Faraday waves in a Hele-Shaw cell

We investigate Faraday waves in a Hele-Shaw cell via experimental, numerical, and theoretical studies. Inspired by the Kelvin-Helmholtz-Darcy theory, we develop the gap-averaged Navier-Stokes equations and end up with the stable standing waves with half frequency of the external forced vibration. To overcome the dependency of a numerical model on the experimental parameter of wave length, we take two-phase flow into consideration and a novel dispersion relation is derived. The numerical results compare well with our experimental data, which effectively validates our proposed mathematical model. Therefore, this model can produce robust solutions of Faraday wave patterns and resolve related physical phenomena, which demonstrates the practical importance of the present study.