Mixed boundary value problems in curvilinear domains relevant to microfluidic cooling using biomimetic structured surfaces

阻碍微处理器芯片发展的一个主要挑战是设计冷却系统，以适应芯片热量产生的增加，从而满足对更快处理速度的持续需求。一种被称为直接液体冷却（DLC）的方法是通过微观通道驱动液体冷却剂直接通过芯片。然而，由于这些通道必须非常窄，所需的驱动压差可能大得不切实际。克服这一问题的一种方法是减少通道壁对冷却剂施加的摩擦力。最近，有人提议通过使用微通道来实现这一点，微通道内衬被称为超疏水表面（SS）。SS是具有微观结构的表面，可以减少流动在其上的液体的摩擦阻力。从本质上讲，结构之间的空腔捕获了空气袋，液体在空气袋上比在固体基板上更自由地流动。这样的表面在自然界中很常见，一个众所周知的例子是荷叶。在各种各样的应用中使用仿生SS的潜力已经有一段时间了，但直到最近，微加工技术的进步才使制造它们成为可能。这导致人们对它们的兴趣激增。然而，在DLC的背景下，虽然SS润滑冷却剂的流动，但很少有人知道它们如何影响热量的流动，这是主要关注的问题。这使得设计带有SS的DLC设备成为一项艰巨的任务。对于工程师来说，最有用的是揭示液体和热量流动对设计参数（例如，微结构的间距）的精确（或接近精确）依赖关系的公式。这样的公式使得精确确定这些参数的值成为可能，这些参数可以比通过数值计算更快地优化冷却。这个项目的主要目的是找到这样的公式。在精确解方面，该领域的基准结果是Philip得出的结果。我们的第一个目标是推导出这些结果的扩展。我们将使用与Philip所使用的技术类似的方法来做这件事，但要加上一套新的数学工具，这些工具是由最近发现的所谓的次级肖特基群产生的。在过去的十年中，使用类似的工具在找到类似问题的精确解决方案方面被证明是非常有效的。然而，菲利普的结果的一个局限性，以及几乎所有其他文献中的结果，是他们假设被困的空气和液体之间的界面或半月板的形状是平坦的。实际上，这个半月板通常是弯曲的，在这些微观尺度上，这种弯曲可以产生重大影响。我们的第二个目标是推导出整个曲率范围的解。使用在平半月板或其他传统方法中应用的工具，以显式公式构建此类解决方案将非常困难。为了克服这个问题，我们将利用一种强大的新技术，它基于所谓的变换，它是专门为解决这类涉及弯曲边界的问题而设计的。这个提议的多个方面（模拟液体和热的流动，包括固体/液体/气体界面和曲率效应），以及应用工具的复杂性和威力（二次肖特基群，新的变换方法），使这成为一个数学上丰富的项目。此外，SS在DLC中的创新应用，以及当前对SS整体的兴趣，加上所提出的数学技术的新颖性，使得这些研究具有及时的意义。此外，考虑到许多其他完全不同的物理过程由相同的数学方程控制，该项目的结果将产生广泛的影响。

A major challenge hindering the development of microprocessor chips capable of meeting the ongoing demand for faster processing speeds, is designing cooling systems that can accommodate the concomitant increases in chip heat production. One method, known as direct liquid cooling (DLC), is to drive liquid coolant directly through a chip via microscopic channels. However, since these channels are necessarily very narrow, the required driving pressure difference can be impractically large. One way to overcome this is to reduce the frictional forces exerted by the channel walls on the coolant. Very recently it has been proposed to do this by using microchannels lined with what are known as superhydrophobic surfaces (SS's).SS's are surfaces textured with microscopic structures which can reduce the frictional drag forces on liquids flowing over them. Essentially, cavities between the structures trap pockets of air over which the liquid flows more freely than it does over a solid substrate. Such surfaces occur commonly in nature, a well-known example being the lotus leaf. The potential for using biomimetic SS's in a wide variety of applications has been known for some time, but it is only recently that advances in microfabrication technology have made it possible for them to be manufactured. This has led to a surge in interest in them. However, in the context of DLC, whilst SS's lubricate the flow of the coolant, little is known about how they influence the flow of heat, which is the primary concern. This currently makes designing DLC devices with SS's a difficult task. What are of most use to engineers doing this, are formulae which reveal the exact (or close to exact) dependence of the flow of liquid and heat on the design parameters (e.g., the spacing of the microstructures). Such formulae make it possible to pinpoint the values of these parameters which optimise cooling much faster than, say, by numerical computations. The primary aim of this project is to find such formulae.In terms of exact solutions, the benchmark results in the field are those derived by Philip. Our first aim is to derive an extension of these results. We shall do this by using similar techniques to those applied by Philip, but with the important addition of a new set of mathematical tools arising from recently-identified, so-called secondary Schottky groups. The use of similar tools has proved to be highly effective at finding exact solutions to similar problems over the last decade.However, one limitation of Philip's results, and almost all others which have followed in the literature, is that they assume the shape of the interface, or meniscus, between the trapped pockets of air and the liquid to be flat. In reality this meniscus is generally curved, and at these microscopic scales, this curvature can have a significant effect. Our second aim is to derive solutions which account for a full range of this curvature. The construction of such solutions in terms of explicit formulae would be very hard using the tools applied in the case of flat menisci or other, traditional methods. To overcome this we shall make use of a powerful new technique, based on so-called transforms, which is specifically designed for solving problems of this type involving curved boundaries.The multiple facets of this proposal (modelling flows of both liquid and heat involving solid/liquid/gas interfaces and curvature effects), and the complexity and power of the tools to be applied (secondary Schottky groups, the new transform method), make this a mathematically rich project. In addition, the innovative application of SS's to DLC, and the current interest in SS's as a whole, combined with the novelty of the proposed mathematical techniques, make these investigations of timely significance. Furthermore, given that a number of other entirely different physical processes are governed by the same mathematical equations, the project's results will have a broad impact.