A thin layer of ferrofluid sandwiched between two close glass plates (a Hele-Shaw cell) starts as a compact droplet. Switching on a magnetic field H perpendicular to the plates magnetizes the fluid; every point on the fluid/air boundary now carries magnetic surface energy that is lowered by elongating the interface, while surface tension γ still tries to shorten it. This is the same destabilize-vs-stabilize competition behind viscous (Saffman–Taylor) fingering, and it produces a linear stability dispersion relation of the schematic form:
ω(k) ≈ a·μ₀M²H·k − b·γ·k³
k = interface perturbation wavenumber
a·μ₀M²H·k → magnetic term, destabilizing, grows with H
b·γ·k³ → tension term, stabilizing, grows faster at small scale
fastest-growing wavenumber: k_max ∝ √(μ₀M²H / γ)
Wavelengths shorter than a tension-set cutoff are damped (too much curvature energy), longer ones grow too slowly to compete, so a characteristic finger spacing wins and repeats as the pattern branches — this is the real mechanism behind ferrofluid "labyrinthine" patterns first described by Rosensweig, Zahn and Shumovich. The grid below is a growth-front automaton tuned to reproduce that same qualitative competition in real time: cells at sharp convex tips (where field lines concentrate, just like a lightning-rod effect on a magnetizable point) grow fastest when H is high, while cells in concave notches fill in fastest when γ is high — smoothing the front and coarsening the pattern instead of branching it.
- Field strength H — raises the growth odds of thin, isolated tip cells: higher H means finer, more numerous branching fingers.
- Surface tension γ — raises the growth odds of cells filling concave notches: higher γ produces thicker, smoother, less-branched lobes.
- Growth rate — how many automaton steps run per second; purely a playback-speed control, it does not change the pattern that emerges.
- Interface length readout tracks the total boundary edge count — a rough proxy for how "fractal" (branched) the front has become; it climbs fast once branching sets in.
Real-world relevance: labyrinthine ferrofluid instabilities are used to study general pattern-formation physics shared with electrodeposition, bacterial colony growth and viscous fingering, and the same field-tunable branching underlies ferrofluid-based damping and magnetic-fluid seals.