Nonlinear Oscillations of a Bubble

Abstract

The oscillation of a gas-filled bubble in a fluid has been studied previously by several approaches. In general, the fluid is considered incompressible, and the gas is assumed to behave as a linear spring. Surface tension and gravity effects are usually ignored. Radiation losses of such bubbles are often determined experimentally. The analysis presented here includes surface-tension effects. The gas considered is a perfect gas and the fluid an inviscid compressible fluid. The field equations for the fluid are linearized so that the nonlinearities of the problem arise from the following two sources. (1) The boundary condition at the fluid-gas interface has a time dependent radius; and (2) the adiabatic pressure-volume relation for the gas is nonlinear. Free oscillations of the bubble are studied. In order to induce free motion, the gas-fluid interface of the bubble is considered to be initially at rest and then a pulse of pressure is delivered to the interior of the bubble. The ensuing motions are analyzed by a numerical integration scheme. Numerical results are presented for air bubbles in water. The most remarkable result is that the frequency of oscillation is essentially independent of amplitude for the cases studied. The results further indicate, as one would expect, that the numerical integration scheme used here is not practical for the determination of the radiation damping. When the small radiation damping is neglected, the motion becomes periodic and—for cases studied—a harmonic analysis of the pressure field was performed that shows that the radius and pressure variations can differ appreciably from the sinusoidal waves predicted by the linearized theory.