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Observations on the dynamics of bubble cluster in an ultrasonic field

Nonlinear Dynamics volume 72, pages 561–574 (2013)Cite this article

Abstract

The dynamics driven interaction between the bubbles in a cavitation cluster is known to be a complex phenomenon indicative of a highly active nonlinear as well as chaotic behavior in ultrasonic fields. By considering the cluster of encapsulated microbubble with a thin elastic shell in ultrasonic fields, in this paper, the dynamics of microbubbles has been studied via applying the methods of chaos physics. Bifurcation, Lyapunov exponent, and time series are plotted with respect to variables such as amplitude, initial bubble radius, frequency and viscosity. The findings of the study indicate that a bubble cluster undergoes a chaotic unstable region as the amplitude and frequency of ultrasonic pulse are increased mainly due to the period doubling phenomenon. The results of the present study are supported by findings of previous studies.

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Author information

Authors and Affiliations

  1. Department of Physics, Urmia University of Technology, Urmia, Iran

    S. Behnia & M. Yahyavi

  2. Department of Physics, Azarbaijan University of Tarbiat Moallem, Tabriz, Iran

    H. Zahir

  3. Department of Physics, Sharif University of Technology, Tehran, Iran

    A. Barzegar

  4. Department of Mechanical Engineering, Urmia University of Technology, Urmia, Iran

    F. Mobadersani

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  1. S. Behnia
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  2. H. Zahir
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  3. M. Yahyavi
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Correspondence to S. Behnia.

Appendix: Stability Analysis

Appendix: Stability Analysis

The first-order differential equations for the specific example (Eq. 1) are of the following form:

(4)

where

or equivalently:

$$ J{\frac{dV}{dt}}=F(V,{\alpha}) $$
(5)

where θ is the cyclic variable, V(x 1,x 2,x 3,x 4,x 5,x 6,θ) an autonomous vector field and α(R 0,P 0,P a ,f,μ,μ sh,β,ρ,c,Γ,ϵ,χ,σ) is an element of the parameter space. This system generates a flow Φ={Φ T} on the phase space M=R 2 S and there exists a global map:

(6)

with \(T=\frac{1}{\nu}\), θ 0 is a constant determining the Poincaré cross-section and (x 1,x 2,x 3,x 4,x 5,x 6) the coordinates of the attractors in the Poincaré cross-section ∑ c , which is defined as:

(7)

The choice of Poincaré section is arbitrary; the only necessary condition is that the trajectory should cross the section once every acoustic cycle. For driven oscillators like the bubble model, a natural way to define ∑ is to cut the torus like state space M transversally to the cyclic θ direction at a fixed value θ 0 of θ [12].

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Behnia, S., Zahir, H., Yahyavi, M. et al. Observations on the dynamics of bubble cluster in an ultrasonic field. Nonlinear Dyn 72, 561–574 (2013). https://doi.org/10.1007/s11071-012-0734-2

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Keywords

  • Encapsulated microbubble
  • Ultrasound contrast agents
  • Keller–Herring equation
  • Period doubling bifurcations
  • Lyapunov spectrum
  • Intermittency