Mathematical framework for novel non-porous viscous fingering instabilities

在数学史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，开辟了全新的研究领域，随后在未来几年里出现了大量新的科学和工业应用。这一提议旨在为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。在建立了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进不稳定性(NPVFI)。新的不稳定性与20世纪50年代发现的Saffman-Taylor指进不稳定性有关，但又有别于Saffman-Taylor指进不稳定性，也称为粘性指进不稳定性(VFI)。这种不稳定性包括形成复杂的、通常是分形状的图案或手指，当较低粘度的流体侵入多孔介质中较粘性的流体时，这些图案或手指会自发形成。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，都得益于观察到这种指法可以随心所欲地操纵。几十年来，这种指法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发生得更广泛，超越了多孔介质，以NPVFI的形式出现。具体地说，NPVFI涉及在不同粘度的流体的自由表面流动或薄膜流动中形成复杂的指进模式。我们的初步理论工作揭示了易受NPVFI影响的整个自由表面流动家族，我假设，更多类别的流动也会受到NPVFI的影响，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，标志着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，并使大量以前不为人知的工业和环境应用受益，从药物和疫苗的鼻腔给药到软基质的构图。为了利用这个及时的机会，有必要克服NPVFI数学建模和分析方面的根本挑战，目前这一挑战阻碍了进一步的理论发展。这一挑战涉及开发入侵前沿(入侵流体的鼻子，不稳定性的来源)的适当数学模型。由于这一挑战，目前还没有一个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。这项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，并对一系列自由表面流族进行实验验证，从而从基础上建立这一基本发现研究的新领域。我们还将研究如何根据实际应用的需要来处理这些不稳定性。

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.