= Beltrami flow
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A <Beltrami flow> has <vorticity> parallel to its <velocity>, expressed as $\boldsymbol\omega=\nabla\times\mathbf u=\lambda\mathbf u$. The proportionality factor can depend on position. For an incompressible flow, taking the divergence gives $\mathbf u\cdot\nabla\lambda=0$, so the factor is constant along each <streamline>.
A concrete example is $\mathbf u=U(\sin kz,\cos kz,0)$ for constants $U$ and $k$. Direct differentiation gives $\nabla\cdot\mathbf u=0$ and $\nabla\times\mathbf u=k\mathbf u$. The velocity direction varies between horizontal planes, while individual particles remain in their initial plane.
The alignment simplifies the <Euler equations for an inviscid fluid> because $\mathbf u\times\boldsymbol\omega=0$. For a steady, constant-density flow with no body force, the vector form of those equations gives $\nabla(p/\rho+|\mathbf u|^2/2)=0$. The <Bernoulli equation> quantity is then spatially uniform within a connected region, a stronger property than merely being constant along separate streamlines.
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