Well-posedness and applications of classical elastohydrodynamics for a swimming filament

TL;DR

This paper develops a mathematical framework for elastohydrodynamics of inextensible filaments, analyzing their swimming capabilities and introducing new numerical methods for simulation and optimization.

Contribution

It provides a foundational mathematical analysis of filament elastohydrodynamics with inextensibility constraints and demonstrates conditions for filament swimming, including novel numerical techniques.

Findings

Derived conditions for filament swimming based on internal forcing

Developed a new numerical method for simulating inextensible swimmers

Optimized internal fiber forcing to enhance swimming speed

Abstract

We consider a classical elastohydrodynamic model of an inextensible filament undergoing planar motion in $\mathbb{R}^3$. The hydrodynamics are described by resistive force theory, and the fiber elasticity is governed by Euler-Bernoulli beam theory. Our aim is twofold: (1) Serve as a starting point for developing the mathematical analysis of filament elastohydrodynamics, particularly the analytical treatment of an inextensibility constraint, and (2) As an application, prove conditions on internal fiber forcing that allow a free-ended filament to swim. Our analysis of fiber swimming speed is supplemented with a numerical optimization of the internal fiber forcing, as well as a novel numerical method for simulating an inextensible swimmer.