Mathematical modelling of droplet impact on soft substrates

在软衬底上飞溅是一个活跃的研究课题，有着广泛的应用，包括降雨、农药应用、喷墨打印、燃料燃烧、法医学和喷雾涂层。然而，最近的实验表明，飞溅过程比之前认为的要复杂得多，因此关于液滴飞溅的基本物理机制以及衬底变形如何促进或阻止它们仍有许多悬而未决的问题。了解如何消除飞溅是一个相对较新的话题，在飞溅可能导致卫生或安全问题的情况下可能会很有用。当液滴撞击干燥、平坦的衬底时，液滴会以高于临界速度的速度飞溅，这取决于液滴大小、粘度和气压等参数。最近的实验结果[1]表明，衬底硬度也影响飞溅阈值，例如，在最软的衬底上的液滴飞溅所需的动能比在刚性衬底上的多70%以上。能量损失是由于软衬底在撞击的最初几微秒内变形造成的。目前还没有关于衬底硬度对飞溅阈值的影响的理论。该项目的目的是在与牛津大学工程系流体动力学实验室的广泛密切合作的指导下，推导和分析液滴对一系列具有实际意义的软衬底的影响。其目的是：(1)使用匹配的渐近展开方法，系统地推导出碰撞早期阶段的简化模型，这些模型保留了所有基本物理，同时保持易于处理，从而将经典碰撞理论推广到新的制度；(2)开发、测试和验证碰撞早期和后期的最先进的直接数值模拟。简化的渐近模型将被用来提取物理洞察力，并启动和验证数值模拟。数值模拟将被用来描述衬底对溅射阈值的影响。我们将从最简单的基板模型开始，即悬臂，然后讨论更复杂的基板流变性，包括弹性、高粘性、粘弹性基板和振动的软表面。该项目属于EPRSC连续介质力学、流体动力学和空气动力学研究领域。[1]Howland，CJ，Antkowiak，A，Castrejón-Pita，Jr；Howison，SD，Oliver，JM，Style，RW，Castrejón-Pita，AA。更难溅到柔软的固体上。《物理评论通讯》第117期，184502期(2016年)。合作者：阿方索·卡斯特雷洪-皮塔，牛津大学工程系流体动力学实验室。

Splashing on soft substrates is an active research topic with a wide range of applications including rainfall, pesticide application, inkjet printing, fuel combustion, forensic science and spray coating. However, recent experiments have shown that the splashing process is more complex than previously thought, so that there are many outstanding questions still remaining regarding the fundamental physical mechanisms involved in droplet splashing and how they are promoted or prohibited by substrate deformation. Finding out how to eliminate splashing is a relatively new topic, and can be useful in situations where splashing could cause hygiene or safety concerns. Droplets splash when they impact dry, flat substrates above a critical velocity that depends on parameters such as droplet size, viscosity, and air pressure. Recent experimental results [1] have shown that substrate stiffness also affects the splashing threshold, for example, droplets on the softest substrates need over 70% more kinetic energy to splash than they do on rigid substrates. The energy losses are caused by deformations of the soft substrate during the first few microseconds of impact. There is currently no theory for the effect of substrate stiffness on the splashing threshold.The aim of the project is to derive and analyse the impact of a liquid droplet on a range of soft substrates of practical importance, guided by extensive close collaboration with the Fluid Dynamics Laboratory in the Department of Engineering, Oxford. The objectives are to (i) use the method of matched asymptotic expansions to derive systematically reduced models for the early stages of the impact that retain all of the essential physics while remaining tractable, generalising thereby classical impact theory into new regimes; (ii) develop, test and validate state-of-the-art direct numerical simulations for the early and late stages of the impact. The reduced asymptotic models will be used to extract physical insight and both initiate and validate the numerical simulations. The numerical simulations will be used to characterise the effect of the substrate on the splashing threshold. We will begin with the simplest possible model for the substrate, namely a cantilever, before moving onto more complicated substrate rheologies including elastic, highly viscous, viscoelastic substrates and vibrating soft surfaces.This project falls within the EPRSC Continuum Mechanics and Fluid Dynamics and Aerodynamics research areas. [1] Howland, CJ, Antkowiak, A, Castrejón-Pita, JR; Howison, SD, Oliver, JM, Style, RW, Castrejón-Pita, AA. It's Harder to Splash on Soft Solids. Physical Review Letters 117, 184502 (2016).Collaborators: Alfonso Castrejon-Pita, Fluid Dynamics Laboratory in the Department of Engineering, Oxford.