Modelling Fluid--Structure Interaction in an Initially Elliptical Elastic-Walled tube: Improved Onset Criterion for Self-Excited Oscillations

TL;DR

This paper develops a theoretical model for fluid-structure interactions in an elastic-walled tube, providing an improved criterion for predicting the onset of self-excited oscillations, relevant for understanding complex biological and engineering systems.

Contribution

It introduces a novel eigenvalue-based approach to model wall mechanics, avoiding ad-hoc approximations and enabling more accurate predictions of oscillation onset in elastic tubes.

Findings

Derived leading-order steady and oscillatory solutions for fluid-structure interaction.

Provided a formal analysis of errors from neglecting higher azimuthal modes.

Established an improved criterion for the onset of self-excited oscillations.

Abstract

We present a theoretical description of the fluid--structure interaction observed within a Starling resistor. The typical setup consists of a pre-stretched finite length thin-walled elastic tube mounted between two rigid tubes. The collapsible section is enclosed within a pressure chamber and a viscous fluid is driven through the system by imposing an axial volume flux at the downstream end. Valid within a long-wavelength thin-walled regime, we use our own results to model the wall mechanics. These results arise from the solution of a generalised eigenvalue problem, and avoid the need to invoke the ad-hoc approximations made in previous studies. The wall mechanics are then coupled to the fluid mechanics using the Navier--Stokes equations, under the assumption that the oscillations in the tube wall are of small amplitude, long wavelength and high frequency. We derive problems governing the leading-order steady and oscillatory fluid-structure interaction. At leading order, the system permits normal-mode oscillations of constant frequency and amplitude, which are obtained in the form of series solutions. Higher-order corrections govern the slow growth or decay of the oscillations, however (as in previous work) these growth rates can be determined by analysing the system's global energy budget without needing to compute the higher-order terms explicitly. Our results permit the first formal analysis of the errors incurred by neglecting contributions from higher-order azimuthal modes, and enable the determination of improved criterion for the onset of self-excited oscillations in the tube wall.