Inertial focusing dynamics of spherical particles in curved microfluidic ducts with a trapezoidal cross-section

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.