  Experiments have shown that flow in compliant microchannels can become
unstable at a much lower Reynolds number than the corresponding flow in a rigid
conduit. Therefore, it has been suggested that the wall's elastic compliance
can be exploited towards new modalities of microscale mixing. While previous
studies mainly focused on the local instability induced by the fluid--structure
interactions (FSIs) in the system, we derive a one-dimensional (1D) model to
study the FSI's effect on the global instability. The proposed 1D FSI model is
tailored to long, shallow rectangular microchannels with a deformable top wall,
similar to the experiments. Going beyond the usual lubrication flows analyzed
in these geometries, we include finite fluid inertia and couple the reduced
flow equations to a novel reduced 1D wall deformation equation. Although a
quantitative comparison to previous experiments is difficult, the behaviors of
the proposed model show qualitative agreement with the experimental
observations, and capture several key effects. Specifically, we find the
critical conditions under which the inflated base state of the 1D FSI model is
linearly unstable to infinitesimal perturbations. The critical Reynolds numbers
predicted are in agreement with experimental observations. The unstable modes
are highly oscillatory, with frequencies close to the natural frequency of the
wall, 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 can
be applied to other microfluidic systems with similar geometric scale
separation under different operating conditions.
