The formula
- Along a streamline
p + ½ρv² + ρgy = constant- Between two points on the same streamline
p₁ + ½ρv₁² + ρgy₁ = p₂ + ½ρv₂² + ρgy₂- Continuity equation, almost always needed alongside it
A₁v₁ = A₂v₂
What the symbols mean
| Symbol | Meaning | Unit |
|---|---|---|
p | Pressure at the point | Pa |
ρ | Density of the fluid, 1.00 × 10³ kg/m³ for fresh water | kg/m³ |
v | Flow speed at the point | m/s |
g | Acceleration due to gravity | 9.80 m/s² |
y | Height above a chosen reference level | m |
A | Cross-sectional area of the pipe | m² |
When it applies
- Steady, smooth flow of an incompressible fluid with negligible viscosity.
- Both points lie on the same streamline.
- Nothing between the two points adds or removes energy, so no pump, no turbine and no long stretch of rough pipe.
- Pair it with the continuity equation, since the two speeds are usually linked by the pipe areas rather than given outright.
Worked example
Problem. Water flows through a horizontal pipe at 2.0 m/s where the pressure is 1.20 × 10⁵ Pa. The pipe narrows and the water speeds up to 6.0 m/s. What is the pressure in the narrow section?
- The pipe is horizontal, so y₁ = y₂ and the ρgy terms cancel from both sides.
- Rearrange for the unknown: p₂ = p₁ + ½ρ(v₁² - v₂²).
- Evaluate the bracket with ρ = 1000 kg/m³: ½ρ(v₁² - v₂²) = (500)(4.0 - 36) = (500)(-32) = -1.6 × 10⁴ Pa.
- Add it to the known pressure: p₂ = 1.20 × 10⁵ - 1.6 × 10⁴ = 1.04 × 10⁵ Pa.
Answer. 1.04 × 10⁵ Pa, about 16 kPa below the pressure in the wide section.
Common mistakes
- Applying it across a pump, a valve or a long rough pipe, where friction and added energy break the assumptions it rests on.
- Expecting pressure to rise where the fluid moves faster. The fast section is the low-pressure one.
- Ignoring the continuity equation and treating the two speeds as independent.
- Using gauge pressure at one point and absolute pressure at the other.
Related formulas
- Pressure formula:
p = F / A - Work-energy theorem:
W_net = ΔKE = ½mv² - ½mv₀²