Evolution of the shape of a bubble deformed at the zero time in a viscous fluid

Abstract

The boundary-value problem of the shape of a charged bubble in a viscous incompressible insulating fluid as a function of time is solved in an approximation linear in the initial deformation amplitude. Both centrosymmetric pulsations and vibrations at constant volume are considered. It is shown that the time evolution of the shape of the bubble, as well as of the velocity and pressure fields of the fluid in its neighborhood, can be represented by finite sums over the numbers of initially excited modes, which involve two terms. The first term is a sum over the roots of the dispersion relation; the second, an improper integral. In the low-and high-viscosity limits, the relevant analytical expressions simplify, i.e., become free of integrals. It turns out that the damping constant of bubble surface vibrations varies with viscosity nonmonotonically in the case of radial pulsations and this dependence is different in the different limits. The frequencies of radial pulsations and surface vibrations vary with viscosity monotonically.