5.4 Bernoulli’s principle

For a steady, inviscid, incompressible flow, Euler’s equation can be integrated once along a streamline, yielding a conserved combination of pressure, speed, and height. The result — Bernoulli’s equation — is the most-used relation in elementary fluid dynamics, and also the most-misapplied, because its conditions of validity are easy to forget.

Integrating Euler along a streamline

▶ From Euler to Bernoulli Derivation

In steady flow $\partial\mathbf{u}/\partial t = 0$ , so Euler’s equation reduces to

\rho\,(\mathbf{u}\cdot\nabla)\mathbf{u} \;=\; -\nabla p \;+\; \rho\mathbf{g}.

Use the vector identity (refresher: vector calculus →)

(\mathbf{u}\cdot\nabla)\mathbf{u} \;=\; \nabla\!\left(\tfrac12 |\mathbf{u}|^2\right) \;-\; \mathbf{u}\times(\nabla\times\mathbf{u}).

Write gravity as the gradient of a potential, $\mathbf{g} = -\nabla(gz)$ . With $\rho$ constant, the equation becomes

\nabla\!\left(\tfrac12\rho|\mathbf{u}|^2 + p + \rho g z\right) \;=\; \rho\,\mathbf{u}\times(\nabla\times\mathbf{u}).

Now dot both sides with the unit tangent to a streamline, $d\mathbf{s}$ . The right-hand side is perpendicular to $\mathbf{u}$ , hence perpendicular to the streamline direction, so it vanishes:

\nabla\!\left(\tfrac12\rho|\mathbf{u}|^2 + p + \rho g z\right)\cdot d\mathbf{s} \;=\; 0.

The gradient of the bracketed quantity has no component along the streamline, so the quantity is constant along it.

This is Bernoulli’s equation:

\tfrac12\rho|\mathbf{u}|^2 \;+\; p \;+\; \rho g z \;=\; \text{const along a streamline}.

The three terms are the dynamic pressure (kinetic energy per unit volume), the static pressure, and the gravitational potential energy per unit volume. Their sum is the constant total head. The conditions are worth stating plainly: the flow must be steady, inviscid, and incompressible, and the constant holds along a streamline, not across — different streamlines may carry different constants unless the flow is also irrotational.

u(x)2.500

p(x)-1.625

½ρu²3.125

p + ½ρu²1.500

Probe location x = 0.00 Volume flow Q = 1.00 Throat opening = 40% of pipe area

Slide the probe across the Venturi. Where the pipe narrows the velocity rises (continuity: Q = A u = const) and the pressure drops (Bernoulli: p + ½ρu² = const along a streamline). The black total-energy curve is *flat* — the trade-off of pressure for kinetic energy along the streamline is exact for steady, inviscid, incompressible flow.

Slide the probe along the constriction. Where the channel narrows, continuity forces the speed up; Bernoulli then forces the static pressure down, while the total head stays rigorously constant along the streamline. The trade between speed and pressure is the whole content of the principle.

Where it applies

The same constant-head relation underlies a family of classic measurements and effects:

The Venturi meter infers flow rate from the pressure drop at a constriction.
The Pitot tube measures airspeed by comparing stagnation pressure (where $\mathbf{u}=0$ , all head is static) with the free-stream static pressure.
The pressure drop in fast flow explains why pressure falls in a constricted pipe, and — combined with circulation — contributes to aerodynamic lift.

In each case the danger is the same: Bernoulli is exact only when viscosity is negligible. Near a wall, where a thin layer of fluid is sharply sheared, viscosity is never negligible, and that is the subject of the next two lessons.