Mathematical modelling of thin liquid films with variable fluid properties

液体薄膜广泛存在于各种环境中，在许多工业过程和生物功能中也扮演着重要的角色。这些液层可能会屈服于界面的不稳定性，由此产生的流动会产生表面波，导致薄膜厚度的明显不均匀。在许多情况下，这种情况会对电影的预期目的产生不利影响。例如，在涉及涂层应用的制造工艺中，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膜，如覆盖眼睛的泪膜或肺部呼吸道的液体衬里，如果变得太薄或形成孔洞，可能会无效。然而，在某些工业部门，界面不稳定可以优化这一过程。例如，在热交换器和质量交换器中，由于液-气界面的表面积增加，界面波的形成改善了设备的运行，从而促进了传输。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Marangoni效应。表面张力的不均匀可能是由于加热或存在溶解在液体中的化学试剂(表面活性剂)。理论模型可以用来预测导致不稳定的条件，并确定由此产生的流型。一项重要的研究努力是通过捕捉所有相关的物理因素来开发准确反映现实世界问题的模型。例如，加热不仅影响表面张力，而且还影响液体的其他性质，如粘度和质量密度，后者也是溶液浓度的函数。然而，在液态薄膜的模拟中，这一方面仍未得到探索。我建议建立一个考虑流体性质变化的数学流动模型，并研究其对流体膜稳定性的影响。我还将研究与其他建模复杂性的组合效果，如底部地形(流动在非平面基材上的液体呈现周期性波动或波纹)。在许多现实情况下，不均匀的衬底是自然产生的，或者是制造过程中无意的缺陷。在其他情况下，衬底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的不稳定性的开始。另一个模型扩展是流体的底部渗透率，并且在流体膜在多孔材料上扩散的情况下是相关的。这种情况出现在印刷和喷漆过程中，以及纺织品的一般涂层中。液压渗透性也与撕裂覆盖的软性隐形眼镜有关。

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.