The Riemann problem for a weakly hyperbolic two-phase flow model of a dispersed phase in a carrier fluid

Abstract

We consider Riemann problems for a two-phase isothermal flow model of a dispersed phase in a compressible carrier phase. It is a weakly hyperbolic system of conservative partial differential equations. This model is the conservation part of a more complete physical model involving phase transitions in case both phases are of the same material. The purpose of this paper is to better understand the mathematical properties of the simplified model. We investigate the characteristic structure of the Riemann problems and construct their exact solutions. Solutions may contain delta shocks or vaporless states. We give examples for initial data corresponding to a system of water bubbles dispersed in liquid water. The analysis is complicated considerably by the fact that a liquid such as water requires an affine equation of state.