Lagrangian formalism for computing oscillations of spherically symmetric encapsulated acoustic antibubbles

pp. 197-208, vol. 19, 2016

Kristoffer Johansen
School of Engineering, James Watt Building, University of Glasgow, Scotland

Michiel Postema
Institute of Fundamental Technological Research, Polish Academy of Sciences, Poland
School of Electrical and Information Engineering, Chamber of Mines Building, University of the Witwatersrand, South Africa
Department of Physics and Technology, University of Bergen, Norway

Key words: Microbubbles; spatio–temporal bubble dynamics; Rayleigh-Plesset equation

Abstract: Antibubbles are gas bubbles containing a liquid droplet core and, typically, a stabilising outer shell. It has been hypothesised that acoustically driven antibubbles can be used for active leakage detection from subsea production facilities. This paper treats the dynamics of spheri-cally symmetric microscopic antibubbles, building on existing models of bubble dynamics. A more complete understanding of microbubble dynamics demands that the e ects of the trans-lational dynamics is included into the Rayleigh-Plesset equation, which has been the primary aim of this paper. Moreover, it is a goal of this paper to derive a theory that is not based on ad-hoc parameters due to the presence of a shell, but rather on material properties. To achieve a coupled set of di erential equations describing the radial and translational dynamics of an antibubble, in this paper Lagrangian formalism is used, where a Rayleigh-Plesset-like equation allows for the shell to be modelled from first principles. Two shell models are adopted; one for a Newtonian fluid shell, and the other for a Maxwell fluid shell. In addition, a zero-thickness approximation of the encapsulation is presented for both models. The Newtonian fluid shell can be considered as a special case of the Maxwell fluid shell. The equations have been linearised and the natural and damped resonance frequencies have been presented for both shell models.

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