# Ferrofluid Rosensweig Instability — Hexagonal Spike Formation

Apply a vertical magnetic field and watch hexagonal spikes spontaneously emerge above the critical field Bc. A competition between magnetic energy and surface tension/gravity.

> Canonical page: https://elysiatools.com/en/visualizations/ferrofluid-rosensweig-instability

- **Category:** Physics

## Overview

Interactive ferrofluid Rosensweig instability visualization. A vertical magnetic field B applied to a ferrofluid surface causes a regular hexagonal array of sharp spikes to spontaneously emerge above the critical field Bc — a first-order phase transition. Physics model uses linear instability theory: critical wavelength λc = 2π·√(σ/(ρg)) sets spike spacing, critical field Bc = √(2μ₀·√(σρg))/χ marks the threshold, and spike amplitude follows the supercritical bifurcation A ∝ √(1 − Bc/B). Height field h(x,y) = sharpen[cos(kx) + 2cos(kx/2)cos(√3ky/2)] (triangular lattice of three wavevectors at 0°/60°/120°). Three visualization panels: (1) Three.js r128 3D height-field mesh (128×128 vertices, MeshStandardMaterial metalness 0.9 roughness 0.15 with directional + point lighting for liquid-metal specular highlights, mouse/touch orbit rotation); (2) Canvas 2D cross-section showing surface profile h(x) with fluid fill gradient and specular dots on spike tips; (3) Top-down height map via putImageData showing hexagonal pattern and container boundaries. Adjustable parameters: field strength B (0–800 mT), susceptibility χ (0.1–5), surface tension σ (0.01–0.1 N/m), container shape (circle/square/hexagon with edge-attenuation window function). Auto-sweep animation ramps B from 0 to showcase the phase-transition emergence moment. Four presets: Classic (B≈1.2·Bc), Near Bc (B≈1.02·Bc), Strong Field (B≈3·Bc), Square Dish. Real-time statistics: critical field Bc, wavelength λc, B/Bc ratio, max spike height. Educational content covers Rosensweig 1967 theory, capillary-gravity wavelength, supercritical bifurcation, hexagonal mode coupling, and applications (seals, loudspeakers, drug delivery, kinetic art). Multi-language support (zh, en, es, fr, de, ru, pt).

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