Trapping-loss transition via a saddle-node bifurcation in thermophoretic particle transport driven by a time-periodic vortex

Abstract

The transport of inertial particles in unsteady flows is often governed by the competition between multiple migration mechanisms. We investigate the interplay between inertia-induced drift and thermophoretic migration in a time-periodic vortex containing a localized temperature field. Starting from the small-Stokes-number limit of the particle equations of motion, we derive a cycle-averaged radial migration model describing the slow evolution of suspended particles over timescales much longer than the forcing period. The competition between outward inertia-induced drift and inward thermophoretic migration gives rise to stable and unstable fixed points of the reduced radial dynamics, corresponding respectively to particle trapping states and separatrices bounding trapped trajectories. The existence and location of these states are shown to be governed by the dimensionless control parameter $\Pi$, which measures the relative strength of inertia-induced transport to the thermophoretic transport. As $\Pi$ is decreased below a critical value, the stable and unstable fixed points coalesce and disappear, resulting in the loss of particle trapping. Phase portraits, bifurcation diagrams, and local asymptotic analysis demonstrate that trapping is destroyed through a saddle-node bifurcation. The transition is further characterized by the vanishing of the dominant eigenvalue and the associated divergence of the relaxation time, indicative of critical slowing down. Additional calculations employing various other velocity and temperature profiles demonstrate that the trapping-loss mechanism is robust and not specific to a particular profile.