Bernoulli's principle

Remarks

The proof uses the identity $(u \cdot \nabla) u = \nabla (\abs{u}^2 / 2) - u \times \omega$, where $\omega = \nabla \times u$ is the vorticity. With it, the steady Euler equations become $\nabla H = u \times \omega$. The right side is orthogonal to $u$, so $u \cdot \nabla H = 0$ and $H$ is constant along streamlines. If $\omega = 0$, then $\nabla H = 0$ everywhere.

Example: water leaves a small hole at depth $h$ below the free surface of a large tank. Along a streamline from the surface (speed about $0$, atmospheric pressure, $\Phi = g h$) to the jet (atmospheric pressure, $\Phi = 0$), part 1 gives the speed $\sqrt{2 g h}$ (Torricelli's law).

Without the inviscid hypothesis the result fails. In steady viscous flow through a straight pipe, the speed along each streamline is constant while the pressure drops, so $H$ decreases downstream.

There is a version for steady compressible barotropic flow, with $p / \rho$ replaced by $\int dp / \rho$.

Going deeper

For a student

Bernoulli's principle is a first integral of the steady Euler equations. It turns the momentum equation into an algebraic relation between pressure and speed, so once the velocity of a potential flow is known, the pressure follows at once. This is how one computes forces in potential flow, for example in the Kutta–Joukowski theorem and in d'Alembert's paradox. The relation $\nabla H = u \times \omega$ links it to Crocco's theorem, which is its thermodynamic version for compressible flow.