# Universal image systems for non-periodic and periodic Stokes flows above a no-slip wall

## TL;DR

This paper introduces a universal, force-neutral image system for Stokes flows near a wall, enabling efficient and accurate simulations in non-periodic and periodic geometries using fast kernel methods.

## Contribution

A novel force-neutral image system that unifies non-periodic and periodic Stokes flow representations, facilitating black-box fast summation techniques.

## Key findings

- Efficient implementation of the image system with fast kernel methods.
- Accurate representation of periodic and non-periodic Stokes flows.
- Extension to other fundamental solutions like the Rotne-Prager-Yamakawa tensor.

## Abstract

It is well-known that by placing judiciously chosen image point forces and doublets to the Stokeslet above a flat wall, the no-slip boundary condition can be conveniently imposed on the wall [Blake, J. R. Math. Proc. Camb. Philos. Soc. 70(2), 1971: 303.]. However, to further impose periodic boundary conditions on directions parallel to the wall usually involves tedious derivations because single or double periodicity in Stokes flow may require the periodic unit to have no net force, which is not satisfied by the well-known image system. In this work we present a force-neutral image system. This neutrality allows us to represent the Stokes image system in a universal formulation for non-periodic, singly periodic and doubly periodic geometries. This formulation enables the black-box style usage of fast kernel summation methods. We demonstrate the efficiency and accuracy of this new image method with the periodic kernel independent fast multipole method in both non-periodic and doubly periodic geometries. We then extend this new image system to other widely used Stokes fundamental solutions, including the Laplacian of the Stokeslet and the Rotne-Prager-Yamakawa tensor.