# Theoretical justification and error analysis for slender body theory with free ends

## TL;DR

This paper extends the PDE analysis of slender body theory to fibers with free ends, providing error bounds that include endpoint effects, which are more realistic for modeling biological fibers like cilia.

## Contribution

The authors develop a PDE framework for free endpoint slender bodies, deriving error bounds that account for endpoint geometry and force decay, advancing the theoretical understanding of slender body approximations.

## Key findings

- Error bound includes a term proportional to ε|log ε| and an endpoint term proportional to ε.
- Extension from closed-loop to free-end fibers introduces new geometric and analytical challenges.
- Results are relevant for more accurate biological fiber modeling in viscous fluids.

## Abstract

Slender body theory is a commonly used approximation in computational models of thin fibers in viscous fluids, especially in simulating the motion of cilia or flagella in swimming microorganisms. In [23], we developed a PDE framework for analyzing the error introduced by the slender body approximation for closed-loop fibers with constant radius $\epsilon$, and showed that the difference between our closed-loop PDE solution and the slender body approximation is bounded by an expression proportional to $\epsilon|\log\epsilon|$. Here we extend the slender body PDE framework to the free endpoint setting, which is more physically relevant from a modeling standpoint but more technically demanding than the closed loop analysis. The main new difficulties arising in the free endpoint setting are defining the endpoint geometry, identifying the extent of the 1D slender body force density, and determining how the well-posedness constants depend on the non-constant fiber radius. Given a slender fiber satisfying certain geometric constraints at the filament endpoints and a one-dimensional force density satisfying an endpoint decay condition, we show a bound for the difference between the solution to the slender body PDE and the slender body approximation in the free endpoint setting. The bound is a sum of the same $\epsilon|\log\epsilon|$ term appearing in the closed loop setting and an endpoint term proportional to $\epsilon$, where $\epsilon$ is now the maximum fiber radius.