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月实际上在达姆施塔特举行的“2021年液滴”)现在专门讨论这一专题。在过去十年中，三位导师，即斯蒂芬·威尔逊教授和亚历山大·雷博士(斯特拉斯克莱德大学)和克利勒·塞菲安教授(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、自增湿流体和反表面活性剂2方面进行了单独和合作的工作。本项目的目的是在监督员先前关于蒸发液滴的工作和爱丁堡正在进行的蒸发液滴实验研究方案的基础上，探索这一具有科学和实际意义的重要问题的两个令人振奋的新方面。(A)受限液滴的蒸发最近监督员关于多个液滴蒸发的工作表明，大气中水蒸气对邻近液滴的所谓“屏蔽”效应降低了液滴的蒸发速度，从而延长了液滴相对于孤立的相同液滴的蒸发速度。然而，这种屏蔽作用相对较弱，间距超过一到两个半径的液滴被有效隔离。在某些情况下，弱相互作用可能是可取的，但在另一些情况下，强相互作用可能是可取的，而加强相邻液滴之间相互作用的一种方法是限制它们蒸发进入的大气。受此启发，并通过在爱丁堡正在进行的实验，我们将对限制大气对液滴蒸发的影响进行理论研究。我们将从理想化的情况(即薄的二维液滴)开始，然后寻求将我们的分析扩展到更现实的场景。(B)在多孔基质上的蒸发虽然以前关于固着液滴蒸发的绝大多数工作都集中在不透水(即固体)基材的情况下，但在许多应用中(例如，打印到纸和纺织品上)基材是可渗透的，液体进入基材的传输对液滴的演变以及液滴的寿命和沉积起着重要作用。在Davis&Hocking的开创性工作、我们最近在渗透膜上和通过渗透膜的溪流流动方面的工作以及在爱丁堡正在进行的实验工作的基础上，我们将制定和分析一个液滴在多孔衬底上蒸发的模型。

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.