Inertial migration of an electrophoretic rigid sphere in a two-dimensional Poiseuille flow

There has been a recent interest in integrating external fields with inertial microfluidic devices to tune particle focusing. In this work, we analyse the inertial migration of an electrophoretic particle in a two-dimensional Poiseuille flow with an electric field applied parallel to the walls. For a thin electrical double layer, the particle exhibits a slip-driven electrokinetic motion along the direction of the applied electric field, which causes the particle to lead or lag the flow (depending on its surface charge). The fluid disturbance caused by this slip-driven motion is characterized by a rapidly decaying source-dipole field which alters the inertial lift on the particle. We determine this inertial lift using the reciprocal theorem. Assuming no wall effects, we derive an analytical expression for a 'phoretic lift' which captures the modification to the inertial lift due to electrophoresis. We also take wall effects into account, at the leading order, using the method of reflections. We find that for a leading particle, the phoretic lift acts towards the regions of high shear (i.e. walls), while the reverse is true for a lagging particle. Using an order-of-magnitude analysis, we obtain different components of the inertial force and classify them on the basis of the interactions from which they emerge. We show that the dominant contribution to the phoretic lift originates from the interaction of the source-dipole field (generated by the electrokinetic slip at the particle surface) with the stresslet field (generated due to particle's resistance to strain in the background flow). Furthermore, to contrast the slip-driven phenomenon (electrophoresis) from the force-driven phenomenon (buoyancy) in terms of their influence on the inertial migration, we also study a non-neutrally buoyant particle. We show that the gravitational effects alter the inertial lift primarily through the interaction of the background shear with the buoyancy-induced Stokeslet field.