Propagation of shock waves in dilute bubbly liquids (Governing equations, Hugoniot relations, and effect of slippage between two phases)

Propagation of shock waves in dilute bubbly liquids is investigated numerically. Governing equations for the bubbly liquid are formulated with emphasis on the radial and translational motions of the bubbles. The conservation equations for mass, momentum and energy of the bubble interior are solved directly in order to estimate precisely the effects of internal phenomena on the bubble motion. A numerical method, in which individual bubbles are tracked to estimate the effects of their volumetric changes and relative motions on the wave phenomena, is developed. For a wave process in a steady-state condition, simple relations, such as the propagation velocity, C-i, are derived. Numerical results under several conditions reveal that the terminal wave propagation velocity coincides with the propagation velocity in an isothermal equilibrium condition, C-i. The slippage between bubbles and liquid influences the time evolution of propagation velocity of the shock wave and the relaxation structure behind the wave. However, the slippage plays a minor role in the wave propagation process.