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Viscous effects on the oscillations of two equal and deformable bubbles under a step change in pressure



According to linear theory and assuming the liquids to be inviscid and the bubbles to remain spherical, bubbles set in oscillation attract or repel each other with a force that is proportional to the product of their amplitude of volume pulsations and inversely proportional to the square of their distance apart. This force is attractive, if the forcing frequency lies outside the range of eigenfrequencies for volume oscillation of the two bubbles. Here we study the nonlinear interaction of two deformable bubbles set in oscillation in water by a step change in the ambient pressure, by solving the Navier–Stokes equations numerically. As in typical experiments, the bubble radii are in the range 1–1000 μm. We find that the smaller bubbles (~5 μm) deform only slightly, especially when they are close to each other initially. Increasing the bubble size decreases the capillary force and increases bubble acceleration towards each other, leading to oblate or spherical cap or even globally deformed shapes. These deformations may develop primarily in the rear side of the bubbles because of a combination of their translation and harmonic or subharmonic resonance between the breathing mode and the surface harmonics. Bubble deformation is also promoted when they are further apart or when the disturbance amplitude decreases. The attractive force depends on the Ohnesorge number and the ambient pressure to capillary forces ratio, linearly on the radius of each bubble and inversely on the square of their separation. Additional damping either because of liquid compressibility or heat transfer in the bubble is also examined.

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Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Batchelor, G. K. 1987 The stability of a large gas bubble rising through liquid. J. Fluid Mech. 184, 399422.
Bjerknes, V. F. K. 1906 Fields of Force. Columbia University Press.
Bjerknes, V. F. K. 1909 Die Craftfelder. Vieweg.
Blake, J. R., Taib, B. B. & Doherty, G. 1986 Transient cavities near boundaries. Part 1. Rigid boundary. J. Fluid Mech. 170, 479497.
Brennen, C. E. 1995 Cavitation and Bubble Dynamics. Oxford University Press.
Brenner, M. P., Hilgenfeldt, S. & Lohse, D. 2002 Single-bubble sonoluminescence. Rev. Mod. Phys. 74, 425484.
Chapman, R. B. & Plesset, M. S. 1971 Thermal effects in the free oscillation of gas bubble. J. Basic Engng 93, 373376.
Chatzidai, N., Giannousakis, A., Dimakopoulos, Y. & Tsamopoulos, J. 2009 On the elliptic mesh generation in domains containing multiple inclusions and undergoing large deformations. J. Comput. Phys. 228, 19802011.
Chatzidai, N. K. 2008 Motion, deformation and interaction of bubbles due to gravity and/or variation of the pressure of the ambient fluid. PhD thesis, University of Patras.
Chen, L., Li, Y. & Manasseh, R. 1998 The coalescence of bubbles: a numerical study. In Third International Conference on Multiphase Flow, ICMF'98, Lyon, France.
Crum, L. A. 1975 Bjerknes forces on bubbles in a stationary sound field. J. Acoust. Soc. Am. 57, 13631370.
Daglia, R. & Poulain, C. 2010 When sound slows down bubbles. Phys. Fluids 22, 041703.
Davies, R. M. & Taylor, G. I. 1950 The mechanics of large bubbles rising through extended liquids and through liquids in tubes. Proc. R. Soc. Lond. A 200, 275390.
Dimakopoulos, Y. & Tsamopoulos, J. 2003 a A quasi-elliptic transformation for moving boundary problems with large anisotropic deformations. J. Comput. Phys. 192, 494522.
Dimakopoulos, Y. & Tsamopoulos, J. A. 2003 b Transient displacement of a Newtonian fluid by air in straight and suddenly constricted tubes. Phys. Fluids 15 (7), 19731991.
Doinikov, A. A. 1999 Bjerknes forces between two bubbles in a viscous fluid. J. Acoust. Soc. Am. 106, 33053312.
Doinikov, A. A. & Zavtrak, S. T. 1995 On the mutual interaction of two gas bubbles in a sound field. Phys. Fluids 7 (8), 19231930.
Feng, Z. & Leal, G. 1997 Nonlinear bubble dynamics. Annu. Rev. Fluid Mech. 29, 201243.
Foteinopoulou, K., Mavrantzas, V., Dimakopoulos, Y. & Tsamopoulos, J. A. 2006 Numerical simulation of multiple bubbles growing in a Newtonian liquid filament undergoing stretching. Phys. Fluids 18 (4), 042106 (124).
Fujikawa, S. & Akamatsu, T. 1980 Effects of the non-equilibrium condensation of vapour on the pressure wave produced by the collapse of a bubble in a liquid. J. Fluid Mech. 97, 481512.
Gaidamour, J. & Hènon, P. 2008 A parallel direct/iterative solver based on a Schur complement approach. In 11th IEEE Intl Conf. Computational Science and Engineering, Sao Paulo, Brazil, pp. 98105.
Goldberg, B. B., Raichlen, J. S. & Forsberg, F. 2001 Ultrasound Contrast Agents: Basic Principles and Clinical Applications. Dunitz Martin.
Hall, P. & Seminara, G. 1980 Nonlinear oscillations of non-spherical cavitation bubbles in acoustic fields. J. Fluid Mech. 101, 423444.
Hènon, P. & Saad, Y. 2006 A parallel multistage ILU factorization based on a hierarchical graph decomposition. SIAM J. Sci. Comput. 28 (6), 22662293.
Johnsen, E. & Colonius, T. 2009 Numerical simulations for non-spherical bubble collapse. J. Fluid Mech. 629, 231262.
Karapetsas, G. & Tsamopoulos, J. 2006 Transient squeeze flow of viscoplastic materials. J. Non-Newtonian Fluid Mech. 133, 3556.
Keller, J. B. & Miksis, M. 1980 Bubble oscillations of large amplitude. J. Acoust. Soc. Am. 68, 628633.
Kornfeld, M. & Suvorov, L. 1944 On the destructive action of cavitation. J. Appl. Phys. 15, 495506.
Lauterborn, W. 1972 High-speed photography of laser-induced cavities in liquids. In 10th Intl Congress on High Speed Cinematography, Nice, pp. 306–309.
Lauterborn, W. & Bolle, H. 1975 Experimental investigations of cavitation-bubble collapse in the neighbourhood of a solid boundary. J. Fluid Mech. 72, 391399.
Lauterborn, W., Kurz, T., Geisler, R., Schanz, D. & Lindau, O. 2007 Acoustic cavitation, bubble dynamics and sonoluminescence. Ultrason. Sonochem. 14, 484491.
Lauterborn, W., Kurz, T., Mettin, R. & Ohl, C. 1999 Experimental and theoretical bubble dynamics. In Advances in Chemical Physics (ed. Prigogine, I. & Rice, S. A.) vol. 110, Ch. 5, pp. 295380. Wiley
Lehoucq, R. B., Sorensen, D. C. & Young, C. 1998 ARPACK User's Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods. SIAM.
Leighton, T. G. 1994 The Acoustic Bubble. Academic Press.
Li, T., Tachibana, K., Kuroki, M. & Kuroki, M. 2003 Gene transfer with echo-enhanced contrast agents: comparison between Albunex, Optison and Levovist in mice: initial results. Radiology 229, 423428.
Mettin, R., Akhatov, I., Parlitz, U., Ohl, C. D. & Lauterborn, W. 1997 Bjerknes forces between small cavitation bubbles in a strong acoustic field. Phys. Rev. E 56, 29242931.
Miller, C. A. & Scriven, L. E. 1968 The oscillations of a fluid droplet immersed in another fluid. J. Fluid Mech. 32, 417435.
Nayfeh, A. H. & Mook, D. T. 1979 Nonlinear Oscillations. John Willey.
Oguz, H. & Prosperetti, A. 1990 A generalization of the impulse and virial theorems with an application to bubble oscillations. J. Fluid Mech. 218, 143162.
Ohl, C., Kurz, T., Geisler, R., Lindau, O. & Lauterborn, W. 1999 Bubble dynamics, shock waves and sonoluminescence. Proc. R. Soc. Lond. A 357, 269294.
Papanastasiou, T. C., Malamataris, N. & Elwood, K. 1992 A new outflow boundary condition. Intl J. Numer. Meth. Fluids 14, 587608.
Pelekasis, N. A. 1991 A study on drop and bubble dynamics via a hybrid boundary element-finite element methodology. PhD thesis, State University of New York at Buffalo.
Pelekasis, N. A., Gaki, A., Doinikov, A. & Tsamopoulos, J. A. 2004 Secondary Bjerknes forces between two bubbles and the phenomenon of acoustic streamers. J. Fluid Mech. 500, 313347.
Pelekasis, N. A. & Tsamopoulos, J. A. 1993 a Bjerknes forces between two bubbles. Part 1. Response to a step change in pressure. J. Fluid Mech. 254, 467499.
Pelekasis, N. A. & Tsamopoulos, J. A. 1993 b Bjerknes forces between two bubbles. Part 2. Response to an oscillatory pressure field. J. Fluid Mech. 254, 501527.
Plesset, M. 1954 On the stability of fluid flows with spherical symmetry. J. Appl. Phys. 25, 9698.
Plesset, M. S. & Chapman, R. B. 1971 Collapse of an initially spherical vapour cavity in the neighbourhood of a solid boundary. J. Fluid Mech. 47 (2), 283290.
Plesset, M. S. & Prosperetti, A. 1977 Bubble dynamics and cavitation. Annu. Rev. Fluid Mech. 9, 145185.
Popinet, St. & Zaleski, St. 2002 Bubble collapse near a solid boundary: a numerical study of the influence of viscosity. J. Fluid Mech. 464, 137163.
Prosperetti, A. & Lezzi, A. 1986 Bubble dynamics in a compressible liquid. Part 1. First-order theory. J. Fluid Mech. 168, 457478.
Rayleigh, Lord 1917 On the pressure developed in a liquid during the collapse of a spherical cavity. Phil. Mag. 34, 9498.
Toilliez, J. O. & Szeri, A. J. 2008 Optimized translation of microbubbles driven by acoustic fields. J. Acoust. Soc. Am. 123, 19161930.
Tomita, Y. & Shima, A. 1986 Mechanisms of impulse pressure generation and damage pit formation by bubble collapse. J. Fluid Mech. 169, 535564.
Tsamopoulos, J. & Brown, R. A. 1983 Non-linear oscillations of inviscid drops and bubbles. J. Fluid Mech. 127, 519537.
Tsamopoulos, J., Dimakopoulos, Y., Chatzidai, N., Karapetsas, G. & Pavlidis, M. 2008 Steady bubble rise and deformation in Newtonian and Viscoplastic fluids and conditions for their entrapment. J. Fluid Mech. 601, 123164.
Tsiglifis, K. & Pelekasis, N. 2007 Nonlinear oscillations and collapse of elongated bubbles subject to weak viscous effects: effect of internal overpressure. Phys. Fluids 19, 072106 (115).
Tsiglifis, K. & Pelekasis, N. 2008 Nonlinear radial oscillations of encapsulated microbubbles subject to ultrasound: the effect of membrane constitutive law. J. Acoust. Soc. Am. 123 (6), 40594070.
Versluis, M., Goertz, D., Palanchon, P., Heitman, I., van der Meer, S., Dollet, B., de Jong, N. & Lohse, D. 2010 Microbubble shape oscillations excited through ultrasonic parametric driving. Phys. Rev. E 82, 026321.
Vogel, A., Lauterborn, W. & Timm, R. 1989 Optical and acoustic investigations of the dynamics of laser-produced cavitation bubbles near a solid boundary. J. Fluid Mech. 206, 299338.
Yasui, K. 1995 Effects of thermal conduction on bubble dynamics near the sonoluminescence threshold. J. Acoust. Soc. Am. 98 (5), 27722782.
Zabolotskaya, E. A. 1984 Interaction of gas bubbles in a sound wave field. Sov. Phys. Acoust. 30, 365368.
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