Faster flow means lower pressure — the trade every wing and Venturi lives on. Along one streamline the sum p + ½ρv² + ρgh never changes: push the fluid faster and the pressure must fall to pay for it. Down the center, data flows: the geometry goes in, the engine conserves head, the pressure drop comes out. The blue team builds and defends it; the red team tries to break it.
source Daniel Bernoulli, Hydrodynamica, sive de viribus et motibus fluidorum commentarii (Strasbourg, 1738) — archive.org/details/bub_gb_3yRVAAAAcAAJ. Year marked AMBER (18th-c. facsimile). Rendered, not quoted.
Take steady, inviscid, incompressible flow. Along any one streamline the energy per unit volume is constant:
p + ½ρv² + ρgh = H — static pressure + dynamic pressure + elevation head. Nothing is added or removed, so if one term rises another must fall.
Continuity ties the speeds to the geometry: A₁v₁ = A₂v₂. Squeeze the tube (A₂ < A₁) and the flow must speed up — so, on a level streamline, its pressure must drop.
Head budget at inlet vs throat for the current setting:
| station | v (m/s) | p (Pa) | H (Pa) |
|---|
Bernoulli alone is one equation in two unknowns (p and v). It only bites when paired with its neighbour — the-continuity-equation, A₁v₁ = A₂v₂, mass in = mass out.
Continuity fixes the throat speed from the areas; Bernoulli then converts that speed into a pressure drop. Together they are the Venturi meter, the carburettor, the aspirator, the Pitot tube. Each sphere is the next one's premise: geometry → speed → pressure.
The blue team's live check: recompute the pressures from the conserved head and confirm that faster flow reads lower pressure and the drop equals ½ρ(v₂²−v₁²). If red flips the sign, this badge is where it shows.
Feed the engine two numbers: the inlet speed v₁ and the contraction ratio A₂/A₁ of a horizontal Venturi tube. Fluid is water: ρ = 1000 kg/m³, g = 9.81 m/s². The total head H is held fixed along the streamline.
Continuity turns the ratio into a throat speed v₂ = v₁⋅(A₁/A₂); a narrower throat means a faster jet. That speed is what the panel below trades against pressure.
Colour = pressure (cyan high → red low). The throat runs fast and red; the wide ends run slow and blue.
Move any control — every pressure is computed live from the conserved head, never looked up.
What the machine proves: on a level streamline the throat is faster and therefore at lower pressure, and the drop is exactly Δp = ½ρ(v₂² − v₁²) > 0. Bring the flow to rest and it recovers the full stagnation pressure p₀ = p + ½ρv².
The blue team's witness (left) confirms the sign and value live; the red team (right) tries to make faster read higher.
Comparing two different streamlines, or across a rotating rotor, the "faster is lower" rule can fail outright — which is why the panel restricts itself to one horizontal streamline with a fixed head.
"Air travels over the longer top of a wing and must go faster, so lower pressure lifts the plane." Cut. The equal-transit-time story is false; lift comes from circulation and downwash (see Kutta–Joukowski). Bernoulli links speed to pressure — it does not explain why the air speeds up.
"Faster fluid always has lower pressure." Corrected. Only along one streamline, only inviscid/incompressible/steady, only with no energy added. Not between streamlines, not across a pump.
"The equation is p + ρv² = const." Cut. The dynamic term is ½ρv² (kinetic energy per volume); the ρgh term is only dropped when the streamline is level.
The red team's move: flip the sign of the dynamic term to p − ½ρv², so faster flow reports higher pressure. The blue team's witness (window 7) is watching.
Flip the sign and the throat reads a pressure rise instead of a drop — the witness recomputes, sees faster-is-higher, and turns red. Nothing is faked; the attack is real and it is caught.