Limitations of validity and common pitfalls

When Bernoulli’s principle no longer applies directly

• Viscous flow with significant pressure losses (long, rough pipework, friction): a pressure loss term must be added, often denoted by ΔH or J.
• Unsteady flow, for example a water hammer: the generalised equation must include a local acceleration term, which is the integral of the time derivative of v along the path.
• High-speed compressible fluid (Mach number greater than approximately 0.3): the density ρ is no longer constant; a compressible Bernoulli equation or the full equations of gas dynamics must be used.
• Presence of a machine (pump, turbine) between the two points under consideration: the mass work exchanged w must be added, in addition to any losses:

P1 + (1/2)rhov1^2 + rhogz1 + w_pump = P2 + (1/2)rhov2^2 + rhogz2 + losses

Summary table of pitfalls

Pitfall	Consequence	Remedy
Points not on the same streamline	Incorrect constant	Check for irrotationality or choose the same streamline
Viscosity erroneously neglected	Underestimation of losses	Add the pressure drop term
Confusion between static and total pressure	Incorrect velocity measurement	Clearly identify the type of sensor used
High-velocity gas	Error due to compressibility	Use compressible equations

Example

In a long industrial pipework network, neglecting pressure drops can lead to a significant overestimation of the actual flow rate obtained for a given pressure difference between the inlet and outlet of the network.