Viscous Fingering

Fingering of 16.5 mPa s fluid into 1285 mPa s fluid. Both of these fluids are water/glycerol mixtures.

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.

Fingering pattern created by the injection of a less viscous fluid into a more viscous one. The viscosity ratio between the two fluids, ηin/ηout, controls a dramatic transition from long, highly-branched fingers to blunt stable structures and then eventually to a fully stable displacement.

Publications:

"Diffusion-driven transition between two regimes of viscous fingering," T. E. Videbæk, S. R. Nagel, Phys. Rev. Fluids 4, 033902 (2019)

“An island of stability in a sea of fingers: emergent large-scale features of the viscous flow instability,” I. Bischofberger, R. Ramachandran, S. R. Nagel, Soft Matter, 11, 7428-7432 (2015)

“Fingering versus stability in the limit of zero interfacial tension,” Irmgard Bischofberger, Radha Ramachandran and Sidney R. Nagel, Nat. Commun. 5:5265 doi: 10.1038/ncomms6265 (2014).