Capillary hysteresis induced by gap-resolved meniscus dynamics on Faraday instability in Hele-Shaw cells

Existing theoretical analyses on the Faraday instability in Hele-Shaw cells typically adopt gap-averaged governing equations and rely on Hamraoui’s model coming from molecular kinetics theory, thereby oversimplifying essential transverse information, such as contact line velocity and capillary hysteresis, and conflicting with the unsteady meniscus dynamics. In this paper, a gap-resolved approach is developed by directly modelling the transverse gap flow and the contact angle dynamics, which overcomes the aforementioned limitations, ultimately yielding a modified damping with respect to the static contact angle and hysteresis range. A novel amplitude equation for linear Faraday instability is derived that combines this damping and the gap-averaged counterpart based on the oscillatory Stokes boundary layer, with the viscous dissipation preserved. By means of Lyapunov’s first method, an explicit analytical expression for the critical stability boundary is established. Two series of laboratory experiments are performed that focus, respectively, on evolutions of the lateral meniscus and the longitudinal free surface near the Faraday onset, from which key parameters relevant to the theory are precisely measured. Based on the experimental data, the validity of the proposed mathematical model for addressing the Faraday instability problem in Hele-Shaw cells is confirmed, and the generation and development mechanisms of the onset are clarified. In the asymptotic analysis, the inclusion of contact angle dynamics increases the overall damping and thus partially compensates for the frequency detuning introduced by oscillatory Stokes flow approximation.