Bernoulli’s theorem

Statement of Bernoulli’s theorem

For an ideal (non-viscous), incompressible fluid in steady flow along a streamline, Bernoulli’s theorem expresses the conservation of mechanical energy per unit volume:

P + (1/2) * ρ * v² + ρ * g * z = constant

where P is the static pressure, (1/2) * ρ * v² is the dynamic pressure (related to kinetic energy), and ρ * g * z is the position term (potential gravitational energy).

Physical interpretation

This relationship reflects a trade-off between pressure, velocity and altitude: where velocity increases, pressure decreases (at constant altitude), and vice versa. This is the principle that explains the lift generated by an aeroplane wing and the Venturi effect.

Practical example: the Venturi tube

In a horizontal pipe (constant z) that narrows, the velocity increases (continuity equation) and therefore the pressure drops. By measuring this pressure difference, we can work out the flow rate:

P1 + (1/2)rhov1^2 = P2 + (1/2)rhov2^2

Numerical example

Air (ρ = 1.2 kg/m³) at v₁ = 10 m/s and P₁ = 101,325 Pa flows through a constriction to v₂ = 20 m/s. The pressure drop is: deltaP = (1/2) * rho * (v2^2 - v1^2) = 0.6 * (400 - 100) = 180 Pa.

Common pitfalls

• Bernoulli’s equation applies only to a single streamline, in steady flow and without friction losses (ideal fluid).
• It cannot be applied directly between two points connected by a pump or a turbine, or in the presence of significant viscous head losses, without adding a corrective term to the equation.