Dynamics of tethered membranes in inviscid flow

TL;DR

This paper explores the complex oscillatory behaviors of tethered membranes in inviscid fluid flow, analyzing stability, oscillation characteristics, and asymptotic behaviors through multiple models.

Contribution

It introduces a comprehensive analysis of membrane dynamics with tethered boundary conditions, including nonlinear eigenvalue solutions and asymptotic scaling laws.

Findings

Identification of parameter regions with periodic and chaotic oscillations

Derivation of stability criteria and eigenvalue problems

Qualitative similarities across different membrane models

Abstract

We investigate the dynamics of membranes that are held by freely-rotating tethers in fluid flows. The tethered boundary condition allows periodic and chaotic oscillatory motions for certain parameter values. We characterize the oscillations in terms of deflection amplitudes, dominant periods, and numbers of deflection extrema along the membranes across the parameter space of membrane mass density, stretching modulus, pretension, and tether length. We determine the region of instability and the small-amplitude behavior by solving a nonlinear eigenvalue problem. We also consider an infinite periodic membrane model, which yields a regular eigenvalue problem, analytical results, and asymptotic scaling laws. We find qualitative similarities among all three models in terms of the oscillation frequencies and membrane shapes at small and large values of membrane mass, pretension, and tether length/stiffness.