  Small finite-size particles suspended in fluid flow through an enclosed
curved duct can focus to points or periodic orbits in the two-dimensional duct
cross-section. This particle focusing is due to a balance between inertial lift
forces arising from axial flow and drag forces arising from cross-sectional
vortices. The inertial particle focusing phenomenon has been exploited in
various industrial and medical applications to passively separate particles by
size using purely hydrodynamic effects. A fixed size particle in a circular
duct with a uniform rectangular cross-section can have a variety of particle
attractors, such as stable nodes/spirals or limit cycles, depending on the
radius of curvature of the duct. Bifurcations occur at different radii of
curvature, such as pitchfork, saddle-node and saddle-node infinite period
(SNIPER), which result in variations in the location, number and nature of
these particle attractors. By using a quasi-steady approximation, we extend the
theoretical model of Harding et al. (2019) developed for particle dynamics in
circular ducts to spiral duct geometries with slowly varying curvature, and
numerically explore the particle dynamics within. Bifurcations of particle
attractors with respect to radius of curvature can be traversed within spiral
ducts and give rise to rich nonlinear particle dynamics and various types of
tipping phenomena, such as bifurcation-induced tipping (B-tipping),
rate-induced tipping (R-tipping) and a combination of both, which we explore in
detail. We discuss implications of these unsteady dynamical behaviours for
particle separation and propose novel mechanisms to separate particles by size
in a non-equilibrium manner.
