Analytical properties and solutions for the synchronous pulsating bubble clusters distributed on a spherical surface

The present work concerns with the nonlinear dynamics for the synchronous pulsating bubble clusters uniformly distributed on a spherical surface. First, the governing equation for such clusters with 4/6/8/12/20 coupled bubbles are established. Second, the maximum and minimum radii for the gas-filled bubble clusters are analyzed according to the first integral. Third, by introducing suitable nonlocal transformations, two novel equivalent parametric analytical solutions in the form of Weierstrass elliptic function are constructed for the gas-filled bubble clusters for a specific polytropic exponent κ=3/2 without considering the surface tension, and based on which we immediately derive the parametric analytical solution for the corresponding vapor bubble clusters. Further, to consider the case of arbitrary polytropic exponent and surface tension, we develop a direct approach to construct the parametric analytical solution using Jacobi elliptic function for gas-filled bubble clusters. It is shown that, the behaviors and results for the bubble clusters will degenerate to the corresponding ones for single bubbles as the radius of the bubble cluster approaches infinity. In addition, on the basis of the analytical results, dynamic properties and motion laws of the bubble clusters are also discussed.