Bifurcations and dynamics in inertial focusing of particles in curved rectangular ducts

The candidate solution accurately mirrors the paper’s leading‑order small‑Rep model, cross‑sectional ODEs, equilibrium classification via the Jacobian, and the workflow (precompute forces/drag, interpolate, integrate), as set out in Section 2.1 of the paper and immediately following text . Its square‑duct (AR=1) bifurcation sequence, slow‑manifold explanation, and Hopf/limit‑cycle remarks agree with the paper’s Figure 2 narrative and eigenvalue tracks . For AR=2, the candidate correctly states the reversed ordering (subcritical pitchfork on the right edge, then saddle–nodes on top/bottom) and the emergence of two stable spirals for small particles, as in Figure 3(a) ; for larger particles, however, the paper reports a different sequence (five fixed points at large R̃ and no initial pitchfork), so the candidate’s phrase “same qualitative ordering occurs” is imprecise relative to Figure 3(b) . For AR=1/2, the candidate’s supercritical pitchfork (right edge) and eventual pair of unstable spirals with limit cycles plus a right‑side saddle matches Figure 4 and discussion . The parameter‑space maps for stable radial focusing positions and the separation trends are consistent with Figure 5 and its summary text . Finally, the focusing‑dynamics trends and Nθ maps, including the trade‑off between time to focus versus number of turns and sensitivity to initial conditions near saddles/slow manifolds, align with Section 4 and Figure 10 . Net: the paper’s argument is sound for the stated small‑Rep regime; the model answer is correct overall with a minor misstatement for the 2×1, ã=0.15 case.