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

TL;DR

This work addresses the Faraday instability in thin Hele-Shaw cells by developing a gap-resolved framework that explicitly resolves transverse gap flow and meniscus dynamics, thereby capturing capillary hysteresis effects absent in gap-averaged theories. It derives a linear Faraday amplitude equation that combines oscillatory Stokes-flow damping with gap dynamics, and introduces a refined contact-angle hysteresis model based on a gap-resolved analysis, yielding an explicit stability boundary and a damping term that depends on the static contact angle $\theta_s$ and hysteresis range $\Delta$. The theory is validated against two sets of experiments: direct observations of the transverse meniscus and Faraday onset measurements across multiple gap sizes and liquids, with parameters such as $|B'|$, $\theta_s$, and $\Delta$ extracted from data and fed into the model. The results show that capillary-hysteresis damping increases overall dissipation and partially mitigates the frequency detuning caused by the oscillatory Stokes flow, improving agreement with the observed onset conditions and dispersion behavior, though fully nonlinear and end-wall effects remain outside the linear framework. Overall, the gap-resolved approach provides a more faithful description of Faraday onset in confined geometries and highlights the critical role of contact-angle hysteresis in determining instability thresholds in Hele-Shaw cells.

Abstract

Existing theoretical analyses on 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 modeling 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.

Figures (17)

• Figure 1: (a) Sketch of Faraday waves in a Hele-Shaw cell that undergoes a vertical sinusoidal oscillation of acceleration amplitude $a$ and angular frequency $\varOmega$. The free surface elevation denoted by $\zeta' \left ( x',y',t' \right )$ contains both the two-dimensional Faraday wave profile along $x'$-direction and the meniscus in the gap direction. (b) View of the gap between two lateral walls. Here $b$ denotes the gap size of the Hele-Shaw cell and $\theta$ is the contact angle of the liquid on the lateral walls. The meniscus profile is denoted by $\eta'\left ( x',y',t' \right )$.
• Figure 2: (a) Sketch of the dynamic contact angle model \ref{['Hamraoui model']} developed by hamraoui2000can. (b) Sketch of the contact angle hysteresis model \ref{['hysteresis contact model']} developed by viola2018capillary.
• Figure 3: Schematic illustration of the experimental set-up for meniscus experiments, with the camera focused on the meniscus from a side view of the container.
• Figure 4: Experimental snapshots of the meniscus for 80$\%$ ethanol solution with $f=30$ Hz and $a=3.9\ \mathrm{m/s^2}$. The meniscus oscillates synchronously with the external vibration, implying a period $T=1/30$ s. The contact line is nearly fixed, while the contact angle reaches the boundary of the hysteresis range. If the acceleration amplitude is further increased to $a=4.0\ \mathrm{m/s^2}$, the contact line will move rapidly and Faraday waves emerge.
• Figure 5: Temporal variation of the contact angle under 30 Hz external vibration for various liquids in the PVC container. (a) Pure ethanol, with $a=3.4\ \mathrm{m/s^2}$. (b) $80\ \text{vol}\%$ ethanol solution, with $a=3.9\ \mathrm{m/s^2}$. (c) $70\ \text{vol}\%$ ethanol solution, with $a=4.1\ \mathrm{m/s^2}$. Red lines with circles represent for the experimental data in six cycles. In order to make the trend of $\theta$ more obvious, we perform Fourier fitting to the third-order on these data and eliminate local deviation, which are plotted with blue lines. Red filled dots identify the maximum and minimum values of $\theta$ in one cycle and correspond to the advancing and receding contact angles.