Influence of Interfacial Rheology on Viscous Fingering

Fluid-fluid interfaces are common in both nature and industry, arising commonly in foams, respiratory droplets, thin films, coatings, and flotation. Generally, stabilizing agents or 'surfactants' such as fatty acids, alcohols, proteins, and particles are used to stabilize the interface against rupture and coalescence. However, interfacial instabilities can still occur even in the presence of surfactants. These instabilities are often undesirable and present challenges in common industrial processes involving multiphase flows. One such phenomenon – the Saffman-Taylor instability – involves the formation of finger-like protrusions when a less viscous fluid displaces a more viscous fluid. Recent studies have shown that complex surfactant-laden interfaces with surface rheological stresses resist interfacial deformation and alter the fluid dynamics of the system. Despite current progress, the impact of surface rheological stresses on the Saffman-Taylor instability is yet unknown. In this work, we demonstrate, for the first time, the stabilizing effect of surface rheology in radial viscous fingering using linear stability analysis. We analyze the effect of surface rheology on the growth of interfacial perturbations in a radial Hele-Shaw cell and illustrate that surface viscous stresses have a stabilizing effect: the most unstable wavenumber as well as the corresponding growth rate decreases with increasing surface viscosity. Based on these, we postulate that surface viscous resistance slows the growth of the instability and results in thicker fingers. Finally, we also show that incorporating surface rheology in radial viscous fingering potentially better captures past experimental data. Our work suggests that surface rheology should be considered as a potential factor in future models and experiments involving complex surfactant-laden interfaces.