A two-dimensional model for the evolution of the fire line – the interface between burned and unburned regions of a wildfire – is formulated. The fire line normal velocity has three contributions: (i) a constant rate of spread representing convection and radiation effects; (ii) a curvature term that smooths the fire line; and (iii) a Stefan-like term in the direction of the oxygen gradient. While the first two effects are geometrical, (iii) is dynamical and requires the solution of the steady advection–diffusion equation for oxygen, with advection owing to a self-induced ‘fire wind’, modelled by the gradient of a harmonic potential field. The conformal invariance of this coupled pair of partial differential equations, which has the Péclet number $\textit {Pe}$ as its only parameter, is exploited to compute numerically the evolution of both radial and infinitely long periodic fire lines. A linear stability analysis shows that fire line instability is possible, dependent on the ratio of curvature to oxygen effects. Unstable fire lines develop finger-like protrusions into the unburned region; the geometry of these fingers is varied and depends on the relative magnitudes of (i)–(iii). It is argued that for radial fires, the fire wind strength scales with the fire's effective radius, meaning that $\textit {Pe}$ increases in time, so all fire lines eventually become unstable. For periodic fire lines, $\textit {Pe}$ remains constant, so fire line stability is possible. The results of this study provide a possible explanation for the formation of fire fingers observed in wildfires.

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Penguins huddling in a cold wind are represented by a two-dimensional, continuum model. The huddle boundary evolves due to heat loss to the huddle exterior and through the reorganisation of penguins as they seek to regulate their heat production within the huddle. These two heat transfer mechanisms, along with area, or penguin number, conservation, gives a free boundary problem whose dynamics depend on both the dynamics interior and exterior to the huddle. Assuming the huddle shape evolves slowly compared to the advective timescale of the exterior wind, the interior temperature is governed by a Poisson equation and the exterior temperature by the steady advection-diffusion equation. The exterior, advective wind velocity is the gradient of a harmonic, scalar field. The conformal invariance of the exterior governing equations is used to convert the system to a Polubarinova-Galin type equation, with forcing depending on both the interior and exterior temperature gradients at the huddle boundary. The interior Poisson equation is not conformally invariant, so the interior temperature gradient is found numerically using a combined adaptive Antoulas-Anderson and least squares algorithm. The results show that, irrespective of the starting shape, penguin huddles evolve into an egg-like steady shape. This shape is dependent on the wind strength, parameterised by the Péclet number Pe, and a parameter $\beta $ β which effectively measures the strength of the interior self-generation of heat by the penguins. The numerical method developed is applicable to a further five free boundary problems.