Mathematical modelling of thin liquid films with variable fluid properties

在许多环境设置中都发现了薄液体膜，并且在许多工业过程和生物功能中也起着重要作用。这些液体层可能屈服于界面不稳定，从而产生流动，产生表面波，导致薄膜厚度明显不均匀。在许多情况下，这种情况对电影的预期目的产生了不利影响。例如，在涉及涂层应用的制造过程中，界面不稳定性的发展可能导致涂层材料的不规则分布。同样，在生物体中起作用的液体薄膜，例如，覆盖眼睛的泪膜或肺部气道的液体衬里，如果它们变得太薄或出现孔洞，就会失效。然而，在某些工业部门，界面不稳定性可以优化过程。例如，在热交换器和质交换器中，界面波的形成改善了设备的运行，因为液气界面的表面积增加了，这有利于传输。界面不稳定的一个重要因素是表面张力的变化，引起所谓的马兰戈尼效应。表面张力的不均匀可能是由于加热或溶解在液体中的化学试剂（表面活性剂）的存在。理论模型可以用来预测导致不稳定的条件，并确定最终的流型。一项重要的研究努力是通过捕获所有相关的物理因素来开发能够准确地代表现实世界问题的模型。例如，已知加热不仅会影响表面张力，还会影响液体的其他性质，如粘度和质量密度，后者也是溶质浓度的函数。然而，这方面在液体膜的建模中仍未得到探索。我建议实现一个数学流动模型，考虑可变流体性质，并研究对流体膜稳定性的影响。我还将研究与其他建模复杂性的综合效应，如底部地形（液体在非平面基底上流动，表现出周期性波动或波纹）。在许多现实世界的情况下，不均匀的基材是自然发生的，或者是制造过程中意想不到的缺陷。在其他情况下，基材的形状可以是受控参数，并且可以设计特定的轮廓以减轻或利用流体膜中不稳定性的开始。另一个建模扩展是流体的底部渗透率，这与流体膜在多孔材料上扩散的情况有关。这种情况出现在印刷和绘画过程以及纺织品的一般涂层中。水力渗透性也与撕裂覆盖软性隐形眼镜有关。

Thin liquid films are found in many environmental settings and also play an important role in numerous industrial processes and biological functions. These liquid layers can succumb to interfacial instability whereby a flow develops that generates surface waves resulting in a pronounced non uniformity in the thickness of the film. In many cases this occurrence has an adverse effect on the intended purpose of the film. For example, in manufacturing processes involving coating applications the development of interfacial instability can result in an irregular distribution of the coating material. Similarly, liquid films functioning in living organisms such as, for example, the tear film covering the eye or the fluid lining of airways in the lungs can be ineffective if they become too thin or develop holes. In certain industrial sectors, however, interfacial instability can optimize the process. For example, in heat and mass exchangers the formation of interfacial waves improves the operation of the device due to the fact that there is an increase in the surface area of the liquid-gas interface, which facilitates the transport. An important contributor to interfacial instability is variation in surface tension causing what is referred to as the Marangoni effect. Unevenness in surface tension can be due to heating or the presence of chemical agents dissolved in the liquid (surfactants). Theoretical models can be used to predict the conditions that lead to instability and determine the resulting flow pattern. An important research endeavour is to develop models that accurately represent real-world problems by capturing all the relevant physical factors. For instance, heating is known to affect not just surface tension, but also other properties of the liquid such as viscosity and mass density, with the latter also being a function of solutal concentration. This aspect has however remained unexplored in the modelling of liquid films. I propose to implement a mathematical flow model that accounts for variable fluid properties and investigate the effect on the stability of fluid films. I will also study the combined effect with other modelling complexities such as bottom topography (liquid flowing on nonplanar substrates exhibiting periodic undulations or corrugations). In many real-world situations uneven substrates occur naturally, or are an unintended flaw of manufacturing. In other circumstances the shape of the substrate may be a controlled parameter, and a particular profile may be designed so as to mitigate or exploit the onset of instability in the fluid film. Another modelling extension is bottom permeability to the fluid, and is relevant in cases where the fluid film spreads over a porous material. Such situations arise in printing and painting processes as well as the general coating of textiles. Hydraulic permeability is also relevant to tear covered soft contact lenses.