Ensemble phase-averaged equations for bubbly flows

A system of averaged equations describing the motion of a mixture of spherical compressible bubbles in an inviscid liquid is derived by the ensemble-averaging method introduced in an earlier paper Zhang and Prosperetti, J. Fluid Mech. 267, 185 (1994). The averaging procedure introduces new terms in the equations, among which a contribution to the liquid stress tensor is of special interest. An extension of the well-known Rayleigh–Plesset equation to the case of bubbles interacting with the flow is also found. The general system of equations is closed in a rigorous manner in the dilute limit generalizing and correcting earlier averaged equations models. The results are illustrated by considering the problem of linear pressure wave propagation in a nonuniform bubbly liquid. Gradients of the bubble concentration are shown to dampen or amplify the wave strength.