Revisiting steady viscous flow of a generalized Newtonian fluid through a slender elastic tube using shell theory

TL;DR

This paper develops an analytical model for steady viscous flow of generalized Newtonian fluids through elastic tubes, incorporating shell theory and asymptotic methods, validated by numerical simulations, with applications to blood vessel flow.

Contribution

It introduces a coupled analytical framework combining shell theory and lubrication approximation for non-Newtonian fluid flow in elastic tubes, extending classical models to deformable biofluidic channels.

Findings

Derived a generalized Hagen--Poiseuille law for soft microtubes.

Validated analytical predictions against detailed numerical simulations.

Identified boundary and corner layer effects in tube deformation.

Abstract

A flow vessel with an elastic wall can deform significantly due to viscous fluid flow within it, even at vanishing Reynolds number (no fluid inertia). Deformation leads to an enhancement of throughput due to the change in cross-sectional area. The latter gives rise to a non-constant pressure gradient in the flow-wise direction and, hence, to a nonlinear flow rate--pressure drop relation (unlike the Hagen--Poiseuille law for a rigid tube). Many biofluids are non-Newtonian, and are well approximated by generalized Newtonian (say, power-law) rheological models. Consequently, we analyze the problem of steady low Reynolds number flow of a generalized Newtonian fluid through a slender elastic tube by coupling fluid lubrication theory to a structural problem posed in terms of Donnell shell theory. A perturbative approach (in the slenderness parameter) yields analytical solutions for both the flow and the deformation. Using matched asymptotics, we obtain a uniformly valid solution for the tube's radial displacement, which features both a boundary layer and a corner layer caused by localized bending near the clamped ends. In doing so, we obtain a ``generalized Hagen--Poiseuille law'' for soft microtubes. We benchmark the mathematical predictions against three-dimensional two-way coupled direct numerical simulations (DNS) of flow and deformation performed using the commercial computational engineering platform by ANSYS. The simulations show good agreement and establish the range of validity of the theory. Finally, we discuss the implications of the theory on the problem of the flow-induced deformation of a blood vessel, which is featured in some textbooks.