Mathematical Models and Computer Simulations

, Volume 10, Issue 2, pp 209–217 | Cite as

Boundary Element Modeling of Dynamics of a Bubble in Contact with a Solid Surface at Low Reynolds Numbers

  • Yu. A. Pityuk
  • O. A. Abramova
  • N. A. Gumerov
  • I. Sh. Akhatov


In the present study, the dynamics of a bubble attached to the surface and driven by the acoustic field at low Reynolds numbers are considered. The approach is based on the boundary element method (BEM) for Stokes flows, which is especially effective for the numerical solution of problems in the three-dimensional case. However, the dynamics of computing compressible bubbles are difficult to formulate due to the degeneration of the conventional BEM for Stokes equations. In the present approach, an additional relation based on the Lorenz reciprocity principle is used to resolve the problem. To describe the contact line dynamics a semiempirical law of motion is used. Such an approach allows us to bypass the known issue of nonintegrability stresses in the moving triple point. The behavior of a bubble attached to the surface in the cases of a pinned or moving contact line is studied. The developed method can be used for the detailed study of bubble dynamics in contact with a solid wall in order to determine the optimal conditions and parameters of surface cleaning processes.


bubble dynamics solid surface contact angle boundary element method Stokes flow 


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  1. 1.
    E. P. Prokopev, S. P. Timoshenkov, and V. V. Kalugin, “Technology of SOI structures,” Peterb. Zh. Elektron. 1, 8–25 (2000).Google Scholar
  2. 2.
    M. Kornfeld and L. Suvorov, “On the destructive action of cavitation,” J. Appl. Phys. 15, 495(1944).CrossRefGoogle Scholar
  3. 3.
    L. Crum, “Surface oscillations and jet development in pulsating bubbles,” J. Phys. 41, 285–288 (1979).Google Scholar
  4. 4.
    E. B. V. Dussan, “The moving contact line: the slip boundary condition,” J. Fluid Mech. 77, 665–684 (1976).CrossRefzbMATHGoogle Scholar
  5. 5.
    E. B. V. Dussan, “On the spreading of liquids on solid surfaces: static and dynamic contact lines,” Ann. Rev. Fluid Mech. 11, 371–400 (1979).CrossRefGoogle Scholar
  6. 6.
    L. M. Hocking, “The damping of capillary-gravity waves at a rigid boundary,” J. Fluid Mech. 179, 253–263 (1987).CrossRefzbMATHGoogle Scholar
  7. 7.
    L. M. Hocking, “Waves produced by a vertically oscillating plate,” J. Fluid Mech. 179, 267–281 (1987).CrossRefzbMATHGoogle Scholar
  8. 8.
    L. M. Hocking, “The spreading of drops with intermolecular forces,” Phys. Fluids 6, 3224–3228 (1994).CrossRefzbMATHGoogle Scholar
  9. 9.
    P. G. D. Gennes, “Wetting: statics and dynamics,” Rev. Mod. Phys. 57, 827–863 (1985).CrossRefGoogle Scholar
  10. 10.
    Y. D. Shikhmurzaev, “Moving contact lines in liquid/liquid/solid systems,” J. Fluid Mech. 334, 211–249 (1997).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    L. M. Hocking, “A moving fluid interface. The removal of the force singularity by a slip flow,” J. Fluid Mech. 79, 209–229 (1977).CrossRefzbMATHGoogle Scholar
  12. 12.
    C. Huh and S. G. Mason, “The steady movement of a liquid meniscus in a capillary tube,” J. Fluid Mech. 81, 401–419 (1977).CrossRefGoogle Scholar
  13. 13.
    H. P. Greenspan, “On the motion of a small viscous droplet that wets a surface,” J. Fluid Mech. 84, 125–143 (1978).CrossRefzbMATHGoogle Scholar
  14. 14.
    T. Young, “An essay on the cohesion of fluids,” Philos. Trans. R. Soc. London 95, 65–87 (1805).CrossRefGoogle Scholar
  15. 15.
    R. Mettin, P. E. Frommhold, X. Xi, F. Cegla, H. Okorn-Schmidt, A. Lippert, and F. Holsteyns, “Acoustic bubbles: control and interaction with particles adhered to a solid substrate,” in Proceedings of the 11th International Symposium on Ultra Clean Processing of Semiconductor Surfaces, Sept. 17–19, 2012, Gent, Belgium, Vol. 195 of Solid State Phenomena (Switzerland, 2013), pp. 161–164.Google Scholar
  16. 16.
    F. Prabowo and C.-D. Ohl, “Surface oscillation and jetting from surface attached acoustic driven bubbles,” Ultrason. Sonochem. 18, 431–435 (2011).CrossRefGoogle Scholar
  17. 17.
    S. Shklyaev and A. V. Straube, “Linear oscillations of a compressible hemispherical bubble on a solid substrate,” Phys. Fluids 20, 052102 (2008).CrossRefzbMATHGoogle Scholar
  18. 18.
    C. Pozrikidis, Boundary Integral and Singularity Methods for Linearized Viscous Flow (Cambridge Univ. Press, New York, 1992).CrossRefzbMATHGoogle Scholar
  19. 19.
    C. Pozrikidis, “Computation of the pressure inside bubbles and pores in Stokes flow,” J. Fluid Mech. 474, 319–337 (2003).MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Y. J. Liu, “A new fast multipole boundary element method for solving 2-D Stokes flow problems based on a dual BIE formulation,” Eng. Anal. Bound. Elem. 32, 139–151 (2008).CrossRefzbMATHGoogle Scholar
  21. 21.
    H. Power, “The interaction of a deformable bubble with a rigid wall at small Reynolds number: a general approach via integral equations,” Eng. Anal. Bound. Elem. 19, 291–297 (1997).CrossRefGoogle Scholar
  22. 22.
    Yu. A. Itkulova, O. A. Abramova, and N. A. Gumerov, “Boundary element simulations of compressible bubble dynamics in Stokes flow,” in Proceedings of the ASME 2013 International Mechanical Engineering Congress and Exposition (Am. Soc. Mech. Eng., 2013), No. IMECE2013-63200, p. V07BT08A010.Google Scholar
  23. 23.
    O. A. Abramova, I. S. Akhatov, and N. A. Gumerov, and Yu. A. Itkulova (Pityuk), “BEM-based numerical study of three-dimensional compressible bubble dynamics in a Stokes flow,” Comput. Math. Math. Phys. 54, 1481–1488 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    R. I. Nigmatulin, Dynamics of Multiphase Systems (Nauka, Moscow, 1987) [in Russian].Google Scholar
  25. 25.
    V. P. Zhitnikov and N. M. Sherykhalina, Modelling of Weighty Fluid Flow Using Multicomponent Analysis Methods (Nauch. Izdanie, Ufa, 2009) [in Russian].Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  • Yu. A. Pityuk
    • 1
  • O. A. Abramova
    • 1
  • N. A. Gumerov
    • 1
    • 2
  • I. Sh. Akhatov
    • 3
  1. 1.Center for Micro and Nanoscale Dynamics of Dispersed SystemsBashkir State UniversityUfaRussia
  2. 2.Institute of Advanced Computer StudyUniversity of MarylandCollege ParkUSA
  3. 3.Center for Design, Manufacturing & MaterialsSkolkovo Institute of Science & Engineering (Skoltech)MoscowRussia

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