 Bernoulli function

A Bernoulli function collects the terms that become constant along streamlines under the conditions of the Bernoulli equation. For a barotropic inviscid flow with conservative body force $-\nabla \Phi$, a useful specific-energy convention isHere $h_b$ is the barotropic enthalpy function. At constant density, $h_b=p/\rho$ up to a reference constant.

For steady flow, the Euler momentum equation gives $\mathbf{u}\cdot \nabla B=0$: $B$ is constant along each streamline. It need not have the same value on distinct streamlines in a rotational flow. Stating the normalization matters because multiplying by a constant density produces a pressure-unit form, whereas the displayed function has energy-per-mass units. Comparing a pressure term directly with specific kinetic energy without that normalization gives an inconsistent balance.