Bernoulli's Equation

Because the total energy along the streamline is fixed. If the fluid speeds up, its kinetic energy density ½ρv² rises, and since the sum P + ½ρv² + ρgh must stay constant, the static pressure P must fall by exactly the same amount. Physically, the pressure gradient along the streamline is what accelerates the fluid in the first place — a favourable pressure gradient (pressure decreasing downstream) pushes the fluid forward and speeds it up. This is the principle behind the Venturi meter, the carburettor, the atomiser, and the generation of lift on an aircraft wing.

Whenever the hydrostatic pressure change is small compared with the dynamic and static terms you care about. For air at sea level, ρgh is only about 12 Pa per metre, so a 100 m elevation change contributes roughly 1.2 kPa — often negligible against atmospheric pressure of 101 kPa. For water, ρgh is about 9,800 Pa per metre, so even a 1 m change matters. The rule of thumb: compare ρgh with ½ρv². If ρgh is less than about 5% of the dynamic pressure, you can safely drop it and use P + ½ρv² = constant.

Static pressure P is the pressure the fluid exerts perpendicular to the flow — what a gauge flush with the pipe wall would read. Dynamic pressure q = ½ρv² is not a pressure you can measure directly; it is the kinetic energy per unit volume, expressed in pressure units. Stagnation pressure (also called total pressure) is what you measure at a point where the flow has been brought to rest isentropically, and it equals P + ½ρv² = P + q. A Pitot-static tube measures both static and stagnation pressure, and the difference between them is the dynamic pressure, which gives the airspeed. Note that in this calculator C is the total pressure including the elevation term, so C = P + q + ρgh.

Only partly, and the popular "equal transit time" story is wrong. Bernoulli's equation does correctly describe the pressure field around an aerofoil: where the air accelerates over the upper surface, the static pressure drops, and that lower pressure contributes to the lift. But the acceleration is caused by the aerofoil's shape and angle of attack deflecting the flow, and a complete lift explanation requires the circulation and the Kutta condition. The most robust statement is that lift is the reaction to the downward momentum imparted to the air, and the pressure distribution computed from Bernoulli (plus the viscous boundary condition) accounts for that reaction.

Head is energy per unit weight, expressed in metres. Dividing Bernoulli's equation through by ρg turns each term into a length: P/(ρg) is the pressure head (how high the pressure alone would push a column of the fluid), v²/(2g) is the velocity head, and h is the elevation head. Their sum is the total head H. Working in head is convenient because it is independent of the fluid — a pump rated at 30 m of head will lift water 30 m, or mercury 30 m, at the same volumetric flow rate. It also makes the energy bookkeeping visually intuitive: head is literally a height.

Very accurate for streamline-aligned, low-viscosity, low-Mach flows away from walls, and progressively worse as you approach the four assumptions' limits. Viscous losses in a long pipe can consume most of the available pressure, and they grow with the square of velocity and with pipe length. Compressibility becomes significant above roughly Mach 0.3 in air, where the isentropic relations must be used instead. Rotational flow behind a bluff body invalidates the single-streamline constant. In practice, engineers use Bernoulli as the ideal baseline and add a discharge coefficient or a loss coefficient to account for reality — a Venturi meter, for instance, typically discharges at 95–98% of the Bernoulli-predicted value.