Shock wave propagation in bubbly liquids at small gas volume fractions

Abstract

It is well known that shock wave propagation in liquid media is strongly affected by the presence of gas bubbles that interact with the shock and in turn affect the gas bubbles. An explicit form of a wave equation was obtained from a set of equations for wave propagation in bubbly liquid (Caflisch et al., 1985a) in this study. Shock wave propagation in bubbly mixtures was considered with the solution for the obtained wave equation, of which homogeneous and particular solutions provide the pressure field due to the shock profile and bubble-bubble interaction, respectively. The gas behavior inside a spherical bubble under the shock wave was obtained by a set of homologous solutions for the mass and momentum conservation equations. The energy equation for the gas inside the bubble was solved analytically with help of the homologous solutions. The bubble wall motion in compressible medium was obtained from the Keller-Miksis equation. The heat transfer from/to the bubble was obtained by solving the energy equation for the gas inside the bubble and for the liquid outside the bubble wall. The relaxation oscillations behind the shock front, which were calculated using the Keller-Miksis equation with the solutions of the obtained wave equation, are in close agreement with those obtained in shock tube experiments for a uniform bubbly flow by Kameda et al. (1998). Heat exchange between the gas bubbles and the liquid and the interaction between bubbles were found to be very important factors to affect the relaxation oscillations in the shock front.