Simulating droplet motion on virtual leaf surfaces

Abstract

A curvilinear thin film model is used to simulate the motion of droplets on a virtual leaf surface, with a view to better understand the retention of agricultural sprays on plants. The governing model, adapted from Roy et al. (2002 J. Fluid Mech. 454, 235-261 (doi:10.1017/S0022112001007133)) with the addition of a disjoining pressure term, describes the gravity- and curvature-driven flow of a small droplet on a complex substrate: a cotton leaf reconstructed from digitized scan data. Coalescence is the key mechanism behind spray coating of foliage, and our simulations demonstrate that various experimentally observed coalescence behaviours can be reproduced qualitatively. By varying the contact angle over the domain, we also demonstrate that the presence of a chemical defect can act as an obstacle to the droplet's path, causing break-up. In simulations on the virtual leaf, it is found that the movement of a typical spray size droplet is driven almost exclusively by substrate curvature gradients. It is not until droplet mass is sufficiently increased via coalescence that gravity becomes the dominating force.

Keywords: alternating direction implicit methods; coalescence; curvilinear; liquid drop; thin film.

Figure 1.

(a) The abaxial surface of an avocado leaf is sprayed with water, leading to coalescence and runoff. (b) A surfactant (Du-Wett^®, Etec Crop Solutions Ltd) has been added to the water, causing a coating film to form on the leaf surface. See also the electronic supplementary material.

Figure 2.

(a) The virtual cotton leaf, constructed using a discrete smoothing D²-spline. A photograph of the cotton leaf has been texture-mapped to the surface. (b) For this study, we have chosen a very small, approximately 2.5 mm by 2.5 mm, subset of the leaf surface on which to perform our simulations.

Figure 3.

As several drops slide down a vertical plane, the larger drop consumes smaller ones in its path. Parameter values are γ=0.072 N m⁻¹, ϵ=0.44 and Bo=0.034, contour spacing is Δh=0.2.

Figure 4.

Drops on a horizontal plane spread and coalesce. A single larger drop is formed for a contact angle of θ_e=1° (a), while a continuously spreading film of fluid develops for θ_e=0° (b). Parameter values for both simulations are γ=0.025 N m⁻¹, ϵ=0.0087 and Bo=0.33, contour spacing is Δh=0.2.

Figure 5.

As a drop flows down an inclined plane with equilibrium contact angle θ_e=15°, it encounters a small surface defect with θ_e=45° (blue dashed square). This causes separation into four smaller droplets as the defect obstructs the path of movement. Parameter values are γ=0.072 N m⁻¹, ϵ=0.13 and Bo=0.84, contour spacing is Δh=0.07.

Figure 6.

A (a) 0.014 μl, (b) 14 μl, (c) 16 μl and (d) 65 μl drop has been placed on the virtual cotton leaf surface (black contours), and its movement simulated for a time of t=40 (dimensionless) units. The blue contours represent the final position of the droplet, while the red dotted path illustrates the path it took to arrive there (at intervals of Δt= 2/3). The contour spacing is 0.1H_s mm for the leaf topography and 0.2H mm for the drops. See also the electronic supplementary material.

Figure 7.

Nine (a) 0.014 μl and (b) 14 μl drops are placed on the virtual cotton leaf surface (black contours), and their movement simulated for a time of t=40 (dimensionless) units. The blue contours represent the final positions of the droplets, while the grey dashed contours represent their initial positions. The contour spacing is 0.1H_s mm for the leaf topography and 0.2H mm for the drops. See also the electronic supplementary material.