Bernoulli & Venturi
Where a pipe narrows, the fluid speeds up — and its pressure drops. Find the speed and pressure after a narrowing or a climb, or turn it around: measure the pressure difference in a Venturi meter and get the flow rate.
Why Faster Means Lower Pressure
Everything that flows into a pipe must come out again, so where the pipe is narrower the fluid has to move faster: Q = A·v (continuity). To speed up, it must be pushed from behind, which means the pressure in the narrow part is lower. Bernoulli’s equation keeps the books: pressure, motion energy and height energy together stay constant along the flow.
A Venturi meter uses exactly this. Measure the pressure in the wide pipe and in the throat, and the difference tells you how fast the fluid flows. The same effect makes a garden sprayer suck up liquid and helps lift a plane’s wing.
Pressure and Speed Along the Pipe
pressureNarrowing a Water Pipe
2 m/s in the wide part, level pipe| D₂ / D₁ | Speed in the throat | Pressure drop |
|---|---|---|
| 0.9 | 2.47 m/s | 1.05 kPa |
| 0.7 | 4.08 m/s | 6.32 kPa |
| 0.5 | 8.00 m/s | 29.9 kPa |
| 0.3 | 22.2 m/s | 244 kPa |
Worked Techniques
The speed changes with the square of the diameter ratio.
Bernoulli between the two points: the extra motion energy and any height gained come out of the pressure.
The ideal flow rate from the measured pressure difference. Real meters multiply this by a discharge coefficient of about 0.98.
Sources & Verification
- D. Halliday, R. Resnick & J. Walker, Fundamentals of Physics, 10th ed., Wiley, 2014 — Ch. 14 (Fluids).
- Y. A. Çengel & J. M. Cimbala, Fluid Mechanics: Fundamentals and Applications, 4th ed., McGraw-Hill, 2018 — Ch. 8 (Internal Flow).