Linear wave propagation of fast and slow modes in mixtures of liquid and gas bubbles

Abstract

A new set of space-averaged equations that governs the dynamics of mixtures of liquid and gas bubbles is derived, where a surface-averaged liquid pressure at the gas–liquid interface is used as a variable as well as volume-averaged pressures, and the liquid compressibility is taken into account. To verify the validity of the model equations, the propagation of linear plane wave in a quiescent mixture is studied theoretically and numerically. It is shown that the compressibility of liquid induces two propagation modes, a fast mode and a slow mode, for all real wave numbers, and if one assumes that the liquid is incompressible there only exists the slow mode. Since the incompressibility approximation has been used for liquid phase in previous studies, the fast mode has not been investigated in detail. In the present study, several important characters of the slow and fast modes are clarified. In particular, it is demonstrated that the amplitude of the fast mode is not always small and it can become large when a typical wave number exceeds a critical value.