On the Deformation of a Hyperelastic Tube Due to Steady Viscous Flow Within

TL;DR

This paper models the steady deformation of hyperelastic tubes under viscous flow, revealing how their nonlinear elasticity influences flow characteristics compared to linear elastic models.

Contribution

It introduces a combined fluid-structure interaction model for hyperelastic tubes using Mooney-Rivlin material theory and lubrication approximation, providing new insights into flow-pressure relationships.

Findings

Hyperelastic tubes deform less than linear elastic ones under the same pressure.

Higher pressure drops are needed in hyperelastic tubes to sustain the same flow rate.

The model applies to physiological flows like blood in arteries.

Abstract

In this chapter, we analyze the steady-state microscale fluid--structure interaction (FSI) between a generalized Newtonian fluid and a hyperelastic tube. Physiological flows, especially in hemodynamics, serve as primary examples of such FSI phenomena. The small scale of the physical system renders the flow field, under the power-law rheological model, amenable to a closed-form solution using the lubrication approximation. On the other hand, negligible shear stresses on the walls of a long vessel allow the structure to be treated as a pressure vessel. The constitutive equation for the microtube is prescribed via the strain energy functional for an incompressible, isotropic Mooney--Rivlin material. We employ both the thin- and thick-walled formulations of the pressure vessel theory, and derive the static relation between the pressure load and the deformation of the structure. We harness the latter to determine the flow rate--pressure drop relationship for non-Newtonian flow in thin- and thick-walled soft hyperelastic microtubes. Through illustrative examples, we discuss how a hyperelastic tube supports the same pressure load as a linearly elastic tube with smaller deformation, thus requiring a higher pressure drop across itself to maintain a fixed flow rate.