Pulsations of Bubbles

Abstract

An analytic investigation of the problem of gas bubble pulsations under oscillations of external pressure is obtained; the similarity numbers governing this process are presented. It is shown that the variation of the gas characterisics inside a bubble is determined by two parameters; the adiabatic index for the gas and the Fourier number, whereas the complete solution of the problem is a function of three similarity numbers (the adiabatic index, the Fourier number, and the reduced bubble radius). A conclusion is made that in the presence of heat exchange it is impossible to describe the real process of bubble oscillations in the framework of polytropic approximation. An analysis of the laws of pulsations of a bubble in dimensionless coordinates is given. It is shown that in the presence of heat exchange a real process of bubble oscillations cannot be described “in detail” in the framework of the polytropic approximation; also it is impossible to describe the process of forced oscillations of bubble “in the mean”. If the dependence for polytropic exponent is chosen so as to describe the amplitude of the pulsations of the pressure, it cannot adequately describe the amplitude of the pulsations of temperature, and vice versa. An explicit dependence for the resonance oscillation frequency of a bubble is obtained for the first time. It is shown that under resonance conditions the pressure homogeneity assumption inside a bubble holds practically always, except in the direct vicinity of the thermodynamic critical point. Adiabatic pulsations of bubbles during stepwise variation of pressure are studied extensively. Analytical solutions were obtained by using a dynamic analysis approach as well as by energy balance methods. The shape of pulsation is presented.