  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.
