Viscous Fingering Instability: Fingering Versus Stability in the Limit of Zero Interfacial Tension

The invasion of one fluid into another of higher viscosity in a quasi-two-dimensional geometry typically produces complex fingering patterns. Because interfacial tension suppresses short-wavelength fluctuations, its elimination by using pairs of miscible fluids would suggest an instability producing highly ramified singular structures. Previous studies focused on wavelength selection at the instability onset and overlooked the striking features appearing more globally. Here we investigate the non-linear growth that occurs after the instability has been fully established. We find a rich variety of patterns that are characterized by the viscosity ratio between the inner and the outer fluid, ηin/ηout, as distinct from the most-unstable wavelength, which determines the onset of the instability. As ηin/ηout increases, a regime dominated by long highly-branched fractal fingers gives way to one dominated by blunt stable structures characteristic of proportionate growth. Simultaneously, a central region of complete outer-fluid displacement grows until it encompasses the entire pattern at ηin/ηout ≈ 0.3.

The displacement of a more viscous fluid by a less viscous one in a quasi-two dimensional geometry leads to the formation of complex fingering patterns. This fingering has been characterized by a most unstable wavelength which depends on the viscosity difference between two immiscible fluids and sets the characteristic width of the fingers. How the finger length grows after the instability occurs is an equally important, but previously overlooked, aspect that characterizes the global features of the patterns. Long after the instability onset, once the fingers are growing in a nearly steady-state regime, there is a stable inner region where the outer fluid is completely displaced. We show that the ratio of the finger length to the radius of this stable region depends only on the viscosity ratio of the fluids and is decoupled from the most unstable wavelength.