Hostname: page-component-76fb5796d-r6qrq Total loading time: 0 Render date: 2024-04-26T11:18:21.670Z Has data issue: false hasContentIssue false
  • English
  • Français

Bubble dynamics in a viscoelastic medium with nonlinear elasticity

Published online by Cambridge University Press:  30 January 2015

R. Gaudron ,
M. T. Warnez  and
E. Johnsen
R. Gaudron*
Affiliation:
Mechanical Engineering Department, University of Michigan, Ann Arbor, MI 48109, USA
M. T. Warnez
Affiliation:
Mechanical Engineering Department, University of Michigan, Ann Arbor, MI 48109, USA
E. Johnsen*
Affiliation:
Mechanical Engineering Department, University of Michigan, Ann Arbor, MI 48109, USA
*
†Email addresses for correspondence: renaud.gaudron@centraliens.net, ejohnsen@umich.edu
†Email addresses for correspondence: renaud.gaudron@centraliens.net, ejohnsen@umich.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

In a variety of recently developed medical procedures, bubbles are formed directly in soft tissue and may cause damage. While cavitation in Newtonian liquids has received significant attention, bubble dynamics in tissue, a viscoelastic medium, remains poorly understood. To model tissue, most previous studies have focused on Maxwell-based viscoelastic fluids. However, soft tissue generally possesses an original configuration to which it relaxes after deformation. Thus, a Kelvin–Voigt-based viscoelastic model is expected to be a more appropriate representation. Furthermore, large oscillations may occur, thus violating the infinitesimal strain assumption and requiring a nonlinear/finite-strain elasticity description. In this article, we develop a theoretical framework to simulate spherical bubble dynamics in a viscoelastic medium with nonlinear elasticity. Following modern continuum mechanics formalism, we derive the form of the elastic forces acting on a bubble for common strain-energy functions (e.g. neo-Hookean, Mooney–Rivlin) and incorporate them into Rayleigh–Plesset-like equations. The main effects of nonlinear elasticity are to reduce the violence of the collapse and rebound for large departures from the equilibrium radius, and increase the oscillation frequency. The present approach can readily be extended to other strain-energy functions and used to compute the stress/deformation fields in the surrounding medium.

JFM classification

Drops and Bubbles: Bubble dynamics Drops and Bubbles: Cavitation Drops and Bubbles: Drops and Bubbles
Type
Papers
Information
Journal of Fluid Mechanics , Volume 766 , 10 March 2015 , pp. 54 - 75
Check for updates
Copyright
© 2015 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Allen, J. & Roy, R. 2000a Dynamics of gas bubbles in viscoelastic fluids. I. Linear viscoelasticity. J. Acoust. Soc. Am. 107, 3167–3178.CrossRefGoogle ScholarPubMed
Allen, J. & Roy, R. 2000b Dynamics of gas bubbles in viscoelastic fluids. II. Nonlinear viscoelasticity. J. Acoust. Soc. Am. 108, 1640–1650.CrossRefGoogle ScholarPubMed
Arfken, G. B., Weber, H. J. & Harris, F. E. 2013 Mathematical Method for Physicists, 7th edn. Academic.Google Scholar
Blake, F. G. 1949 The tensile strength of liquids; a review of the literature. Harvard Acou. Res. Lab. Rep. TM9.Google Scholar
Blake, J. R. & Gibson, D. C. 1987 Cavitation bubbles near boundaries. Annu. Rev. Fluid Mech. 19, 99–123.CrossRefGoogle Scholar
Brennen, C. E. 1995 Cavitation and Bubble Dynamics. Oxford University Press.Google Scholar
Brujan, E. A., Ohl, C. D., Lauterborn, W. & Philipp, A. 1996 Dynamics of laser-induced cavitation bubbles in polymer solutions. Acust., Acta Acust. 82, 423–430.Google Scholar
Coussios, C. C. & Roy, R. A. 2008 Applications of acoustics and cavitation to noninvasive therapy and drug delivery. Annu. Rev. Fluid Mech. 40, 395–420.CrossRefGoogle Scholar
Ellis, A. T. & Hoyt, J. W. 1958 Some effects of macromolecules on cavitation inception. In ASME 1968 Cavitation Forum, ASME Fluids Engineering Conference, Philadelphia, PA, pp. 2–3.Google Scholar
Flynn, H. G. 1975 Cavitation dynamics. II. Free pulsations and models for cavitation bubbles. J. Acoust. Soc. Am. 58, 1160–1170.Google Scholar
Fogler, H. S. & Goddard, J. D. 1970 Collapse of spherical cavities in viscoelastic fluids. Phys. Fluids 13, 1135–1141.CrossRefGoogle Scholar
Fogler, H. S. & Goddard, J. D. 1971 Oscillations of a gas bubble in viscoelastic liquids subject to acoustic and impulsive pressure variations. J. Appl. Phys. 42, 259–263.CrossRefGoogle Scholar
Foteinopoulou, K. & Laso, M. 2010 Numerical simulation of bubble dynamics in a Phan-Thien–Tanner liquid: non-linear shape and size oscillatory response under periodic pressure. Ultrasonics 50, 758–776.CrossRefGoogle Scholar
Fung, Y. C. 1993 Biomechanics: Mechanical Properties of Living Tissues, 2nd edn. Springer.Google Scholar
Fuster, D., Dopazo, C. & Hauke, G. 2011 Liquid compressibility effects during the collapse of a single cavitating bubble. J. Acoust. Soc. Am. 129 (1), 122–131.CrossRefGoogle ScholarPubMed
Gent, A. N. & Tompkins, D. A. 1969 Nucleation and growth of gas bubbles in elastomers. J. Appl. Phys. 40, 2520–2525.CrossRefGoogle Scholar
Gilmore, F. R.1952 The collapse and growth of a spherical bubble in viscous compressible liquid. Report No. 26-4, California Institute of Technology.Google Scholar
Goldberg, B. B., Liu, J. B. & Forsberg, F. 1994 Ultrasound contrast agents: a review. Ultrasound Med. Biol. 20, 319–333.CrossRefGoogle ScholarPubMed
ter Haar, G., Sinnett, D. & Rivens, I. 1989 High intensity focused ultrasound – a surgical technique for the treatment of discrete liver tumours. Phys. Med. Biol. 34, 1743–1750.CrossRefGoogle ScholarPubMed
Hammitt, F. G. 1980 Cavitation and Multiphase Flow Phenomena. McGraw-Hill.Google Scholar
Hara, S. K. & Schowalter, W. R. 1984 Dynamics of non-spherical bubbles surrounded by viscoelastic fluid. J. Non-Newtonian Fluid Mech. 14, 249–264.Google Scholar
Herring1941 Theory of the pulsations of the gas bubbles produced by an underwater explosion. OSRD Report, Division of National Defense Research 236, Columbia University.Google Scholar
Hickling, R. & Plesset, M. S. 1964 Collapse and rebound of a spherical bubble in water. Phys. Fluids 7, 7–14.Google Scholar
Hoger, A. & Johnson, B. E. 1995 Linear elasticity for constrained materials: incompressibility. J. Elast. 38, 69–93.Google Scholar
Holzapfel, G. 2000 Nonlinear Solid Mechanics. John Wiley & Sons, Ltd.Google Scholar
Horgan, C. O. & Polignone, D. A. 1995 Cavitation in nonlinearly elastic solids: a review. Appl. Mech. Rev. 48, 471–485.Google Scholar
Hua, C. & Johnsen, E. 2013 Nonlinear oscillations following the Rayleigh collapse of a gas bubble in a linear viscoelastic (tissue-like) medium. Phys. Fluids 25, 083101.Google Scholar
Jimenez-Fernandez, J. & Crespo, A. 2006 The collapse of gas bubbles and cavities in a viscoelastic fluid. Intl J. Multiphase Flow 32, 1294–1299.Google Scholar
Johnsen, E. & Colonius, T. 2009 Numerical simulations of non-spherical bubble collapse. J. Fluid Mech. 629, 231–262.Google Scholar
Keller, J. B. & Miksis, M. 1980 Bubble oscillations of large amplitude. J. Acoust. Soc. Am. 68 (2), 628–633.Google Scholar
Khismatullin, D. B. & Nadim, A. 2002 Radial oscillations of encapsulated microbubbles in viscoelastic liquids. Phys. Fluids 14, 3534–3557.Google Scholar
Lauterborn, W. 1976 Numerical investigation of nonlinear oscillations of gas bubbles in liquids. J. Acoust. Soc. Am. 59, 283–293.Google Scholar
Lauterborn, W. & Kurz, T. 2010 Physics of bubble oscillations. Rep. Prog. Phys. 73, 106501.Google Scholar
Lind, S. J. & Phillips, T. N. 2010 Spherical bubble collapse in viscoelastic fluids. J. Non-Newtonian Fluid Mech. 165, 56–64.Google Scholar
Lingeman, J. E. 1997 Extracorporeal shock wave lithotripsy: development, instrument, and current status. Urol. Clin. North Am. 24, 185–211.CrossRefGoogle ScholarPubMed
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
Madsen, E. L., Sathoff, H. J. & Zagzebski, H. J. 1983 Ultrasonic shear wave properties of soft tissues and tissuelike materials. J. Acoust. Soc. Am. 74, 1346–1355.Google Scholar
Maxwell, A. D., Cain, C. A., Hall, T. L., Fowlkes, J. B. & Xu, Z. 2013 Probability of cavitation for single ultrasound pulses applied to tissues and tissue-mimicking materials. Ultrasound Med. Biol. 39, 449–465.Google Scholar
Mooney, M. 1940 A theory of large elastic deformation. J. Appl. Phys. 11, 582–592.Google Scholar
Mukundakrishnan, K., Eckmann, D. M. & Ayyaswamy, P. S. 2009 Bubble motion through a generalized power-law fluid flowing in a vertical tube. Ann. New York Acad. Sci. 1161, 256–267.CrossRefGoogle Scholar
Naude, J. & Mendez, F. 2008 Periodic and chaotic oscillations of a bubble gas immersed in an upper convective Maxwell fluid. J. Non-Newtonain Fluid Mech. 155, 30–38.CrossRefGoogle Scholar
Ogden, R. W. 1972 Large deformation isotropic elasticity – on the correlation of theory and experiments for incompressible rubberlike solids. Proc. R. Soc. Lond. A 328, 565–584.Google Scholar
Papanastasiou, A. C., Scriven, L. E. & Macosko, C. W. 1984 Bubble growth and collapse in viscoelastic liquids analyzed. J. Non-Newtonian Fluid Mech. 16, 53–75.Google Scholar
Parsons, J. E., Cain, C. A., Abrams, G. D. & Fowlkes, J. B. 2006 Pulsed cavitational ultrasound therapy for controlled tissue homogenization. Ultrasound Med. Biol. 32, 115–129.Google Scholar
Plesset, M. S. 1949 The dynamics of cavitation bubbles. Trans. ASME J. Appl. Mech. 16, 277–282.Google Scholar
Plesset, M. S. & Prosperetti, A. 1977 Bubble dynamics and cavitation. Annu. Rev. Fluid. Mech. 9, 145–185.CrossRefGoogle Scholar
Prosperetti, A. & Lezzi, A. 1986 Bubble dynamics in a compressible liquid. Part 1. First-order theory. J. Fluid Mech. 168 (2), 457–478.Google Scholar
Rayleigh, J. W. S. 1917 On the pressure developed in a liquid during the collapse of a spherical cavity. Phil. Mag. 34, 94–98.Google Scholar
Rivlin, R. S. 1948 Large elastic deformations of isotropic materials. I. Fundamental concepts. Phil. Trans. R. Soc. A 240 (822), 459–490.Google Scholar
Roberts, W. W. 2014 Develpment and translation of histotripsy: current status and future directions. Curr. Opin. Urol. 24, 104–110.Google Scholar
Roberts, W. W., Hall, T. L., Ives, K., Wolf, J. S. Jr, Fowlkes, J. B. & Cain, C. A. 2006 Pulsed cavitational ultrasound: a noninvasive technology for controlled tissue ablation (histotripsy) in the rabbit kidney. J. Urol. 175, 734–738.Google Scholar
Shima, A. & Tsujino, T. 1976 The behavior of bubbles in polymer solutions. Chem. Engng Sci. 31, 863–869.Google Scholar
Stricker, L., Prosperetti, A. & Lohse, D. 2011 Validation of an approximate model for the thermal behavior in acoustically driven bubbles. J. Acoust. Soc. Am. 130, 3243–3251.Google Scholar
Treloar, L. R. G. 1943a The elasticity of a network of long-chain molecules. I. J. Chem. Soc. Faraday Trans. 39, 36–41.CrossRefGoogle Scholar
Treloar, L. R. G. 1943b The elasticity of a network of long-chain molecules. II. J. Chem. Soc. Faraday Trans. 39, 241–246.Google Scholar
Verner, J. H. 1978 Explicit Runge–Kutta methods with estimates of the local truncation error. SIAM J. Numer. Anal. 15, 772–790.CrossRefGoogle Scholar
Vlaisavljevich, E., Maxwell, A., Warnez, M., Johnsen, E., Cain, C. A. & Xu, Z. 2014 Histotripsy-induced cavitation cloud initiation thresholds in tissues of different mechanical properties. IEEE Trans. Ultrason. Ferroelectr. 61, 341–352.Google Scholar
Wells, P. N. & Liang, H. D. 2011 Medical ultrasound: imaging of soft tissue strain and elasticity. J. R. Soc. Interface 8, 1521–1549.Google Scholar
Yang, X. & Church, C. C. 2005 A model for the dynamics of gas bubbles in soft tissues. J. Acoust. Soc. Am. 118 (6), 3595–3606.Google Scholar