Inertial migration of non-neutrally buoyant particles in uni-directional shearing flows

The science behind the separation of particles of different sizes and densities via the action of shear-induced lift forces, has various practical applications. The present line of research stems from the need to extend recent investigations on the inertial migration of neutrally buoyant particles (buoyancy parameter B_y = 0) in plane Poiseuille flow and Couette flow to non-neutrally buoyant particles (B_y ≠ 0) by including the effects of gravity. The study is conducted at small particle Reynolds numbers and confinement ratios, but for arbitrary B_y and channel Reynolds numbers (Re_c). The objective is to organize the equilibrium positions and relevant scaling regimes in the B_y-Re_c plane. One may have different orientations of gravity, and we focus on the flow-aligned case. For plane Poiseuille flow, as B_y increases, we observe the migration of the off-center equilibria from the original Segre-Silberberg locations at B_y = 0 towards the centerline, albeit in a non-monotonic fashion. For plane Couette flow, the recently discovered supercritical pitchfork bifurcation for B_y = 0 transforms to a broken pitchfork resembling a disfigured 's' for small but finite B_y, that then opens up with increasing B_y. Our calculations should provide a rational basis for particle separation protocols.