  Inertial focusing in curved microfluidic ducts exploits the interaction of
drag force from the Dean flow with the inertial lift force to separate
particles or cells laterally across the cross-section width according to their
size. Experimental work has identified that using a trapezoidal cross-section,
as opposed to a rectangular one, can enhance the sized based separation of
particles/cells over a wide range of flow rates. Using our model, derived by
carefully examining the way the Dean drag and inertial lift forces interact at
low flow rates, we calculate the leading order approximation of these forces
for a range of trapezoidal ducts, both vertically symmetric and non-symmetric,
with increasing amount of skew towards the outside wall. We then conduct a
systematic study to examine the bifurcations in the particle equilbira that
occur with respect to a shape parameter characterising the trapezoidal
cross-section. We reveal how the dynamics associated with particle migration
are modified by the degree of skew in the cross-section shape, and show the
existence of cusp bifurcations (with the bend radius as a second parameter).
Additionally, our investigation suggests an optimal amount of skew for the
trapezoidal cross-section for the purposes of maximising particle separation
over a wide range of bend radii.
