Modelling Radial Oscillations of a Bubble in a Spherical Liquid-Filled Elastic Solid


  • Dana A. Mohammedameen Department of Mathematics, College of Science, Salahaddin University-Erbil, Kurdistan Region, Iraq
  • Kawa M.A. MANMI Department of Mathematics, College of Science, Salahaddin University-Erbil, Kurdistan Region, Iraq
  • Waleed H. Aziz Department of Mathematics, College of Science, Salahaddin University-Erbil, Kurdistan Region,



Spherical bubble, Bubble oscillations, Acoustic wave, Elastic solid, Rayleigh-Plesset equation


In this paper, equation of a spherical bubble oscillations in Newtonian and viscous fluid with considering compressible fluid flow in a spherical liquid-filled elastic solid was developed. The exact solution of the linearisation and the numerical solution of the equation have been derived and compared.  Then numerical calculations are performed to investigate the effects of compressibility parameter  for gas and acoustic bubbles. We conclude that the amplitude of bubble radius oscillations increases with  and the effects of  is accumulating. Further, the discrepancy between the exact and numerical solutions increases with time when . However, the difference was not changed when the  is smaller than . Dual frequency is also considered in the acoustic environment.


ACHESON, D. J. 1991. Elementary fluid dynamics. Acoustical Society of America.

ARıTAN, S. 2006. Bulk Modulus.

BATCHELOR, C. K. & BATCHELOR, G. 2000. An introduction to fluid dynamics, Cambridge University Press.

BOSSCHERS, J. 2018. Propeller tip-vortex cavitation and its broadband noise.


COCHARD, H. 2006. Cavitation in trees. Comptes Rendus Physique, 7, 1018-1026.

CRUM, L. A., MASON, T. J., REISSE, J. L. & SUSLICK, K. S. 1998. Sonochemistry and sonoluminescence, Springer Science & Business Media.

DOINIKOV, A. A., DOLLET, B. & MARMOTTANT, P. 2018a. Cavitation in a liquid-filled cavity surrounded by an elastic medium: Intercoupling of cavitation events in neighboring cavities. Physical Review E, 98, 013108.

DOINIKOV, A. A., DOLLET, B. & MARMOTTANT, P. 2018b. Model for the growth and the oscillation of a cavitation bubble in a spherical liquid-filled cavity enclosed in an elastic medium. Physical Review E, 97, 013108.

DRYSDALE, C., DOINIKOV, A. A. & MARMOTTANT, P. 2017. Radiation dynamics of a cavitation bubble in a liquid-filled cavity surrounded by an elastic solid. Physical Review E, 95, 053104.

DURYEA, A. P., ROBERTS, W. W., CAIN, C. A. & HALL, T. L. Optimization of histotripsy for kidney stone erosion. 2010 IEEE International Ultrasonics Symposium, 2010. IEEE, 342-345.

DURYEA, A. P., ROBERTS, W. W., CAIN, C. A. & HALL, T. L. 2015. Removal of residual cavitation nuclei to enhance histotripsy erosion of model urinary stones. IEEE transactions on ultrasonics, ferroelectrics, and frequency control, 62, 896-904.

FRANC, J.-P. & MICHEL, J.-M. 2006. Fundamentals of cavitation, Springer science & Business media.

HU, A., LI, Y. & ZHENG, J. 2019. Dual-frequency ultrasonic effect on the structure and properties of starch with different size. LWT, 106, 254-262.

JENSEN, K. H., BERG-SØRENSEN, K., BRUUS, H., HOLBROOK, N. M., LIESCHE, J., SCHULZ, A., ZWIENIECKI, M. A. & BOHR, T. 2016. Sap flow and sugar transport in plants. Reviews of modern physics, 88, 035007.

KELLER, J. B. & KOLODNER, I. I. 1956. Damping of underwater explosion bubble oscillations. Journal of applied physics, 27, 1152-1161.

KLAPCSIK, K. 2021. GPU accelerated numerical investigation of the spherical stability of an acoustic cavitation bubble excited by dual-frequency. Ultrason Sonochem, 77, 105684.

KORKUT, E. & ATLAR, M. 2012. An experimental investigation of the effect of foul release coating application on performance, noise and cavitation characteristics of marine propellers. Ocean Engineering, 41, 1-12.

LARTER, M., BRODRIBB, T. J., PFAUTSCH, S., BURLETT, R., COCHARD, H. & DELZON, S. 2015. Extreme aridity pushes trees to their physical limits. Plant Physiology, 168, 804-807.

LAUTERBORN, W. & KURZ, T. 2010. Physics of bubble oscillations. Reports on progress in physics, 73, 106501.

LIU, Y., WANG, Q. & ZHANG, A.-M. 2018. Surface stability of a bubble in a liquid fully confined by an elastic solid. Physics of Fluids, 30, 127106.

MARION, M. & TEMAM, R. 1998. Navier-Stokes equations: Theory and approximation. Handbook of numerical analysis, 6, 503-689.

MATHWORKS 2018. Matlab R2018b User’s Guide.

NADIR, N. N. & MANMI, K. M. 2020. Comparative Investigation of the Spherical Acoustic Microbubble Models in an Unbounded Liquid. Zanco Journal of Pure and Applied Sciences, 32, 82-88.

NIGMATULIN, R., AKHATOV, I. S., VAKHITOVA, N. & LAHEY, R. 1999. Hydrodynamics, acoustics and transport in sonoluminescence phenomena. Sonochemistry and Sonoluminescence. Springer.

PLESSET, M. S. & PROSPERETTI, A. 1977. Bubble dynamics and cavitation. Annual review of fluid mechanics, 9, 145-185.

RAYLEIGH, L. 1917. VIII. On the pressure developed in a liquid during the collapse of a spherical cavity. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 34, 94-98.

SAEED, R. & SADEEQ, M. I. 2017. Radial Basis Function-Pseudospectral Method for Solving Non-Linear Whitham-Broer-Kaup Model. Sohag Journal of Mathematics, 4, 13-18.

SAITOH, S., IZUMI, M. & MINE, Y. 1995. A dual frequency ultrasonic probe for medical applications. IEEE transactions on ultrasonics, ferroelectrics, and frequency control, 42, 294-300.

STROOCK, A. D., PAGAY, V. V., ZWIENIECKI, M. A. & HOLBROOK, N. M. 2014. The physicochemical hydrodynamics of vascular plants. Annu. Rev. Fluid Mech, 46, 615-642.

TYREE, M. T. & SPERRY, J. S. 1989. Vulnerability of xylem to cavitation and embolism. Annual review of plant biology, 40, 19-36.

WANG, Q. 2017. Oscillation of a bubble in a liquid confined in an elastic solid. Physics of Fluids, 29, 072101.

WANG, Q. & MANMI, K. 2014. Three dimensional microbubble dynamics near a wall subject to high intensity ultrasound. Physics of Fluids, 26, 032104.

YASUI, K. 2018. Acoustic cavitation and bubble dynamics, Springer.

YI, J. & LU, Y. 2017. Effects of vapour pressure on the motion of cavitation bubble. Physics and Chemistry of Liquids, 55, 1-10.

YUSOF, N. S. M., ANANDAN, S., SIVASHANMUGAM, P., FLORES, E. M. & ASHOKKUMAR, M. 2022. A correlation between cavitation bubble temperature, sonoluminescence and interfacial chemistry–A minireview. Ultrasonics Sonochemistry, 105988.

ZHOU, M., YUSOF, N. S. M. & ASHOKKUMAR, M. 2013. Correlation between sonochemistry and sonoluminescence at various frequencies. RSC advances, 3, 9319-9324.





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