Singular perturbation theory applied to the collective oscillation of gas bubbles in a liquid

Abstract

By using the technique of multiple scaling a theory of gas-bubble oscillations in a liquid is developed. Collections of bubbles of arbitrary shape under the action of surface tension, buoyancy and solid surfaces are considered. In the absence of thermal conduction in the bubbles and compressibility of the liquid a conserved ‘action’ is defined for each of the modes of oscillation. The equation governing the decay of the action with time is found by carrying the analysis to second order. The geometrical configuration of the bubbles, in which the oscillations take place, evolves in time under convection by an underlying ‘basic’ flow for which the governing equations are derived. The bubble pulsations influence the development of the basic motion. Later in the work a source of gas bubbles is brought in and its effects on the oscillations discussed. The results of the interaction of pulsating bubbles with the liquid surface are also briefly considered. The determination of the amplitude of oscillations induced by the splitting up of bubbles and by the generation of bubbles from the gas source is described. Finally, several applications of the theory to specific problems are given.