  Existing theoretical analyses of Faraday waves in Hele-Shaw cells rely on the
Darcy approximation and assume a parabolic flow profile in the narrow
direction. However, Darcy's model is known to be inaccurate when convective or
unsteady inertial effects are important. In this work, we propose a
gap-averaged Floquet theory accounting for inertial effects induced by the
unsteady terms in the Navier-Stokes equations, a scenario that corresponds to a
pulsatile flow where the fluid motion reduces to a two-dimensional oscillating
Poiseuille flow, similarly to the Womersley flow in arteries. When
gap-averaging the linearized Navier-Stokes equation, this results in a modified
damping coefficient, which is a function of the ratio between the Stokes
boundary layer thickness and the cell's gap, and whose complex value depends on
the frequency of the wave response specific to each unstable parametric region.
We first revisit the standard case of horizontally infinite rectangular
Hele-Shaw cells by also accounting for a dynamic contact angle model. A
comparison with existing experiments shows the predictive improvement brought
by the present theory and points out how the standard gap-averaged model often
underestimates the Faraday threshold. The analysis is then extended to the less
conventional case of thin annuli. A series of dedicated experiments for this
configuration highlights how Darcy's thin-gap approximation overlooks a
frequency detuning that is essential to correctly predict the locations of the
Faraday tongues in the frequency-amplitude parameter plane. These findings are
well rationalized and captured by the present model.
