Case 1
- Upstream area A₁
- 0.1m²
- Upstream velocity v₁
- 2m/s
- Downstream area A₂
- 0.025m²
8.00m/s
Open with these valuesContinuity Equation Calculator
8.00m/s
Result: 8.00 m/sThe same volume passes every cross-section each second, so a narrower pipe means a faster fluid: v₂ = A₁ × v₁ ÷ A₂. Quarter the area and the velocity quadruples. Both areas must use the same unit; the answer comes back in the unit of the upstream velocity.
Worked examples
Case 2
- Upstream area A₁
- 0.05m²
- Upstream velocity v₁
- 4m/s
- Downstream area A₂
- 0.01m²
20.00m/s
Open with these valuesCase 3
- Upstream area A₁
- 0.2m²
- Upstream velocity v₁
- 3m/s
- Downstream area A₂
- 0.6m²
1.00m/s
Open with these valuesHow it's calculated
v₂ = (A₁ × v₁) ÷ A₂
- StepMeasure the cross-section at the inlet and at the outlet in the same unit.
- StepEnter the speed the fluid has at the inlet.
- ResultRead the outlet speed — it rises exactly as much as the area shrinks.
Reference table
| A₁, v₁, A₂ | Pipe | v₂ (m/s) |
|---|---|---|
| 0.1, 2, 0.025 | Narrows to a quarter | 8 |
| 1, 1, 1 | Unchanged | 1 |
| 0.05, 4, 0.01 | Narrows to a fifth | 20 |
| 0.2, 3, 0.6 | Widens threefold | 1 |
| 0.0314, 1.5, 0.00785 | Diameter halved | 6 |
Questions
How do I calculate velocity with the continuity equation?
Multiply the upstream area by the upstream velocity, then divide by the downstream area: v₂ = (A₁ × v₁) ÷ A₂. A pipe narrowing from 0.1 m² to 0.025 m² with fluid entering at 2 m/s gives 8 m/s.
What is the continuity equation?
The continuity equation, A₁v₁ = A₂v₂, expresses conservation of mass for an incompressible fluid: the same volume passes every cross-section each second. Because area times velocity stays constant, a smaller area has to be matched by a higher velocity.
Why does fluid speed up when a pipe narrows?
Because the volume passing each point per second is fixed. Halve the area and the velocity doubles; cut it to a quarter and the velocity quadruples. It is the same effect that makes a partly covered hose shoot water farther.
What units should I use?
Square metres for both areas and metres per second for the upstream velocity, which returns the downstream velocity in metres per second. The area units cancel, so any consistent pair works — but mixing m² with cm² gives a wrong answer.
When does the continuity equation hold?
For an incompressible fluid in steady flow with no leaks and no branches, so that mass is conserved between the two sections. For a compressible gas at high speed the density changes and the mass form ρ₁A₁v₁ = ρ₂A₂v₂ is needed instead.
Sources and last check
Information, not professional advice.