Bernoulli Pressure Change
Downstream pressure from an upstream state, two velocities, and an elevation change.
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The engineering
Bernoulli is an energy budget with three accounts — pressure, velocity, height — and a fixed total along a streamline (no pumps, no friction, incompressible). Speed the flow up through a contraction and the pressure account pays for it; that trade is the venturi meter, the carburetor, and the aspirator in one line.
The fine print matters: this is the *lossless* answer, an upper bound on p₂. Real pipe runs lose head to friction (see the Darcy-Weisbach card), and if p₂ dives toward the vapor pressure, the fluid will cavitate rather than honor your algebra.
Where this math comes from
Daniel Bernoulli published the pressure-velocity trade in Hydrodynamica (1738), reportedly to his father Johann's competitive fury — Johann backdated his own Hydraulica to claim priority. The clean equation on this card, though, is Leonhard Euler's 1757 formulation; Bernoulli had the physics, Euler the calculus.
Giovanni Battista Venturi showed in 1797 that a smooth contraction-diffuser measures flow by its pressure dip, and Clemens Herschel turned that into the commercial venturi meter in 1887 — the equation earning its keep in waterworks, then in every pitot-static system aloft.
- 1738Daniel BernoulliHydrodynamica — the pressure-velocity-height budget.
- 1757Leonhard EulerThe equation in its modern form.
- 1797G. B. VenturiThe converging-diverging meter principle.
- 1887Clemens HerschelCommercial venturi meter for waterworks.
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