Newton's second law, written for a fluid. Momentum in, momentum out: ρ(∂v/∂t + v·∇v) = −∇p + μ∇²v, with ∇·v = 0. Set the viscosity to zero and it collapses to Euler; drop the inertia at low Reynolds number and it collapses to Stokes; plane Poiseuille flow is an exact solution you can watch balance. Whether smooth solutions always exist is a million-dollar open problem. Terms go in, the balance is computed, the residual comes out. The blue team builds it; the red team attacks it.
source Navier (1822), Mém. Acad. Sci. 6, 389; Stokes (1845), “On the theories of the internal friction of fluids in motion”, Trans. Camb. Phil. Soc. 8, 287–319 — archive.org · full text. AMBER: pre-DOI printings; page-image scan, no stable canonical URL. Rendered, not quoted.
Every parcel obeys F = ma, per unit volume. Read the equation term by term:
ρ ∂v/∂t unsteady acceleration. ρ(v·∇)v the convective term — nonlinear, quadratic in v, the source of turbulence and of the smoothness question. −∇p the pressure force. μ∇²v viscous diffusion of momentum. And ∇·v = 0 — incompressibility, the constraint that fixes the pressure.
Live terms for the current profile (plane Poiseuille, exact):
| term | value | status |
|---|
Navier-Stokes is Newton for a continuum: ρ Dv/Dt = −∇p + μ∇²v. Two neighbouring spheres are its special cases, not rivals.
Kill viscosity (μ→0) and integrate along a steady streamline — you get the-bernoulli-equation. Keep viscosity, drop inertia, drive a channel by pressure — you get the exact solution the-poiseuille-flow. This sphere is the trunk both branch from.
The blue team's live check: re-derive the Euler and Stokes limits, re-verify that Poiseuille's exact profile drives the steady residual to zero, and confirm the convective term scales quadratically. If red drops the viscous term, this badge is where it shows.
Four numbers feed the balance: density ρ, dynamic viscosity μ, the driving pressure gradient G = −dp/dx, and the channel half-gap H. From them the machine builds the exact plane-Poiseuille profile
vx(y) = (G / 2μ)·(H² − y²)
a parabola pinned to zero at both walls (y = ±H), peaking at the centreline. Its curvature is exactly what the pressure gradient demands — that is what the engine below verifies, and what the Reynolds number tells you which regime it lives in.
FULL: all terms live. Poiseuille is fully-developed, so time and convective terms vanish identically and the viscous term alone must balance G.
Move any slider — the profile, the residual and the Reynolds number are computed on the spot from the equation, never looked up.
What the machine proves: the exact Poiseuille profile makes the steady momentum residual G + μvx″(y) vanish to a finite-difference tolerance of 1e−9 at every y; setting μ=0 removes the viscous term exactly (Euler); at low Reynolds number the inertial-to-viscous ratio equals Re and vanishes (Stokes); and the convective term scales as a² under v → a·v — nonlinear, by measurement.
The blue team's witness (left) re-checks these live; the red team (right) tries to make them wrong.
The plane-Poiseuille profile is also only stable below a critical Reynolds number (linear theory ≈ 5772; experiment transitions far lower). Above it the parabola is a mathematical solution the real flow abandons — so “solution” and “what the fluid does” part ways. AMBER: the transition Re is regime- and disturbance-dependent, not a single constant.
“Viscous flow is just inviscid flow with a little friction.” Cut. The viscous term carries the highest derivative; dropping it changes the equation's order and its boundary conditions (no-slip dies). Euler is a singular limit, not a small correction — d'Alembert's paradox is the receipt.
“Doubling the pressure drop doubles the flow, always.” Kept, corrected. True only while laminar: here Q ∝ G exactly, and Poiseuille through a pipe gives Q ∝ r⁴. Once turbulent, the linear law breaks.
“Navier and Stokes derived the same thing together.” Cut. Navier (1822) reached the equations from a flawed molecular model; Stokes (1845) gave the continuum-stress derivation that survives. Same equations, different and decades-apart roads.
The red team's move: drop the viscous term μ∇²v — treat the viscous channel flow as if it were inviscid. Now nothing balances the pressure gradient. The blue team's witness (window 7) is watching.
Delete the viscous term and Poiseuille's residual jumps from ~0 to G — the profile no longer solves the equation. The witness recomputes, disagrees with the known zero, and turns red. Nothing is faked; the attack is real and it is caught.