A Vlasov equation for pressure wave propagation in bubbly uids

The derivation of eeective equations for pressure wave propagation in a bubbly uid at very low void fractions is examined. A Vlasov-type equation is derived for the probability distribution of the bubbles in phase space instead of computing eeective equations in terms of averaged quantities. This provides a more general description of the bubble mixture and contains previously derived eeective equations as a special case. This Vlasov equation allows for the possibility that locally bubbles may oscillate with diierent phases or amplitudes or may have diierent sizes. The linearization of this equation recovers the dispersion relation derived by Carstensen & Foldy. The initial value problem is examined for both ideal bubbly ows and situations where the bubble dynamics have damping mechanisms. In the ideal case, it is found that the pressure waves will damp to zero whereas the bubbles continue to oscillate but with the oscillations becoming incoherent. This damping mechanism is similar to Landau damping in plasmas. Nonlinear eeects are considered by using the Hamiltonian structure. It is proven that there is a damping mechanism due to the nonlinearity of single bubble motion. The Vlasov equation is modiied to include eeects of liquid viscosity and heat transfer. It is shown that the pressure waves have two damping mechanisms, one from the eeects of size distribution and the other from single bubble damping eeects. Consequently, the pressure waves can damp faster than bubble oscillations.