Johnson, H. Gadelha, and D. J. Smith, “Spermatozoa scattering by a microchannel feature: An elastohydrodynamic model,” Roy. Soc. Open Sci. 2, 140475–140475 (2015).CrossRefGoogle Scholar
  9. 9.
    A. Bukatin, I. Kukhtevich, N. Stoop, N. J. Dunkel, and V. Kantsler, Bimodal rheotactic behavior reflects flagellar beat asymmetry in human sperm cells. Proc. Natl. Acad. Sci. U.S.A. 112, 15904 (2015).CrossRefGoogle Scholar
  10. 10.
    B. Rodenborn, C-H. Chen, H. L. Swinney, B. Liu, and H. P. Zhang, “Propulsion of microorganisms by a helical flagellum,” Proc. Natl. Acad. Sci. U.S.A. 110, E338 (2013).CrossRefGoogle Scholar
  11. 11.
    D. R. Brumley, K. Y. Wan, M. Polin, and R. E. Goldstein, “Flagellar synchronization through direct hydrodynamic interactions,” eLife 3, e02750 (2014).CrossRefGoogle Scholar
  12. 12.
    F. Alouges, A. DeSimone, L. Giraldi, and M. Zoppello, “Self-propulsion of slender micro-swimmers by curvature control: N-link swimmers,” Int. J. Nonlin. Mech. 56, 132–141 (2013).CrossRefGoogle Scholar
  13. 13.
    F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods (Springer, New York, 1991).CrossRefzbMATHGoogle Scholar
  14. 14.
    T. J. R. Huges, The Finite Element Method (Prentice-Hall, New Jersy, 1987).Google Scholar
  15. 15.
    N. I. Drobyshevskii,"Modified quadrilateal finite element for solving two-dimensional problems of nonlinear deformation of structures," Izv. Akad. Nauk, Mekh. Tverd. Tela, No. 2, 152–162 (1996).Google Scholar
  16. 16.
    O. Zienkiewicz, The Finite Element Method in Engineering Science (Wiley, New York, 1971; Mir, Moscow, 1975).Google Scholar
  17. 17.
    S. Pissanetzky, Sparse Matrix Technology (Academic, London, 1984; Mir, Moscow, 1988).Google Scholar
  18. 18.
    Program Intel® Math Kernel Library. https://software.intel.com/en-us/articles/intel-math-kernel-library-documentation/Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Nuclear Safety Institute, Russian Academy of SciencesMoscowRussia
  2. 2.Queen Mary University of LondonMile EndLondonUK

Personalised recommendations