Elastic stresses in an encapsulated gas bubble

Abstract

The main objective of the present work is to study the elastic behaviour and the non-linear radial oscillations of a thin spherical shell encapsulating a perfect gas under the influence of an external acoustic pressure field. Our basic motivation is to derive a full elastic description of the above conjugate fluid mechanics problem. The mathematical model is reduced to solve a modified Rayleigh–Plesset equation to predict the external radius together with a Cauchy equation needed to predict the deformations and elastic stresses of the spherical thin shell. We impose conjugate boundary and initial conditions to predict the radius, elastic deformations and stresses and the resulting governing equations are solved numerically. Using dimensionless variables, we introduce, among others, three characteristic parameters that dictate the dynamic response of the encapsulated bubble: β that represents the competition between acoustic forces and induced stresses into the elastic solid, πx is the ratio between the equilibrium pressure and the elasticity modulus of the shell and βA defines the competition between the characteristic pressures of the system given lines below. The numerical predictions reveal that this last dimensionless parameter βA regulates clearly a pronounced cavitation limit and a non-linear oscillating behaviour. The most relevant physical variables of the problem such as the external radius of the shell, the elastic deformations and stresses are obtained as functions of time and different values of the above dimensionless parameters.