Remarks on regularized Stokeslets in slender body theory

TL;DR

This paper analyzes the effects of regularization parameters in slender body theory for Stokes flow, showing that regularized SBT does not converge to classical SBT as fiber radius shrinks, but the practical impact may be limited.

Contribution

It derives bounds on the difference between regularized and classical SBT and clarifies how regularization affects flow accuracy around slender fibers.

Findings

Regularized SBT differs from classical SBT by a term involving log(δ/ε).

Choosing δ proportional to ε results in an O(1) discrepancy as ε→0.

Numerical results confirm the theoretical discrepancy and suggest limited practical impact.

Abstract

We remark on the use of regularized Stokeslets in the slender body theory (SBT) approximation of Stokes flow about a thin fiber of radius $\epsilon>0$. Denoting the regularization parameter by $\delta$, we consider regularized SBT based on the most common regularized Stokeslet plus a regularized doublet correction. Given sufficiently smooth force data along the filament, we derive $L^\infty$ bounds for the difference between regularized SBT and its classical counterpart in terms of $\delta$, $\epsilon$, and the force data. We show that the regularized and classical expressions for the velocity of the filament itself differ by a term proportional to $\log(\delta/\epsilon)$ -- in particular, $\delta=\epsilon$ is necessary to avoid an $O(1)$ discrepancy between the theories. However, the flow at the surface of the fiber differs by an expression proportional to $\log(1+\delta^2/\epsilon^2)$, and any choice of $\delta\propto\epsilon$ will result in an $O(1)$ discrepancy as $\epsilon\to 0$. Consequently, the flow around a slender fiber due to regularized SBT does not converge to the solution of the well-posed \emph{slender body PDE} which classical SBT is known to approximate. Numerics verify this $O(1)$ discrepancy but also indicate that the difference may have little impact in practice.