Tracing the flow…

NAVIER–STOKES / A VISUAL STUDY

The shape
of a singularity.

A fluid spirals inward, spins faster,
and stretches along its axis.
Follow the motion into the core.

ANGULAR ROTATION
SlowerFaster
MOTION BALANCE · ACTUAL EQUATION ∂tu + (u · ∇)u − νΔu + ∇p = f Acceleration, transport, viscosity and pressure balance a smooth force.
INCOMPRESSIBILITY + CORE SHAPE ∇ · u = 0  ·   ℓr ≍ τ1/2   ℓz ≍ τ1/2−h Radial inflow is balanced by axial outflow; the core becomes slender.
WHY SMOOTHNESS BREAKS · τ = 1 − t → 0 |uθ|, |uz| ≍ τ−1/2−h → ∞ Peak velocity diverges at t = 1 while shrinking volume keeps energy finite.
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Smooth forcingRadial inflowVortex stretchingVelocity growth

BEHIND THE VISUALIZATION

A picture of the motion.

This study brings the composition of OpenAI’s vortex illustration into motion: a broad teal inflow, blue spirals, and an orange core flowing in both axial directions.

Each strand is a trajectory trail. It begins as one point, then grows behind a leading tip carried by the three-dimensional, incompressible velocity field. Once the trail reaches its maximum history length, its tail advances along the same path.

New strands enter near the middle as older ones leave along the axis. As the timeline advances, the swirl core and the region supplying new strands narrow. Color follows each point’s angular rotation about the axis; it does not represent temperature or total speed.

The optional vector field samples that same velocity function at fixed points throughout the visible flow volume. The arrows rotate in place and grow brighter and longer as the field evolves. Their lengths are capped so they remain legible without covering the strands.

This is an explanatory animation, not a numerical reconstruction of the proof. The timing, profiles, colors, and aspect ratios are illustrative. The fine oscillatory corrections in the full construction are omitted, and the animation ends before a singularity.

In the paper, with τ = 1 − t and 0 < h < 1/100, the core scales as ℓr ≍ τ1/2 and ℓz ≍ τ1/2−h, while characteristic azimuthal and axial speeds scale as τ−1/2−h. Its energy is of order τ1/2−3h, so it can remain bounded even while peak velocity diverges.

Read the research ↗Explore the formalization ↗