Subgrid CFD film-drainage modelling: Application to buoyancy-driven droplet-wall collisions in emulsions

The buoyancy-driven interaction between a free droplet and solid wall is simulated. The presented model makes use of the continuum-surface-force (CSF) framework to couple an interface-capturing technique to a lower-dimensional lubrication reduction of the thin-film momentum equation. Droplet-scale fluid transport was modelled using an axisymmetric level-set method, including a reinitialisation procedure. Through comparison of model results with experimental data from the literature, we demonstrate the effectiveness of this approach in simulating the challenging problem of small-length-scale drainage in computational fluid dynamics (CFD) modelling. Film-thickness profiles were modelled up until the onset of asymmetric filmdrainage. Results presented will aid in the future improvement of simulation techniques for droplet interactions in liquid–liquid systems. Introduction A detailed understanding of droplet–interface and droplet– droplet interactions is required in the design and optimisation of multi-phase industrial processing equipment. Droplet collisions, however, pose unique experimental and numerical challenges due to the large differences in length scales present. Experimental studies have focused on (i) droplet–wall [5], (ii) droplet–home-phase [2] and (iii) droplet–droplet [4] collision scenarios. Using a capillary, Hartland [6] generated free droplets of known volume that were allowed to fall under buoyancy and collide with a solid surface. Profiles of the film thickness were obtained via tracing magnified photographs of the collision sequence. The refractive indices of the two fluids were matched to minimise optical distortion. A positive pressure was reasoned to exist near the centre of the film in order to drive viscous film drainage, consistent with the observed “dimpling” of the film-thickness profiles. Similar droplet–wall systems have been studied using high-speed video and interferometric techniques [11, 5]. Modelling studies have applied film-thickness based adaptive mesh refinement and moving-mesh interface-tracking methods [14]. To capture the collision dynamics completely, a discretisation mesh is required to span scales from the droplet diameter (typically of order 1–10 mm) to the thickness of the draining film, which can be as low as 10–100 nm prior to the onset of wetting. Alternative approaches have included the coupling of droplet-scale dynamics, via force balances [11] or experimental imaging [12], to semi-analytical thin-film drainage models [3]. In this study, we demonstrate the efficacy of a multi-scale modelling approach that alleviates the need for computationally taxing mesh refinement. A level-set method is coupled to a concurrent numerical solution of the Reynolds lubrication equations using the finite-volume solver arb [9]; the coupling is achieved via incorporating the film pressure into the continuum–surface–force (CSF) framework of Brackbill et al. [1] A volume-of-fluid based version of the modelling framework has previously been applied to Leidenfrost [10] and binary coalescence [13] dynamics in liquid–gas spray systems. Here, we demonstrate its applicability to droplet–wall collisions in an emulsion system. Direct comparisons are made to the buoyancy-driven collision experiments of Hartland [6].