For steady, inviscid, incompressible flow along a streamline (neglecting shaft work and heat transfer), the sum of static pressure p, dynamic pressure ½ρV² and hydrostatic term ρ g z is constant: p + ½ρV² + ρ g z = constant; this relation expresses conservation of mechanical energy per unit volume along the streamline under its assumptions.

Principle

An increase in local flow speed along the same streamline corresponds to a decrease in static pressure (and vice versa), provided the flow meets the Bernoulli assumptions; thus changes in geometry or elevation that accelerate the flow produce predictable static pressure changes along that streamline.

Demonstration

Illustrative scenario → Steady incompressible flow through a horizontal converging duct. Recognition → Continuity increases V in the constriction. Action → According to Bernoulli (p + ½ρV² = constant) static pressure measured at a tap in the constriction is lower than upstream. Consequence → This pressure difference can drive flow instrumentation readings or be used to estimate velocity from pressure measurements when assumptions hold.

Misapplication

Applying Bernoulli between two points connected by different streamlines, across a shock or separated region, or in flows where viscous losses, unsteady acceleration, heat addition, or compressibility are significant; the error is ignoring the theorem's assumptions and therefore attributing pressure differences solely to velocity changes when other processes dominate.

Consequence

Correct application gives a simple tool for relating velocity and pressure for design and measurement (pitot/static systems, simple venturi meters). Misapplication causes incorrect velocity or force estimates, flawed instrumentation interpretation, and unsafe design decisions when viscous losses or compressibility are non‑negligible.

Reversal

When the flow is compressible (high Mach), viscous (boundary layers, significant head loss), unsteady, rotational (vorticity), or includes work/heat exchange, the simple Bernoulli form fails and must be replaced by energy equations, compressible Bernoulli variants, or the full Navier–Stokes treatment with loss terms included.

Boundary

Clearly within: steady, incompressible, inviscid, single‑streamline flows without heat/work exchange (e.g., low‑Re external potential flow regions away from boundaries). Boundary case: moderate Reynolds number boundary‑layer regions where local Bernoulli may approximate outer flow but fails in the viscous sublayer. Clearly outside: flows with shocks, strong viscous dissipation, separated flows, or significant unsteadiness.

Semantic Tension

Bernoulli's simplification competes with the Navier–Stokes description: Bernoulli offers intuition and simple relations under restrictive assumptions, while Navier–Stokes provides comprehensive but more complex and often numerical descriptions; choosing between them balances tractability against fidelity.

Synthesis

Bernoulli condenses mechanical energy conservation into a compact relation that is powerful for conceptual reasoning and simple measurements, but its correct use requires explicit verification of its assumptions and, where they fail, supplementation by loss terms or a more complete fluid dynamic model.