Using geometric algebra to represent curvature in shell theory with applications to Starling resistors

TL;DR

This paper introduces a geometric algebra approach using rotors to represent curvature changes in shell theory, applying it to analyze the mechanics of lung-like flexible tubes and validating findings with empirical imaging data.

Contribution

It presents a new rotor-based decomposition of the curvature change tensor in shell theory, enabling detailed analysis of bending and stretching in biological tubes.

Findings

Bending energy is dominated by stretching energy in observed oscillations.

Scaling analysis shows this dominance persists in lung airways.

Empirical stereographic imaging validates the theoretical analysis.

Abstract

We present a novel application of rotors in geometric algebra to represent the change of curvature tensor, that is used in shell theory as part of the constitutive law. We introduce a new decomposition of the change of curvature tensor, which has explicit terms for changes of curvature due to initial curvature combined with strain, and changes in rotation over the surface. We use this decomposition to perform a scaling analysis of the relative importance of bending and stretching in flexible tubes undergoing self excited oscillations. These oscillations have relevance to the lung, in which it is believed that they are responsible for wheezing. The new analysis is necessitated by the fact that the working fluid is air, compared to water in most previous work. We use stereographic imaging to empirically measure the relative importance of bending and stretching energy in observed self excited oscillations. This enables us to validate our scaling analysis. We show that bending energy is dominated by stretching energy, and the scaling analysis makes clear that this will remain true for tubes in the airways of the lung.