Faraday Instability in Hele-Shaw Cell with Two Commensurate Frequencies Forcing

We investigate the Faraday instability with an air-liquid interface in a Hele-Shaw cell subjected to a periodic vertical oscillation with two commensurate frequencies, \(\varOmega _1\) and \(\varOmega _2\) . The linear stability formulation allows to reduce the governing equations to a damped Mathieu equation describing the evolution of the interface amplitude. Thereafter, spectral methods are used to establish the marginal stability curves in terms of reduced critical forcing amplitude as a function of wavenumber, which allows us to examine the effects of frequency ratios, the amplitudes of the excitations, and the effect of phase shift of the two superimposed accelerations. The instability diagrams are presented in the form of tongues which are resonances that can be harmonic or sub-harmonic with the existence of several bi-critical points. We show that the phase shift has a stabilizing effect. Also, for each frequency ratio, \(\omega =\frac{\varOmega _2}{\varOmega _1}\) , the instability has maximum at a finite frequency \(\varOmega _1\) while for the low and high fresquencies stabilizing effect is observed. The decrease of the superimposed acceleration amplitudes has also a stabilizing effect.