Continuity
A short lesson with diagrams, a worked example, a problem to try, and a quick check. Then quiz yourself on 5 practice questions.
An incompressible fluid can’t pile up or thin out, so the same volume passes every point each second. That volume flow rate is Av, so A1v1 = A2v2: where the pipe narrows, the fluid speeds up.
1. Narrower means faster
2. It’s conservation of mass
Mass can’t appear or disappear inside the pipe. For a liquid, which doesn’t compress, that means equal volumes in and out every second.
Putting your thumb over a garden hose makes the water squirt faster for the same reason.
Watch for radius or diameter instead of area. Area goes as r², so a pipe with half the radius has a quarter of the area, and the fluid moves 4 times as fast, not 2.
3. Example, then try it
A hose has a cross-sectional area of 3.0 × 10⁻⁴ m², and water moves through it at 2.0 m/s. How long does it take to fill a 12 L (0.012 m³) bucket?
Flow rate: Av = (3.0 × 10⁻⁴)(2.0) = 6.0 × 10⁻⁴ m³/s.
Time: 0.012 ÷ (6.0 × 10⁻⁴) = 20 s.
Water flows at 1.5 m/s through a pipe of radius 2.0 cm. The pipe narrows to a radius of 1.0 cm. How fast does the water move in the narrow part? What is the volume flow rate?
Area ratio: A ∝ r², so A1/A2 = (2.0/1.0)² = 4.
Speed: v2 = v1(A1/A2) = (1.5)(4) = 6.0 m/s.
Flow rate: A1 = π(0.020)² ≈ 1.26 × 10⁻³ m², so Av ≈ (1.26 × 10⁻³)(1.5) ≈ 1.9 × 10⁻³ m³/s (about 1.9 L each second), the same in both parts.
Answer: 6.0 m/s; about 1.9 × 10⁻³ m³/s.
4. Common mistakes
Water flows through a pipe whose diameter is cut in half. How does the water’s speed in the narrow part compare with the wide part?
Half the diameter gives a quarter of the area (A ∝ d²). Since Av is constant, the speed is 4 times as large.
Ready to practice?
5 questions on Continuity, each with a hint and an explanation for every wrong answer.