Squeeze the pipe and the fluid must speed up — mass cannot vanish. For incompressible steady flow the volume rate Q = A·v is constant along the tube, so A₁v₁ = A₂v₂; in differential form div v = 0. Rendered, not quoted.
SOURCE Leonhard Euler, “Principes généraux du mouvement des fluides” (1757), Mém. Acad. Sci. Berlin — the continuity equation / conservation of mass. Eneström E226. AMBER historical facsimile.
Take a tube of varying cross-section carrying an incompressible fluid in steady flow. No fluid piles up and none is created, so whatever volume enters per second must leave per second.
Halve the area and the speed must double. The local statement of the same law is the divergence: div v = ∂u/∂x + ∂w/∂y = 0 everywhere.
Mass cannot vanish — that is the whole content of continuity. A₁v₁ = A₂v₂ and div v = 0 are the incompressibility that lets the-bernoulli-equation trade speed for pressure: continuity says the narrow section is faster, Bernoulli then says it is at lower pressure.
Live re-check of the engine's invariants. Confirms green; flips red the instant window 6 tampers with the conservation law.
Inlet area A₁, inlet speed v₁, and an outlet area A₂. Drag the throat to contract the tube.
Live engine: it computes v₂ from Q = A·v and paints a divergence-free field. Faster flow = denser, longer streamlines through the throat.
Proven result — the flow rate that came in is the flow rate that goes out.
“Continuity always means faster in a narrow pipe.” Only if incompressible. For a compressible gas it is ρ₁A₁v₁ = ρ₂A₂v₂ — mass, not volume. Past the sonic throat of a converging–diverging nozzle, area increase speeds supersonic flow up. Volume continuity is the incompressible special case.
“Conserve A + v: what the throat takes from area it hands to speed.”
Correction: the conserved quantity is the product A·v, not the sum. Halving A must double v (product fixed at 6), not just add back the lost area. The tamper in window 6 plants exactly this error.
“div v = 0 means the fluid never rotates.”
Correction: zero divergence means volume-preserving; it says nothing about curl (vorticity). A field can have div v = 0 and still spin.
Planted void: swap the conserved product A·v for the sum A+v. A contraction then fails to speed the flow correctly — and the witness in window 7 catches it live.