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Bernoulli's equation is valid only for inviscid, incompressible, steady flow along a streamline. Real fluids have viscosity that dissipates mechanical energy as head loss, making Bernoulli applicable only between points close enough that losses are negligible. Understanding when Bernoulli breaks down prevents dangerous design errors in pipes, channels, and pumping systems.
Compare Bernoulli predictions to measured pressures in real pipe flow. Quantify discrepancies and relate them to Reynolds number and flow distance.
Bernoulli applies to all steady flow. Bernoulli applies to turbulent flow. Losses are negligible over any distance. Compressibility effects are always small.
Bernoulli's equation — which relates pressure, velocity, and elevation along a streamline — is derived with three assumptions baked in: the fluid is inviscid (zero viscosity), incompressible (constant density), and the flow is steady. In ideal-fluid theory, these assumptions let you trade kinetic energy for pressure energy and back again with no losses. The equation is remarkably useful precisely because it's simple. But real fluids violate each assumption to varying degrees, and knowing when the violations are tolerable is the practical skill this topic builds.
The most important departure is viscosity. In a real fluid, internal friction between fluid layers dissipates mechanical energy as heat. This means that as you move along a streamline in a pipe, the total mechanical energy — the sum of pressure head, velocity head, and elevation head — decreases in the direction of flow. The amount of mechanical energy lost per unit weight of fluid is called head loss, and it accumulates with distance. Bernoulli with viscosity ignored predicts the same total head everywhere along a pipe; a real pipe shows total head declining continuously downstream. The further apart your two measurement points, the larger the error in a Bernoulli-only analysis.
Compressibility becomes relevant when flow speeds approach the speed of sound. At Mach numbers above ~0.3, density changes become significant and the incompressible Bernoulli equation breaks down. Unsteady flow — changing with time — violates the steady-flow assumption; in rapidly accelerating or decelerating flows, the ∂V/∂t term in the full Euler equation cannot be dropped. Finally, Bernoulli applies along a streamline, not across them; applying it between two points on different streamlines requires the additional condition that the flow be irrotational.
The practical design consequence is this: when using Bernoulli for pressure calculations between two points, always ask "how far apart are they, and what is the Reynolds number?" Short distances in high-Re laminar flow or in streamlined geometries — converging nozzles, Venturi meters — make Bernoulli highly accurate. Long pipe runs, bends, fittings, and high-turbulence zones accumulate significant head loss. The corrected version of Bernoulli — the extended Bernoulli equation or mechanical energy balance — adds a head-loss term h_L to account for viscous dissipation, and a pump-head or turbine-head term when machinery is present. This extended form is the equation used for virtually all real engineering pipe-system analysis.