The lab

Viscous Fingering

Squeeze two glass plates together with viscous oil between them, then inject water at the center. The flat oil-water interface is unstable: any perturbation grows. Fingers advance, split at their tips, compete for space, and branch into fractal structures. This is the Saffman-Taylor instability — the same physics that drives dendritic crystal growth, groundwater invasion, and the branching of river networks.

Darcy flow · Hele-Shaw cell · Saffman & Taylor, 1958 · DLA-like in the limit of high viscosity contrast

fingers: 0

coverage: 0%

step 0

growing

regime

interface noise 0.08

surface tension 0.30

The Hele-Shaw cell is a deceptively simple device: two parallel glass plates separated by a thin gap, filled with viscous fluid. When a less viscous fluid (water, air) is injected to displace the more viscous one, the physics of Darcy flow in the thin gap makes the pressure field behave like an electrostatic potential. The interface between fluids follows the pressure gradient — exactly like a potential-flow problem.

The flat interface is linearly unstable. A small bump advances faster than its neighbors because the pressure gradient is concentrated at the tip. The bump grows. It sharpens. Eventually, surface tension at the tip creates a curvature pressure that competes with the growth — and the finger settles into a steady shape. But two nearby fingers compete for the same pressure gradient: the leading one shades the trailing one, which slows and eventually stops. This tip-splitting and shielding dynamic is what generates the characteristic fractal branching.

In the limit where surface tension goes to zero, the fingering pattern becomes mathematically equivalent to Diffusion-Limited Aggregation (DLA) — the same fractal dimension (approximately 1.71), the same shielding geometry. Both are governed by Laplace's equation with a moving boundary. The same mathematics describes dendritic snowflake growth, dielectric breakdown, electrodeposition, and the branching of lightning.

Surface tension is the stabilizing influence: it suppresses the shortest-wavelength instabilities and selects a preferred finger width. The ratio of viscous to surface tension forces — the capillary number Ca = μU/γ — determines the regime. At low Ca, fingers are fat and stable; at high Ca, fingers are thin and fractal. In between: the natural branching patterns you see in inkblots pressed between glass.

The simulation uses a pressure-field based DLA variant with curvature regularization at fingertips to model surface tension. Reduce surface tension for wilder fractal branching; increase it for fewer, fatter, more stable fingers.