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

TL;DR

This paper proves the well-posedness of an N-link mechanical model for slender rods in viscous fluids and demonstrates its convergence to classical elastohydrodynamics equations, validating a discretization approach for flexible filaments.

Contribution

It establishes existence, uniqueness, and convergence results for the N-link model, providing a mathematical foundation for discretizing flexible filaments in viscous flows.

Findings

Existence and uniqueness of solutions for the N-link model.

Convergence of the N-link model to classical elastohydrodynamics equations.

Validation of the N-link discretization strategy for flexible filaments.

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.