Viscous fingering with partially miscible fluids

Viscous fingering—the fluid-mechanical instability that takes place when a low-viscosity fluid displaces a high-viscosity fluid—has traditionally been studied under either fully miscible or fully immiscible fluid systems. Here we study the impact of partial miscibility (a common occurrence in practice) on the fingering dynamics. Through a careful design of the thermodynamic free energy of a binary mixture, we develop a phase-field model of fluid-fluid displacements in a Hele-Shaw cell for the general case in which the two fluids have limited (but nonzero) solubility into one another. We show, by means of high-resolution numerical simulations, that partial miscibility exerts a powerful control on the degree of fingering: fluid dissolution hinders fingering while fluid exsolution enhances fingering. We also show that, as a result of the interplay between compositional exchange and the hydrodynamic pattern-forming process, stronger fingering promotes the system to approach thermodynamic equilibrium more quickly.