  We present a reduced order model for three dimensional unsteady
pressure-driven flows in micro-channels of variable cross-section. This fast
and accurate model is valid for long channels, but allows for large variations
in the channel's cross-section along the axis. It is based on an asymptotic
expansion of the governing equations in the aspect ratio of the channel. A
finite Fourier transform in the plane normal to the flow direction is used to
solve for the leading order axial velocity. The corresponding pressure and
transverse velocity are obtained via a hybrid analytic-numerical scheme based
on recursion. The channel geometry is such that one of the transverse velocity
components is negligible, and the other component, in the plane of variation of
channel height, is obtained from combination of the corresponding momentum
equation and the continuity equations, assuming a low degree polynomial Ansatz
of the pressure forcing. A key feature of the model is that it puts no
restriction on the time dependence of the pressure forcing, in terms of shape
and frequency, as long as the advective component of the inertia term is small.
This is a major departure from many previous expositions which assume harmonic
forcing. The model reveals to be accurate for a wide range of parameters and is
two orders of magnitude faster than conventional three dimensional CFD
simulations.
