STUDY #33 · 2026 · IN OBSERVATION

Ferrofluid — Rosensweig Instability

A model-driven visual study of a liquid surface sculpted by a magnetic field.

MOVING IMAGE — THE ROUND-TRIP ARC B 1.93 → 2.18 → 1.90 · leap at B_c · memory below B_c · e3 0.6 · obsidian

WHAT IS THIS

A ferrofluid is a liquid that magnetizes: nanoparticles of magnetite suspended in oil, black because the particles absorb light across the visible band. Held flat under a growing vertical magnetic field, its surface stays mirror-still until a critical field strength — then it leaps, all at once, into a hexagonal lattice of spikes. This is the Rosensweig (normal-field) instability. The field sculpts the liquid without touching it: a spike gathers magnetic flux, and gathered flux raises the spike, while gravity guards the long waves and surface tension the short ones. The field first breaks through at exactly one wavelength between them — the capillary length sets the spacing of the thorns, about a centimetre in a real dish.

This study runs a volume-conserving gradient flow of the interface energy — gravity, surface tension, and the nonlocal magnetic term, the operator |k|, a half-derivative — in real time on the GPU, with the magnetic dial B as the only protagonist. In these units the critical values are exact numbers: B_c = 2, k_c = 1. And because the leap is subcritical, the spikes persist below the critical field on the way back down. The pattern remembers.

the leap lattice — just above B_c, the hexagonal answer B 2.06 · e3 0.6 · k_c 1 · spacing 2πλ_c · 384² · obsidian

Motif ferrofluid / normal-field (Rosensweig) instability / hexagonal spike lattice / hysteresis

Method A small simulator was generated and modified with AI assistance, then ported to a real-time GPU (GLSL) renderer. The visual output was selected through parameter exploration.

Observation Below B_c = 2 the surface refuses every ripple; crossing it, a hexagonal spike lattice leaps up at the capillary wavelength (k_c = 1). The leap is subcritical: lowering the field, the thorns persist past B_c and let go together at the saddle 2 − e3²/g_eff. One dial of asymmetry swaps peaks for stripes for a honeycomb of wells; deep fields stretch the lattice into a labyrinth.

Reference M. D. Cowley & R. E. Rosensweig, "The interfacial stability of a ferromagnetic fluid," Journal of Fluid Mechanics, vol.30, 671-688 (1967); A. Gailitis, "Formation of the hexagonal pattern on the surface of a ferromagnetic fluid in an applied magnetic field," Journal of Fluid Mechanics, vol.82, 401-413 (1977).

Tools Python / NumPy / three.js / React / GLSL / ffmpeg / AI coding assistant

Year 2026

This is not a scientific simulation result, but a visual interpretation of the phenomenon.

TWO LIQUID SURFACES, A DIFFERENT CLOCK

Faraday's waves must keep dancing. Rosensweig's thorns stand still.

Study #29 — Faraday Waves Study #33 — Ferrofluid

The drive the floor shakes — energy pumped in every half period a static field — time plays no role at all

What stands standing waves that die the moment the shaking stops an energy minimum — thorns that outlive their making

The wavelength set by the drive frequency through the dispersion relation set by the liquid itself — the capillary length, whoever applies the field

Memory none — below threshold the mirror always returns hysteresis — the lattice survives below B_c, down to the saddle

PARAMETERS EXPLORED

param meaning effect on the image

B magnetic Bond number — the square of the magnetization, the dial below 2 every ripple dies; at 2 the lattice leaps; higher, deeper thorns — and on the way down the collapse waits at the saddle 2 − e3²/g_eff

e3 up-down asymmetry — sharp peaks gather more flux than round valleys + peak hexagons · 0 stripes · − a honeycomb of wells; the hysteresis width grows as e3²

e4 saturation stops the spike height; smaller means taller and sharper

seed initial disturbance the lattice's grain — orientation, grain boundaries, dislocations

Each image below records its exact parameter set.

THE MATHEMATICS the model behind the images

The whole study is one energy and its steepest descent. The magnetic term carries the operator |k| — neither a derivative nor an integral, the half-derivative that a deep magnetizable liquid presents to its own surface.

E[h] = \int \Big[ \tfrac{1}{2}h^2 + \tfrac{1}{2}|\nabla h|^2 - \tfrac{B}{2}\, h\,\Lambda h - \tfrac{\varepsilon_3}{3} h^3 + \tfrac{\varepsilon_4}{4} h^4 \Big]\, d\mathbf{x}, \qquad \partial_t h = -\Big( \tfrac{\delta E}{\delta h} - \big\langle \tfrac{\delta E}{\delta h} \big\rangle \Big)

the interface energy and its volume-conserving gradient flow; Λ = |k| is the nonlocal magnetic (Dirichlet-to-Neumann) operator

\omega(k) = Bk - 1 - k^2, \qquad B_n(k) = \frac{1+k^2}{k}, \qquad k_c = 1, \; B_c = 2

linear growth rate and neutral curve in capillary units — the Cowley–Rosensweig critical condition

\dot{A} = \varepsilon A + 2\varepsilon_3 A^2 - g_{\mathrm{eff}} A^3, \qquad B_{\mathrm{sn}} = 2 - \frac{\varepsilon_3^2}{g_{\mathrm{eff}}}

the resonant-triad amplitude equation; the saddle below B_c is where the thorns finally let go

The linear physics and the neutral curve are exact statements of the model; the amplitude equation is the standard weakly nonlinear reduction (Gailitis 1977), with the cubic coefficient renormalized by the adiabatically eliminated second-harmonic and difference modes.

SELECTED STILLS — 4

the leap lattice — onset's hexagonal answer B 2.06 · e3 0.6 · settled from noise · obsidian

stripes — up-down symmetry restored B 2.4 · e3 0 · rolls instead of thorns · obsidian

the inverted lattice — a honeycomb of wells B 2.3 · e3 −0.7 · obsidian

the memory — thorns standing below the critical field film frame 432 · B 1.966 < B_c · rms 0.138

PROCESS — PARAMETER SWEEPS

The measured round trip behind the film — hexagon amplitude A against the dial B at e3 = 0.6. The ascending branch hugs the floor and leaps at B_c = 2; the descending branch rides the upper solution past B_c and survives to the saddle at 2 − 0.040, tracking the renormalized branch A₊(B). The thumbnails are the surface's states along the loop: on the way down at B = 1.96 the lattice still stands where, on the way up, the same dial met a mirror.

the hysteresis round trip, measured e3 0.6 · saddle 2 − 0.040 · orientation-free ring measure · sweep sheet3

SIGNATURE — THE MEMORY OF FORM

The way up and the way down are different roads.

The leap is subcritical. Up-down asymmetry — a sharp peak gathers more flux than a round valley loses — feeds the hexagons a quadratic term, and the bifurcation overhangs: below B_c a spiked solution coexists with the flat mirror. Raising the dial, the surface leaps at B_c = 2; lowering it, the thorns ride the upper branch past B_c and let go together only at the saddle B_sn = 2 − e3²/g_eff. Between the two roads lies the memory of form.

The film performs that loop. On the way up, at B = 2.008, the surface is a mirror (rms 0.005); on the way down at B = 1.998 — the same region of the dial — it still carries the full lattice (rms 0.238), and it holds through a long beat at B = 1.966 < B_c before the saddle takes it. The measured hysteresis width matches the amplitude theory only after the slaved harmonics are eliminated: the naive cubic overshoots by tens of percent, the renormalized one lands within 1%.

below the critical field on the way down — the thorns holding B 1.966 < B_c = 1.990 (engine units) · rms 0.138 · film frame 432

COLOUR = INTERFACE OPTICS ON A BLACK LIQUID

The liquid is painted black because ferrofluid is black: its magnetite nanoparticles absorb light across the visible band. Everything legible in the frame — the steel highlights, the cold rim, the zenith gleam on each crown — is interface optics on a mirror-dark skin, which is exactly how a real Rosensweig dish presents itself.

The spike spacing follows the capillary length √(σ/ρg) — about 1.6 mm for a typical oil-based ferrofluid, spikes about a centimetre apart — and the hexagonal packing, the stripe and honeycomb variants, and the hysteresis of the lattice are the model's own selections, not staged.

the same optics on the inverted lattice — wells instead of thorns B 2.3 · e3 −0.7 · steel speculars · ice rim

The colours are artistic approximations of the real material, not measurements.

REFERENCES

M. D. Cowley, R. E. Rosensweig. "The interfacial stability of a ferromagnetic fluid." Journal of Fluid Mechanics, vol.30, 671-688 (1967).
A. Gailitis. "Formation of the hexagonal pattern on the surface of a ferromagnetic fluid in an applied magnetic field." Journal of Fluid Mechanics, vol.82, 401-413 (1977).
R. Friedrichs, A. Engel. "Pattern and spike selection in the Rosensweig instability." Physical Review E, vol.64, 021406 (2001).

INTERACTIVE STUDY

A magnetizable liquid surface under a vertical field. Below the critical field B_c = 2 every ripple dies; cross it and the surface leaps into a lattice of spikes. Now bring the dial back down — the thorns outlive the field that made them, all the way to the golden saddle below B_c, where they let go at once. The gauge under the plate draws that round trip live: the way up and the way down are different roads. Click the surface to plant a seed — inside the hysteretic window it grows a lattice from the point you touched. The |k| operator here is exact (a built-in FFT), so the threshold is real, not staged. It is a deliberately simplified instrument, capped at a 128² spectral field with a few curated knobs and no export, separate from the full engine used to author the finished works.

SIMPLIFIED INSTRUMENT · 128² SPECTRAL FIELDROSENSWEIG · |k| EXACT · B_c = 2 TRUE THRESHOLD

This interactive study is not intended as a scientifically validated reproduction. It is a visual interpretation generated from an implemented model and curated parameter exploration — and it is a deliberately simplified instrument, separate from the full engine used to author the finished works.