Continuity Equation
Conservation of flow in an incompressible fluid.
Formula
Variables
How it works
For an incompressible fluid, the amount of fluid entering a pipe each second must equal the amount leaving it, so the volume flow rate stays constant. Since flow rate is area times speed, a smaller cross-sectional area forces the fluid to move faster, and a larger area lets it slow down. This result comes from conservation of mass applied to a steady flow with no leaks or storage in the pipe.
When to use it
Use this when fluid flows steadily through pipes, hoses, nozzles, blood vessels, or channels and the fluid can be treated as incompressible, such as water. It is especially useful for finding how speed changes when the pipe narrows or widens.
Worked example, step by step
Water flows through a pipe that narrows from 0.020 m^2 to 0.0050 m^2. The speed in the wider section is 1.5 m/s. Use A1v1 = A2v2. (0.020)(1.5) = (0.0050)v2 0.030 = 0.0050v2 v2 = 0.030 / 0.0050 = 6.0 m/s Final answer: the water speed in the narrower section is 6.0 m/s.
Common mistakes to avoid
- Using the formula for compressible gases without checking whether density changes are important
- Mixing up area and speed, or assuming the smaller area has the smaller speed
- Forgetting that the two sections must be part of the same steady flow, so the volumetric flow rate is the same
Did You Know?
The continuity equation is why pinching a hose nozzle makes the water shoot out faster.
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