Beltrami flow

A Beltrami flow has vorticity parallel to its velocity, expressed as $\mathbf{\omega }=\nabla \times \mathbf{u}=\lambda \mathbf{u}$. 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 $\mathbf{u}\times \mathbf{\omega }=0$. 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.