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Experiments have shown that flow in compliant microchannels can becomeunstable at a much lower Reynolds number than the corresponding flow in a rigidconduit. Therefore, it has been suggested that the wall's elastic compliancecan be exploited towards new modalities of microscale mixing. While previousstudies mainly focused on the local instability induced by the fluid--structureinteractions (FSIs) in the system, we derive a one-dimensional (1D) model tostudy the FSI's effect on the global instability. The proposed 1D FSI model istailored to long, shallow rectangular microchannels with a deformable top wall,similar to the experiments. Going beyond the usual lubrication flows analyzedin these geometries, we include finite fluid inertia and couple the reducedflow equations to a novel reduced 1D wall deformation equation. Although aquantitative comparison to previous experiments is difficult, the behaviors ofthe proposed model show qualitative agreement with the experimentalobservations, and capture several key effects. Specifically, we find thecritical conditions under which the inflated base state of the 1D FSI model islinearly unstable to infinitesimal perturbations. The critical Reynolds numberspredicted are in agreement with experimental observations. The unstable modesare highly oscillatory, with frequencies close to the natural frequency of thewall, suggesting that the observed instabilities are resonance phenomena.Furthermore, during the start-up from an undeformed initial state,self-sustained oscillations can be triggered by FSI. Our modeling framework canbe applied to other microfluidic systems with similar geometric scaleseparation under different operating conditions.

Reduced modelling and global instability of finite-Reynolds-number flow in compliant rectangular channels