Equation of Continuity
Simple Explanation
For an incompressible fluid flowing through a pipe, the equation of continuity — A₁v₁=A₂v₂ — says the same volume of fluid must pass every cross-section each second, so the fluid speeds up wherever the pipe narrows.
Why Do We Need It?
This single conservation principle explains why water shoots out faster when you cover part of a hose nozzle, and it is the essential first step before applying Bernoulli's equation to any pipe of varying width.
See It
A pipe that starts wide and narrows to a smaller cross-section, with A1 v1 labelled in the wide section and A2 v2 labelled in the narrow section, and arrows showing the flow direction
Formula
Equation of Continuity
A₁v₁ = A₂v₂
For an incompressible fluid flowing through a pipe, the same volume of fluid must pass every cross-section every second — so where the pipe narrows, the fluid must speed up.
- A₁, A₂
- — cross-sectional area at two points along the pipe, in square metres (m²)
- v₁, v₂
- — flow speed at those same two points, in metres per second (m/s)
When to use it: Whenever fluid flows through a pipe or channel that changes cross-sectional area, and the speed at one point needs to be found from the speed and areas elsewhere.
Worked Example
Find a fluid's speed using the equation of continuity
A pipe narrows from A₁=0.02 m² (where v₁=2 m/s) to A₂=0.005 m². Find the speed v₂ in the narrow section.
Why Does This Work?
The volume of fluid passing any cross-section per second is Av (area times speed) — since the fluid is incompressible, it cannot pile up or vanish anywhere along the pipe, so this volume flow rate must be identical everywhere along the pipe, giving A₁v₁=A₂v₂.
Real-Life Example
Covering part of a garden hose's nozzle
Partially covering the end of a garden hose with a thumb makes the water shoot out much faster and farther.
Covering the opening reduces its cross-sectional area — by the equation of continuity, the same volume of water must now pass through a smaller area every second, so its speed increases.
Practice
A pipe narrows from A₁=0.03 m² (v₁=1.5 m/s) to A₂=0.01 m². Find v₂.
MediumCommon mistake
Assuming a narrower pipe means slower flow (like a traffic jam) — for an incompressible fluid, it is the opposite: a smaller area forces the same volume through faster, not slower.
Quick Review
- A₁v₁ = A₂v₂ — volume flow rate stays constant along a pipe.
- A narrower section means faster flow.
- Only valid for an incompressible fluid.