A numerical study on radial Hele-Shaw flow: influence of fluid miscibility and injection scheme

Fingering instability triggered by injection of a less viscous fluid in a Hele-Shaw cell is numerically investigated. Simulations are based on a diffuse-interface method, and the formulation, capable of dealing with immiscible and miscible interfaces, is presented in details. Three miscibility conditions, including immiscible with surface tension, partially miscible with effective interfacial tension and fully miscible without interfacial stresses, are simulated to verify generality of an optimal linear injection scheme proposed by Dias et al. (Phys Rev Lett 109:144502, 2012) in the limit of infinite viscosity contrast. stabilizing effects of this linear injection scheme are universally confirmed for the interfaces with the presence of interfacial stresses, such as immiscible and partially miscible conditions. On the other hand, the linear injection scheme in a fully miscible interface leads to contradictory results. Even the fingering pattern appears qualitatively more stable without the secondary phenomenon of finger merging, relevant quantitative measurements, such as longer channeling zone and interfacial length, indicate enhancement of fingering prominence. The inconsistent behaviors suggest that the coupling effects with the interfacial stresses are crucial in the applications of the optimal linear injection scheme.