  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.
