How stagnation lines move, persist and break up in fluid flows
Abstract
Stagnation curves of arbitrary shape (zeros of v(x,t)) are non-generic yet common; we derive their motion. Differentiating v=0 gives LU=−a (U curve velocity, a=vt, L=∇v kills tangent, acts on normal plane via 2×2 block K); dynamics enters only via a. Normal velocity: −K−1 times normal a; tangential solvability: scalar condition Σ=0 per point (Σ: part of a outside range L). Potential flow: K is strain rate, normal velocity pressure gradient over principal strain rate ε, persistence needs uniform pressure on it. Stokes flow: tangential vorticity enters K (centre lines), Σ involves normal vorticity. Navier-Stokes: Σ set by instantaneous flow via nonlocal pressure, viscosity. If Σ≠0, line becomes, to first order in t, isolated points at zeros of Σ (critical points of φt 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≡0. Incompressibility enters via trK=0; for inviscid irrotational homentropic compressible flow, line dilatation is φtt/c2, so lines can be nodes, persistence needs uniform enthalpy, singular K arises at ordinary points, acoustic standing waves give exact examples (node line breaking into alternating sinks, saddles); also degenerate K (pairs born at distance O(√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.
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BibTeX
@misc{rajamanickam2026stagnation,
title = {{How stagnation lines move, persist and break up in fluid flows}},
author = {Prabakaran Rajamanickam},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26164},
eprint = {2610.26164},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26164},
}
RIS
TY - UNPB
TI - How stagnation lines move, persist and break up in fluid flows
AU - Prabakaran Rajamanickam
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26164
UR - https://arxiv.org/abs/2610.26164
AB - Stagnation curves of arbitrary shape (zeros of v(x,t)) are non-generic yet common; we derive their motion. Differentiating v=0 gives LU=−a (U curve velocity, a=v_t, L=∇v kills tangent, acts on normal plane via 2×2 block K); dynamics enters only via a. Normal velocity: −K^−1 times normal a; tangential solvability: scalar condition Σ=0 per point (Σ: part of a outside range L). Potential flow: K is strain rate, normal velocity pressure gradient over principal strain rate ε, persistence needs uniform pressure on it. Stokes flow: tangential vorticity enters K (centre lines), Σ involves normal vorticity. Navier-Stokes: Σ set by instantaneous flow via nonlocal pressure, viscosity. If Σ≠0, line becomes, to first order in t, isolated points at zeros of Σ (critical points of φ_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≡0. Incompressibility enters via trK=0; for inviscid irrotational homentropic compressible flow, line dilatation is φ_tt/c^2, so lines can be nodes, persistence needs uniform enthalpy, singular K arises at ordinary points, acoustic standing waves give exact examples (node line breaking into alternating sinks, saddles); also degenerate K (pairs born at distance O(√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.
ER -