The European Physical Journal Special Topics

, Volume 225, Issue 11–12, pp 2269–2285 | Cite as

Stationary shapes of deformable particles moving at low Reynolds numbers

  • Horst-Holger Boltz
  • Jan KierfeldEmail author
Regular Article Artificial Microswimmers
Part of the following topical collections:
  1. Microswimmers – From Single Particle Motion to Collective Behaviour


We introduce an iterative solution scheme in order to calculate stationary shapes of deformable elastic capsules which are steadily moving through a viscous fluid at low Reynolds numbers. The iterative solution scheme couples hydrodynamic boundary integral methods and elastic shape equations to find the stationary axisymmetric shape and the velocity of an elastic capsule moving in a viscous fluid governed by the Stokes equation. We use this approach to systematically study dynamical shape transitions of capsules with Hookean stretching and bending energies and spherical resting shape sedimenting under the influence of gravity or centrifugal forces. We find three types of possible axisymmetric stationary shapes for sedimenting capsules with fixed volume: a pseudospherical state, a pear-shaped state, and buckled shapes. Capsule shapes are controlled by two dimensionless parameters, the Föppl-von-Kármán number characterizing the elastic properties and a Bond number characterizing the driving force. For increasing gravitational force the spherical shape transforms into a pear shape. For very large bending rigidity (very small Föppl-von-Kármán number) this transition is discontinuous with shape hysteresis. The corresponding transition line terminates, however, in a critical point, such that the discontinuous transition is not present at typical Föppl-von-Kármán numbers of synthetic capsules. In an additional bifurcation, buckled shapes occur upon increasing the gravitational force.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    D. Barthès-Biesel, Curr. Opin. Colloid Interface Sci. 16, 3 (2011)CrossRefGoogle Scholar
  2. 2.
    D.A. Fedosov, H. Noguchi, G. Gompper, Biomech. Model. Mechanobiol. 13, 239 (2014)CrossRefGoogle Scholar
  3. 3.
    H. Stone, Annu. Rev. Fluid Mech. 46, 67 (2014)CrossRefGoogle Scholar
  4. 4.
    D. Abreu, M. Levant, V. Steinberg, U. Seifert, Adv. Colloid Interface Sci. 208, 129 (2014)CrossRefGoogle Scholar
  5. 5.
    Z. Huang, M. Abkarian, A. Viallat, New J. Phys. 13, 035026 (2011)ADSCrossRefGoogle Scholar
  6. 6.
    R. Clift, J. Grace, M. Weber, Bubbles, Drops, and Particles (Academic Press, New York, 1978)Google Scholar
  7. 7.
    H.A. Stone, Annu. Rev. Fluid Mech. 26, 65 (1994)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    G. Coupier, A. Farutin, C. Minetti, T. Podgorski, C. Misbhah, Phys. Rev. Lett. 108, 178106 (2012)ADSCrossRefGoogle Scholar
  9. 9.
    Y. Lefebvre, E. Leclerc, D. Barthès-Biesel, J. Walter, F. Edwards-Lévy, Phys. Fluids 20, 123102 (2008)ADSCrossRefGoogle Scholar
  10. 10.
    A. Mietke, O. Otto, S. Girardo, P. Rosendahl, A. Taubenberger, S. Golfier, E. Ulbricht, S. Aland, J. Guck, E. Fischer-Friedrich, Biophys. J. 109, 2023 (2015)ADSCrossRefGoogle Scholar
  11. 11.
    V.F. Geyer, F. Jülicher, J. Howard, B.M. Friedrich, Proc. Natl. Acad. Sci. U.S.A. 110, 18058 (2013)ADSCrossRefGoogle Scholar
  12. 12.
    H.-H. Boltz, J. Kierfeld, Phys. Rev. E 92, 033003 (2015)ADSCrossRefGoogle Scholar
  13. 13.
    H.-H. Boltz, J. Kierfeld, Phys. Rev. E 92, 069904 (2015)ADSCrossRefGoogle Scholar
  14. 14.
    U. Seifert, Adv. Phys. 46, 13 (1997)ADSCrossRefGoogle Scholar
  15. 15.
    G. Boedec, M. Leonetti, M. Jaeger, J. Comput. Phys. 230, 1020 (2011)ADSCrossRefGoogle Scholar
  16. 16.
    G. Boedec, M. Jaeger, M. Leonetti, Phys. Rev. E 88, 010702 (2013)ADSCrossRefGoogle Scholar
  17. 17.
    I. Rey Suàrez, C. Leidy, G. Tèllez, G. Gay, A. Gonzalez-Mancera, PLoS ONE 8, e68309 (2013)ADSCrossRefGoogle Scholar
  18. 18.
    W.D. Corry, H.J. Meiselman, Biophys. J. 21, 19 (1978)CrossRefGoogle Scholar
  19. 19.
    J.F. Hoffman, S. Inoué, Proc. Natl. Acad. Sci. USA 103, 2971 (2006)ADSCrossRefGoogle Scholar
  20. 20.
    M. Peltomäki, G. Gompper, Soft Matter 9, 8346 (2013)ADSCrossRefGoogle Scholar
  21. 21.
    G.K. Batchelor, An introduction to fluid dynamics (Cambridge University Press, Cambridge, 2000)Google Scholar
  22. 22.
    A. Libai, J. Simmonds, The Nonlinear Theory of Elastic Shells (Cambridge University Press, Cambridge 1998)Google Scholar
  23. 23.
    C. Pozrikidis, Modeling and Simulation of Capsules and Biological Cells (CRC Press, Boca Raton, 2003)Google Scholar
  24. 24.
    S. Knoche, D. Vella, E. Aumaitre, P. Degen, H. Rehage, P. Cicuta, J. Kierfeld, Langmuir 29, 12463 (2013)CrossRefGoogle Scholar
  25. 25.
    S. Knoche, J. Kierfeld, Phys. Rev. E 84, 046608 (2011)ADSCrossRefGoogle Scholar
  26. 26.
    L. Landau, E. Lifshitz, Theory of Elasticity (Pergamon, New York, 1986)Google Scholar
  27. 27.
    J. Happel, H. Brenner, Low Reynolds Number Hydrodynamics: With Special Applications to Particulate Media (Martinus Nijhoff Publishers, The Hague, 1983)Google Scholar
  28. 28.
    C. Pozrikidis, Boundary Integral and Singularity Methods for Linearized Viscous Flow (Cambridge University Press, Cambridge, 1992)Google Scholar
  29. 29.
    H.P. Langtangen, K.-A. Mardal, R. Winther, Adv. Water Resour. 25, 1125 (2002)ADSCrossRefGoogle Scholar
  30. 30.
    E. Lauga, T.R. Powers, Rep. Prog. Phys. 72, 096601 (2009)ADSCrossRefGoogle Scholar
  31. 31.
    P. Degen, Curr. Opin. Colloid Interface Sci. 19, 611 (2014)CrossRefGoogle Scholar
  32. 32.
    M. Lighthill, Comm. Pure Appl. Math. 5, 109 (1952)MathSciNetCrossRefGoogle Scholar
  33. 33.
    J. Blake, J. Fluid Mech. 46, 199 (1971)ADSCrossRefGoogle Scholar

Copyright information

© EDP Sciences and Springer 2016

Authors and Affiliations

  1. 1.Physics Department, TU Dortmund UniversityDortmundGermany

Personalised recommendations