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Bernoulli’s equation, explained

In Hebrew: משוואת ברנולי

Along a streamline in steady, inviscid, incompressible flow, pressure, kinetic energy and height trade off with their sum held constant. That is Bernoulli’s equation.

What it says

Each term is an energy per unit volume. p is the static pressure, ½ρV² is the dynamic pressure, and ρgz is the hydrostatic term. Divide through by ρg and every term becomes a height, called head: pressure head, velocity head and elevation head. Where the flow speeds up, pressure drops. Where it climbs, pressure drops too.

Their sum, the total or stagnation pressure, is what a probe facing straight into the flow measures. The difference between that and the static pressure is what tells you the speed.

The assumptions, and why they matter

Four conditions: steady flow, no viscous losses, constant density, and the equation is applied along one streamline. If the flow is also irrotational, the constant is the same on every streamline and you can compare any two points. There is no pump, turbine or heat transfer between the two points.

Real pipes have friction, and real systems have pumps. The engineering version keeps Bernoulli’s structure and adds two terms: a pump head h_p on the inlet side and a head loss h_L on the outlet side. That extended energy equation is what you actually use to size a pump.

Where you meet it

A Pitot tube reads the stagnation pressure and a static port reads p; the difference gives the airspeed on every aircraft. A Venturi meter narrows a pipe and measures the pressure drop to get the flow rate. A tank draining through a hole at depth h gives Torricelli’s result, V = √(2gh), the same speed as a stone dropped from that height.

A worked number

Water drains from an open tank through a hole 5 m below the surface. Both the surface and the jet are at atmospheric pressure, and the surface barely moves, so V = √(2 × 9.81 × 5) ≈ 9.9 m/s. A real orifice delivers about 60% of the ideal flow rate, because the jet contracts and friction takes the rest. That factor is the discharge coefficient.

Common mistakes

  • Applying it between two streamlines in a rotational flow, such as across a boundary layer.
  • Writing it across a pump or a fan without adding the pump head.
  • Using it for gas above roughly Mach 0.3, where density changes are no longer small.
  • Explaining wing lift with “equal transit time”. Air over the top does not have to rejoin the air underneath, and it does not.

Mechanical engineering, explained