Bernoulli's Equation
Bernoulli's equation P + ½ρv² + ρgh = constant relates pressure, velocity, and elevation in fluid flow.
Essential for fluid dynamics.
The Formula
Bernoulli's equation is one of the most important principles in fluid mechanics. It states that for a steady, incompressible, non-viscous flow, the total mechanical energy along a streamline remains constant. The equation was published in 1738 by Daniel Bernoulli, a Swiss mathematician, in his book Hydrodynamica.
The three terms represent different forms of energy per unit volume. The first term, P, is the static pressure — the pressure exerted by the fluid at rest. The second term, ½ρv², is the dynamic pressure — the kinetic energy per unit volume due to the fluid's motion. The third term, ρgh, is the hydrostatic pressure — the potential energy per unit volume due to elevation. When one form of energy increases, another must decrease to keep the total constant.
This tradeoff between pressure and velocity explains many everyday phenomena. When fluid speeds up through a narrow section of pipe, the pressure drops. This is the Venturi effect, used in carburetors, atomizers, and flow measurement devices. Airplane wings are shaped so air moves faster over the top surface, creating lower pressure above the wing than below, generating lift.
Bernoulli's equation applies strictly to ideal fluids with no viscosity or compressibility. For real-world applications with friction losses, engineers add correction terms. Despite this limitation, it remains an excellent first approximation for many practical problems in pipe systems, aerodynamics, and hydraulics.
Variables
| Symbol | Meaning |
|---|---|
| P | Static pressure (pascals, Pa) |
| ρ | Fluid density (kilograms per cubic meter, kg/m³) — water ≈ 1,000 kg/m³, air ≈ 1.225 kg/m³ |
| v | Flow velocity (meters per second, m/s) |
| g | Gravitational acceleration (9.81 m/s²) |
| h | Height above a reference point (meters, m) |
Example 1
Water flows through a horizontal pipe that narrows from 10 cm to 5 cm diameter. The velocity in the wide section is 2 m/s and the pressure is 200,000 Pa. What is the pressure in the narrow section?
Find velocity in narrow section using continuity: A1v1 = A2v2
v2 = v1 × (d1/d2)² = 2 × (10/5)² = 2 × 4 = 8 m/s
Apply Bernoulli (horizontal, so h terms cancel): P1 + ½ρv1² = P2 + ½ρv2²
P2 = 200,000 + ½(1,000)(2²) − ½(1,000)(8²) = 200,000 + 2,000 − 32,000
P2 = 170,000 Pa — the pressure drops by 30,000 Pa as velocity increases
Example 2
A water tank has a small hole 3 meters below the surface. What is the velocity of water exiting the hole? (This is Torricelli's theorem, a special case of Bernoulli's equation.)
At the surface: v ≈ 0 (large tank), P = atmospheric. At the hole: P = atmospheric, h = 0.
Bernoulli simplifies to: ρgh = ½ρv², so v = √(2gh)
v = √(2 × 9.81 × 3) = √(58.86)
v ≈ 7.67 m/s — the water exits as fast as if it had fallen 3 meters in free fall
When to Use It
Use Bernoulli's equation when analyzing steady, incompressible fluid flow where friction losses are small or can be accounted for separately.
- Calculating pressure drops in pipe systems with changing diameters
- Designing Venturi meters and flow measurement devices
- Estimating lift forces on airplane wings and airfoils
- Analyzing water flow from tanks, reservoirs, and nozzles
- Understanding how carburetors, atomizers, and spray nozzles work