Fluid Continuity

Water flows through a pipe that narrows and widens again. Change the cross-sections and the flow rate, and watch tracer particles speed up in the narrow section while the pressure there drops: the continuity equation and Bernoulli's principle together.

Speed

How the model works

For an incompressible fluid, whatever volume enters a section per second must leave it. So the volume flow rate is the same everywhere along the pipe:

Q = A₁ v₁ = A₂ v₂ ⇒ v₂ = v₁ · A₁ / A₂

Halve the area and the speed doubles. Along a streamline of steady, frictionless flow, Bernoulli's equation links pressure and speed (horizontal pipe):

P + ½ ρ v² = constant

The fluid speeds up only if something pushes it forward, so pressure must fall where the pipe narrows. That is the Venturi effect. The vertical manometer tubes show the pressure as a height of water, h = P/(ρg), measured relative to the outlet.

Ideal fluid: incompressible, no viscosity, no turbulence, uniform velocity across each section. Real pipes lose pressure to friction, and sharp narrowings cause turbulence.

Guided experiments

Halve the throat. Inlet 20 cm², throat 10 cm², Q = 2 L/s. Prediction: v goes from 1 m/s to 2 m/s, and the pressure drop is ½ρ(2² − 1²) = 1.5 kPa, about 15 cm of water.
Double the flow. Same pipe, Q = 4 L/s. Prediction: all speeds double and the pressure drop quadruples to 6 kPa, because it depends on v².
Garden hose nozzle. Throat 1 cm². Prediction: water exits 20 times faster than in the inlet. Squeezing a hose works this way.

Challenge

A Venturi meter has an inlet of 50 cm² and a throat of 20 cm². The manometers differ by 30 cm of water. What is the flow rate?

Reveal the explanation

ΔP = ρgh = 1000 × 9.81 × 0.3 ≈ 2943 Pa. With v₂ = 2.5 v₁: ΔP = ½ρ v₁²(2.5² − 1) = 2625 v₁², so v₁ ≈ 1.06 m/s and Q = A₁v₁ ≈ 0.0050 × 1.06 ≈ 5.3 L/s. This is how Venturi flow meters work.

References

OpenStax, University Physics Volume 1, §14.5 “Fluid Dynamics” and §14.6 “Bernoulli's Equation”.