  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.
