Reduced models of unidirectional flows in compliant rectangular ducts at finite Reynolds number

TL;DR

This paper develops simplified one-dimensional models for steady Newtonian flow in compliant rectangular ducts, capturing the effects of wall elasticity and flow inertia, and reveals how compliance influences flow resistance.

Contribution

It introduces a reduced model for unidirectional flow in compliant ducts at finite Reynolds number, incorporating flow-structure interaction and providing analytical expressions for flow characteristics.

Findings

Poiseuille number varies along the duct due to compliance.

Compliance can increase flow resistance by up to four times.

The model accurately captures flow and deformation coupling in compliant channels.

Abstract

Soft hydraulics, which addresses the interaction between an internal flow and a compliant conduit, is a central problem in microfluidics. We analyze Newtonian fluid flow in a rectangular duct with a soft top wall at steady state. The resulting fluid--structure interaction (FSI) is formulated for both vanishing and finite flow inertia. At the leading-order in the small aspect ratio, the lubrication approximation implies that the pressure only varies in the streamwise direction. Meanwhile, the compliant wall's slenderness makes the fluid--solid interface behave like a Winkler foundation, with the displacement fully determined by the local pressure. Coupling flow and deformation and averaging across the cross-section leads to a one-dimensional reduced model. In the case of vanishing flow inertia, an effective deformed channel height is defined rigorously to eliminate the spanwise dependence of the deformation. It is shown that a previously-used averaged height concept is an acceptable approximation. From the one-dimensional model, a friction factor and the corresponding Poiseuille number are derived. Unlike the rigid duct case, the Poiseuille number for a compliant duct is not constant but varies in the streamwise direction. Compliance can increase the Poiseuille number by a factor of up to four. The model for finite flow inertia is obtained by assuming a parabolic vertical variation of the streamwise velocity. To satisfy the displacement constraints along the edges of the channel, weak tension is introduced in the streamwise direction to regularize the Winkler-foundation-like model. Matched asymptotic solutions of the regularized model are derived.