  Particles suspended in a fluid flow through a curved duct can focus to
specific locations within the duct cross-section. This particle focusing is a
result of a balance between two dominant forces acting on the particle: (i) the
inertial lift force arising from small but non-negligible inertia of the fluid,
and (ii) the secondary drag force due to the cross-sectional vortices induced
by the curvature of the duct. By adopting a simplified particle dynamics model
developed by Ha et al.~[1], we investigate both analytically and numerically,
the particle equilibria and their bifurcations when a small particle is
suspended in low-flow-rate fluid flow through a curved duct having a $2\times1$
and a $1\times2$ rectangular cross-section. In certain parameter regimes of the
model, we analytically obtain the particle equilibria and deduce their
stability, while for other parameter regimes, we numerically calculate the
particle equilibria and stability. Moreover, we observe a number of different
bifurcations in particle equilibria such as saddle-node, pitchfork and Hopf, as
the model parameters are varied. These results may aid in the design of
inertial microfluidic devices aimed at particle separation by size.
