= Solution
{c}
In steady flow, the supplied <Euler equations for an inviscid fluid> gives
$$
\nabla H=-\rho\,\boldsymbol\omega\times\mathbf u.
$$
Taking its scalar product with $\mathbf u$ gives $\mathbf u\cdot\nabla H=0$. Along a <streamline> with tangent $\mathbf u$, this is precisely zero derivative of $H$. Hence \b[$H$ is constant along each <streamline>], with possibly different constants on different <streamlines>.
If the flow is also irrotational, $\boldsymbol\omega=0$ and thus $\nabla H=0$. Therefore \b[$H$ has one constant value throughout each connected fluid region]. The stronger conclusion uses vanishing of the entire <gradient>, not just its component along the <velocity>.
Back to article page
