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Geometric Field Theory for Elastohydrodynamics of Cosserat Rods

Mingjia Yan, Mohamed Warda, Balázs Németh, Lukas Kikuchi, Ronojoy Adhikari

TL;DR

This work addresses the nonlinear elastohydrodynamics of slender Cosserat rods in slow viscous flows by casting the governing equations as a geometric field theory on the Lie group SE(3) via Cartan's moving frames. By formulating kinematics and dynamics with differential forms and Lie algebra-valued fields, the authors obtain coordinate-free, isometry-invariant expressions that unify kinematics, dynamics, and constitutive modeling, and derive integrability conditions to distinguish passive from active responses. The framework yields a closed 12-dimensional system, provides linearization and coordinatized reductions (including a planar SE(2) case), and demonstrates a beam-limit reduction to Euler-Bernoulli theory in appropriate limits, while preserving geometric structure for numerical integration. This geometric perspective offers a robust foundation for actuation, non-conservative forces, and structure-preserving discretizations with potential applications in soft robotics and complex filamentary systems. Key mathematical ingredients include the SE(3) configuration phi, deformation and velocity variables in se(3), the Maurer–Cartan form dphi = phi xi, and balance laws written as D*Sigma + j du = 0, together with energy-based constitutive relations and integrability conditions on the left-invariant vector fields. The results provide a unified, coordinate-free description of slender-body elastohydrodynamics that naturally respects material indifference and supports efficient numerical schemes that preserve underlying geometric structure.

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.

Geometric Field Theory for Elastohydrodynamics of Cosserat Rods

TL;DR

This work addresses the nonlinear elastohydrodynamics of slender Cosserat rods in slow viscous flows by casting the governing equations as a geometric field theory on the Lie group SE(3) via Cartan's moving frames. By formulating kinematics and dynamics with differential forms and Lie algebra-valued fields, the authors obtain coordinate-free, isometry-invariant expressions that unify kinematics, dynamics, and constitutive modeling, and derive integrability conditions to distinguish passive from active responses. The framework yields a closed 12-dimensional system, provides linearization and coordinatized reductions (including a planar SE(2) case), and demonstrates a beam-limit reduction to Euler-Bernoulli theory in appropriate limits, while preserving geometric structure for numerical integration. This geometric perspective offers a robust foundation for actuation, non-conservative forces, and structure-preserving discretizations with potential applications in soft robotics and complex filamentary systems. Key mathematical ingredients include the SE(3) configuration phi, deformation and velocity variables in se(3), the Maurer–Cartan form dphi = phi xi, and balance laws written as D*Sigma + j du = 0, together with energy-based constitutive relations and integrability conditions on the left-invariant vector fields. The results provide a unified, coordinate-free description of slender-body elastohydrodynamics that naturally respects material indifference and supports efficient numerical schemes that preserve underlying geometric structure.

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.
Paper Structure (12 sections, 82 equations, 2 figures, 1 table)

This paper contains 12 sections, 82 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Cosserat rods
  3. moving frames
  4. Geometric field theory
  5. Lie groups
  6. Differential forms
  7. Elastohydrodynamics
  8. Work, Constitutive laws and boundary conditions
  9. Linearization
  10. Coordinatized equations
  11. Beam limits
  12. Discussion and Conclusion

Figures (2)

  • Figure 1: Kinematics of the Cosserat rod showing the centerline $\boldsymbol{r}(u,t)$ and the orthonormal frames $\boldsymbol{e}_{i}(u,t)$ rigidly attached to the cross-section at material parameter $u$ at time $t$.
  • Figure 2: Examples of deformations of the Cosserat rod. (a) A straight Cosserat rod with no deformation. (b-f) Stretching, shearing, twisting, bending deformation modes, as well as the combined deformation mode. The transparent light blue body is the bulk of the Cosserat rod, with the centerline (black line) going through its radial centre. The material frames and cross-sections are displayed at some intermediate points along the centreline, where the frame vectors are $\boldsymbol{e}_{1}$ (red), $\boldsymbol{e}_{2}$ (blue), and $\boldsymbol{e}_{3}$ (green).