# Adapting the Teubner reciprocal relations for stokeslet objects

## TL;DR

This paper presents a numerical method to predict the motion of asymmetric self-propelled colloids with arbitrary surface velocity profiles using a stokeslet representation, adapting the Lorentz Reciprocal Theorem for discrete sources.

## Contribution

It introduces a novel numerical approach combining the Teubner method and the Lorentz Reciprocal Theorem for modeling colloidal swimmer motion with arbitrary surface velocities.

## Key findings

- Method accurately predicts swimmer velocities based on surface velocity profiles.
- The approach simplifies calculations by using linear operations on the Oseen matrix.
- Applicable to nonuniformly charged bodies in electrophoretic mobility studies.

## Abstract

Self-propelled colloidal swimmers move by pushing the adjacent fluid backwards. The resulting motion of an asymmetric body depends on the profile of pushing velocity over its surface. We describe a method of predicting the motion arising from arbitrary velocity profiles over a given body shape, using a discrete-source "stokeslet" representation. The net velocity and angular velocity is a sum of contributions from each point on the surface. The contributions from a given point depend only on the shape. We give a numerical method to find these contributions in terms of the stokeslet positions defining the shape. Each contribution is determined by linear operations on the Oseen interaction matrix between pairs of stokeslets. We first adapt the Lorentz Reciprocal Theorem to discrete sources. We then use the theorem to implement the method of Teubner[1] to determine electrophoretic mobilities of nonuniformly charged bodies.