Harris, Samuel Joseph; (2025) Wildfires, flowers, and huddling penguins: mathematical modelling of two-dimensional free boundary problems in nature. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

Natural phenomena are modelled as two-dimensional interfacial problems, and their equilibria, stability and evolution are considered analytically and numerically in this work. Wildfire spread is studied, where the harmonic pyrogenic wind induced by the fire and the resulting steady transport of oxygen to the fire line must be solved exterior to the wildfire. The corresponding coupled pair of conformally invariant governing equations - the Laplace and steady advection-diffusion equations - motivates the use of conformal mapping to determine the stability and nonlinear evolution of the fire line. The same conformal mapping method is applied to a free boundary problem with reciprocal growth law which offers a deterministic wildfire basic rate of spread effect rather than an ad-hoc constant traditionally used. Analytical solutions to this problem for growing ellipses and a class of polynomial lemniscates are also derived. Penguin huddle evolution is, from a mathematical modelling perspective, strikingly similar to wildfire spread, involving a steady transport of temperature by the wind. The penguin huddle interior is also modelled as a continuum, where the penguins rearrange themselves in the huddle to spread their heat. The resulting Poisson equation is conformally variant and so interior temperature is instead solved using the adaptive Antoulas–Anderson--least squares (AAA-LS) algorithm. This algorithm also solves Laplace problems in multiply connected domains, enabling the spread and merger of multiple wildfires to be computed by this method. Both ambient and pyrogenic winds are included into the single and multiple wildfire models and the results are compared, with good agreement, against experimental data and an existing wildfire model. A new algorithm based on the AAA-LS method is developed to solve mixed boundary value two-domain problems and to examine the electrostatic interaction between flowers and their pollinators. The two-domain AAA-LS algorithm is also applied to problems in vortex dynamics.

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