Effective Equations for Sound and Void Wave Propagation in Bubbly Fluids

Effective equations that describe both sound wave and void wave propagation for bubbly flows at high Reynolds numbers are derived in this paper. First ideal bubble flows are considered, and a new method for solving Laplace’s equation for the velocity potential is presented. This approach is based on a generalization of the method of images and also yields a precise definition of the ambient field experienced by a bubble. With the velocity potential known, the Lagrangian is then computed, and equations of motion for a finite number of bubbles using the Euler–Lagrange equations are derived. The continuum limit is then used to obtain our effective equations. Our expressions for the sound wave and void wave speeds agree well with previous investigations. The effects of gravity and viscosity on void waves are considered. Viscous effects are incorporated using a dissipation function. The steady rise speed and void wave speed for a column of rising bubbles are computed and found to agree well with experiments.