NAVIER–STOKES, IN PLAIN ENGLISH
Inside a swirl.
Can a tiny part of a fluid become infinitely fast?
Five short chapters.
Follow the motion.
No advanced math needed.
A FIELD GUIDE TO BIG IDEASRead the paper Inside the spinning core
Preparing the fluid view…
Inner pathsNearby pathsDrag to turn
Core enlarged for visibility. Readouts follow the paper’s scaling laws.
INWARD → AROUND → OUT
Follow a dot through the swirl.
Follow the flowA local view
Press play and follow the moving dots.
WHAT ARE NAVIER–STOKES EQUATIONS?
Rules for how fluids move.
They connect changes in motion to pressure, internal friction, and outside forces. The challenge is whether smooth motion can always stay smooth.Go deeperThe equation, the scaling, and what this shows
The equation, translated
∂ₜu + (u · ∇)u = −∇p + ν∇²u + f
- Change in motion
- Local change + motion carrying motion.
- Pressure
- Pushes fluid from higher toward lower pressure.
- Viscosity
- Smooths differences in nearby fluid motion.
- Outside force
- The carefully constructed forcing in the paper.
The extra condition ∇ · u = 0 means fluid keeps its volume. The shrinking core is a region fluid passes through, not a sealed parcel of fluid.
The numbers behind the bars
Let τ mean time left before the singularity. Section 2.1 gives the following leading scales, up to constant factors. We use h = 0.005, within the stated range 0 < h < 0.01.
Radius ∝ τ½Height ∝ τ½−hSpeed ∝ τ−½−hVolume ∝ τ³⁄₂−hCore energy ∝ τ½−3h
Core energy scales like volume × speed². Its positive exponent makes it decrease as τ approaches zero. The bars normalize these power laws to an earlier reference moment. They describe the approach to the singularity, not startup from rest.
What the animation shows
The paths explain inward spiraling and outward escape. The ring colors distinguish two pulse families. Geometry, wavelength, and playback speed use visual exaggeration. The model does not solve the fluid equations, measure a physical experiment, or verify the proof. It stops before the singularity.
What the paper claims
For every positive viscosity, the authors construct a three-dimensional flow starting at rest with a smooth, compactly supported external force. Its peak velocity becomes unbounded in finite time, while total kinetic energy stays bounded. The paper targets alternatives C and D of the Millennium problem.