Bernoulli's Equation
Bernoulli's equation relates pressure, velocity, and height in fluid flow.
Learn the formula with practical worked examples.
The Formula
Bernoulli's equation states that in a steady, incompressible, frictionless flow, the total mechanical energy per unit volume remains constant along a streamline. As fluid speed increases, its pressure decreases, and vice versa.
This principle was published by Swiss mathematician Daniel Bernoulli in 1738. It forms the foundation of aerodynamics and explains how airplane wings generate lift.
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| P | Static pressure of the fluid | pascals (Pa) |
| rho (ρ) | Fluid density | kg/m³ |
| v | Flow velocity | m/s |
| g | Gravitational acceleration (9.81) | m/s² |
| h | Height above a reference point | m |
Between Two Points
Example 1 — Water Pipe Narrowing
Water flows through a horizontal pipe. At point 1 the pressure is 200,000 Pa and velocity is 2 m/s. At a narrower point 2 the velocity is 5 m/s. Find the pressure at point 2. (Water density = 1000 kg/m³)
Since the pipe is horizontal, h₁ = h₂, so the height terms cancel.
P₁ + ½ρv₁² = P₂ + ½ρv₂²
P₂ = P₁ + ½ρ(v₁² - v₂²)
P₂ = 200,000 + ½(1000)(4 - 25)
P₂ = 200,000 + 500(-21) = 200,000 - 10,500
P₂ = 189,500 Pa
Example 2 — Water Tank Drain Speed
An open water tank has a small hole 3 m below the surface. Find the speed of water leaving the hole. (This is Torricelli's theorem, a special case of Bernoulli.)
At the surface: v₁ = 0 (large tank), P₁ = atmospheric. At the hole: P₂ = atmospheric, h₂ = 0.
The pressure terms and v₁ cancel, leaving: ρgh = ½ρv₂²
v₂ = sqrt(2gh) = sqrt(2 x 9.81 x 3) = sqrt(58.86)
v₂ = 7.67 m/s
When to Use It
Use Bernoulli's equation when:
- Calculating pressure changes in pipes with varying cross-sections
- Analyzing lift on airplane wings and airfoils
- Designing Venturi meters and flow measurement devices
- Estimating drain speed from tanks and reservoirs
- Understanding how carburetors and atomizers work