Solvability of a moving contact-line problem with interface formation for an incompressible viscous fluid

We investigate the free-boundary problem of a steadily advancing meniscus in a circular capillary tube. The problem is described using the “interface formation model,” which was originally introduced with the aim of avoiding the singularities that arise when classical hydrodynamics is applied to problems with a moving contact line. We prove the existence of an axially symmetric solution in weighted Hölder spaces for low meniscus speeds.

Surface textures suppress viscoelastic braking on soft substrates

A gravity-driven droplet will rapidly flow down an inclined substrate, resisted only by stresses inside the liquid. If the substrate is compliant, with an elastic modulusG< 100 kPa, the droplet will markedly slow as a consequence of viscoelastic braking. This phenomenon arises due to deformations of the solid at the moving contact line, enhancing dissipation in the solid phase. Here, we pattern compliant surfaces with textures and probe their interaction with droplets. We show that the superhydrophobic Cassie state, where a droplet is supported atop air-immersed textures, is preserved on soft textured substrates. Confocal microscopy reveals that every texture in contact with the liquid is deformed by capillary stresses. This deformation is coupled to liquid pinning induced by the orientation of contact lines atop soft textures. Thus, compared to flat substrates, greater forcing is required for the onset of drop motion when the soft solid is textured. Surprisingly, droplet velocities down inclined soft or hard textured substrates are indistinguishable; the textures thus suppress viscoelastic braking despite substantial fluid–solid contact. High-speed microscopy shows that contact line velocities atop the pillars vastly exceed those associated with viscoelastic braking. This velocity regime involves less deformation, thus less dissipation, in the solid phase. Such rapid motions are only possible because the textures introduce a new scale and contact-line geometry. The contact-line orientation atop soft pillars induces significant deflections of the pillars on the receding edge of the droplet; calculations confirm that this does not slow down the droplet.

Boundary conditions for dynamic wetting - A mathematical analysis

The moving contact line paradox discussed in the famous paper by Huh and Scriven has lead to an extensive scientific discussion about singularities in continuum mechanical models of dynamic wetting in the framework of the two-phase Navier–Stokes equations. Since the no-slip condition introduces a non-integrable and therefore unphysical singularity into the model, various models to relax the singularity have been proposed. Many of the relaxation mechanisms still retain a weak (integrable) singularity, while other approaches look for completely regular solutions with finite curvature and pressure at the moving contact line. In particular, the model introduced recently in [A.V. Lukyanov, T. Pryer, Langmuir 33, 8582 (2017)] aims for regular solutions through modified boundary conditions. The present work applies the mathematical tool of compatibility analysis to continuum models of dynamic wetting. The basic idea is that the boundary conditions have to be compatible at the contact line in order to allow for regular solutions. Remarkably, the method allows to compute explicit expressions for the pressure and the curvature locally at the moving contact line for regular solutions to the model of Lukyanov and Pryer. It is found that solutions may still be singular for the latter model.

Steady moving contact line of water over a no-slip substrate

The movement of the triple contact line plays a crucial role in many applications such as ink-jet printing, liquid coating and drainage (imbibition) in porous media. To design accurate computational tools for these applications, predictive models of the moving contact line are needed. However, the basic mechanisms responsible for movement of the triple contact line are not well understood but still debated. We investigate the movement of the contact line between water, vapour and a silica-like solid surface under steady conditions in low capillary number regime. We use molecular dynamics (MD) with an atomistic water model to simulate a nanoscopic drop between two moving plates. We include hydrogen bonding between the water molecules and the solid substrate, which leads to a sub-molecular slip length. We benchmark two continuum methods, the Cahn–Hilliard phase-field (PF) model and a volume-of-fluid (VOF) model, against MD results. We show that both continuum models reproduce the statistical measures obtained from MD reasonably well, with a trade-off in accuracy. We demonstrate the importance of the phase-field mobility parameter and the local slip length in accurately modelling the moving contact line.