8.4 Fluid Flow Continuity

Continuity Equation

For an incompressible fluid flowing through a pipe, mass flow rate is constant: A₁v₁ = A₂v₂, where A is cross-sectional area and v is fluid speed. This is a statement of conservation of mass — the same volume of fluid must pass every point per second since fluid can't be created or destroyed inside the pipe.

Speed Depends on Area

Since A₁v₁ = A₂v₂, speed and area are inversely proportional. Where a pipe narrows, fluid speeds up; where it widens, fluid slows down. This is why water shoots faster from a narrow nozzle than a wide hose opening — the volumetric flow rate stays the same.

Volume Flow Rate

The quantity Av represents volume flow rate (m³/s), the volume of fluid passing a cross-section per unit time. Continuity states this flow rate is the same at every point along a single, non-branching pipe with incompressible, steady flow — no fluid leaks in or out.

Key terms

Continuity equation

A₁v₁ = A₂v₂ — for incompressible, steady flow, the product of cross-sectional area and speed is constant along a pipe.

Volume flow rate

Av, measured in m³/s; the volume of fluid passing a given cross-section per unit time, constant along a non-branching pipe.

Effect of narrowing a pipe

Fluid speed increases because the smaller area requires higher velocity to maintain the same volume flow rate.

Effect of widening a pipe

Fluid speed decreases since a larger area needs lower velocity to keep Av constant.

Physical basis of continuity

Conservation of mass — for an incompressible fluid, mass entering a pipe segment per second must equal mass leaving it.

Try an AP-style question

Water flows through a pipe that narrows to half its cross-sectional area. What happens to its speed?

1. It doubles
2. It halves
3. It stays the same
4. It quadruples