The Euler equations and Bernoulli's theorem

An ideal fluid has no viscous stress, \(\sigma = -pI\). Cauchy's equation with constant density becomes the incompressible Euler equations (Euler, 1757): \[ \frac{\partial\mathbf{u}}{\partial t} + (\mathbf{u}\cdot\nabla)\mathbf{u} = -\frac{1}{\rho}\nabla p + \mathbf{f}, \qquad \nabla\cdot\mathbf{u} = 0. \] Four equations for four unknowns, the three velocity components and the pressure. At a solid wall only the normal velocity is constrained, \(\mathbf{u}\cdot\mathbf{n} = 0\): an ideal fluid may slide along a boundary.

The acceleration has a useful rewriting, the Lamb form. The vector identity \[ (\mathbf{u}\cdot\nabla)\mathbf{u} = \nabla\!\left(\tfrac12|\mathbf{u}|^2\right) - \mathbf{u}\times\boldsymbol{\omega} \] (check it component by component) turns the Euler equations with a potential force \(\mathbf{f} = -\nabla\Phi\) into \[ \frac{\partial\mathbf{u}}{\partial t} - \mathbf{u}\times\boldsymbol{\omega} = -\nabla H, \qquad H = \frac{p}{\rho} + \tfrac12|\mathbf{u}|^2 + \Phi. \]

Bernoulli's theorem follows at once. In steady flow, dot with \(\mathbf{u}\): since \(\mathbf{u}\times\boldsymbol{\omega}\) is perpendicular to \(\mathbf{u}\), we get \(\mathbf{u}\cdot\nabla H = 0\), so \(H\) is constant along each streamline (and, dotting with \(\boldsymbol{\omega}\), along each vortex line). If in addition the flow is irrotational, \(\nabla H = 0\) and \(H\) is the same constant everywhere. For unsteady irrotational flow with \(\mathbf{u} = \nabla\phi\) the version is \(\partial_t\phi + \tfrac12|\nabla\phi|^2 + p/\rho + \Phi = F(t)\). Daniel Bernoulli's Hydrodynamica (1738) contains the idea; the equation in this form is due to Euler.

The theorem explains a lot with one line. Water leaving a hole at depth \(h\) below the surface of a large tank has \(\tfrac12\rho v^2 = \rho g h\), so \(v = \sqrt{2gh}\) (Torricelli's law, 1643, a century earlier). In a pipe that narrows, continuity makes the speed rise and Bernoulli makes the pressure fall: the Venturi meter. A Pitot tube facing the flow measures the stagnation pressure \(p + \tfrac12\rho v^2\) and so the speed. The limits matter too: Bernoulli ignores viscosity, so it cannot predict drag or losses in long pipes.

One more feature is essential for everything later. Take the divergence of the Euler equations and use \(\nabla\cdot\mathbf{u} = 0\): \[ -\frac{1}{\rho}\Delta p = \partial_i u_j\,\partial_j u_i. \] The pressure is not an independent unknown with its own evolution; it is determined at each instant by the velocity, through a Poisson equation, and depends on the velocity everywhere at once.

Picture it: along a streamline, energy per unit volume moves between three accounts (pressure, kinetic, potential) and the total never changes. Think it: Bernoulli is conservation of energy for an ideal fluid restricted to a streamline, and the Poisson equation for pressure is why incompressible flow is non-local: a push here is felt everywhere instantly.