Non-Newtonian fluid-structure interaction: Flow of a viscoelastic Oldroyd-B fluid in a deformable channel

TL;DR

This paper develops a theoretical framework to analyze how viscoelasticity of an Oldroyd-B fluid influences flow and deformation in a deformable channel, revealing that viscoelasticity and compliance have opposing effects on pressure drop and deformation.

Contribution

It provides a novel analytical model for non-Newtonian fluid-structure interaction in deformable channels, incorporating viscoelastic effects up to first order in Deborah number.

Findings

Viscoelasticity decreases pressure drop in the flow.

Channel compliance increases deformation, while fluid viscoelasticity decreases it.

Analytical expressions relate pressure, deformation, and flow rate in the weakly viscoelastic regime.

Abstract

We analyze the steady non-Newtonian fluid-structure interaction between the flow of an Oldroyd-B fluid and a deformable channel. Specifically, we provide a theoretical framework for calculating the leading-order effect of the fluid's viscoelasticity on the flow rate-pressure drop relation and on the deformation of the channel's elastic wall. We first identify the characteristic scales and dimensionless parameters governing the fluid-structure interaction in slender and shallow channels. Applying the lubrication approximation for the flow and employing a perturbation expansion in powers of the Deborah number $De$, we derive a closed-form expression for the pressure as a function of the non-uniform shape of the channel in the weakly viscoelastic limit up to $\mathrm{O}(De)$. Coupling the hydrodynamic pressure to the elastic deformation, we provide the leading-order effect of the interplay between the viscoelasticity of the fluid and the compliance of the channel on the pressure and deformation fields, as well as on the flow rate-pressure drop relation. For the flow-rate-controlled regime and in the weakly viscoelastic limit, we show analytically that both the compliance of the deforming top wall and the viscoelasticity of the fluid decrease the pressure drop. Furthermore, we reveal a trade-off between the influence of compliance of the channel and the fluid's viscoelasticity on the deformation. While the channel's compliance increases the deformation, the fluid's viscoelasticity decreases it.