  The recent advent of advanced microfabrication capabilities of microfluidic
devices has driven attention towards the behavior of particles in inertial
flows within microchannels for applications related to the separation and
concentration of bio-particles. The phenomena of inertial focusing has been
demonstrated to be a robust technique in such applications, where the flow of
particles in a curvilinear geometry has proven to be particularly advantageous,
not only because the geometry can reduce the foot-print of a lab-on-chip
device, but also because the coupling of secondary Dean flows to inertial
forces allows for exquisite particle manipulations. However, the ability to
design a curvilinear channel for a specific application is often based on
empirical results, as theoretical models to date typically do not include the
effects of a finite sized particle within the flow. Here we present a complete
numerical model that directly simulates a particle within a confide curvilinear
flow and using this model we investigate the three dimensional focusing
behavior of inertial particles as well as the applicability of the point
particle assumptions previous researchers have proposed. Finally, we propose a
new model that takes into account the full physics, but relies on a
perturbation expansion of the lateral forces, where the perturbation parameter
is the curvature ratio of the channel. This simple model can be used to predict
the behavior of particles in complex channel geometries where the curvature may
not be constant.
