Viscous fingering

Viscous fingering is the formation of patterns in a morphologically unstable interface between two fluids in a porous medium or in a Hele-Shaw cell. It occurs when a less viscous fluid is injected displacing a more viscous one (in the inverse situation, with the more viscous displacing the other, the interface is stable and no patterns form). It can also occur driven by gravity (without injection) if the interface is horizontal separating two fluids of different densities, being the heavier one above the other. In the rectangular configuration the system evolves until a single finger (the Saffman–Taylor finger) forms. In the radial configuration the pattern grows forming fingers by successive tip-splitting.

The mathematical description of viscous fingering is the Darcy's law for the flow in the bulk of each fluid, and a boundary condition at the interface accounting for surface tension.

Most experimental research on viscous fingering has been performed on Hele-Shaw cells. The two most common set-ups are the channel configuration, in which the less viscous fluid is injected by an end of the channel, and the radial one, in which the less viscous fluid is injected by the center of the cell. Instabilities analogous to viscous fingering can also be self-generated in biological systems.

Simulations methods for viscous fingering problems include boundary integral methods, phase field models, etc.

References

• Viscous Fingering at Center for Nonlinear Dynamics

• P. G. Saffman and G. Taylor. The penetration of a fluid into a medium or hele-shaw cell containing a more viscous liquid. Proc. Soc. London, Ser A, 245:312-329, 1958.