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In inertial microfluidics colloidal particles in a Poiseuille flow experiencethe Segr\'e-Silberberg lift force, which drives them to specific positions inthe channel cross section. Due to the Saffman effect an external force appliedalong the microchannel induces a cross-streamline migration to a newequilibrium position. We apply optimal control theory to design the timeprotocol of the axial control force in order to steer a single particle asprecisely as possible from a channel inlet to an outlet at a chosen targetposition. We discuss the influence of particle radius and channel length andshow that optimal steering is cheaper than using a constant control force.Using a single optimized control-force protocol, we demonstrate that even apulse of particles spread along the channel axis can be steered to a target andthat particles of different radii can be separarted most efficiently.

Optimal control of colloidal trajectories in inertial microfluidics using the Saffman effect