Abstract
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.
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References
E. P. Prokopev, S. P. Timoshenkov, and V. V. Kalugin, “Technology of SOI structures,” Peterb. Zh. Elektron. 1, 8–25 (2000).
M. Kornfeld and L. Suvorov, “On the destructive action of cavitation,” J. Appl. Phys. 15, 495(1944).
L. Crum, “Surface oscillations and jet development in pulsating bubbles,” J. Phys. 41, 285–288 (1979).
E. B. V. Dussan, “The moving contact line: the slip boundary condition,” J. Fluid Mech. 77, 665–684 (1976).
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).
L. M. Hocking, “The damping of capillary-gravity waves at a rigid boundary,” J. Fluid Mech. 179, 253–263 (1987).
L. M. Hocking, “Waves produced by a vertically oscillating plate,” J. Fluid Mech. 179, 267–281 (1987).
L. M. Hocking, “The spreading of drops with intermolecular forces,” Phys. Fluids 6, 3224–3228 (1994).
P. G. D. Gennes, “Wetting: statics and dynamics,” Rev. Mod. Phys. 57, 827–863 (1985).
Y. D. Shikhmurzaev, “Moving contact lines in liquid/liquid/solid systems,” J. Fluid Mech. 334, 211–249 (1997).
L. M. Hocking, “A moving fluid interface. The removal of the force singularity by a slip flow,” J. Fluid Mech. 79, 209–229 (1977).
C. Huh and S. G. Mason, “The steady movement of a liquid meniscus in a capillary tube,” J. Fluid Mech. 81, 401–419 (1977).
H. P. Greenspan, “On the motion of a small viscous droplet that wets a surface,” J. Fluid Mech. 84, 125–143 (1978).
T. Young, “An essay on the cohesion of fluids,” Philos. Trans. R. Soc. London 95, 65–87 (1805).
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.
F. Prabowo and C.-D. Ohl, “Surface oscillation and jetting from surface attached acoustic driven bubbles,” Ultrason. Sonochem. 18, 431–435 (2011).
S. Shklyaev and A. V. Straube, “Linear oscillations of a compressible hemispherical bubble on a solid substrate,” Phys. Fluids 20, 052102 (2008).
C. Pozrikidis, Boundary Integral and Singularity Methods for Linearized Viscous Flow (Cambridge Univ. Press, New York, 1992).
C. Pozrikidis, “Computation of the pressure inside bubbles and pores in Stokes flow,” J. Fluid Mech. 474, 319–337 (2003).
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).
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).
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.
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).
R. I. Nigmatulin, Dynamics of Multiphase Systems (Nauka, Moscow, 1987) [in Russian].
V. P. Zhitnikov and N. M. Sherykhalina, Modelling of Weighty Fluid Flow Using Multicomponent Analysis Methods (Nauch. Izdanie, Ufa, 2009) [in Russian].
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Original Russian Text © Yu.A. Pityuk, O.A. Abramova, N.A. Gumerov, I.Sh. Akhatov, 2017, published in Matematicheskoe Modelirovanie, 2017, Vol. 29, No. 9, pp. 77–89.
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Pityuk, Y.A., Abramova, O.A., Gumerov, N.A. et al. Boundary Element Modeling of Dynamics of a Bubble in Contact with a Solid Surface at Low Reynolds Numbers. Math Models Comput Simul 10, 209–217 (2018). https://doi.org/10.1134/S2070048218020102
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DOI: https://doi.org/10.1134/S2070048218020102