A revised gap-averaged model of Faraday waves in Hele-Shaw cells

A. Bongarzone (speaker) and F. Gallaire

Previous theoretical analyses of Faraday waves in Hele-Shaw cells typically rely on the Darcy approximation, which is based on the parabolic flow profile assumption in the narrow direction and that translates into a damping coefficient $\gamma=12\nu/b^2$, with $\nu$ the fluid kinematic viscosity and $b$ the cell’s gap-size. However, Darcy’s model is known to be inaccurate whenever inertia is not negligible, e.g. in unsteady flows. In this work, we propose a revised gap-averaged linear model that accounts for inertial effects induced by the unsteady terms in the Navier-Stokes equations. Such a scenario corresponds to a pulsatile flow where the fluid’s motion reduces to a two-dimensional oscillating Poiseuille flow. This results in a modified damping coefficient, $\gamma=\chi \nu/b^2$, with $\chi=\chi_r+i\,\chi_i$ complex-valued, which is a function of the ratio between the Stokes boundary layer thickness $\delta=\sqrt{2\nu/\Omega}$ and the cell’s gap-size $b$, and whose value depends on the frequency of the system’s response, $\Omega$, specific to each unstable parametric Faraday tongue, i.e. $\Omega=n\omega/2$ ($n=1,2,…$) with $\omega=2\pi f$ the driving frequency. We consider the case of Faraday waves in a thin annulus, for which the system’s natural frequencies obey the non-dimensional dispersion relation $\omega_m^2=\left(m+m^3/Bo\right)\tanh{\left(m\,h/R\right)}$, where $Bo$ is the Bond number and $m$ is an integer representing the azimuthal wavenumber. We show that Darcy’s model typically underestimates the Faraday threshold and overlooks a frequency detuning introduced by $\chi_i$, which appears essential to correctly predict the location of the Faraday tongue in the frequency spectrum. The latter aspect is confirmed by a full numerical eigenvalue calculation of the actual system’s natural frequencies, which provides the driving frequencies around which the Faraday tongues are centered, hence proving the predictive power of the present revised gap-averaged model.