Past exam of the options I course of the University of Oxford / 2007 / K / K1

For steady flow the given equation gives $\nabla H=-\rho \omega \times u$. Dotting with $u$ yields $u\cdot \nabla H=0$, so $H$ is constant along each streamline. It can have different values on different streamlines. Dotting with $\omega$ also makes it constant along vortex lines where they are defined.

If the flow is additionally irrotational, $\omega =0$ and hence $\nabla H=0$. Thus $H$ is one constant on each connected fluid region, the stronger global Bernoulli conclusion.

Take the curl. A gradient has zero curl, and the supplied identity, with $\nabla \cdot u=0$ and $\nabla \cdot \omega =\nabla \cdot (\nabla \times u)=0$, givesTherefore $D\omega /Dt=(\omega \cdot \nabla )u$. In planar flow $\omega =(0,0,{\partial }_{x}v-{\partial }_{y}u)$ and ${\partial }_{z}u=0$, so the stretching term vanishes. ConsequentlyVorticity is carried unchanged with each fluid particle. It need not be spatially uniform or independent of time at a fixed spatial point.