Ewald summation of doubly periodic 2- and 3-dimensional Stokes multipoles
Abstract
In spite of the qualitative differences between Stokes flows in 2 and 3 dimensions, most particle-resolved computational studies of microswimmers and other colloidal systems dominated by low-Re hydrodynamic interactions are carried out in triply periodic geometries, while the corresponding continuum descriptions are frequently solved in 2-dimensional doubly periodic domains. Furthermore, most experimental examples of biological and synthetic microswimmers are realised in confined, quasi-2-dimensional geometries with a finite thickness in one of the three dimensions. To enable studying the non-trivial differences between these system geometries, in this paper we systematically derive Ewald representations for periodic Stokes force multipoles in geometries with two periodic directions, treating both three-dimensional Stokes flow with one unbounded coordinate and intrinsically two-dimensional singularities. Starting from the periodic Stokeslet, we obtain explicit expressions for force dipoles and quadrupoles and analyse the non-trivial zero-wavevector contribution arising under partial periodicity. The analytical expressions are independently validated by comparing with results from an explicit lattice Boltzmann solver in each geometry.
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BibTeX
@misc{pol2026ewald,
title = {{Ewald summation of doubly periodic 2- and 3-dimensional Stokes multipoles}},
author = {Teunike A. van de Pol and Zhan Ma and Alexander Morozov and Joakim Stenhammar},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00308},
eprint = {2611.00308},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00308},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Ewald summation of doubly periodic 2- and 3-dimensional Stokes multipoles
AU - Teunike A. van de Pol
AU - Zhan Ma
AU - Alexander Morozov
AU - Joakim Stenhammar
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00308
UR - https://arxiv.org/abs/2611.00308
AB - In spite of the qualitative differences between Stokes flows in 2 and 3 dimensions, most particle-resolved computational studies of microswimmers and other colloidal systems dominated by low-Re hydrodynamic interactions are carried out in triply periodic geometries, while the corresponding continuum descriptions are frequently solved in 2-dimensional doubly periodic domains. Furthermore, most experimental examples of biological and synthetic microswimmers are realised in confined, quasi-2-dimensional geometries with a finite thickness in one of the three dimensions. To enable studying the non-trivial differences between these system geometries, in this paper we systematically derive Ewald representations for periodic Stokes force multipoles in geometries with two periodic directions, treating both three-dimensional Stokes flow with one unbounded coordinate and intrinsically two-dimensional singularities. Starting from the periodic Stokeslet, we obtain explicit expressions for force dipoles and quadrupoles and analyse the non-trivial zero-wavevector contribution arising under partial periodicity. The analytical expressions are independently validated by comparing with results from an explicit lattice Boltzmann solver in each geometry.
ER -