Numerical solutions for thin film flow down the outside and inside of a vertical cylinder

Abstract

We consider a model for thin film flow down the outside and inside of a vertical cylinder. Our focus is the effect the curvature of the cylinder has on the gravity driven instability of the advancing contact line and to simulate the resulting fingering patterns that form due to this instability. The governing partial differential equation is fourth order with a nonlinear degenerate diffusion term that represents the stabilising effect of surface tension. We present numerical solutions obtained by implementing an efficient alternating direction implicit scheme. When compared to the problem of flow down a vertical plane, we find that increasing substrate curvature tends to increase the fingering instability for flow down the outside of the cylinder, whereas flow down the inside of the cylinder substrate curvature has the opposite effect. Further,

Simulating droplet motion on virtual leaf surfaces

“…While ADI methods have been popular in solving thin-film-type equations like ours [18,24,29], more recent work has shown that parallelized multigrid methods may offer the greatest potential for efficiency [55,56]. However, we find that an ADI approach is sufficiently fast for the scale of the presented simulations.…”

“…The coalescence and separation simulations depicted in figures 3-5 have been presented for comparison with real observations like those in figure 1. Similar droplet behaviours have been studied before [16,18,19,24,29,32,34], but have rarely been studied in the context of spray droplets on leaves.…”

“…As the numerical scheme must operate on a significantly larger scale, the usage of the disjoining pressure in the current context can be interpreted as a physically motivated device to prescribe an equilibrium contact angle and also to allow de-pinning of the receding contact line [17]. The two-term form of disjoining pressure has been used effectively in many previous studies to model droplet movement down a plane [18,19,22,23], on simply curved surfaces like a cylinder [29], and on more complex surfaces including surface topography [32,35].…”

“…Other variations of (2.1) incorporate the effect of substrate curvature to model thin film flow on simply curved substrates such as cylinders [26][27][28][29][30] and spheres [31]. Mayo et al [29] simulated droplets sliding down the outer wall of a vertical cylinder, observing the same regimes as those on an inclined plane [18][19][20][21].…”

“…Other variations of (2.1) incorporate the effect of substrate curvature to model thin film flow on simply curved substrates such as cylinders [26][27][28][29][30] and spheres [31]. Mayo et al [29] simulated droplets sliding down the outer wall of a vertical cylinder, observing the same regimes as those on an inclined plane [18][19][20][21]. Other authors have employed a surface profile function to consider the effect of topographical features on droplet movement, such as the channels and microscopic plant surface characteristics presented by Glass et al [32].…”