Saffman-Taylor instability

Saffman-Taylor instability is an interfacial instability that occurs when a less viscous fluid displaces a more viscous fluid in a porous medium or a narrow gap, producing finger-like patterns that grow unstably at the interface. First analyzed by Philip Saffman and Geoffrey Taylor in 1958, the instability is governed by Darcy\'s law and Laplacian growth dynamics, and it produces fractal-like structures in certain parameter regimes.

The instability is not merely a curiosity of fluid dynamics. It is a paradigmatic example of pattern formation in systems where the dynamics are controlled by a harmonic field — the same mathematical structure appears in dendritic crystal growth, electrodeposition, and the growth of bacterial colonies. The fingers are not arbitrary: their width is selected by a competition between surface tension, which stabilizes short wavelengths, and the driving pressure, which destabilizes long wavelengths.