Pulsations of a gas bubble in the acoustic field (Resonance and limits of the polytropic approximation)

The problem of pulsations of a gas bubble that are excited by oscillations of the external pressure has been solved analytically. The similarity numbers controlling the process under consideration have been obtained. It has been shown that variations of the parameters inside the bubble are determined by the adiabatic exponent (gas specific heat ratio) and the Fourier number, whereas the complete solution of an external problem is a function of three similarity numbers (the adiabatic exponent, the Fourier number, and the dimensionless radius of the bubble). The characteristics of pulsations of the bubble in dimensionless coordinates have been analyzed in detail. It has been shown that the real process of pulsations of the bubble in the presence of heat exchange cannot be described in the frames of polytropic approximation. A formula for the calculation of the frequency of resonant pulsations has been obtained for the first time. It has been shown that the pressure distribution inside the bubble under resonance conditions can almost always be treated as uniform except for the nearest vicinity of the thermodynamic critical point.