Mathematical framework for novel non-porous viscous fingering instabilities

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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.