H. Takahira et al., DYNAMICS OF A CLUSTER OF BUBBLES IN A LIQUID (THEORETICAL-ANALYSIS), JSME international journal. Series B, fluids and thermal engineering, 37(2), 1994, pp. 297-305
The present work investigates the dynamics of a cluster of bubbles in
a liquid by means of a series expansion of spherical harmonics. The go
verning equations of three-dimensional motions for an arbitrary config
uration of N bubbles are derived by taking account of the translationa
l motion and the deformation of each bubble induced by mutual interact
ions among the bubbles. These equations are exact to the order of (R(I
c)/L(IJ0))5 for inviscid terms, where R(Ic) is the characteristic radi
us 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 con
sidered up to the first order for quantities of the translational moti
on and the deformation, on the basis of the potential solution. The eq
uations involve previous results for a single and two bubbles as speci
al cases. The characteristic equation for oscillations of N spherical
bubbles is also obtained, and natural frequencies are calculated for s
pecified configurations of the bubbles. It is shown that the lowest na
tural frequency of the bubbles is much lower than the frequency of an
isolated bubble.