The resonant mechanism of subdivision of a gas bubble in a fluid

摘要

Small nonlinear oscillations of a bubble in a fluid at the resonance of the frequencies of the radial mode and an arbitrary deformational mode 2: 1 are considered. The deformational mode is determined by the associated Legendre polynomial with indices n = 2, 3, ?, m = 0, 1, ?, n. The energy transfer from the radial mode into the Legendre deformational mode is described by the method of invariant normalization. An analogy is established with oscillations of the material point on the string with the frequency ratio of the vertical mode to the horizontal one of 2. During the transfer, the amplitude of the Legendre mode with indices n, m exceeds the amplitude of radial oscillations by a factor of 3n at m = 0. As index m increases, the transfer time increases considerably and the maximal amplitude of the Legendre mode increases insignificantly in this case. From here, it is concluded that the deformational Legendre mode with indices n, m = n has the greatest probability to rise. The considered effect can serve as a mechanism of subdivision of gas bubbles under varying the external pressure in the fluid.