Motion of several slender rigid filaments in a Stokes flow

TL;DR

This paper analyzes the limiting behavior of multiple slender rigid filaments in a Stokes flow as their thickness approaches zero, deriving simplified decoupled equations and quantifying convergence rates.

Contribution

It provides a rigorous derivation of the limit equations for slender filaments in Stokes flow, including explicit convergence rates and fluid-structure interaction effects.

Findings

Limit equations are decoupled first-order ODEs depending on the filaments' limit curves and background flow.

Convergence rate of the filament trajectories is of order O(|log ε|^{-1/2}).

Identifies initial exponential relaxation in velocities within O(ε^2 |log ε|) time.

Abstract

We investigate the dynamics of several slender rigid bodies moving in a flow driven by the three-dimensional steady Stokes system in presence of a smooth background flow. More precisely we consider the limit where the thickness of these slender rigid bodies tends to zero with a common rate $\varepsilon$, while their volumetric mass density is held fixed, so that the bodies shrink into separated massless curves. While for each positive $\varepsilon$, the bodies' dynamics are given by the Newton equations and correspond to some coupled second-order ODEs for the positions of the bodies, we prove that the limit equations are decoupled first-order ODEs whose coefficients only depend on the limit curves and on the background flow. These coefficients appear through appropriate renormalized Stokes resistance tensors associated with each limit curve, and through renormalized Fax\'en-type force and torque associated with the limit curves and the background flow. We establish a rate of convergence of the curves of order $O( \vert \log \varepsilon \vert^{-1/2})$. We also determine the limit effect due to the limit curves on the fluid, in the spirit of the immersed boundary method. Both for the convergence of the filament velocities and the fluid velocities we identify an initial exponential relaxation within a $O(\varepsilon^2 \vert \log \varepsilon \vert)$ time.