Remarks on regularized Stokeslets in slender body theory

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.