Mathematical Modelling and Analysis of the Evaporation of Confined Droplets and the Evaporation of Droplets on Porous Substrates

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1. Background The evaporation of sessile droplets is an area of very active international research, with new publications appearing on a daily basis and entire conferences (such as "Droplets 2021" held virtually in Darmstadt in September 2021) now dedicated to the topic. Over the last decade, the three supervisors, namely Professor Stephen Wilson and Dr Alexander Wray (University of Strathclyde) and Professor Khellil Sefiane (School of Engineering, University of Edinburgh and a Visiting Professor at the University of Strathclyde), have worked both singly and in collaboration on various aspects of droplet evaporation, as well as on bubble dynamics, self-rewetting fluids, and anti-surfactants.2. The Present ProjectThe aim of the present project is to build on the supervisors' previous work on evaporating droplets, and an ongoing programme of experimental investigations of evaporating droplets in Edinburgh, to explore two exciting new aspects of this scientifically and practically important problem.(a) The Evaporation of Confined DropletsRecent work by the supervisors on the evaporation of multiple droplets shows that the so-called "shielding" effect of the vapour in the atmosphere from neighbouring droplets decreases the evaporation rate, and hence increases the lifetime, of a droplet in proximity to other droplets relative to that of the same droplet in isolation. However, this shielding effect is relatively weak, and droplets spaced more than one or two radii apart are effectively isolated. In some situations weak interactions may be desirable, but in others strong interactions may be preferred, and one way to enhance the interactions between neighbouring droplets is to confine the atmosphere into which they evaporate. Motivated by this, and by ongoing experiments in Edinburgh, we will undertake a theoretical investigation of the effect of confining the atmosphere on droplet evaporation. We will begin in an idealised situation (namely, a thin, two-dimensional droplet) before seeking to extend our analysis to more realistic scenarios.(b) Evaporation on a Porous SubstrateWhile the vast majority of previous work on the evaporation of sessile droplets has focused on the case of impermeable (i.e., solid) substrates, in many applications (e.g., printing onto paper and textiles) the substrate is permeable, and transport of liquid into the substrate plays a significant role in the evolution, and hence the lifetime of and any deposit from, the droplet. Building in the pioneering work of Davis & Hocking, our recent work on rivulet flow over and through a permeable membrane, and ongoing experimental work in Edinburgh, we will formulate and analyse a model for the evaporation of a droplet on a porous substrate.