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On the interaction of two encapsulated bubbles in an ultrasound field

Published online by Cambridge University Press:  31 August 2016

Yunqiao Liu ,
Kazuyasu Sugiyama  and
Shu Takagi
Yunqiao Liu*
Affiliation:
MOE Key Laboratory of Hydrodynamics, Department of Engineering Mechanics, Shanghai Jiao Tong University, Shanghai 200240, China
Kazuyasu Sugiyama
Affiliation:
Department of Mechanical Science and Bioengineering, Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama, Toyonaka, Osaka 560-0043, Japan
Shu Takagi
Affiliation:
Department of Mechanical Engineering, School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
*
†Email address for correspondence: [email protected]
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Abstract

We establish a theoretical model for the radial oscillations, translational motions and deformations of two interacting encapsulated bubbles. The flow field outside the bubbles is approximated by a potential flow with a viscous correction. The in-plane stresses and bending moments of the viscoelastic membranes are balanced by the hydrodynamic tractions at the interfaces of the bubbles. Since the material points move along the membranes accompanied by their movements in the radial direction when the encapsulated bubbles undergo deformations, stress balance in both the tangential and normal directions and the no-velocity-jump condition at the bubble surface are applied. The derived expression for the viscous drag includes the quasisteady drag force and the history force, which is validated by the solution of the unsteady Stokes equation. With an appropriate choice of the interface parameters, the present model is suitable for bubbles with free-slip, viscoelastic or no-slip interfaces. The viscous correction and the potential part of our solution are validated, respectively, by comparing them with previous experimental observations. The encapsulated bubble shows more stability in resisting shape oscillation. The attractive or repulsive movements of the two bubbles subjected to a driving frequency are consistent with the prediction by Bjerknes’ theory. For gas bubbles, the drag is mainly from the quasisteady component of the flow. For encapsulated bubbles, the no-velocity-jump condition enhances viscous dissipation, and thus contributes significantly to the history force in the viscous drag, generating more damping in the translational motion.

JFM classification

Drops and Bubbles: Bubble dynamics Biological Fluid Dynamics: Membranes
Type
Papers
Information
Journal of Fluid Mechanics , Volume 804 , 10 October 2016 , pp. 58 - 89
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Copyright
© 2016 Cambridge University Press 

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References

Akhatov, I., Parlitz, U. & Lauterborn, W. 1994 Pattern formation in acoustic cavitation. J. Acoust. Soc. Am. 96 (6), 3627–3635.Google Scholar
Barbat, T., Ashgriz, N. & Liu, C. 1999 Dynamics of two interacting bubbles in an acoustic field. J. Fluid Mech. 389, 137–168.CrossRefGoogle Scholar
Barthès-Biesel, D. & Rallison, J. M. 1981 The time-dependent deformation of a capsule freely suspended in a linear shear flow. J. Fluid Mech. 113, 251–267.Google Scholar
Barthès-Biesel, D. & Sgaier, H. 1985 Role of membrane viscosity in the orientation and deformation of a spherical capsule suspended in shear flow. J. Fluid Mech. 160, 119–135.Google Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Benjamin, T. B. & Ellis, A. T. 1990 Self-propulsion of asymmetrically vibrating bubbles. J. Fluid Mech. 212, 65–80.Google Scholar
Bjerknes, V. F. K. 1906 Fields of Force. Columbia University Press.Google Scholar
Bui, T. T., Ong, E. T., Khoo, B. C., Klaseboer, E. & Hung, K. C. 2006 A fast algorithm for modeling multiple bubbles dynamics. J. Comput. Phys. 216 (2), 430–453.Google Scholar
Chahine, G. L. & Duraiswami, R. 1992 Dynamical interactions in a multi-bubble cloud. Trans. ASME J. Fluids Engng 114 (4), 680–686.Google Scholar
Chatzidai, N., Dimakopoulos, Y. & Tsamopoulos, J. 2011 Viscous effects on the oscillations of two equal and deformable bubbles under a step change in pressure. J. Fluid Mech. 673, 513–547.Google Scholar
Crum, L. A. 1975 Bjerknes forces on bubbles in a stationary sound field. J. Acoust. Soc. Am. 57 (6), 1363–1370.Google Scholar
Crum, L. A. 1994 Sonoluminescence. Phys. Today 47 (9), 22–30.CrossRefGoogle Scholar
Diaz, A., Pelekasis, N. A. & Barthès-Biesel, D. 2000 Transient response of a capsule subjected to varying flow conditions: effect of internal fluid viscosity and membrane elasticity. Phys. Fluids 12, 948–957.Google Scholar
Doinikov, A. A. 1999 Bjerknes forces between two bubbles in a viscous fluid. J. Acoust. Soc. Am. 106 (6), 3305–3312.Google Scholar
Doinikov, A. A. 2001 Translational motion of two interacting bubbles in a strong acoustic field. Phys. Rev. E 64 (2), 026301.Google Scholar
Doinikov, A. A. 2002 Viscous effects on the interaction force between two small gas bubbles in a weak acoustic field. J. Acoust. Soc. Am. 111 (4), 1602–1609.CrossRefGoogle Scholar
Doinikov, A. A. & Zavtrak, S. T. 1995 On the mutual interaction of two gas bubbles in a sound field. Phys. Fluids 7 (8), 1923–1930.Google Scholar
Eller, A. I. & Crum, L. A. 1970 Instability of the motion of a pulsating bubble in a sound field. J. Acoust. Soc. Am. 47 (3B), 762–767.Google Scholar
Garbin, V., Dollet, B., Overvelde, M., Cojoc, D., Di Fabrizio, E., Van Wijngaarden, L., Prosperetti, A., De Jong, N., Lohse, D. & Versluis, M. 2009 History force on coated microbubbles propelled by ultrasound. Phys. Fluids 21 (9), 092003.Google Scholar
Green, A. E. & Adkins, J. E. 1960 Large Elastic Deformations and Non-Linear Continuum Mechanics. Oxford Clarenden Press.Google Scholar
Harkin, A., Kaper, T. J. & Nadim, A. 2001 Coupled pulsation and translation of two gas bubbles in a liquid. J. Fluid Mech. 445, 377–411.Google Scholar
Hejazialhosseini, B., Rossinelli, D. & Koumoutsakos, P. 2013 Vortex dynamics in 3D shock-bubble interaction. Phys. Fluids 25 (11), 110816.Google Scholar
Hilgenfeldt, S., Lohse, D. & Brenner, M. 1996 Phase diagrams for sonoluminescing bubbles. Phys. Fluids 8, 2808–2826.Google Scholar
Hobson, E. W. 1931 The Theory of Spherical and Ellipsoidal Harmonics. Cambridge University Press.Google Scholar
Hoff, L. 2001 Acoustic Characterization of Contrast Agents for Medical Ultrasound Imaging. Springer.Google Scholar
Ida, M. 2003 Alternative interpretation of the sign reversal of secondary Bjerknes force acting between two pulsating gas bubbles. Phys. Rev. E 67 (5), 056617.Google Scholar
Kang, I. S. & Leal, L. G. 1988 The drag coefficient for a spherical bubble in a uniform streaming flow. Phys. Fluids 31 (2), 233–237.Google Scholar
Khismatullin, D. B. 2004 Resonance frequency of microbubbles: effect of viscosity. J. Acoust. Soc. Am. 116 (3), 1463–1473.Google Scholar
Kobelev, Y. A., Ostrovskii, L. A. & Sutin, A. M. 1979 Effect of self-clearing for acoustic waves in a liquid with gas bubbles. Pis ma Zh. Eksp. Teor. Fiz 30, 423–425.Google Scholar
Lac, E., Barthès-Biesel, D., Pelekasis, N. A. & Tsamopoulos, J. 2004 Spherical capsules in three-dimensional unbounded Stokes flows: effect of the membrane constitutive law and onset of buckling. J. Fluid Mech. 516, 303–334.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1987 Fluid Mechanics. Elsevier.Google Scholar
Lauterborn, W. & Bolle, H. 1975 Experimental investigations of cavitation-bubble collapse in the neighbourhood of a solid boundary. J. Fluid Mech. 72 (02), 391–399.Google Scholar
Levich, V. G. 1949 The motion of bubbles at high Reynolds numbers. Zh. Eksp. Teor. Fiz. 19, 18–24.Google Scholar
Leyrat-Maurin, A. & Barthès-Biesel, D. 1994 Motion of a deformable capsule through a hyperbolic constriction. J. Fluid Mech. 279, 135–163.Google Scholar
Li, X. Z., Barthès-Biesel, D. & Helmy, A. 1988 Large deformations and burst of a capsule freely suspended in an elongational flow. J. Fluid Mech. 187, 179–196.Google Scholar
Lindner, J. R. 2004 Microbubbles in medical imaging: current applications and future directions. Nat. Rev. Drug Discov. 3 (6), 527–533.Google ScholarPubMed
Liu, Y., Sugiyama, K., Takagi, S. & Matsumoto, Y. 2011 Numerical study on the shape oscillation of an encapsulated microbubble in ultrasound field. Phys. Fluids 23 (4), 041904.Google Scholar
Liu, Y., Sugiyama, K., Takagi, S. & Matsumoto, Y. 2012 Surface instability of an encapsulated bubble induced by an ultrasonic pressure wave. J. Fluid Mech. 691, 315–340.Google Scholar
Love, A. E. H. 1888 The small free vibrations and deformation of a thin elastic shell. Phil. Trans. R. Soc. Lond. A 179, 491–546.Google Scholar
Magnaudet, J. & Eames, I. 2000 The motion of high-Reynolds-number bubbles in inhomogeneous flows. Annu. Rev. Fluid Mech. 32 (1), 659–708.Google Scholar
Magnaudet, J. & Legendre, D. 1998 The viscous drag force on a spherical bubble with a time-dependent radius. Phys. Fluids 10 (3), 550–554.Google Scholar
van der Meer, S. M., Dollet, B., Voormolen, M. M., Chin, C. T., Bouakaz, A., de Jong, N., Versluis, M. & Lohse, D. 2007 Microbubble spectroscopy of ultrasound contrast agents. J. Acoust. Soc. Am. 121, 648–656.Google Scholar
Mei, C. C. & Zhou, X. 1991 Parametric resonance of a spherical bubble. J. Fluid Mech. 229, 29–50.Google Scholar
Mooney, M. 1940 A theory of large elastic deformation. J. Appl. Phys. 11, 582–592.Google Scholar
Oguz, H. N. & Prosperetti, A. 1990 A generalization of the impulse and virial theorems with an application to bubble oscillations. J. Fluid Mech. 218, 143–162.Google Scholar
Pelekasis, N. A., Gaki, A., Doinikov, A. & Tsamopoulos, J. 2004 Secondary Bjerknes forces between two bubbles and the phenomenon of acoustic streamers. J. Fluid Mech. 500, 313–347.Google Scholar
Pelekasis, N. A. & Tsamopoulos, J. A. 1993a Bjerknes forces between two bubbles. Part 1. Response to a step change in pressure. J. Fluid Mech. 254, 467–499.CrossRefGoogle Scholar
Pelekasis, N. A. & Tsamopoulos, J. A. 1993b Bjerknes forces between two bubbles. Part 2. Response to an oscillatory pressure field. J. Fluid Mech. 254, 501–527.Google Scholar
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 (02), 283–290.Google Scholar
Pozrikidis, C. 2001 Effect of membrane bending stiffness on the deformation of capsules in simple shear flow. J. Fluid Mech. 440, 269–291.Google Scholar
Pozrikidis, C. 2003a Deformed shapes of axisymmetric capsules enclosed by elastic membranes. J. Engng Maths 45 (2), 169–182.CrossRefGoogle Scholar
Pozrikidis, C. 2003b Numerical simulation of the flow-induced deformation of red blood cells. Ann. Biomed. Engng 31 (10), 1194–1205.Google Scholar
Pozrikidis, C. 2005 Axisymmetric motion of a file of red blood cells through capillaries. Phys. Fluids 17, 031503.Google Scholar
Prosperetti, A. 1977 Viscous effects on perturbed spherical flows. Q. Appl. Maths 34, 339–352.Google Scholar
Prosperetti, A. 1991 The thermal behaviour of oscillating gas bubbles. J. Fluid Mech. 222, 587–616.Google Scholar
Quéguiner, C. & Barthès-Biesel, D. 1997 Axisymmetric motion of capsules through cylindrical channels. J. Fluid Mech. 348, 349–376.Google Scholar
Strasberg, M. & Benjamin, T. B. 1958 Excitation of oscillations in the shape of pulsating gas bubbles; experimental work. J. Acoust. Soc. Am. 30 (7), 697–697.Google Scholar
Takahira, H., Akamatsu, T. & Fujikawa, S. 1991 Dynamics of two nonspherical bubbles in a visous liquid. Trans. JSME B 57 (534), 51–59.Google Scholar
Tsiglifis, K. & Pelekasis, N. A. 2011 Parametric stability and dynamic buckling of an encapsulated microbubble subject to acoustic disturbances. Phys. Fluids 23, 012102.Google Scholar
Unger, E. C., Hersh, E., Vannan, M., Matsunaga, T. O. & Mccreery, T. 2001 Local drug and gene delivery through microbubbles. Prog. Cardiovascular Disc. 44 (1), 45–54.Google Scholar
Zabolotskaya, E. A. 1984 Interaction of gas-bubbles in a sound field. Sov. Phys. Acoust. 30 (5), 365–368.Google Scholar