Eeective Equations for Sound and Void Wave Propagation in Bubbly Uids

EEective equations that describe both sound wave and void wave propagation for bubbly ows at high Reynolds numbers are derived in this paper. First ideal bubble ows 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 deenition of the ambient eld experienced by a bubble. With the velocity potential known, the Lagrangian is then computed and equations of motion for a nite number of bubbles using the Euler-Lagrange equations are derived. The continuum limit is then used to obtain our eeective equations. Our expressions for the sound wave and void wave speeds agree well with previous investigations. The eeects of gravity and viscosity on void waves are considered. Viscous eeects 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.