Qualitative analysis and analytical solution for higher dimensional gas-filled hyper-spherical bubbles in an ideal fluid

Abstract

The present work concerns with the higher dimensional Rayleigh–Plesset equation for describing the nonlinear dynamics of gas-filled hyper-spherical bubbles in an ideal fluid. A strict qualitative analysis is made by means of the bifurcation theory of dynamic system, indicating that the bubble oscillation type is periodic. An analytical approach based on elliptic function is suggested to construct parametric analytical solution with arbitrary space dimension $N$, polytropic exponent $\kappa$ and surface tension $\sigma$ to the normalized higher dimensional Rayleigh–Plesset equation. The new obtained analytical solution extends the known ones for arbitrary (or some special cases of) $N$ and $\kappa$ without considering the effect of surface tension. In addition, we also discuss the dynamic characteristics for the oscillating hyper-spherical bubbles.