Traveling Waves, Riemann Problems and Computations of a Model of the Dynamics of Liquid/Vapor Phase Transitions

Abstract

A model for the liquid/vapor phase transitions in a shock tube is discussed. Computations for the one-dimensional isothermal case is carried out to show that this model exhibits one-dimensional wave patterns observed in actual experiments. The existence of traveling waves under two different scalings are studied. For the first scaling, where the diffusion of different phases is very small relative to typical reaction time for the growth of phases while the viscosity is comparable to the reaction time, complete phase diagrams of traveling waves are obtained. Many of these traveling waves are undercompressive. Some of compressive traveling waves are unstable. For the second scaling, where viscosity, typical reaction time, and diffusion of different phases are comparable, the existence of traveling wave profiles of liquefaction and evaporation shocks when shock speeds are larger than some number is proved. These shocks are undercompressive. Evaporation shocks are rarefaction shocks. The nonexistence of these undercompressive shocks when the shock speed is smaller than some number is proved. It is observed through numerical computation that most of these traveling waves are unstable while some of them are stable or metastable. Riemann problems are considered. Admissibility of shocks involving phase changes under the second scaling is discussed. Solutions of Riemann problems with wave patterns observed in actual experiments are presented. Nonuniqueness of solutions of some Riemann problems is discussed.