Rheotaxis of a Microswimmer in Poiseuille Flow

Abstract

We investigate the motion of a spherical microswimmer in a planar Poiseuille flow using analytical calculations and numerical simulations. Our results show that the microswimmer's orientation undergoes periodic oscillations governed by pendulum-like dynamics. The oscillation period is proportional to the channel width and to the complete elliptic integral of the first kind evaluated at the maximum orientation angle, and inversely proportional to the square root of the product of the maximum flow speed and the self-propulsion speed. Based on the net displacement along the flow direction within one oscillation period and the signs of the maximum and minimum velocities, we identify five motion states: upstream, oscillatory upstream, zero-drift oscillatory, oscillatory downstream, and downstream motion. The net displacement is jointly determined by the ratio of the self-propulsion speed to the maximum flow speed and the complete elliptic integrals of the first and second kinds. The signs of the maximum and minimum velocities are determined by the speed ratio and the cosine of the orientation angle. Flow nonuniformity promotes upstream migration, whereas inertial lift produces negligible radial displacement during one oscillation period. These findings identify the key parameters governing the motion of spherical microswimmers and provide significant implications for understanding microorganism motility and designing microrobots with prescribed motion states.