Assumptions and derivation of Bernoulli’s equation

The framework of the model

Bernoulli’s equation describes the flow of an ideal (non-viscous), incompressible fluid in steady-state conditions along a streamline. These four assumptions are essential: as soon as even one of them is no longer satisfied, the classical equation no longer applies as it stands.

Derivation from the kinetic energy theorem

By applying the kinetic energy theorem to a slice of fluid between two cross-sections of a flow tube, and taking into account the work done by pressure forces and weight, we obtain the following between two points 1 and 2 on the same streamline:

P1 + (1/2) * ρ * v1^2 + ρ * g * z1 = P2 + (1/2) * ρ * v2^2 + ρ * g * z2

where P is the static pressure (Pa), ρ is the density (kg/m³), v is the velocity (m/s), g is the acceleration due to gravity (approximately 9.81 m/s²) and z is the height (m).

A constant quantity along a streamline

This sum is conserved along a single streamline:

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

Term	Physical meaning
P	pressure energy per unit volume
(1/2) * ρ * v²	kinetic energy per unit volume
ρ * g * z	gravitational potential energy per unit volume

Common pitfalls

A frequent pitfall is to apply Bernoulli’s equation between two points that are not located on the same streamline, without having established that the flow is irrotational (in which case the constant is the same throughout the fluid). Another common pitfall is forgetting that the fluid must remain incompressible, which is not the case for a gas travelling at high speed (high Mach number).