Volume flow rate and continuity for incompressible flow

v is a speed in m/s. Q is volume per time in m³/s. The relation Q=A v̄ shows that the same mean speed gives a larger flow rate through a larger pipe area. Speed and volume flow are therefore not interchangeable and should be stated with their own units. Use mean velocity normal to each section. Volume-flow conservation assumes steady incompressible flow with no side branches; mass conservation is more general. The equation does not imply uniform local speed across a real pipe profile. Finally check units, assumptions, measurement conditions and reference quantities against the problem before using the result.

For steady incompressible flow, A v̄ remains constant. When area decreases, mean speed must rise so the same volume passes in the same time. This follows from mass conservation, not from Bernoulli alone. Real losses affect pressure but not the continuity balance. Use mean velocity normal to each section. Volume-flow conservation assumes steady incompressible flow with no side branches; mass conservation is more general. The equation does not imply uniform local speed across a real pipe profile. Finally check units, assumptions, measurement conditions and reference quantities against the problem before using the result.

If density may vary along the flow, as in compressible gases, mass flow ρA v̄ is conserved while volume flow need not be. Then ρ₁A₁v̄₁=ρ₂A₂v̄₂ links sections. For liquids under moderate conditions, constant density is often a good approximation. Use mean velocity normal to each section. Volume-flow conservation assumes steady incompressible flow with no side branches; mass conservation is more general. The equation does not imply uniform local speed across a real pipe profile. Finally check units, assumptions, measurement conditions and reference quantities against the problem before using the result.

Write the quantity and unit from the variable list next to every value: Q in m³/s; A in m²; v̄ in m/s. Substitute only after converting to a consistent unit system, and state sign or reference direction explicitly. The example Water flows through A₁=4 cm² at 1.5 m/s. If A₂=2 cm², continuity gives v̄₂=3.0 m/s. is a calculation check: result, unit and order of magnitude must fit the described situation. Round after intermediate calculations so small contributions are not lost prematurely. Use mean velocity normal to each section. Volume-flow conservation assumes steady incompressible flow with no side branches; mass conservation is more general. The equation does not imply uniform local speed across a real pipe profile. Finally check units, assumptions, measurement conditions and reference quantities against the problem before using the result.

The background states this starting condition: Q=A v̄ defines the flow rate for a suitable mean velocity. The continuity form follows from mass conservation. That limits when the result is a valid model prediction. Compare the problem’s boundary conditions with this assumption before substituting values; real components, material properties or loads can introduce additional effects. The cited reference supports the relation used here but does not remove the need to check whether the specific case fits the model. Use mean velocity normal to each section. Volume-flow conservation assumes steady incompressible flow with no side branches; mass conservation is more general. The equation does not imply uniform local speed across a real pipe profile. Finally check units, assumptions, measurement conditions and reference quantities against the problem before using the result.