Mathematical and Numerical Modeling of Two-Phase Compressible Flows with Micro-Inertia

A new model with full coupling between micro- and macroscale motion is developed for compressible multiphase mixtures. The equations of motion and the coupling microstructural equation (an analogue of the Rayleigh–Lamb equation) are obtained by using the Hamilton principle of stationary action. In the particular case of bubbly fluids, the resulting model contains eight partial differential equations (one-dimensional case) and is unconditionally hyperbolic. The equations are solved numerically by an adapted Godunov method. The model and methods are validated for two very different test problems. The first one consists of a wave propagating in a liquid containing a small quantity of gas bubbles. Computed oscillating shock waves fit perfectly the experimental data. Then the one-dimensional multiphase model is used as a reduction tool for the multidimensional interaction of a shock wave with a large bubble. Good agreement is again obtained.