Stability and hysteresis of Faraday waves in Hele-Shaw cells

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.

Marangoni-enhanced capillary wetting in surfactant-driven superspreading

Superspreading is a phenomenon such that a drop of a certain class of surfactant on a substrate can spread with a radius that grows linearly with time much faster than the usual capillary wetting. Its origin, in spite of many efforts, is still not fully understood. Previous modelling and simulation studies (Karapetsas et al. J. Fluid Mech., vol. 670, 2011, pp. 5–37; Theodorakis et al. Langmuir, vol. 31, 2015, pp. 2304–2309) suggest that the transfer of the interfacial surfactant molecules onto the substrate in the vicinity of the contact line plays a crucial role in superspreading. Here, we construct a detailed theory to elaborate on this idea, showing that a rational account for superspreading can be made using a purely hydrodynamic approach without involving a specific surfactant structure or sorption kinetics. Using this theory it can be shown analytically, for both insoluble and soluble surfactants, that the curious linear spreading law can be derived from a new dynamic contact line structure due to a tiny surfactant leakage from the air–liquid interface to the substrate. Such a leak not only establishes a concentrated Marangoni shearing toward the contact line at a rate much faster than the usual viscous stress singularity, but also results in a microscopic surfactant-devoid zone in the vicinity of the contact line. The strong Marangoni shearing then turns into a local capillary force in the zone, making the contact line in effect advance in a surfactant-free manner. This local Marangoni-driven capillary wetting in turn renders a constant wetting speed governed by the de Gennes–Cox–Voinov law and hence the linear spreading law. We also determine the range of surfactant concentration within which superspreading can be sustained by local surfactant leakage without being mitigated by the contact line sweeping, explaining why only limited classes of surfactants can serve as superspreaders. We further show that spreading of surfactant spreaders can exhibit either the $1/6$ or $1/2$ power law, depending on the ability of interfacial surfactant to transfer/leak to the bulk/substrate. All these findings can account for a variety of results seen in experiments (Rafai et al. Langmuir, vol. 18, 2002, pp. 10486–10488; Nikolov & Wasan, Adv. Colloid Interface Sci., vol. 222, 2015, pp. 517–529) and simulations (Karapetsas et al. 2011). Analogy to thermocapillary spreading is also made, reverberating the ubiquitous role of the Marangoni effect in enhancing dynamic wetting driven by non-uniform surface tension.

Mechanisms of dynamic wetting failure in the presence of soluble surfactants

A hydrodynamic model and flow visualization experiments are used to understand the mechanisms through which soluble surfactants can influence the onset of dynamic wetting failure. In the model, a Newtonian liquid displaces air in a rectangular channel in the absence of inertia. A Navier-slip boundary condition and constant contact angle are used to describe the dynamic contact line, and surfactants are allowed to adsorb to the interface and moving channel wall (substrate). The Galerkin finite element method is used to calculate steady states and identify the critical capillary number $Ca^{crit}$ at which wetting failure occurs. It is found that surfactant solubility weakens the influence of Marangoni stresses, which tend to promote the onset of wetting failure. Adsorption of surfactants to the substrate can delay the onset of wetting failure due to the emergence of Marangoni stresses that thicken the air film near the dynamic contact line. The experiments indicate that $Ca^{crit}$ increases with surfactant concentration. For the more viscous solutions used, this behaviour can largely be explained by accounting for changes to the mean surface tension and static contact angle produced by surfactants. For the lowest-viscosity solution used, comparison between the model predictions and experimental observations suggests that other surfactant-induced phenomena such as Marangoni stresses may play a more important role.

Dynamic wetting failure and hydrodynamic assist in curtain coating

Dynamic wetting failure in curtain coating of Newtonian liquids is studied in this work. A hydrodynamic model accounting for air flow near the dynamic contact line (DCL) is developed to describe two-dimensional (2D) steady wetting and to predict the onset of wetting failure. A hybrid approach is used where air is described by a one-dimensional model and liquid by a 2D model, and the resulting hybrid formulation is solved with the Galerkin finite element method. The results reveal that the delay of wetting failure in curtain coating – often termed hydrodynamic assist – mainly arises from the hydrodynamic pressure generated by the inertia of the impinging curtain. This pressure leads to a strong capillary-stress gradient that pumps air away from the DCL and thus increases the critical substrate speed for wetting failure. Although the parameter values used in the model are different from those in experiments due to computational limitations, the model is able to capture the experimentally observed non-monotonic behaviour of the critical substrate speed as the feed flow rate increases (Blake et al., Phys. Fluids, vol. 11, 1999, p. 1995–2007). The influence of insoluble surfactants is also investigated, and the results show that Marangoni stresses tend to thin the air film and increase air-pressure gradients near the DCL, thereby promoting the onset of wetting failure. In addition, Marangoni stresses reduce the degree of hydrodynamic assist in curtain coating, suggesting a possible mechanism for experimental observations reported by Marston et al. (Exp. Fluids, vol. 46, 2009, pp. 549–558).

Classifying dynamic contact line modes in drying drops

Evaporating droplets of both PEO polymer solutions and blood at low pressure are observed to form pillar-like deposits when dried. We use normalised h–R plots to illustrate that this behaviour is volume-independent and find scaling arguments to support our measurements that the contact line recedes with a speed inversely proportional to the droplet radius.

Dynamics of sessile drops. Part 1. Inviscid theory

A sessile droplet partially wets a planar solid support. We study the linear stability of this spherical-cap base state to disturbances whose three-phase contact line is (i) pinned, (ii) moves with fixed contact angle and (iii) moves with a contact angle that is a smooth function of the contact-line speed. The governing hydrodynamic equations for inviscid motions are reduced to a functional eigenvalue problem on linear operators, which are parameterized by the base-state volume through the static contact angle and contact-line mobility via a spreading parameter. A solution is facilitated using inverse operators for disturbances (i) and (ii) to report frequencies and modal shapes identified by a polar $k$ and azimuthal $l$ wavenumber. For the dynamic contact-line condition (iii), we show that the disturbance energy balance takes the form of a damped-harmonic oscillator with ‘Davis dissipation’ that encompasses the dynamic effects associated with (iii). The effect of the contact-line motion on the dissipation mechanism is illustrated. We report an instability of the super-hemispherical base states with mobile contact lines (ii) that correlates with horizontal motion of the centre-of-mass, called the ‘walking’ instability. Davis dissipation from the dynamic contact-line condition (iii) can suppress the instability. The remainder of the spectrum exhibits oscillatory behaviour. For the hemispherical base state with mobile contact line (ii), the spectrum is degenerate with respect to the azimuthal wavenumber. We show that varying either the base-state volume or contact-line mobility lifts this degeneracy. For most values of these symmetry-breaking parameters, a certain spectral ordering of frequencies is maintained. However, because certain modes are more strongly influenced by the support than others, there are instances of additional modal degeneracies. We explain the physical reason for these and show how to locate them.

The motion, stability and breakup of a stretching liquid bridge with a receding contact line

The complex behaviour of drop deposition on a hydrophobic surface is considered by looking at a model problem in which the evolution of a constant-volume liquid bridge is studied as the bridge is stretched. The bridge is pinned with a fixed diameter at the upper contact point, but the contact line at the lower attachment point is free to move on a smooth substrate. Experiments indicate that initially, as the bridge is stretched, the lower contact line slowly retreats inward. However, at a critical radius, the bridge becomes unstable, and the contact line accelerates dramatically, moving inward very quickly. The bridge subsequently pinches off, and a small droplet is left on the substrate. A quasi-static analysis, using the Young–Laplace equation, is used to accurately predict the shape of the bridge during the initial bridge evolution, including the initial onset of the slow contact line retraction. A stability analysis is used to predict the onset of pinch-off, and a one-dimensional dynamical equation, coupled with a Tanner law for the dynamic contact angle, is used to model the rapid pinch-off behaviour. Excellent agreement between numerical predictions and experiments is found throughout the bridge evolution, and the importance of the dynamic contact line model is demonstrated.