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

## TL;DR

This paper analyzes how electrophoretic particles migrate in a 2D Poiseuille flow, revealing the effects of electrokinetic slip and wall interactions on inertial lift, with analytical expressions and comparisons to buoyancy effects.

## Contribution

It provides an analytical model for electrophoretic inertial migration considering wall effects and distinguishes slip-driven from force-driven phenomena.

## Key findings

- Phoretic lift acts towards high shear regions for leading particles.
- Slip-driven source-dipole interacts with stresslet to dominate lift.
- Gravity modifies inertial lift via buoyancy and shear interactions.

## Abstract

In this work, we analyze the inertial migration of an electrophoretic particle in a 2-D 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 and find that the analytical expression is valid away from the walls. 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 which contribute to 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 slip-driven source-dipole field with the stresslet field (generated due to particle's resistance to strain in the background flow). Furthermore, to contrast the slip-driven phenomenon from a force-driven phenomenon 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 background shear with buoyancy induced stokeslet field.