🫧 Saffman-Taylor — Viscous Fingering

The Saffman-Taylor instability is what happens when a low-viscosity fluid is forced into a high-viscosity fluid inside a narrow gap: the flat interface between them cannot advance smoothly and instead breaks into branching fingers. This simulation reproduces it numerically by relaxing a pressure field each frame and growing the interface fastest where the pressure gradient is steepest — exactly the mechanism identified by Philip Saffman and Geoffrey Taylor in 1958.

A Hele-Shaw cell is two flat plates separated by a very thin gap; fluid squeezed between them behaves like a two-dimensional version of flow through a porous medium, obeying Darcy's law u = -(k/μ)∇p. The simulation's grid represents this cell from above: filled cells are the injected fluid, empty cells are the viscous fluid whose pressure the code solves for with a grid relaxation of Laplace's equation every frame.

Capillary number scales how strongly a cell's growth chance depends on the local pressure drop, raised to the power 1/Ca — lower Ca means only the steepest gradients grow, giving wide fingers. Viscosity ratio multiplies that same growth chance directly, so a bigger contrast between the fluids speeds up fingering overall. Surface tension both lowers the pressure assigned to convex, filled tip-cells (Gibbs-Thomson) and subtracts extra growth chance from concave dimples, which smooths the interface and sets the minimum finger width.

With surface tension near zero the code stops damping sharp tips, so growth is governed purely by the pressure gradient of a Laplace field — the same rule that drives diffusion-limited aggregation (DLA), where particles stick where a diffusing field's gradient is steepest. Both processes are examples of Laplacian growth, so at low surface tension the simulation produces the same kind of self-similar branching fractal that DLA clusters show, with a fractal dimension near 1.7.

Every 30 frames the code checks how many grid cells near the top and bottom edges have been filled by the injected fluid. If more than a quarter of the cells along those edges are filled, it calls the same seed() function used by the Reset button, clearing the grid and starting a fresh disc at the centre — this keeps the pattern from growing into an uninteresting solid block that has hit the boundary of the simulated cell.