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Center For Applied and Computational Mathematics

Nonlinear Dynamics of Gas Bubbles in Liquids

Bubble Cluster

Faculty: Anthony Harkin


Cavitation can be roughly described as the violent, nonlinear collapse of a tiny gas or vapor bubble immersed in a liquid.  The energy released by cavitating bubbles may, over time, result in significant erosion to the surfaces of nearby solid objects.  This can be seen in the corrosive effects of cavitating bubbles on maritime propeller blades and on pumps.  Cavitation erosion has long been exploited for industrial and laboratory applications.  Examples include cleaning baths, cell disruptors and endosonic techniques in dentistry.  Cavitation is also thought to occur in vivo during lithotripsy, a medical procedure employed to break up kidney stones.  Due to the widespread use of diagnostic ultrasound in medicine, the bioeffects of ultrasonic induced cavitation have been the focus of much research.  Sonochemistry is the use of cavitation phenomenon to enhance chemical reaction rates, and has been used to facilitate the absorption of drugs through the skin. Another fascinating aspect of cavitation is that, under controlled conditions, a stable cavitating bubble can be made to emit a pulse of light upon each collapse.  This phenomenon, known as single bubble sonoluminescence (SBSL), continues to be a topic of intense research.

Acoustic Cavitation and Nonlinear Dynamics of an Oscillating Microbubble


Gas-vapor bubbles immersed in a liquid will cavitate when a sufficiently strong acoustic field is applied.  For acoustic pressure fields that are varying very slowly in time, the liquid pressure changes very slowly and the bubble size will undergo quasistatic motion.  For this quasistatic motion, there is a pressure amplitude, known as the Blake threshold, at which the bubble begins to cavitate.  The focus of this research is to refine the Blake threshold by studying the effects of a rapidly varying time-periodic acoustic pressure field on tiny gas-vapor bubbles in liquids.  The equation we use as governing the spherical oscillations of a bubble immersed in a liquid is the classical Rayleigh-Plesset equation.  The Rayleigh-Plesset equation is highly nonlinear, and we develop a dynamic cavitation threshold from it by employing techniques from dynamical systems theory, including normal form theory, Poincare sections and Melnikov theory.


  1. On acoustic cavitation of slightly subcritical bubbles, A. Harkin, A. Nadim and T.J. Kaper, Physics of Fluids 11(2), 274-287 (1999).

The Motion of Two Interacting Bubbles in a Liquid


More than a century ago, the Norwegian scientists C.A. Bjerknes and his son V.F.K. Bjerknes observed that two nearby, pulsating gas bubbles in a liquid will either attract or repel each other depending on whether their radial pulsations are in, or out, of phase.  They also observed that the magnitude of the mutual force between the bubbles obeys an inverse square law.  This hydrodynamic phenomenon between two bubbles has since come to be known as the secondary Bjerknes force.  In this work, we derive and analyze a system of coupled, nonlinear ODE's describing the pulsations and translations of two interacting bubbles.


  1. Coupled pulsation and translation of two gas bubbles in a liquid, A. Harkin, T.J Kaper and A. Nadim,, Journal of Fluid Mechanics 445, 377-411 (2001).

Nonlinear Stability of Nonspherical Bubbles


When an acoustic field is driving the oscillations of a bubble, both the spherical (volume) mode and shape modes can be activated.  If the shape mode oscillations become too large, the bubble may break apart.  Our goal is to understand the parametric stability of a nonspherical bubble by deriving an equation for the shape modes that is similar to the Matheiu equation.  Then the question of shape mode stability can be addressed by Floquet analysis and calculating the boundaries of Arnold tongues that emanate from resonances.


  1. Energy transfer and parametric stability of nonspherical bubbles, A. Harkin, A. Nadim and T.J. Kaper, (in progress).


Ali Nadim (Keck Graduate School)
Tasso Kaper (Boston University)