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Register a new userSurfactant spreading in a two-dimensional cavity and emergent contact-line singularities
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We model the advective Marangoni spreading of insoluble surfactant at the free surface of a viscous fluid that is confined within a two-dimensional rectangular cavity. Interfacial deflections are assumed small, with contact lines pinned to the walls of the cavity, and inertia is neglected. Linearizing the surfactant transport equation about the equilibrium state allows a modal decomposition of the dynamics, with eigenvalues corresponding to decay rates of perturbations. Computation of the family of mutually orthogonal two-dimensional eigenfunctions reveals singular flow structures near each contact line, resulting in spatially oscillatory patterns of wall shear stress and a pressure field that diverges logarithmically. These singularities at a stationary contact line are associated with dynamic compression of the surfactant monolayer; we show how they can be regularized by weak surface diffusion. Their existence highlights the need for careful treatment in computations of unsteady advection-dominated surfactant transport in confined domains.
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We study the adhesion of a droplet with insoluble surfactant laid on its capillary surface to a textured substrate. In this process, the surfactant-dependent surface tension dominates the behaviors of the whole dynamics, particularly the moving contact lines. This allows us to derive the full dynamics of the droplets laid by the insoluble surfactant: (i) the moving contact lines, (ii) the evolution of the capillary surface, and (iii) the surfactant dynamics on this moving surface with a boundary condition at the contact lines. Our derivations base on Onsagers principle with Rayleigh dissipation functionals for either the viscous flow inside droplets or the motion by mean curvature of the capillary surface. We also prove the Rayleigh dissipation functional for the viscous flow case is stronger than the one for the motion by mean curvature. After incorporating the textured substrate profile, we design numerical schemes based on unconditionally stable explicit boundary updates and moving grids, which enable efficient computations for many challenging examples showing the significant contributions of the surfactant.
The hydrodynamics of a liquid-vapour interface in contact with an heterogeneous surface is largely impacted by the presence of defects at the smaller scales. Such defects introduce morphological disturbances on the contact line and ultimately determine the force exerted on the wedge of liquid in contact with the surface. From the mathematical point of view, defects introduce perturbation modes, whose space-time evolution is governed by the interfacial hydrodynamic equations of the contact line. In this paper we derive the response function of the contact line to such generic perturbations. The contact line response may be used to design simplified 1+1 dimensional models accounting for the complexity of interfacial flows coupled to nanoscale defects, yet offering a more tractable mathematical framework to include thermal fluctuations and explore thermally activated contact line motion through a disordered energy landscape.
The dynamics of receding contact lines is investigated experimentally through controlled perturbations of a meniscus in a dip coating experiment. We first characterize stationary menisci and their breakdown at the coating transition. It is then shown that the dynamics of both liquid deposition and long-wavelength perturbations adiabatically follow these stationary states. This provides a first experimental access to the entire bifurcation diagram of dynamical wetting, confirming the hydrodynamic theory developed in Part 1. In contrast to quasi-static theories based on a dynamic contact angle, we demonstrate that the transition strongly depends on the large scale flow geometry. We then establish the dispersion relation for large wavenumbers, for which we find that sigma is linear in q. The speed dependence of sigma is well described by hydrodynamic theory, in particular the absence of diverging time-scales at the critical point. Finally, we highlight some open problems related to contact angle hysteresis that lead beyond the current description.
Building on the recent theoretical work of Wray, Duffy and Wilson [J. Fluid Mech. 884, A45 (2020)] concerning the competitive diffusion-limited evaporation of multiple thin sessile droplets in proximity to each other, we obtain theoretical predictions for the spatially non-uniform densities of the contact-line deposits (often referred to as coffee stains or ring stains) left on the substrate after such droplets containing suspended solid particles have completely evaporated. Neighbouring droplets interact via their vapour fields, which results in a spatially non-uniform shielding effect. We give predictions for the deposits from a pair of identical droplets, which show that the deposit is reduced the most where the droplets are closest together, and demonstrate excellent quantitative agreement with experimental results of Pradhan and Panigrahi [Coll. Surf. A 482, 562-567 (2015)]. We also give corresponding predictions for a triplet of identical droplets arranged in an equilateral triangle, which show that the effect of shielding on the deposit is more subtle in this case.
Extremely small amounts of surface-active contaminants are known to drastically modify the hydrodynamic response of the water-air interface. Surfactant concentrations as low as a few thousand molecules per square micron are sufficient to eventually induce complete stiffening. In order to probe the shear response of a water-air interface, we design a radial flow experiment that consists in an upward water jet directed to the interface. We observe that the standard no-slip effect is often circumvented by an azimuthal instability with the occurence of a vortex pair. Supported by numerical simulations, we highlight that the instability occurs in the (inertia-less) Stokes regime and is driven by surfactant advection by the flow. The latter mechanism is suggested as a general feature in a wide variety of reported and yet unexplained observations.
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