Mathematical framework for novel non-porous viscous fingering instabilities

新型无孔粘性指法不稳定性的数学框架

• 批准号: EP/Y021959/1
• 负责人: Katarzyna Kowal
• 金额: $ 44.49万
• 依托单位: University of Glasgow
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2024
• 资助国家: 英国
• 起止时间: 2024 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FY021959%2F1
• 关键词: Mathematical; framework; novel; non; porous

New instabilities have been discovered every few decades or centuries in the history of mathematics, and science more generally, opening completely new areas of research followed by a wealth of new scientific and industrial applications for years to come. This proposal aims to establish a new line of fundamental research on one of these new instabilities, discovered in surprising recent experiments. In developing the first mathematical framework for understanding its origins, we named the new instability the non-porous viscous fingering instability (NPVFI).The new instability is related to, yet distinct from, what is known as a Saffman-Taylor fingering instability, also referred to as a viscous fingering instability (VFI), discovered in the 1950s. Such instabilities involve the formation of complex, often fractal-like, patterns, or fingers, which form spontaneously when a less viscous fluid intrudes into a more viscous fluid in a porous medium. An abundance of scientific and technological applications followed this discovery, ranging from enhanced oil recovery to microfluidics, all benefitting from the observation that such fingering can be manipulated as desired. For decades, such fingering instabilities have been thought to occur in porous media only. What was unknown up until recently is that the fundamental mechanism of such instabilities in fact occurs much more widely, beyond porous media, in the form of NPVFI. Specifically, NPVFI involves the formation of complex fingering patterns in the free-surface flow, or thin-film flow, of fluids of unequal viscosity. Our preliminary theoretical work revealed whole families of free-surface flows susceptible to NPVFI, and I hypothesise that further classes of flow are susceptible to NPVFI as well, all beyond porous media and beyond the original experiments revealing NPVFI. As with the pioneering of VFI in the 1950s, the theoretical exploration of NPVFI marks an opportunity to open an exciting new area of research in applied mathematics and continuum mechanics, and to benefit a wealth of previously unexplained industrial and environmental applications, ranging in diversity from the nasal delivery of drugs and vaccines to the patterning of soft substrates.To make use of this timely opportunity, it is necessary to overcome a fundamental challenge in the mathematical modelling and analysis of NPVFI, which is currently hindering further theoretical developments. This challenge involves developing an appropriate mathematical model of the intrusion front (the nose of the intruding fluid, where the instability originates). Because of this challenge, there is currently no mathematical framework to explain the number of fingers seen in experiments and their growth, and no framework to control - suppress or enhance - the instability. The proposed research will shed light on these open questions and establish this new area of fundamental discovery research from the foundation by developing a new mathematical framework for modelling the front and validating it experimentally for a suite of families of free-surface flows. We will also investigate how to manipulate these instabilities as desired for practical applications.

在数学和更广泛的科学史上，每隔几十年或几个世纪就会发现新的不稳定性，从而开辟全新的研究领域，并在未来几年带来大量新的科学和工业应用。该提案旨在针对最近令人惊讶的实验中发现的这些新的不稳定性之一建立一条新的基础研究路线。在开发第一个数学框架来理解其起源时，我们将这种新的不稳定性命名为无孔粘性指法不稳定性 (NPVFI)。这种新的不稳定性与 20 世纪 50 年代发现的萨夫曼-泰勒指法不稳定性（也称为粘性指法不稳定性 (VFI)）相关，但又有所不同。这种不稳定性涉及复杂的、通常是分形的图案或指状物的形成，当多孔介质中粘性较小的流体侵入粘性较高的流体时，这些图案或指状结构会自发形成。这一发现带来了大量的科学和技术应用，从提高石油采收率到微流体技术，所有这些都受益于可以根据需要操纵这种指进的观察。几十年来，这种指法不稳定性被认为只发生在多孔介质中。直到最近我们还不知道的是，这种不稳定性的基本机制实际上以 NPVFI 的形式在多孔介质之外更广泛地发生。具体来说，NPVFI 涉及在不等粘度流体的自由表面流或薄膜流中形成复杂的指状图案。我们的初步理论工作揭示了整个系列的自由表面流易受 NPVFI 影响，并且我假设其他类别的流也易受 NPVFI 影响，所有这些都超出了多孔介质和揭示 NPVFI 的原始实验。与 20 世纪 50 年代 VFI 的先驱一样，NPVFI 的理论探索标志着一个机会，可以在应用数学和连续介质力学领域开启一个令人兴奋的新研究领域，并使大量以前无法解释的工业和环境应用受益，范围从药物和疫苗的鼻腔输送到软基质的图案化。要利用这个及时的机会，有必要克服 NPVFI 数学建模和分析中的一个根本挑战，目前阻碍了进一步的理论发展。这一挑战涉及开发入侵前沿（入侵流体的鼻部，不稳定的根源）的适当数学模型。由于这一挑战，目前没有数学框架来解释实验中看到的手指数量及其增长，也没有框架来控制（抑制或增强）不稳定性。拟议的研究将阐明这些悬而未决的问题，并通过开发一个新的数学框架来建模前沿并针对一系列自由表面流进行实验验证，从基础上建立这一基础发现研究的新领域。我们还将研究如何根据实际应用的需要操纵这些不稳定性。