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

Darcy's law

Darcy's law is a phenomenologically derived constitutive equation that describes the flow of a fluid through a porous medium. The law was formulated by Henry Darcy based on the results of experiments on the flow of water through beds of sand...

 for the flow in the bulk of each fluid, and a boundary condition at the interface accounting for surface tension

Surface tension

Surface tension is a property of the surface of a liquid that allows it to resist an external force. It is revealed, for example, in floating of some objects on the surface of water, even though they are denser than water, and in the ability of some insects to run on the water surface...

.

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.

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

Phase field models

A phase field model is a mathematical model for solving interfacial problems. It has mainly been applied to solidification dynamics, but it has also been applied to other situations such as viscous fingering, fracture dynamics, vesicle dynamics, etc....

, etc.