Evaporation of Interacting Droplets on an Inclined Surface

The proposed project aims to tackle novel aspects of evaporating drops on an inclined plane by using a combination of numerical simulations and asymptotic theory.
The problem set up for studying evaporation of a sessile, possibly containing suspended solutes. The cross-section of the drop at time t = 0 is circular with an initially prescribed contact angle. The droplet evaporates, converting liquid to vapour, as a result of which the drop loses mass and shrinks. The local mass flux is simply given by the surface gradient of the vapour concentration. The vapour concentration in the bulk is assumed to satisfy Laplace's equation, i.e. it simply diffuses. Analytical progress can be made by making some simplifying assumptions. The drop is assumed to be thin, i.e. it always maintains a flat disc shape. Consequently, analytical expressions for the vapour concentration problem are obtained by using asymptotics. Similarly, the thin drop assumption makes it possible to solve the hydrodynamics of the drop using well-established lubrication theory. Finally, the particles concentration satisfies advection-diffusion equation which is coupled to the flow problem which is solved numerically. Hence, there are three entities that interact with each other nonlinearly, the vapour concentration, flow velocity, and solute particles concentration giving rise to very rich dynamics. The final outcome of the model of practical interest is the solute concentration that is left behind after the drop has completely evaporated.
Recently, black-box solvers like Surface Evolver have been used to obtain steady drop shapes by minimising the Helmholtz free energy and then the evaporation rate is computed by solving Laplace's equation for the vapor concentration field via a finite-volume method. To the best of our knowledge, there are no simulations that make use of boundary element method (BEM) to solve the problem of evaporating droplets even though it is the natural choice for numerical simulations as it is most suitable for solving Laplace's and Stokes equation. We first propose to fill this gap in literature by performing BEM simulations of an evaporating sessile droplet. The results will be compared against lubrication theory described above. The solute particles concentration satisfy advection equation making standard BEM inapplicable and hence, a dual-reciprocity BEM method will have to be employed. However, this problem can be circumvented easily with a Langrangian description for the particles that are simply advected with the flow field generated inside the droplet.
The problem of evaporating drops on inclined and vertical planes is of practical significance in many industrial settings. Inclining the plane allows gravitational forces to take a central role in the final deposit patterns. This problem has been solved numerically recently in two- and three-dimensions, respectively. However, the natural choice for simulations, BEM, has not employed to study this problem so far and transient drop dynamics remains unaddressed. We propose to extend the results by including the effect of wall inclination. Finally, the most practical situation where a pair drops on an inclined plane are interacting with each other through the vapour field will be tackled. This again can be easily achieved using BEM. The problem of interacting evaporating drops has not been studied numerically so far.