Theory of Natural Frequencies of Two Pulsating Bubbles in Infinite Liquid

The volume oscillation of bubbles is a fundamental problem associated with the behavior of coolant in nuclear reactors and with reactor safety. Many studies have been reported on the oscillation frequency of a single bubble. Based on theory governing the behavior of single bubble, an accurate theory is derived on the natural frequencies of two pulsating spherical bubbles placed in an infinite liquid. The problem is reduced to calculation of the effect of a bubble on the inertial constant of the other bubble. Bispherical coordinates are introduced to calculate the velocity potential. The effects of liquid compressibilty and viscosity on the natural frequency are shown to he negligible. The most prominent result is that two kinds of frequencies are found to exist when two bubbles oscillate in phase and 180° out of phase. Examination further reveal that the analysis presented here includes Shima's solution as a special case. Numerical calculations substantiate the prediction that the resonant frequency of many bubbles is smaller than that of a single bubble.