How stagnation lines move, persist and break up in fluid flows

Abstract: Stagnation curves of arbitrary shape (zeros of $\mathbf v(\mathbf x,t)$) are non-generic yet common; we derive their motion. Differentiating $\mathbf v=\mathbf0$ gives $\mathbf L\mathbf U=-\mathbf a$ ($\mathbf U$ curve velocity, $\mathbf a=\mathbf v_t$, $\mathbf L=\nabla\mathbf v$ kills tangent, acts on normal plane via $2\times2$ block $\mathbf K$); dynamics enters only via $\mathbf a$. Normal velocity: $-\mathbf K<sup>{-1}$ times normal $\mathbf a$; tangential solvability: scalar condition $Σ=0$ per point ($Σ$: part of $\mathbf a$ outside range $\mathbf L$). Potential flow: $\mathbf K$ is strain rate, normal velocity pressure gradient over principal strain rate $ε$, persistence needs uniform pressure on it. Stokes flow: tangential vorticity enters $\mathbf K$ (centre lines), $Σ$ involves normal vorticity. Navier-Stokes: $Σ$ set by instantaneous flow via nonlocal pressure, viscosity. If $Σ\ne0$, line becomes, to first order in $t$, isolated points at zeros of $Σ$ (critical points of $\varphi_t$ if potential; maybe none in Stokes flow: loops vanish). Symmetry (line an orbit of rigid-motion group, no velocity along it) suffices for persistence, not necessary: flows steady in rigidly moving frame have $Σ\equiv0$. Incompressibility enters via $\mathrm{tr}\mathbf K=0$; for inviscid irrotational homentropic compressible flow, line dilatation is $\varphi_{tt}/c<sup>2$, so lines can be nodes, persistence needs uniform enthalpy, singular $\mathbf K$ arises at ordinary points, acoustic standing waves give exact examples (node line breaking into alternating sinks, saddles); also degenerate $\mathbf K$ (pairs born at distance $O(\sqrt t)$), curvature, torsion evolution, closure for moving-singularity-driven flows, helical and elliptically conical stagnation-point flow, exact Navier-Stokes example: saddle, centre lines split into four points per period.