Hybrid asymptotic-numerical schemes for exponentially small selection mechanisms

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[1] There are a number of problems in fluid mechanics and the wider physical sciences that involve the asymptotic analysis of nonlinear differential equations studied in some singular limit, where an underlying selection mechanism determines a sequence of discrete countably infinite eigenvalues. In certain challenging cases, this mechanism is governed by exponentially small terms beyond-all-orders and the resultant analysis demands specialised techniques. [2] One classic problem for which this occurs is in the context of Saffman-Taylor viscous fingering where the selection of the finger width is determined by terms exponentially small in the surface tension parameter. The resolution of the Saffman-Taylor problem pioneered modern methods in exponential asymptotics. However, the problem contains certain niceties that render the analysis tractable. In particular, the leading-order surface-tension-free solution is known in closed form and this turns out to be a crucial component in the application of exponential asymptotics. Similar selection mechanisms governing viscous fingering are expected to apply in generalisations to time-dependent flows or flows in complex geometries, but it remains an open challenge to adopt the asymptotic techniques to these extensions. [3] This thesis will focus on problems where the asymptotic or perturbative solutions cannot be determined in closed form, even at leading order. In such cases, it is essential to consider the extension of classical exponential asymptotic methodologies to hybrid schemes where numerical methods are used in conjunction with analytical theory. These numerical methods involve, for example, solutions of ordinary or partial differential equations and subsequent analytic continuation of real-valued solutions to higher-dimensional complex-valued spaces. The study of analytic continuation then yields key properties of solutions near singular points, which is then encoded into the asymptotic schemes.[4] The PhD will be focused on the development of these analytical and numerical methods, and their applications to several open problems in continuum and fluid mechanics. Applications will include some or all of the following: (i) the study of jet separation in a two-dimensional nozzle; (ii) jet separation in a three-dimensional or axi-symmetric nozzle; and (iii) Saffman-Taylor viscous fingering in a wedge.