Bernoulli Equation

It is energy conservation for flowing liquids and gases. Three energy densities share a fixed budget: the static pressure p (what a sensor drifting along would measure), the dynamic pressure ½ρv² (motion share) and the elevation pressure ρgh (height share). Their sum is constant along a streamline. This yields the famous core statement: where the flow speeds up, for instance in a constriction, the static pressure must drop, and vice versa. This seems paradoxical at first; many expect higher pressure in the narrow section. In fact it is precisely the pressure difference that accelerates the fluid into the constriction, pushing it from high towards low pressure.

Choose two points on the same streamline and write the equation for both: p₁ + ½ρv₁² + ρgh₁ = p₂ + ½ρv₂² + ρgh₂. Cancel what is equal, for horizontal flow the height terms. Missing speeds come from the continuity equation A₁·v₁ = A₂·v₂. Example: water at p₁ = 200 kPa flows at 2 m/s into a pipe that narrows so that v₂ = 8 m/s. Then p₂ = p₁ + ½ρ(v₁² − v₂²) = 200,000 + 500 × (4 − 64) = 170,000 Pa = 170 kPa. The pressure drops by 30 kPa although nothing is pumped, purely due to the acceleration into the constriction.

Torricelli describes the efflux speed from an open tank: v = √(2gh). It follows as a special case: compare the calm water surface (point 1) with the hole at depth h (point 2). Atmospheric pressure acts at both places, so the pressure terms cancel; for a large tank the surface sinks negligibly slowly, so v₁ ≈ 0. What remains is ρgh = ½ρv², solved as v = √(2gh). At h = 5 m the water exits at √(2 × 9.81 × 5) ≈ 9.9 m/s, exactly as fast as if it had fallen freely from 5 m. That is no coincidence: both calculations are the same energy conservation.

Partly, and the popular short version is often told wrongly. What is correct: air flows faster over the curved upper surface of the wing, so by Bernoulli the static pressure there is lower than below, and the pressure difference carries the aircraft. What is wrong is the widespread claim that two neighbouring air parcels must arrive at the trailing edge simultaneously; demonstrably they do not. The deeper cause of the faster flow lies in the circulation around the profile and the angle of attack: the wing deflects air downwards, and by Newton third law the reaction force acts upwards. Bernoulli and the momentum view are two consistent perspectives on the same physics, not competitors.

The equation assumes four idealisations: steady flow (no changes in time), absence of friction, incompressibility and validity along a streamline. In narrow, long pipes viscosity dominates and the real pressure drop exceeds the calculated one (Hagen-Poiseuille law). For gases Bernoulli only works while the density change stays small, as a rule of thumb up to about 0.3 times the speed of sound (roughly 100 m/s in air); beyond that compressible flow theory is needed. In turbulent eddies and across streamlines the equation likewise does not apply directly. For exams this means: Bernoulli for short, smooth flow paths with water or slow air, not for capillaries or supersonic flow.