Bernoulli's Equation

Take a small fluid element on a streamline. The forces on it are the pressure difference across its ends and the component of its weight along the path. Write F = ma and integrate along the streamline. That integration is Euler's equation, and Bernoulli is its result for constant density.

It is not a new principle. It is Newton's second law, written for a fluid element and integrated along its own path.

Dividing through by ρg gives every term the dimension of length, which is why they can be plotted on one axis and read off a manometer.

Pressure head, p/ρg

The height this pressure could lift the fluid to. What a static tapping reads.

Velocity head, v²/2g

The height this speed could carry the fluid to. What appears when the flow is brought to rest.

Elevation head, z

Where the fluid already is, measured from any convenient datum.

Their sum — the total head — is constant along a streamline. Multiply through by ρg instead and it becomes an energy statement in joules per cubic metre, which is the same equation in different clothes.

Since the total is fixed, a rise in one head is paid for by a fall in another. Constrict a pipe and continuity raises the velocity, so pressure must drop. The fluid gains no energy; it converts pressure energy into kinetic energy.

A pitot tube brings fluid to rest at its opening, so the velocity head converts entirely to pressure. Subtract the static reading and the difference is ½ρv². Every airspeed indicator works this way — and every pitot icing incident is this equation being fed a bad input.

• Steady flow — no transients, no water hammer, no starting a pump.
• Incompressible — liquids, or gases below about Mach 0.3.
• Frictionless — no viscous loss between the two points.
• Along one streamline — unless the flow is also irrotational, in which case it holds between any two points.
• No shaft work — no pump or turbine between the two sections.

The streamline restriction is the sneakiest of the five. Two points in the same pipe may not lie on the same streamline, and the equation only extends across them if the flow is irrotational as well.

Real flow dissipates energy as heat through viscosity, so the total head falls along the pipe. The working form adds a loss term — always on the downstream side, since losses are irreversible and never run backwards.

Adding pump head positively and turbine head negatively completes it, and with those three additions the equation describes an actual installation rather than an idealisation. Computing h_L is exactly what Darcy-Weisbach exists for.

Bernoulli says fast flow accompanies low pressure. It does not say why the flow was fast — and the popular aerofoil story invents a reason.

• The "equal transit time" claim — that air must rejoin at the trailing edge — is simply false. The upper flow arrives well ahead, as smoke-tunnel photographs show.
• A spinning ball does curve, and there is a real pressure difference, but the mechanism is the boundary layer being dragged around the ball and separation shifting asymmetrically.
• Lift is better argued from circulation, or from the momentum given downward to the air. Both give the right answer for the right reason.

The relationship holds; the causal story is the part to be careful with. That is a good habit with any equation, and this is the one where it is most often abandoned.