# Theory of the flow-induced deformation of shallow compliant microchannels with thick walls

## TL;DR

This paper presents a theoretical model for the steady-state deformation of shallow, thick-walled microchannels under fluid flow, providing a new flow rate-pressure drop relation that aligns with experimental data.

## Contribution

It develops a parameter-free analytical model for thick-walled microchannel deformation using lubrication and elasticity theories, extending previous models.

## Key findings

- Deformation can be considered independently at each cross-section.
- The top wall behaves as a simply supported rectangle under uniform pressure.
- Derived flow rate-pressure drop relation matches experimental results.

## Abstract

Long, shallow microchannels embedded in thick soft materials are widely used in microfluidic devices for lab-on-a-chip applications. However, the bulging effect caused by fluid--structure interactions between the internal viscous flow and the soft walls has not been completely understood. Previous models either contain a fitting parameter or are specialized to channels with plate-like walls. This work is a theoretical study of the steady-state response of a compliant microchannel with a thick wall. Using lubrication theory for low-Reynolds-number flows and the theory for linearly elastic isotropic solids, we obtain perturbative solutions for the flow and deformation. Specifically, only the channel's top wall deformation is considered, and the ratio between its thickness $t$ and width $w$ is assumed to be $(t/w)^2 \gg 1$. We show that the deformation at each stream-wise cross-section can be considered independently, and that the top wall can be regarded as a simply supported rectangle subject to uniform pressure at its bottom. The stress and displacement fields are found using Fourier series, based on which the channel shape and the hydrodynamic resistance are calculated, yielding a new flow rate--pressure drop relation without fitting parameters. Our results agree favorably with, and thus rationalize, previous experiments.