Continuity Equation
Conservation of mass in incompressible flow.
Formula
Variables
Example
Narrow pipe speeds flow
How it works
The continuity equation says that for an incompressible fluid, the amount of fluid passing any cross-section each second must stay the same. If the pipe gets narrower, the fluid has to move faster so the same volume still gets through. This comes from conservation of mass: fluid is not created or destroyed as it flows, so the volumetric flow rate remains constant.
When to use it
Use this formula when analyzing flow in pipes, nozzles, ducts, or channels where the fluid density is effectively constant. It is especially useful for finding an unknown speed or area when the other section of the flow is known.
Worked example, step by step
A garden hose has a cross-sectional area of 3.0 cm^2 and water flows through it at 2.0 m/s. The hose is fitted with a nozzle that reduces the area to 0.75 cm^2. Use A1v1 = A2v2. Substitute: (3.0 cm^2)(2.0 m/s) = (0.75 cm^2)(v2) Calculate the left side: 6.0 = 0.75v2 Solve for v2: v2 = 6.0 / 0.75 = 8.0 m/s Final answer: the water speed in the nozzle is 8.0 m/s.
Common mistakes to avoid
- Mixing units for area or speed without converting them consistently
- Using the formula for compressible fluids without checking whether density is constant
- Reversing the areas and speeds, which gives the wrong unknown value
Did You Know?
The continuity principle guarantees that what flows into a pipe must flow out, unless it leaks.
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