Numerical Investigation of Nonlinear Oscillations and Compression-Only Behavior of a Coated Microbubble Near an Elastic Wall

Document Type: Original Article


Mechanical Engineering Department, Shahid Rajaei Teacher Training University, Tehran, Iran


During the ultrasound imaging process, the ultrasound contrast agents (UCAs) are beating near the blood vessel wall. Therefore, the purpose of the present simulation study is to investigate the effect of the presence of an elastic wall on the radial and frequency acoustic response of a UCA microbubble oscillating in a nonlinear regime. For this reason, the numerical simulation of the dynamic behavior of a coated microbubble was performed using coding in MATLAB and a Rayleigh-Plesset equation modified by Doinikov. To study the nonlinear bubble oscillations, its compression-only behavior and the sub-harmonic nonlinear component are taken from a nonlinear shell model presented by Marmottant et al. Initially, coated bubble oscillations in two linear and nonlinear regimes were investigated for two types of shell models, and it was observed that presence of the elastic wall affects the bubble's compression-only behavior. Finally, due to the importance of the subharmonic component in the nonlinear oscillation of the coated bubble, the threshold of the appearance of subharmonic components for a coated bubble near an elastic wall was investigated using the Fast Fourier Transform (FFT) and compared with the oscillation in the infinite fluid.


1.     De Jong, N., Hoff, L., Skotland, T., and Bom, N. “Absorption and scatter of encapsulated gas filled microspheres: Theoretical considerations and some measurements.” Ultrasonics, Vol. 30, No. 2, (1992), 95–103.
2.     Church, C. C. “The effects of an elastic solid surface layer on the radial pulsations of gas bubbles.” Journal of the Acoustical Society of America, Vol. 97, No. 3, (1995), 1510–1521.
3.     Hoff, L., Sontum, P. C., and Hovem, J. M. “Oscillations of polymeric microbubbles: Effect of the encapsulating shell.” The Journal of the Acoustical Society of America, Vol. 107, No. 4, (2000), 2272–2280.
4.     De Jong, N., Bouakaz, A., and Frinking, P. “Basic acoustic properties of microbubbles.” Echocardiography. Vol. 19, No. 3, (2002), 229-240.
5.     Brennen, C. Cavitation and bubble dynamics. Oxford University Press, (1995).
6.     Leighton, T. G. “The inertial terms in equations of motion for bubbles in tubular vessels or between plates.” The Journal of the Acoustical Society of America, Vol. 130, No. 5, (2011), 3333–3338.
7.     Blake, J. R., and Gibson, D. C. “Cavitation Bubbles Near Boundaries.” Annual Review of Fluid Mechanics, Vol. 19, No. 1, (1987), 99–123.
8.     Herring, C. “Theory of the pulsations of the gas bubble produced by an underwater explosion.” In Technical Report 236,  Columbia University, Division of National Defense Research, (1941).
9.     Strasberg, M. “The Pulsation Frequency of Nonspherical Gas Bubbles in Liquids.” Journal of the Acoustical Society of America, Vol. 25, No. 3, (1953), 536–537.
10.   Blue, J. E. “Resonance of a Bubble on an Infinite Rigid Boundary.” The Journal of the Acoustical Society of America, Vol. 41, No. 2, (1967), 369–372.
11.   Doinikov, A. A., Zhao, S., and Dayton, P. A. “Modeling of the acoustic response from contrast agent microbubbles near a rigid wall.” Ultrasonics, Vol. 49, No. 2, (2009), 195–201.
12.   Tomita, Y., and Shima, A. “Mechanisms of impulsive pressure generation and damage pit formation by bubble collapse.” Journal of Fluid Mechanics, Vol. 169, (1986), 535–564.
13.   Doinikov, A. A., Aired, L., and Bouakaz, A. “Acoustic response from a bubble pulsating near a fluid layer of finite density and thickness.” The Journal of the Acoustical Society of America, Vol. 129, No. 2, (2011), 616–621.
14.   Garbin, V., Cojoc, D., Ferrari, E., Di Fabrizio, E., Overvelde, M. L. J., Van Der Meer, S. M., De Jong, N., Lohse, D., and Versluis, M. “Changes in microbubble dynamics near a boundary revealed by combined optical micromanipulation and high-speed imaging.” Applied Physics Letters, Vol. 90, No. 11, (2007), 114103.
15.   Doinikov, A. A., and Bouakaz, A. “Interaction of an ultrasound-activated contrast microbubble with a wall at arbitrary separation distances.” Physics in Medicine and Biology, Vol. 60, No. 20, (2015), 7909–7925.
16.   Overvelde, M., Garbin, V., Dollet, B., De Jong, N., Lohse, D., and Versluis, M. “Dynamics of Coated Microbubbles Adherent to a Wall.” Ultrasound in Medicine and Biology, Vol. 37, No. 9, (2011), 1500–1508.
17.   Aired, L., Doinikov, A. A., and Bouakaz, A. “Effect of an elastic wall on the dynamics of an encapsulated microbubble: A simulation study.” Ultrasonics, Vol. 53, No. 1, (2013), 23–28.
18.   Garashchuk, I. R., Sinelshchikov, D. I., and Kudryashov, N. A. “Nonlinear Dynamics of a Bubble Contrast Agent Oscillating near an Elastic Wall.” Regular and Chaotic Dynamics, Vol. 23, No. 3, (2018), 257–272.
19.   Paul, S., Katiyar, A., Sarkar, K., Chatterjee, D., Shi, W. T., and Forsberg, F. “Material characterization of the encapsulation of an ultrasound contrast microbubble and its subharmonic response: Strain-softening interfacial elasticity model.” The Journal of the Acoustical Society of America, Vol. 127, No. 6, (2010), 3846–3857.
20.   Paul, S. “Acoustic characterization of ultrasound contrast microbubbles and echogenic liposomes: applications to imaging and drug-delivery”, Doctoral Dissertations, University of Delaware, USA, (2013). Retrieved from
21.   Marmottant, P., van der Meer, S., Emmer, M., Versluis, M., de Jong, N., Hilgenfeldt, S., and Lohse, D. “A model for large amplitude oscillations of coated bubbles accounting for buckling and rupture.” The Journal of the Acoustical Society of America, Vol. 118, No. 6, (2005), 3499–3505.
22.   Löfstedt, R., Barber, B. P., and Putterman, S. J. “Toward a hydrodynamic theory of sonoluminescence.” Physics of Fluids A, Vol. 5, No. 11, (1992), 2911–2928.