Motion of a model swimmer near a weakly deforming interface

Abstract

Locomotion of microswimmers near an interface has attracted recent attention and has several applications related to synthetic swimmers and microorganisms. In this work, we study the motion of a model swimmer called the ‘squirmer’ with an arbitrary time-dependent swimming gait near a weakly deforming interface. We first obtain an exact solution of the governing equations for the motion of the swimmer near a plane interface using the bipolar coordinate method, and then an approximate solution using the method of reflections. We thereby derive the velocity of a swimmer due to small interface deformations using the domain perturbation method and Lorentz reciprocal theorem. We use our solution to study the dynamics of a swimmer with steady, as well as time-reversible, squirming gaits. The long-time dynamics of a time-reversible swimmer is such that it either moves towards or away from the interface. Thus, we divide its phase space into regions of attraction (repulsion) towards (from) the interface. The long-time orientation of a time-reversible swimmer that is moving towards the interface depends on its initial orientation. Additionally, we find that the method of reflections is accurate to $O(1)$ distances of the swimmer from the interface.