DYNAMICS OF A CLUSTER OF BUBBLES IN A LIQUID (THEORETICAL-ANALYSIS)

The present work investigates the dynamics of a cluster of bubbles in a liquid by means of a series expansion of spherical harmonics. The governing equations of three-dimensional motions for an arbitrary configuration of N bubbles are derived by taking account of the translational motion and the deformation of each bubble induced by mutual interactions among the bubbles. These equations are exact to the order of (R(Ic)/L(IJ0))5 for inviscid terms, where R(Ic) is the characteristic radius of a specified bubble I and L(IJ0) the initial distance between the centers of the bubbles I and J. Viscous effects of the liquid are considered up to the first order for quantities of the translational motion and the deformation, on the basis of the potential solution. The equations involve previous results for a single and two bubbles as special cases. The characteristic equation for oscillations of N spherical bubbles is also obtained, and natural frequencies are calculated for specified configurations of the bubbles. It is shown that the lowest natural frequency of the bubbles is much lower than the frequency of an isolated bubble.