Geometric Field Theory for Elastohydrodynamics of Cosserat Rods

Abstract

Slender structures are ubiquitous in biological and physical systems, from bacterial flagella to soft robotic arms. The Cosserat rod provides a mathematical framework for slender bodies that can stretch, shear, twist and bend. In viscous fluid environments at low Reynolds numbers - as encountered in soft matter physics, biophysics, and soft continuum robotics - inertial effects become negligible, and hydrodynamic forces are well approximated by Stokes friction. We demonstrate that the resulting elastohydrodynamic equations of motion, when formulated using Cartan's method of moving frames, possess the structure of a geometric field theory in which the configuration field takes values in SE(3), the Lie group of rigid body motions. This geometric formulation yields coordinate-independent equations that are manifestly invariant under spatial isometries and naturally suited to constitutive modeling based on Curie's principle. We derive integrability conditions that determine when constitutive laws can be derived from an energy functional, thereby distinguishing between passive and active material responses. We also obtain the beam limit for small deformations. Our results establish a unified geometric framework for the nonlinear mechanics of slender structures in slow viscous flow and enable efficient numerical solutions.