# Kelvin-Helmholtz Instability — Two-Phase Shear Flow

Two fluid layers sliding past each other destabilize at their interface into the characteristic KH billows. Tune density ratio, shear, surface tension, and toggle magnetic (MHD) stabilization.

> Canonical page: https://elysiatools.com/en/visualizations/kelvin-helmholtz-instability

- **Category:** Physics

## Overview

Interactive two-phase Kelvin-Helmholtz (KH) instability visualization focusing on the physical mechanism and linear stability of a sheared fluid interface, complementing the existing single-phase numerical vortex-sheet simulation. Two fluid layers (heavy lower fluid ρ₁, light upper fluid ρ₂) slide past each other with shear ΔU; the interface destabilizes into the characteristic KH billows when the shear overcomes the stabilizing effects. Full linear dispersion relation solved analytically: (ρ₁+ρ₂)ω² − 2(ρ₁U₁+ρ₂U₂)kω + (ρ₁U₁²+ρ₂U₂² − B²/μ₀)k² − (ρ₁−ρ₂)gk − σk³ = 0, giving complex roots ω = ωᵣ − iγ whose growth rate γ(k) > 0 marks the unstable band. Three visualization panels: (1) Main animated scene showing the two stratified fluid layers (blue heavy / orange light) with the evolving interface (built from the most-unstable mode and harmonics via η(x,t) = A·exp(γt)·cos(kx+ωᵣt)), a live perturbed velocity vector field grid, and advected streamline tracer particles showing the shear-driven roll-up; MHD mode overlays horizontal magnetic field lines; a status banner flags stable vs unstable. (2) Growth Rate γ(k) plot showing the imaginary part of ω across wavenumbers 0.05–8.0, with the unstable band shaded amber, the most-unstable mode k* marked, and a stable-band message when γ=0 everywhere. (3) Dispersion ωᵣ(k) plot showing the real frequency, color-coded cyan for stable gravity-capillary-Alfvén waves and dashed amber where the band is unstable. Physics: KH mechanism (Bernoulli pressure dip at crests amplifies corrugation), three stabilizers — stable density stratification (ρ₁>ρ₂ gravity term on long waves, why the sea needs a critical wind speed), surface tension σk³ (high-k cutoff, why ripples are smooth), and horizontal magnetic field B²k²/μ₀ Alfvén tension (can fully suppress KH, why the solar-wind magnetopause stays sharp). Adjustable parameters: density ratio ρ₂/ρ₁ (0.1–2.0), shear ΔU (0–3), gravity g (0–2), surface tension σ (0–1), and an MHD toggle with magnetic tension B²/μ₀ (0–2). Five scenario presets: Classic Shear (equal density pure shear, all k unstable), Wind over Sea (light upper fluid + gravity + tension, ocean-wave regime), Jupiter Bands (near-equal density, strong shear, gas-giant atmosphere), Magnetopause (MHD mode, magnetic stabilization of the solar-wind boundary), Stratified (heavy stratification + low shear + strong gravity/tension, near-stable). Real-time statistics: regime (KH / Stratified-KH / MHD-KH / Stable / MHD-Stable), growth rate γ, most-unstable wavenumber k*, e-folding time 1/γ. Educational content covers the KH mechanism, the three stabilization mechanisms, and applications (ocean waves, Jupiter's Great Red Spot shear boundary, Earth's magnetopause plasma leakage, billow clouds, jet exhaust, scramjet combustion, supernova-remnant and accretion-disk turbulence). Multi-language support (zh, en, es, fr, de, ru, pt).

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