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.
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