Persistence of nonplanar stagnation lines under suitable driving

Determine which suitable drivings can maintain a stagnation line of arbitrary nonplanar shape on a finite time interval, and in particular establish whether the necessary persistence conditions are sufficient for nonplanar curves.

Background

The paper derives necessary conditions for a stagnation line in potential flow to persist: the time derivative of the potential must be constant along the line, and a second-order condition involving φtt\varphi_{tt}, the normal acceleration components, and the strain rate must also hold. The authors explain that these conditions form a hierarchy and constrain the driving rather than the geometry of the line.

An exact construction is provided for analytic planar curves, showing that suitable harmonic potentials can produce persisting lines of arbitrary planar shape, including shapes that are not rigid-motion orbits. The unresolved issue is whether an analogous construction or sufficiency result holds for nonplanar curves.

References

The converse question is when a stagnation line of arbitrary shape can be maintained on a finite time interval by a suitable driving. The conditions of \cref{prop:potlaw}(ii) and eq:second are necessary, and at every order they only involve the driving. They are also sufficient in at least one non-trivial class: \cref{ex:ellipse} gives, for any analytic planar curve f(x,y,t)=0 with \nabla f\neq0, an exact (locally convergent, and for polynomial f global) harmonic \varphi with a persisting line of that shape and motion, which is not an orbit of any rigid motion. For curves that are not planar, we have not checked this and do not use it.

— How stagnation lines move, persist and break up in fluid flows  (2610.26164 - Rajamanickam, 19 Oct 2026) in Remark [Open problem], Section 3, subsection “Loss of a line” (following Corollary \ref{cor:potbreak})