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.

在数学史和科学史上，每隔几十年或几个世纪就会发现新的不稳定性，从而开辟了全新的研究领域，并在未来几年内产生了大量新的科学和工业应用。这项提议旨在建立一个新的基础研究路线，对这些新的不稳定性之一，在令人惊讶的最近的实验中发现。在发展第一个数学框架来理解它的起源时，我们将这种新的不稳定性命名为无孔粘性指进不稳定性（NPVFI）。这种新的不稳定性与20世纪50年代发现的Saffman-Taylor指进不稳定性（也称为粘性指进不稳定性（VFI））有关，但又不同。这种不稳定性涉及复杂的，通常是分形状的图案或指状物的形成，当粘性较低的流体侵入多孔介质中的粘性较高的流体时，这些图案或指状物自发形成。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，所有这些都得益于这种指进可以根据需要进行操纵的观察。几十年来，这种指进不稳定性一直被认为只发生在多孔介质中。直到最近还不知道的是，这种不稳定性的基本机制实际上以NPVFI的形式出现在多孔介质之外的更广泛的地方。具体地说，NPVFI涉及在不等粘度流体的自由表面流动或薄膜流动中形成复杂的指状图案。我们的初步理论工作揭示了整个家庭的自由表面流易受NPVFI，我假设，进一步类的流量也易受NPVFI，所有超出多孔介质和超出原来的实验揭示NPVFI。与20世纪50年代VFI的开创性一样，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中开辟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，从药物和疫苗的鼻腔输送到软基底的图案化。有必要克服NPVFI的数学建模和分析中的一个根本性挑战，该挑战目前阻碍了进一步的理论发展。这一挑战涉及开发一个适当的数学模型的入侵前沿（鼻子的入侵流体，不稳定性的起源）。由于这一挑战，目前还没有数学框架来解释实验中看到的手指数量及其生长，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开放的问题，并通过开发一个新的数学框架来建模前沿，并通过实验验证一系列自由表面流，从而从基础上建立这一新的基础发现研究领域。我们还将研究如何操纵这些不稳定性的实际应用所需的。