On the General Analytical Solution of the Kinematic Cosserat Equations

TL;DR

This paper derives a comprehensive analytical solution to the kinematic Cosserat equations for elastic rods using Lie symmetry analysis, enabling more efficient simulations of fluid-structure interactions like microswimmers.

Contribution

It introduces a general analytical solution to the Cosserat equations and develops a hybrid solver for fluid-rod problems, reducing numerical stiffness in simulations.

Findings

Closed-form solution depends on two arbitrary vector functions.

Solution is analytical except in a specific domain with an algebraic relation.

Hybrid solver enables high-fidelity simulations of flagellated microswimmers.

Abstract

Based on a Lie symmetry analysis, we construct a closed form solution to the kinematic part of the (partial differential) Cosserat equations describing the mechanical behavior of elastic rods. The solution depends on two arbitrary analytical vector functions and is analytical everywhere except a certain domain of the independent variables in which one of the arbitrary vector functions satisfies a simple explicitly given algebraic relation. As our main theoretical result, in addition to the construction of the solution, we proof its generality. Based on this observation, a hybrid semi-analytical solver for highly viscous two-way coupled fluid-rod problems is developed which allows for the interactive high-fidelity simulations of flagellated microswimmers as a result of a substantial reduction of the numerical stiffness.