A boundary element regularised Stokeslet method applied to cilia and flagella-driven flow

TL;DR

This paper introduces a boundary element regularised Stokeslet method for simulating cilia and flagella-driven flows, improving computational efficiency and accuracy by discretising forces with basis functions and analyzing biological flow patterns.

Contribution

It presents a boundary element implementation of the regularised Stokeslet method that reduces degrees of freedom and decouples mesh size from regularisation, enhancing biological flow modeling.

Findings

Reducing cilia spacing decreases transport.

Increasing cilia number increases transport up to a plateau.

Lorica structure affects flow outside but not force on flagellum.

Abstract

A boundary element implementation of the regularised Stokeslet method of Cortez is applied to cilia and flagella-driven flows in biology. Previously-published approaches implicitly combine the force discretisation and the numerical quadrature used to evaluate boundary integrals. By contrast, a boundary element method can be implemented by discretising the force using basis functions, and calculating integrals using accurate numerical or analytic integration. This substantially weakens the coupling of the mesh size for the force and the regularisation parameter, and greatly reduces the number of degrees of freedom required. When modelling a cilium or flagellum as a one-dimensional filament, the regularisation parameter can be considered a proxy for the body radius, as opposed to being a parameter used to minimise numerical errors. Modelling a patch of cilia, it is found that: (1) For a fixed number of cilia, reducing cilia spacing reduces transport. (2) For fixed patch dimension, increasing cilia number increases the transport, up to a plateau at $9\times 9$ cilia. Modelling a choanoflagellate cell it is found that the presence of a lorica structure significantly affects transport and flow outside the lorica, but does not significantly alter the force experienced by the flagellum.