Skip to main content
SpringerLink
Log in

Creeping motion of a solid particle inside a spherical elastic cavity

  • Regular Article
  • Published:
The European Physical Journal E Aims and scope Submit manuscript

Abstract.

On the basis of the linear hydrodynamic equations, we present an analytical theory for the low-Reynolds-number motion of a solid particle moving inside a larger spherical elastic cavity which can be seen as a model system for a fluid vesicle. In the particular situation where the particle is concentric with the cavity, we use the stream function technique to find exact analytical solutions of the fluid motion equations on both sides of the elastic cavity. In this particular situation, we find that the solution of the hydrodynamic equations is solely determined by membrane shear properties and that bending does not play a role. For an arbitrary position of the solid particle within the spherical cavity, we employ the image solution technique to compute the axisymmetric flow field induced by a point force (Stokeslet). We then obtain analytical expressions of the leading-order mobility function describing the fluid-mediated hydrodynamic interactions between the particle and the confining elastic cavity. In the quasi-steady limit of vanishing frequency, we find that the particle self-mobility function is higher than that predicted inside a rigid no-slip cavity. Considering the cavity motion, we find that the pair-mobility function is determined only by membrane shear properties. Our analytical predictions are supplemented and validated by fully resolved boundary integral simulations where a very good agreement is obtained over the whole range of applied forcing frequencies.

Graphical abstract

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. R.B. Bird, W.E. Stewart, E.N. Lightfoot, Transport Phenomena (John Wiley & Sons, New York, 2007)

  2. R.B. Schoch, J. Han, P. Renaud, Rev. Mod. Phys. 80, 839 (2008)

    Article  ADS  Google Scholar 

  3. D. Chowdhury, A. Schadschneider, K. Nishinari, Phys. Life Rev. 2, 318 (2005)

    Article  ADS  Google Scholar 

  4. J. Panyam, V. Labhasetwar, Adv. Drug Deliv. Rev. 55, 329 (2003)

    Article  Google Scholar 

  5. L.M. Bareford, P.W. Swaan, Adv. Drug Deliv. Rev. 59, 748 (2007)

    Article  Google Scholar 

  6. J. Bereiter-Hahn, M. Vöth, Microsc. Res. Tech. 27, 198 (1994)

    Article  Google Scholar 

  7. B. ten Hagen, S. van Teeffelen, H. Löwen, J. Phys.: Condens. Matter 23, 194119 (2011)

    ADS  Google Scholar 

  8. W. Wang, S. Li, L. Mair, S. Ahmed, T. Jun, Angew. Chem. Int. Ed. 53, 3201 (2014)

    Article  Google Scholar 

  9. B. Liebchen, M.E. Cates, D. Marenduzzo, Soft Matter 12, 7259 (2016)

    Article  ADS  Google Scholar 

  10. C. Bechinger, R. Di Leonardo, H. Löwen, C. Reichhardt, G. Volpe, G. Volpe, Rev. Mod. Phys. 88, 045006 (2016)

    Article  ADS  Google Scholar 

  11. A.M. Menzel, Phys. Rep. 554, 1 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  12. F. Rühle, J. Blaschke, J.-T. Kuhr, H. Stark, New J. Phys. 20, 025003 (2018)

    Article  ADS  Google Scholar 

  13. S. Kim, S.J. Karrila, Microhydrodynamics: Principles and Selected Applications (Courier Corporation, Mineola, 2013)

  14. L.G. Leal, Annu. Rev. Fluid Mech. 12, 435 (1980)

    Article  ADS  Google Scholar 

  15. I.F. Sbalzarini, P. Koumoutsakos, J. Struct. Biol. 151, 182 (2005)

    Article  Google Scholar 

  16. N. Gal, D. Lechtman-Goldstein, D. Weihs, Rheol. Acta 52, 425 (2013)

    Article  Google Scholar 

  17. Y. Li, J. Schnekenburger, M.H.G. Duits, J. Biomed. Opt. 14, 064005 (2009)

    Article  ADS  Google Scholar 

  18. D. Ott, P.M. Bendix, L.B. Oddershede, ACS Nano 7, 8333 (2013)

    Article  Google Scholar 

  19. É. Fodor, M. Guo, N.S. Gov, P. Visco, D.A. Weitz, F. van Wijland, EPL 110, 48005 (2015)

    Article  ADS  Google Scholar 

  20. T.J. Lampo, S. Stylianidou, M.P. Backlund, P.A. Wiggins, A.J. Spakowitz, Biophys. J. 112, 532 (2017)

    Article  ADS  Google Scholar 

  21. S. Yamada, D. Wirtz, S.C. Kuo, Biophys. J. 78, 1736 (2000)

    Article  Google Scholar 

  22. D.T. Chen, E.R. Weeks, J.C. Crocker, M.F. Islam, R. Verma, J. Gruber, A.J. Levine, T.C. Lubensky, A.G. Yodh, Phys. Rev. Lett. 90, 108301 (2003)

    Article  ADS  Google Scholar 

  23. A. El Kaffas, D. Bekah, M. Rui, J.C. Kumaradas, M.C. Kolios, Phys. Med. Biol. 58, 923 (2013)

    Article  Google Scholar 

  24. J. Happel, H. Brenner, Low Reynolds Number Hydrodynamics: With Special Applications to Particulate Media, Vol. 1 (Springer Science & Business Media, The Netherlands, 2012)

  25. C.W. Oseen, Neuere Methoden und Ergebnisse in der Hydrodynamik (Leipzig Akademische verlagsgesellschaft m.b.h., Leipzig, Germany, 1928)

  26. S.F.J. Butler, A note on Stokes's stream function for motion with a spherical boundary, in Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 49 (Cambridge University Press, 1953) pp. 169--174

  27. W.D. Collins, Mathematika 1, 125 (1954)

    Article  MathSciNet  Google Scholar 

  28. W.D. Collins, Mathematika 5, 118 (1958)

    Article  MathSciNet  Google Scholar 

  29. H. Hasimoto, J. Phys. Soc. Jpn. 11, 793 (1956)

    Article  ADS  Google Scholar 

  30. H. Hasimoto, J. Phys. Soc. Jpn. 61, 3027 (1992)

    Article  ADS  Google Scholar 

  31. H. Hasimoto, Phys. Fluids 9, 1838 (1997)

    Article  ADS  MathSciNet  Google Scholar 

  32. R. Shail, Q. J. Mech. App. Math. 40, 223 (1987)

    Article  MathSciNet  Google Scholar 

  33. R. Shail, S.H. Onslow, Mathematika 35, 233 (1988)

    Article  MathSciNet  Google Scholar 

  34. A. Sellier, Comput. Model. Eng. Sci. 25, 165 (2008)

    Google Scholar 

  35. C. Maul, S. Kim, Phys. Fluids 6, 2221 (1994)

    Article  ADS  Google Scholar 

  36. C. Maul, S. Kim, Image of a point force in a spherical container and its connection to the Lorentz reflection formula, in The Centenary of a Paper on Slow Viscous Flow by the Physicist HA Lorentz (Springer, 1996) pp. 119--130

  37. C. Pozrikidis, J. Comput. Phys. 169, 250 (2001)

    Article  ADS  Google Scholar 

  38. Y.O. Fuentes, S. Kim, D.J. Jeffrey, Phys. Fluids 31, 2445 (1988)

    Article  ADS  Google Scholar 

  39. Y.O. Fuentes, S. Kim, D.J. Jeffrey, Phys. Fluids 1, 61 (1989)

    Article  ADS  Google Scholar 

  40. B.U. Felderhof, A. Sellier, J. Chem. Phys. 136, 054703 (2012)

    Article  ADS  Google Scholar 

  41. C. Aponte-Rivera, R.N. Zia, Phys. Rev. Fluids 1, 023301 (2016)

    Article  ADS  Google Scholar 

  42. C. Aponte-Rivera, Y. Su, R.N. Zia, J. Fluid Mech. 836, 413 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  43. D. Tsemakh, O.M. Lavrenteva, A. Nir, Int. J. Multiphase Flow 30, 1337 (2004)

    Article  Google Scholar 

  44. S.Y. Reigh, L. Zhu, F. Gallaire, E. Lauga, Soft Matter 13, 3161 (2017)

    Article  ADS  Google Scholar 

  45. V.A. Shaik, V. Vasani, A.M. Ardekani, J. Fluid Mech. 851, 187 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  46. Y. Zhan, J. Wang, N. Bao, C. Lu, Anal. Chim. 81, 2027 (2009)

    Article  Google Scholar 

  47. L. Zhu, F. Gallaire, Phys. Rev. Lett. 119, 064502 (2017)

    Article  ADS  Google Scholar 

  48. A. Daddi-Moussa-Ider, A. Guckenberger, S. Gekle, Phys. Fluids 28, 071903 (2016)

    Article  ADS  Google Scholar 

  49. A. Daddi-Moussa-Ider, S. Gekle, J. Chem. Phys. 145, 014905 (2016)

    Article  ADS  Google Scholar 

  50. B. Rallabandi, B. Saintyves, T. Jules, T. Salez, C. Schönecker, L. Mahadevan, H.A. Stone, Phys. Rev. Fluids 2, 074102 (2017)

    Article  ADS  Google Scholar 

  51. A. Daddi-Moussa-Ider, S. Gekle, Eur. Phys. J. E 41, 19 (2018)

    Article  Google Scholar 

  52. A. Daddi-Moussa-Ider, M. Lisicki, S. Gekle, A.M. Menzel, H. Löwen, J. Chem. Phys. 149, 014901 (2018)

    Article  ADS  Google Scholar 

  53. R. Skalak, A. Tozeren, R.P. Zarda, S. Chien, Biophys. J. 13, 245 (1973)

    Article  ADS  Google Scholar 

  54. T.W. Secomb, Annu. Rev. Fluid Mech. 49, 443 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  55. W. Helfrich, Z. Naturfocsch. C 28, 693 (1973)

    Article  Google Scholar 

  56. S. Timoshenko, S. Woinowsky-Krieger, S. Woinowsky-Krieger, Theory of Plates and Shells, Vol. 2 (McGraw-hill New York, 1959)

  57. H. Zhou, C. Pozrikidis, J. Fluid Mech. 283, 175 (1995)

    Article  ADS  Google Scholar 

  58. J.K.G. Dhont, An Introduction to Dynamics of Colloids (Elsevier, 1996)

  59. T. Krüger, F. Varnik, D. Raabe, Comput. Math. Appl. 61, 3485 (2011)

    Article  MathSciNet  Google Scholar 

  60. T. Krüger, Computer Simulation Study of Collective Phenomena in Dense Suspensions of Red Blood Cells Under Shear (Springer Science & Business Media, 2012)

  61. D. Barthès-Biesel, Annu. Rev. Fluid Mech. 48, 25 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  62. S. Gekle, Biophys. J. 110, 514 (2016)

    Article  ADS  Google Scholar 

  63. A. Guckenberger, A. Kihm, T. John, C. Wagner, S. Gekle, Soft Matter 14, 2032 (2018)

    Article  ADS  Google Scholar 

  64. T. Bickel, Eur. Phys. J. E 20, 379 (2006)

    Article  Google Scholar 

  65. A. Guckenberger, M.P. Schraml, P.G. Chen, M. Leonetti, S. Gekle, Comput. Phys. Comm. 207, 1 (2016)

    Article  ADS  Google Scholar 

  66. C. Pozrikidis, J. Fluid Mech. 440, 269 (2001)

    Article  ADS  Google Scholar 

  67. A. Guckenberger, S. Gekle, J. Phys.: Condens. Matt. 29, 203001 (2017)

    ADS  Google Scholar 

  68. A. Daddi-Moussa-Ider, S. Gekle, Phys. Rev. E 95, 013108 (2017)

    Article  ADS  Google Scholar 

  69. A. Daddi-Moussa-Ider, Diffusion of nanoparticles nearby elastic cell membranes: A theoretical study, PhD Thesis, University of Bayreuth, Germany (2017)

  70. S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, Vol. 1 (Interscience Publishers, New York, 1963)

  71. M. Deserno, Chem. Phys. Lipids 185, 11 (2015)

    Article  Google Scholar 

  72. M. Stimson, G.B. Jeffery, Proc. R. Soc. London, Ser. A 111, 110 (1926)

    Article  ADS  Google Scholar 

  73. A. Daddi-Moussa-Ider, A. Guckenberger, S. Gekle, Phys. Rev. E 93, 012612 (2016)

    Article  ADS  Google Scholar 

  74. B.U. Felderhof, J. Chem. Phys. 125, 124904 (2006)

    Article  ADS  Google Scholar 

  75. A. Daddi-Moussa-Ider, M. Lisicki, S. Gekle, Phys. Rev. E 95, 053117 (2017)

    Article  ADS  Google Scholar 

  76. V.A. Shaik, A.M. Ardekani, Phys. Rev. Fluids 2, 113606 (2017)

    Article  ADS  Google Scholar 

  77. H. Lamb, Hydrodynamics (Cambridge University Press, 1932)

  78. R.G. Cox, J. Fluid Mech. 37, 601 (1969)

    Article  ADS  Google Scholar 

  79. M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, Vol. 1 (New York, Dover, 1972)

  80. C. Misbah, Phys. Rev. Lett. 96, 028104 (2006)

    Article  ADS  Google Scholar 

  81. Jonathan B. Freund, Annu. Rev. Fluid Mech. 46, 67 (2014)

    Article  ADS  Google Scholar 

  82. L. Zhu, Simulation of individual cells in flow, PhD Thesis, KTH Royal Institute of Technology (2014)

  83. H. Noguchi, G. Gompper, Proc. Natl. Acad. Sci. U.S.A. 102, 14159 (2005)

    Article  ADS  Google Scholar 

  84. B. Kaoui, T. Krüger, J. Harting, Soft Matter 8, 9246 (2012)

    Article  ADS  Google Scholar 

  85. B. Kaoui, J. Harting, Rheol. Acta 55, 465 (2016)

    Article  Google Scholar 

  86. A. Nait-Ouhra, A. Farutin, O. Aouane, H. Ez-Zahraouy, A. Benyoussef, C. Misbah, Phys. Rev. E 97, 012404 (2018)

    Article  ADS  Google Scholar 

  87. H.A. Lorentz, Abh. Theor. Phys. 1, 23 (1907)

    Google Scholar 

  88. S.H. Lee, R.S. Chadwick, L.G. Leal, J. Fluid Mech. 93, 705 (1979)

    Article  ADS  Google Scholar 

  89. S.H. Lee, L.G. Leal, J. Fluid Mech. 98, 193 (1980)

    Article  ADS  Google Scholar 

  90. T. Bickel, Phys. Rev. E 75, 041403 (2007)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, Universitätsstraße 1, 40225, Düsseldorf, Germany

    Abdallah Daddi-Moussa-Ider & Hartmut Löwen

  2. Biofluid Simulation and Modeling, Theoretische Physik VI, Universität Bayreuth, Universitätsstraße 30, 95440, Bayreuth, Germany

    Stephan Gekle

Authors
  1. Abdallah Daddi-Moussa-Ider
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hartmut Löwen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Stephan Gekle
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Abdallah Daddi-Moussa-Ider.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Daddi-Moussa-Ider, A., Löwen, H. & Gekle, S. Creeping motion of a solid particle inside a spherical elastic cavity. Eur. Phys. J. E 41, 104 (2018). https://doi.org/10.1140/epje/i2018-11715-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epje/i2018-11715-7

Keywords

Search

Navigation