Bernoulli's Theorem

More subtle and more powerful than the continuity equation is the theorem that ties together pressure, velocity and height of a fluid along a single flow line. It is the conceptual core of elementary fluid dynamics and explains, with a single idea, phenomena that seem quite different from one another.

Law — Bernoulli's theorem

For an ideal fluid (non-viscous, incompressible) in steady motion, along a flow line $\ev{P + \tfrac{1}{2}\,\rho\,v^2 + \rho\,g\,h = \text{cost.}}$ The sum of the static pressure $P$, the dynamic pressure $\tfrac{1}{2}\rho v^2$ and the gravitational term $\rho g h$ is the same at every point of the flow line.

Key formula

$P + \tfrac{1}{2}\rho v^2 + \rho g h = \text{cost.}$ This is energy conservation written per unit volume of fluid.

The interpretation of the theorem is indeed energetic. Each term is an energy per unit volume: $P$ is the compression energy, $\tfrac{1}{2}\rho v^2$ is the kinetic energy and $\rho g h$ is the gravitational potential energy, all referred to the unit volume of fluid. That their sum stays constant along a single flow line is simply the principle of energy conservation applied to the moving fluid (Lauga 2022; see Riferimenti bibliografici).

An immediate, and perhaps counterintuitive, consequence is obtained when the height does not change.

Principle — Venturi effect

At constant height ($h_1 = h_2$), where the fluid velocity increases the pressure decreases: $\Delta P = \tfrac{1}{2}\,\rho\,(v_2^2 - v_1^2)$

Where the fluid runs faster, then, the pressure is lower. This inverse relationship between velocity and pressure is the key to understanding the two examples that follow.

Lift on an aircraft wing

An aircraft wing has a curved upper surface and an almost flat lower one. The air flow passing above must travel a longer path in the same time, and so runs faster than the flow passing below. By Bernoulli’s theorem, above the wing the pressure is lower and below it is higher: the pressure difference, multiplied by the wing area, gives the lift force that holds the aircraft up.

For a Boeing 747 (mass of order $4\cdot 10^5\;\text{kg}$, wing area about $525\;\text{m}^2$, cruising speed $v \approx 250\;\text{m/s}$) the lift per unit area must equal $\rho_a (v_{\text{sopra}}^2 - v_{\text{sotto}}^2)/2 \approx 7\,500\;\text{N/m}^2$. With $\rho_a \approx 1{,}2\;\text{kg/m}^3$ and $v_{\text{sotto}} = 250\;\text{m/s}$ it follows that $v_{\text{sopra}} \approx 265\;\text{m/s}$: a speed difference of only about $6\%$ is enough to keep $400$ tonnes airborne.

Sheet of paper that "lifts"

Hold a floppy sheet of paper in your hand, parallel to the floor, and blow horizontally across its upper face: the sheet rises. Above, the air speed is high (your breath), below it is practically zero; by Bernoulli, above the pressure is lower and below it stays atmospheric, and the difference pushes the sheet upward. It is a classroom-bench experiment that shows the Venturi effect live.

Links

Topics: Fluidostatica e fluidodinamica Concepts: Teorema di Bernoulli · Pressione Objects: Ala d’aereo

Related exercises: Esercizio svolto — Pressa per la frutta · Esercizio svolto — Pressione del cuore · Esercizio svolto — La colonna d’aria che schiaccia i nostri piedi