Genuine Two-Phase Flow Dynamics with a Free Interface Separating Gas-Liquid Mixture from Gas

Abstract

In this work we deal with the no-slip drift-flux model for gas-liquid flow dynamics. We focus on a situation where there is a free interface separating the gas-liquid mixture from a pure gas region which takes a positive pressure $p^*$. This situation is highly relevant for gas-liquid flow in the context of wellbore operations. Previous works have assumed that there is vacuum, i.e., the pressure $p^*$ is zero. The positive pressure $p^*>0$ creates a boundary term that must be treated in a consistent manner throughout the analysis. We derive time-independent estimates and make some observations related to the role played by $p^*$. The estimates allow us to discuss the long-time behavior of the two-phase flow system. In particular, it is shown that the stationary solution connecting the gas-liquid mixture to the pure gas region with the specified pressure $p^*$ in a continuous manner is asymptotically stable for sufficiently small initial perturbations. The analysis clearly shows how this perturbation directly depends on the size of the outer pressure $p^*$. A higher pressure $p^*$ allows for larger initial perturbations from steady state. One ingredient in the analysis is the rate at which the liquid mass decays to zero at the free interface. Insight into mechanisms that control the decay rate of the liquid mass at the free interface is also of interest since such transition zones often are associated with instabilities in numerical discretizations of two-phase models.