The fluid mechanics of poohsticks

The paper derives the relative-velocity equation dA/dt = −(J + (2/3)St^{-1} I)A for neutrally buoyant particles (A = v − u), diagonalizes it to obtain instantaneous growth rates λ − (2/3)St^{-1} and −λ − (2/3)St^{-1}, and links λ^2 = Q = (s^2 − ω^2)/4, concluding that in regions with Q > 4/9 St^{-2} the particle–flow trajectories separate exponentially (Eqs. (2.4)–(2.11) and discussion) . The paper notes this diagonalization-based argument is heuristic and points to a rigorous invariant-manifold treatment by Sapsis and Haller, which corroborates the threshold and shows the estimate is a lower bound . The model reproduces the same derivation, uses the Okubo–Weiss parameter, obtains the same threshold Q > 4/9 St^{-2}, and adds a standard exponential-dichotomy justification (Coppel) to formalize the nonautonomous case. Hence both are correct and follow substantially the same proof, with the model supplying a standard rigorization of the paper’s heuristic step.