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Dynamical model of an oscillating shelled microbubble

Cowley, James and Mulholland, Anthony J. and Stewart, Iain W. and Gachagan, Anthony (2015) Dynamical model of an oscillating shelled microbubble. [Report]

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Abstract

There is considerable interest at the moment on using shelled microbubbles as a transportation mechanism for localised drug delivery, specifically in the treatment of various cancers. In this report a theoretical model is proposed which predicts the dynamics of an oscillating shelled microbubble. A neo-Hookean, compressible strain energy density function is used to model the potential energy per unit volume of the shell. The shell is then stressed by applying a series of small radially directed stress steps to the inner surface of the shell whilst setting the outer surface’s stress to zero. The spatial profiles of the Cauchy radial and angular (hoop) stresses that are created within the shell during this quasistatic inflationary process are then stored as the shelled microbubble is inflated. The shelled microbubble is then allowed to collapse by setting the stress at the inner surface to zero. The model which results is then used to predict the dynamics of the shelled microbubble as it oscillates about its equilibrium state. A linear approximation is then used to allow analytical insight into both the quasistatic inflationary and oscillating phases of the shelled microbubble. Numerical results from the full nonlinear model are produced which show the influence of the shell’s thickness, Poisson ratio and shear modulus on the rate of oscillation of the shelled microbubble and these are compared to the approximate analytical solution. The theoretical model’s collapse time is compared to published experimental results.