Flow rate--pressure drop relation for deformable shallow microfluidic channels

TL;DR

This paper develops a parameter-free theoretical model for the flow rate-pressure drop relation in deformable shallow microfluidic channels, confirming nonlinear behavior and matching experimental data without fitting parameters.

Contribution

It introduces a perturbation approach based on plate bending theory and lubrication approximation, removing the need for empirical fitting parameters in modeling deformation effects.

Findings

Flow rate varies as the cube of the channel height in deformation.

The model predicts a quartic polynomial relation between flow rate and pressure drop.

The theoretical results agree well with experimental measurements.

Abstract

Laminar flow in devices fabricated from soft materials causes deformation of the passage geometry, which affects the flow rate--pressure drop relation. For a given pressure drop, in channels with narrow rectangular cross-section, the flow rate varies as the cube of the channel height, so deformation can produce significant quantitative effects, including nonlinear dependence on the pressure drop [{Gervais, T., El-Ali, J., G\"unther, A. \& Jensen, K.\ F.}\ 2006 Flow-induced deformation of shallow microfluidic channels.\ \textit{Lab Chip} \textbf{6}, 500--507]. Gervais et. al. proposed a successful model of the deformation-induced change in the flow rate by heuristically coupling a Hookean elastic response with the lubrication approximation for Stokes flow. However, their model contains a fitting parameter that must be found for each channel shape by performing an experiment. We present a perturbation approach for the flow rate--pressure drop relation in a shallow deformable microchannel using the theory of isotropic quasi-static plate bending and the Stokes equations under a lubrication approximation (specifically, the ratio of the channel's height to its width and of the channel's height to its length are both assumed small). Our result contains no free parameters and confirms Gervais et. al.'s observation that the flow rate is a quartic polynomial of the pressure drop. The derived flow rate--pressure drop relation compares favorably with experimental measurements.