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The N-link model for slender rods in a viscous fluid: well-posedness and convergence to classical elastohydrodynamics equations

Clément Moreau, François Alouges, Aline Lefebvre-Lepot, Jessie Levillain

TL;DR

The paper establishes the well-posedness of the discrete $N$-link elastohydrodynamics model for planar, inextensible filaments in a viscous fluid and proves convergence of its solutions to the classical elastohydrodynamics equations as the discretization becomes finer. Central to the analysis is an energy-dissipation law that yields uniform bounds and enables compactness arguments, linking the discrete and continuous formulations. The results provide a rigorous justification for using the $N$-link discretization as a faithful discrete approximation of a continuous filament, and they imply the global existence of solutions for the elastohydrodynamics model under suitable initial data. The framework sets the stage for extensions to nonlocal hydrodynamics, active filaments, three-dimensional motion, and multi-filament interactions, with potential implications for bio-inspired micro-robotics and cellular micromechanics.

Abstract

Flexible fibers at the microscopic scale, such as flagella and cilia, play essential roles in biological and synthetic systems. The dynamics of these slender filaments in viscous flows involve intricate interactions between their mechanical properties and hydrodynamic drag. In this paper, considering a 1D, planar, inextensible Euler-Bernoulli rod in a viscous fluid modeled by Resistive Force Theory, we establish the existence and uniqueness of solutions for the $N$-link model, a mechanical model, designed to approximate the continuous filament with rigid segments. Then, we prove the convergence of the $N$-link model's solutions towards the solutions to classical elastohydrodynamics equations of a flexible slender rod. This provides an existence result for the limit model, comparable to those by Mori and Ohm [Nonlinearity, 2023], in a different functional context and with different methods. Due to its mechanical foundation, the discrete system satisfies an energy dissipation law, which serves as one of the main ingredients in our proofs. Our results provide mathematical validation for the discretization strategy that consists in approximating a continuous filament by the mechanical $N$-link model, which does not correspond to a classical approximation of the underlying PDE.

The N-link model for slender rods in a viscous fluid: well-posedness and convergence to classical elastohydrodynamics equations

TL;DR

The paper establishes the well-posedness of the discrete -link elastohydrodynamics model for planar, inextensible filaments in a viscous fluid and proves convergence of its solutions to the classical elastohydrodynamics equations as the discretization becomes finer. Central to the analysis is an energy-dissipation law that yields uniform bounds and enables compactness arguments, linking the discrete and continuous formulations. The results provide a rigorous justification for using the -link discretization as a faithful discrete approximation of a continuous filament, and they imply the global existence of solutions for the elastohydrodynamics model under suitable initial data. The framework sets the stage for extensions to nonlocal hydrodynamics, active filaments, three-dimensional motion, and multi-filament interactions, with potential implications for bio-inspired micro-robotics and cellular micromechanics.

Abstract

Flexible fibers at the microscopic scale, such as flagella and cilia, play essential roles in biological and synthetic systems. The dynamics of these slender filaments in viscous flows involve intricate interactions between their mechanical properties and hydrodynamic drag. In this paper, considering a 1D, planar, inextensible Euler-Bernoulli rod in a viscous fluid modeled by Resistive Force Theory, we establish the existence and uniqueness of solutions for the -link model, a mechanical model, designed to approximate the continuous filament with rigid segments. Then, we prove the convergence of the -link model's solutions towards the solutions to classical elastohydrodynamics equations of a flexible slender rod. This provides an existence result for the limit model, comparable to those by Mori and Ohm [Nonlinearity, 2023], in a different functional context and with different methods. Due to its mechanical foundation, the discrete system satisfies an energy dissipation law, which serves as one of the main ingredients in our proofs. Our results provide mathematical validation for the discretization strategy that consists in approximating a continuous filament by the mechanical -link model, which does not correspond to a classical approximation of the underlying PDE.

Paper Structure

This paper contains 16 sections, 8 theorems, 94 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem formulation and main results
  3. Continuous model
  4. $N$-link model
  5. Matrix expression
  6. Main results
  7. Energy dissipation
  8. Proofs
  9. Proof of Theorem \ref{['thm: discrete well-posedness']}
  10. Proof of Theorem \ref{['thm: convergence']}
  11. Proof of Corollary \ref{['corollary']}
  12. Discussion
  13. Non-local and non-Newtonian hydrodynamics.
  14. Active filament.
  15. Three-dimensional motion.
  16. ...and 1 more sections

Key Result

Theorem 2.1

The discrete system eq: matrixAXF, given a set of initial conditions $r_1(0)=r_1^0 \in \mathbb{R}^{2}, \, \theta_i(0)=\theta_i^0 \in \mathbb{R}$ for $1\leq i \leq N$, admits a unique global solution $r_1 \in C^1(\mathbb{R}_+), \, (\theta_i)_{1\leq i \leq N}\in C^1(\mathbb{R}_+)^{N}$, $(m_i^z)_{1\leq

Figures (1)

  • Figure 1: Diagram and notations for (a) the continuous elastohydrodynamic flagellum model and (b) the $N$-link model.

Theorems & Definitions (19)

  • Remark 2.1
  • Remark 2.2
  • Theorem 2.1: Well-posedness for the $N$-link swimmer
  • Theorem 2.2: Convergence
  • Corollary 2.1: Existence of solutions to the continuous system
  • Proposition 3.1: Energy dissipation
  • proof
  • Remark 3.1
  • Remark 3.2
  • Lemma 4.1: Bounds on $(r_i)_i$ and $(\theta_i)_i$
  • ...and 9 more