  We develop a model of the forces on a spherical particle suspended in flow
through a curved duct under the assumption that the particle Reynolds number is
small. This extends an asymptotic model of inertial lift force previously
developed to study inertial migration in straight ducts. Of particular interest
is the existence and location of stable equilibria within the cross-sectional
plane towards which particles migrates. The Navier-Stokes equations determine
the hydrodynamic forces acting on a particle. A leading order model of the
forces within the cross-sectional plane is obtained through the use of a
rotating coordinate system and a perturbation expansion in the particle
Reynolds number of the disturbance flow. We predict the behaviour of neutrally
buoyant particles at low flow rates and examine the variation in focusing
position with respect to particle size and bend radius, independent of the flow
rate. In this regime, the lateral focusing position of particles approximately
collapses with respect to a dimensionless parameter dependent on three length
scales, specifically the particle radius, duct height, and duct bend radius.
Additionally, a trapezoidal shaped cross-section is considered in order to
demonstrate how changes in the cross-section design influence the dynamics of
particles.
