Bistability of buoyancy-driven exchange flow in vertical conduits: a dynamical approach

Buoyancy-driven exchange flows are common to a variety of natural and engineering processes, ranging from persistently active volcanoes to the cementing process in wells. In these systems, gravity is the only driving force which triggers the counter-flow of the flowing phases. However, even in laminar flow conditions, the effects of gravity on exchange flows are not yet completely understood. We use a core-annular flow solution for vertical conduits and reveal the existence of two steady-state solutions: one with fast flow in a thin core and another one with relatively slow flow in a thick core. By analysis of laboratory experiments, we interpret the existence of two equilibrium configurations in the framework of bistability. Specifically, we show the existence of two regimes: for viscosity ratios of order one, the solution is unique and can be identified by maximizing the flux; instead, for viscosity ratios above a critical value, the system tends to minimize the potential energy. Finally, we develop a simplified dynamic model for predicting the system evolution from a non-equilibrium state in order to elucidate the impact that initial conditions have on the two equilibrium configurations.