A simple model for frothy fluids

A mathematical model for the flow of a frothy fluid is developed in this paper, based on a two-component system containing air and water. For simplicity, attention is focused upon uni-directional pipe flow, and it is shown that there is a deep analogy between this model and classical gas dynamics. The system can be reduced to two governing partial differential equations. An effective “equation of state” is derived for the bubbly fluid and given in an analogous form to classical gas dynamics. It predicts the much-reduced sound speed observed experimentally. The jump conditions at a normal shock are presented, and a novel shock-capturing spectral method is derived and applied to an example of shock formation for uni-directional flow in a frothy fluid.