A nonlocal curve evolution for an immersed elastic filament: global existence and convergence to resistive force theory

TL;DR

This paper studies a nonlocal curve evolution model for an elastic filament in Stokes flow, proving global existence and showing it converges to resistive force theory as the filament radius shrinks.

Contribution

It establishes global well-posedness of the model and demonstrates the emergence of resistive force theory from a more detailed nonlocal model.

Findings

Proves global well-posedness in the natural energy space.

Shows convergence to resistive force theory as filament radius approaches zero.

Models fluid effects via a pseudodifferential operator interpolating between theories.

Abstract

We consider a nonlocal curve evolution belonging to a hierarchy of models for the dynamics of an inextensible elastic filament in a 3D Stokes fluid. This model captures the principal part of a full free boundary problem for an elastic filament in Stokes flow. The fluid effects on the filament evolution are encoded in a pseudodifferential force-to-velocity operator which may be regarded as an interpolation between resistive force theory at low wavenumbers and a Stokes boundary value problem at high wavenumbers. Here the curve is considered to be the centerline of a 3D filament with constant cross sectional radius $\epsilon>0$. We show global well-posedness for the curve evolution in the natural energy space. This loosely suggests that the full evolution may be globally well-posed if the large-scale geometry is controlled. Furthermore, we prove convergence to resistive force theory dynamics as $\epsilon\to 0$, which illustrates how resistive force theory emerges from more detailed models.