Effect of a mass flux at bubble-liquid interface on propagation properties of weakly nonlinear waves

Abstract

Toward the establishment and realization of medical applications accompanied with a phase change such as a drug delivery system, a formulation of the effect of mass transfer (i.e., mass flux) at the bubble-liquid interface on a ultrasound propagation process has been desired. Since a large amplitude ultrasound is utilized in such an application, the consideration of both the mass flux and the wave nonlinearity is essential. Although Fuster and Montel [J. Fluid Mech.(2015)] investigated linear wave propagation, nonlinear analysis has not been performed. In this paper, we theoretically clarify the effect of mass flux at the bubble-liquid interface on weakly nonlinear (i.e., finite but small amplitude) propagation of pressure waves in bubbly liquids. The effect of total evaporation or condensation as a mass flux across the bubble interface is taken into account as a simple classical model. Although various types of dissipation are installed, the effect of viscosity of gas inside the bubble is neglected. The set of basic equations for bubbly flows is composed of the conservation laws of mass, momentum, and energy, equation of motion for radial oscillations of bubble wall, equations of state, and so on. From the method of multiple scales, some nonlinear wave equations (e.g., KdV-Burgers equation) are derived. We then discuss the effects of mass flux on the propagation processes of nonlinear waves.