Spectral analysis for elastica 3-dimensional dynamics in a shear flow

TL;DR

This paper performs spectral analysis on the three-dimensional dynamics of elastic filaments in shear flow, revealing stability characteristics and buckling behaviors similar to previous two-dimensional and compressional flow studies.

Contribution

It introduces a spectral analysis approach for 3D elastica in shear flow, extending stability analysis methods to three dimensions with novel eigenvalue solutions.

Findings

Eigenvalues and eigenfunctions derived for 3D elastica stability.

Similar buckling behaviors as in 2D elastica and compressional flow.

Dependence of buckled shapes on bending-to-hydrodynamic force ratio.

Abstract

We present the spectral analysis of three-dimensional dynamics of an elastic filament in a shear flow of a viscous fluid at a low Reynolds number in the absence of Brownian motion. The elastica model is used. The fiber initially is almost straight at an arbitrary orientation, with small perpendicular perturbations in the shear plane and out-of-plane. To analyze the stability of both perturbations, equations for the eigenvalues and eigenfunctions are derived and solved by the Chebyshev spectral collocation method. It is shown that their crucial features are the same as in the case of the two-dimensional elastica dynamics in shear flow [Becker and Shelley, Phys. Rev. Lett. 2001] and the three-dimensional elastica dynamics in the compressional flow [Chakrabarti et al., Nat. Phys., 2020]. We find a similar dependence of the buckled shapes on the ratio of bending to hydrodynamic forces as in the simulations for elastic fibers of a nonzero thickness [Slowicka et al., New J. Phys., 2022].