Stability and hysteresis of Faraday waves in Hele-Shaw cells

Abstract

The instability of Faraday waves in Hele-Shaw cells is investigated experimentally and theoretically. A novel hydrodynamic model involving capillary action is proposed to capture the variation of the dynamic contact line between two close walls of narrow containers. The amplitude equations are derived from the gap-averaged model. By means of Lyapunov’s first method, a good prediction of the onset threshold of forcing acceleration is obtained, which shows the model’s validity for addressing the stability problem for Faraday waves in Hele-Shaw cells. It is found that the effect of the dynamic contact line is much greater than that of Poiseuille assumption of velocity profile for the cases under investigation. A new dispersion relation is obtained, which agrees well with experimental data. However, we highly recommend the conventional dispersion relation for gravity–capillary waves, which can generally meet common needs. Surface tension is found to be a key factor of interface flows in Hele-Shaw cells. According to our experimental observations, a liquid film is found on the front wall of the Hele-Shaw cell when the wave is falling. As a property of the friction coefficient from molecular kinetics, wet and dry plates show different wetting procedures. Unlike some authors of previous publications, we attribute the hysteresis to the out-of-plane interface shape rather than to detuning, i.e. the difference between natural frequency and response frequency.

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

“…A vis-à-vis comparison with experiments by Li et al. (2019) points out how the standard Darcy model often underestimates the Faraday threshold. In contrast, the present theory can explain and close the gap with these experiments.…”

“…The capillary force in the -direction becomes important only at large enough wavenumbers, although the associated term can be retained in the analysis so as to retrieve the well-known dispersion relation (Saffman & Taylor 1958; Chuoke, van Meurs & van der Poel 1959; McLean & Saffman 1981; Park & Homsy 1984; Schwartz 1986; Afkhami & Renardy 2013; Li et al. 2019). With the introduction of the Floquet ansatzes (2.6 b )–(2.17) and by recalling the -expansion of the interface and pressure as , the averaged normal stress equation becomes where the decomposition has also been used to decompose the right-hand side into the th and th harmonics.…”

“…This simplified treatment neglects the contact line dynamics and may lead to miscalculations in certain situations. Advances in this direction were made by Li, Li & Liao (2019), who found that the out-of-plane capillary forces associated with the meniscus curvature across the thin-gap direction should be retained in order to improve the description of the wave dynamics, as experimental evidence suggests. By employing a more sophisticated model, coming from molecular kinetics theory (Blake 1993; Hamraoui et al.…”

A height function based momentum balance model to simulate contact angle dynamics with hysteresis

“…To accurately simulate this physical process will help people gain a more in-depth look into many complex flow phenomena and applications. This includes cleaning and decontamination of surfaces, 1 textiles moisture, 2 printing, 3,4 drop impact, 5,6 capillary flows in porous media, 7,8 microfluids, 9 biological fluids, 10 and even disease transmission and pharmaceutical manufacturing. 11,12 Most of these systems have relevance in the moving contact lines where the contact angle changes.…”

Sediment motion induced by Faraday waves in a Hele-Shaw cell

“…The authors show that the dimensionless wave heigth h w /∆h (h w is wave height and ∆h is liquid depth) determines the extent of the nonlinearity: as h w /∆h lowers down and f / √ (a/∆h) increases (f and a are forcing frequency and acceleration, respectively), the dispersion relations work well. Recently, the stability problem for Faraday waves in Hele-Shaw cells was addresed via a novel hydrodynamic model involving capillary action, capturing the variation of the dynamic contact line between two close walls, showing that the effect of the dynamic contact line is much greater than that of Poiseuille assumption [14]. Also recently, experiments were conducted in a Hele-Shaw cell to investigate the formation of Faraday waves in a triple layer of fluids (air, pure ethanol and silicone oil) [15].…”