Analytical investigation of the nonlinear dynamics of empty spherical multi-bubbles in hydrodynamic cavitation

The nonlinear dynamics of empty multi-bubbles with the same distance and initial conditions are studied analytically through a modified Rayleigh–Plesset equation. The collapse time and analytical solution are derived under various initial conditions. In particular, when considering a positive initial vibration velocity, the exact analytical expression for the maximal radius is obtained by solving a cubic algebraic equation. To the best of our knowledge, this is the first time that a parabolic function has been used to construct the parametric analytical solution for this case. This type of function is able to simulate the collapse motion whereby the bubble radius first grows to the maximal radius and then decays to zero. The limiting behavior of the resulting analytical results for multi-bubbles (including the collapse time, analytical solution, and maximal radius) is also investigated, enabling the corresponding analytical results for single bubbles to be deduced in the limit as the distance between the multi-bubbles approaches infinity. In addition, the dynamical characteristics and qualitative analysis of these bubbles and the effects of the relevant physical parameters are studied.