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Equation of continuity
Intermediate
Mass is conserved, so speed increases in narrower pipes.
A₁v₁ = A₂v₂For incompressible steady flow A₁v₁ = A₂v₂ (volume flow rate is constant). The fluid speeds up in narrow sections.
A pipe narrowing from 10 cm² to 2 cm²: speed ×5.
Blocking a hose end with a thumb speeds up the jet.
In more detail
The equation of continuity says that for incompressible flow, the product of cross-sectional area and speed is constant, A1v1 = A2v2. Narrow pipes make fluid flow faster.
Real-world uses
- A garden hose speeds up water when its nozzle is narrowed.
- Blood flow speed changes in arteries of different widths.
- A garden hose is narrowed at the end to make the water shoot out faster, because the same volume of water must pass through a smaller area.
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Frequently asked questions
Why does narrowing increase speed?
Because the same volume of fluid must pass through a smaller area each second.
Does it apply to gases?
Only when the gas speed is low enough that its density stays roughly constant.
Does the volume flow rate change along a pipe?
For an incompressible fluid in steady flow, the flow rate is constant, so a narrow section means a faster velocity.
Related topics in Fluids & Elasticity
Hydrometer · Fluid dynamics · Streamline and turbulent flow · Bernoulli's principle · Applications of Bernoulli · Torricelli's theorem