Diffuse-interface modelling of droplet impact

Abstract

The impact of micron-size drops on a smooth, flat, chemically homogeneous solid surface is studied using a diffuse-interface model (DIM). The model is based on the Cahn–Hilliard theory that couples thermodynamics with hydrodynamics, and is extended to include non-90° contact angles. The (axisymmetric) equations are numerically solved using a combination of finite- and spectral-element methods. The influence of various process and material parameters such as impact velocity, droplet diameter, viscosity, surface tension and wettability on the impact behaviour of drops is investigated. Relevant dimensionless parameters are defined and, depending on the values of the Reynolds number, the Weber number and the contact angle, which for the cases considered here range from 1.3 to 130, 0.43 to 150 and 45° to 135°, respectively, the model predicts the spreading of a droplet with or without recoil or even rebound of the droplet, totally or partially, from the solid surface. The wettability significantly affects the impact behaviour and this is particularly demonstrated with an impact at Re = 130 and We = 1.5, where for θ < 60° the droplet oscillates a few times before attaining equilibrium while for θ ≥ 60° partial rebound of the droplet occurs, i.e. the droplet breaks into two unequal sized drops. The size of the part that remains in contact with the solid surface progressively decreases with increasing θ until at a value θ ≈ 120° a transition to total rebound happens. When the droplet rebounds totally, it has a top-heavy shape.

Phase-Field Models for Multi-Component Fluid Flows

“…For the density, harmonic interpolation is used and linear interpolation is used for viscosity in [54] …”

“…In the phase-field model, on the domain boundary ∂Ω, we have the following conditions where f w (φ)=ǫ(φ 3 −3φ)/(3 √ 2)cosθ is the specific wall free energy, which depends only on the concentration at the solid surface and the contact angle θ [48,54]. This arises from the total Helmholtz free energy functional…”

“…The surface integral term W (φ) represents the contribution of solid-fluid interactions; it is also used in [54] for diffuse-interface modeling of droplet impacts on solid surfaces. Ding and Spelt [29] proposed a geometric formulation and we briefly describe the geometric formulation: The normal vector to the interface can be written in terms of the gradient of φ as n s = ∇φ/|∇φ|.…”

“…In interface tracking methods (volume-of-fluid [42], front-tracking [40], and immersed boundary [49,50,80,81,96]), Lagrangian particles are used to track the interfaces and are advected by the velocity field. In interface capturing methods such as level-set [24,78,79,85,91,92] and phase-field methods [4,6,13,29,46,47,54,56,74,86,90,102], the interface is implicitly captured by a contour of a particular scalar function. The governing equations of unsteady, viscous, incompressible, and immiscible two fluid systems in three-dimensional space are the Navier-Stokes equations:…”

A mesh-dependent model for applying dynamic contact angles to VOF simulations

“…Precursor models assume that a pre-existing thin liquid film covers the surface ahead of the contact line, so that the no-slip condition can be applied over the entire solid surface [7][8][9]; these models are particularly suited for highly wetting situations. Diffuse interface models treat the interface as a finite layer across which fluid properties vary abruptly but smoothly [10][11][12]; the contact line slips due to the diffusive fluxes between the fluids. Finally, slip models are used to relax the stress singularity at the contact line [1,2,13,14]; these allow independent imposition of a non-zero contact angle and are suitable for partially wetting contact lines that are advancing or receding over a solid surface.…”