Regularized Stokeslet segments

TL;DR

This paper introduces a specialized variation of the regularized Stokeslet method for modeling forces on filaments, enabling efficient and accurate simulations of biological structures like flagella with fewer discretization points.

Contribution

The paper develops a new formulation that decouples the regularization parameter from segment length, allowing for simplified and more efficient filament modeling in Stokes flow.

Findings

Flagella can be accurately modeled with as few as 11 segments.

The regularization parameter influences swimming speed and filament behavior.

Doubling segments does not significantly affect results if forces are well-resolved.

Abstract

We present a variation of the method of regularized Stokeslet (MRS) specialized for the case of forces and torques distributed over filaments in three dimensions. The new formulation is based on the exact solution of Stokes equation generated by a linear continuous distribution of regularized forces along a line segment. Therefore, a straight filament with linearly varying forces does not require discretization. A general filament is approximated by a piecewise linear curve in three dimensions where the length of each line segment is chosen only based on the variation of the force field and the desired accuracy of its piecewise linear approximation. The most significant advantage of this formulation is that the values of the regularization parameter $\epsilon$ and the length of the segments $h$ are decoupled as long as $\epsilon<h$ so that $\epsilon$ can be selected as a proxy for the radius of the filament and $h$ is chosen to discretize the forces and torques. We analyze the performance on test problems and present biological applications of sperm motility based on existing models of swimming flagella in open space and near a plane wall. The results show, for example, that because the forces along the flagellum vary mildly, a flagellum can be approximated with as few as 11 segments of length $h$ while fixing the regularization parameter to $\epsilon=h/30$, overcoming the need for hundreds of discretization nodes required by the MRS when $\epsilon$ is small. The filament behaves like a slender cylindrical tube of radius $\approx 0.97\epsilon$ so that the value of $\epsilon$ influences the flagellum's swimming speed. For fixed regularization, doubling the number of line segments does not affect the results significantly as long as the force field is resolved. Examples that require rotlets and potential dipoles along the filament are also presented.