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

1.背景固着液滴的蒸发是一个非常活跃的国际研究领域，每天都有新的出版物出现，整个会议（如2021年9月在达姆施塔特虚拟举行的“Droplets 2021”）现在都致力于该主题。在过去的十年中，三位导师，即Stephen Wilson教授和亚历山大雷博士（斯特拉斯克莱德大学）和Khellil Sefiane教授（爱丁堡大学工程学院和斯特拉斯克莱德大学客座教授），在液滴蒸发的各个方面以及气泡动力学，自再润湿流体和抗表面活性剂方面进行了单独和合作。本项目本项目的目的是建立在主管以前的工作蒸发液滴，蒸发液滴在爱丁堡的实验研究正在进行的计划，探索这一科学和实践上的重要问题的两个令人兴奋的新方面。(a)受限制液滴的蒸发最近由多个液滴蒸发的监督者进行的工作表明，大气中的蒸汽对相邻液滴的所谓“屏蔽”效应降低了蒸发速率，因此相对于孤立的同一液滴，与其他液滴接近的液滴的寿命增加。然而，这种屏蔽效果相对较弱，并且间隔开多于一个或两个半径的液滴被有效地隔离。在某些情况下，弱相互作用可能是期望的，但在其他情况下，强相互作用可能是优选的，并且增强相邻液滴之间的相互作用的一种方式是限制它们蒸发到其中的大气。受此启发，并正在进行的实验在爱丁堡，我们将进行理论研究的影响，限制大气液滴蒸发。我们将开始在一个理想化的情况下（即，薄，二维液滴），然后寻求扩展我们的分析，以更现实的情况。(b)在多孔基底上的蒸发虽然绝大多数关于固着液滴蒸发的先前工作集中在不可渗透的情况（即，固体）基材，在许多应用中（例如，印刷到纸和纺织品上）时，基底是可渗透的，并且液体向基底中的输送在液滴的演变中起重要作用，因此液滴的寿命和来自液滴的任何存款都起重要作用。在戴维斯和霍金的开创性工作，我们最近的工作溪流流动和通过渗透膜，并正在进行的实验工作在爱丁堡，我们将制定和分析一个模型的液滴在多孔基板上的蒸发。

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.