Abstract
We study the relaxation towards equilibrium of a liquid barrel—a partially wetting droplet in a wedge geometry—using a diffuse-interface approach. We formulate a hydrodynamic model of the motion of the barrel in the framework of the Navier–Stokes and Cahn–Hilliard equations of motion. We present a lattice-Boltzmann method to integrate the diffuse-interface equations, where we introduce an algorithm to model the dynamic wetting of the liquid on smooth solid boundaries. We present simulation results of the over-damped dynamics of the liquid barrel. We find that the relaxation of the droplets is driven by capillary forces and damped by friction forces. We show that the friction is determined by the contribution of the bulk flow, the corner flow near the contact lines and the motion of the contact lines by comparing simulation results for the relaxation time of the barrel. Our results are in broad agreement with previous analytical predictions based on a sharp interface model.
Highlights
Introduction ceDroplets in wedge geometries appear in many natural environments
Liquid barrels form when the wetting angle of the droplet satisfies the relation θe > 90◦ + β [15]. They differ from edge spreads, edge blobs (90◦ − β < θe ≤ 90◦ + β), and free drops in that they equilibrate into a truncated-sphere shape away from apex of the wedge [15, 16, 17]. When displaced from their equilibrium position, they relax back driven by capillary forces and damped by frictional forces [18]
The structure of the flow near the contact line, shown in the close-up of figure 8(c), is consistent with the generic corner flow of wetting dynamics predicted by Cox and Voinov [53, 40], which results in a tread-milling motion of the interface as documented in experiments by [54]
Summary
We describe the system using a diffuse-interface model where the liquid barrel, referred to as the inner phase, and the surrounding fluid, or outer phase, are identified using a phase field φ(x). Within the diffuse-interface model, the motion of the contact line occurs by virtue of diffusive currents caused by a local imbalance in the chemical potential field [21, 24]. This is because, while the velocity field vanishes at the solid-fluid interface by virtue of Eq (13), the diffusive term in Eq (7) does not. This regularises the singularity that stems from the no-slip boundary condition [23]. We expect that the friction coefficient obeys ηin ζ0 ∝ M −1 ∝ ∗
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