Mathematical framework for novel non-porous viscous fingering instabilities

在数学的历史上，每隔几十年或几个世纪就会发现新的不稳定性，更广泛地说，科学史上的不稳定性，开辟了全新的研究领域，随后是未来几年里大量新的科学和工业应用。这项提议的目的是在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。在开发理解其起源的第一个数学框架时，我们将这种新的不稳定性命名为无孔粘性指指不稳定性（NPVFI）。这种新的不稳定性与20世纪50年代发现的所谓的Saffman-Taylor指法不稳定性（也称为粘性指法不稳定性（VFI））有关，但又不同。这种不稳定性包括复杂的，通常是分形的，图案或指状的形成，当粘性较低的流体侵入多孔介质中粘性较强的流体时，这些图案或指状会自发形成。在这一发现之后，大量的科学和技术应用，从提高石油采收率到微流体，都受益于这种指法可以按需操作的观察。几十年来，这种指法不稳定性被认为只发生在多孔介质中。直到最近人们才知道，这种不稳定性的基本机制实际上以NPVFI的形式更广泛地存在于多孔介质之外。具体来说，NPVFI涉及不等粘度流体在自由表面流动或薄膜流动中形成复杂指指模式。我们的初步理论工作揭示了整个家族的自由表面流动易受NPVFI的影响，我假设进一步的流动类别也易受NPVFI的影响，所有这些都超出了多孔介质和原始实验揭示的NPVFI。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，在应用数学和连续介质力学中开辟了一个令人兴奋的新研究领域，并受益于大量以前无法解释的工业和环境应用，范围从药物和疫苗的鼻腔输送到软基质的图案。为了利用这一及时的机会，有必要克服NPVFI数学建模和分析中的一个基本挑战，这一挑战目前阻碍了进一步的理论发展。这一挑战包括建立侵入前沿（侵入流体的前端，不稳定性的起源）的适当数学模型。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悬而未决的问题，并通过开发一个新的数学框架来模拟锋面并在一系列自由表面流的实验中验证它，从基础上建立这个新的基础发现研究领域。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。

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.