Gravitational fingering and droplet formation during the phase separation of a partially miscible binary mixture in a vertical Hele-Shaw cell

Abstract

In this study, the phase separation process of partially miscible and miscible binary systems in a vertical Hele-Shaw cell is numerically analyzed. The governing equations of a partially miscible binary system for momentum and mass transfer under buoyancy-driven fluid motion are newly implemented by taking into account both the Korteweg force and thermodynamic stability. The thermodynamic instability is reflected in the mass transfer using the modified Cahn-Hilliard equation and the Flory-Huggins Gibbs free energy for mixing. The onset of fingering and drop formation are demonstrated under buoyancy-driven motion depending on the repulsive interaction parameter (χ). When χ is higher than χc, droplet formation in a metastable or unstable region induced by convective mixing ultimately results in enhanced phase separation. Meanwhile, for a hydrodynamic stable system, the diffusional mixing is not strong enough to induce drop formation in stable and metastable regions. The Korteweg force resists the drop formation process induced by convection motion, determining the drop size and wavelength on the long fingers. For a metastable or unstable binary system, high Korteweg force results in the larger size of the droplet. Even in a fully miscible system, i.e., χ<χc, the Korteweg force still affects the system by making it stable, especially with an increase in the gradient interaction parameter (κ). This numerical study provides a deep understanding of phase separation in a thermodynamic binary system under buoyance-driven convection motion.