arXiv subjects

Matthew R. Turner

Publications and source records attributed to Matthew R. Turner.

4 recordsLinked to original sources

arXiv2026

Global bifurcation of localised 2D patterns emerging from spatial heterogeneity

We present a general approach to prove the existence, both locally and globally in amplitude, of fully localised multi-dimensional patterns in partial differential equations containing a compact spatial heterogeneity. While one-dimensional localised patterns induced by spatial heterogeneities have been well-studied, proving the existence of fully localised patterns emerging from a Turing instability in higher dimensions remains a key open problem in pattern formation. In order to demonstrate the approach, we consider the two-dimensional Swift--Hohenberg equation, whose linear bifurcation parameter is perturbed by a radially-symmetric potential function. In this case, the trivial state is unstable in a compact neighbourhood of the origin and linearly stable outside. We prove the existence of local bifurcation branches of fully localised patterns, characterise their stability and bifurcation structure, and then rigorously continue solutions to large amplitude via analytic global bifurcation theory. Notably, the primary bifurcating branch in the Swift--Hohenberg equation alternates between an axisymmetric spot and a non-axisymmetric `dipole' pattern, depending on the width of the spatial heterogeneity.

nlin.PS↗

arXiv2025

Critical inertia for particle capture is determined by surface geometry at forward stagnation point

Aerosols are ubiquitous, and particle capture from particle-laden air as it flows past an obstacle is of widespread practical importance. Neglecting diffusion, previous work has shown that for a smooth curved surface in both Stokes flow and inviscid flow, only particles with inertia above a threshold value (quantified by the nondimensional Stokes number) collide with the surface. Here we show that the critical Stokes number decreases with increasing Reynolds number of the air flow, and the mechanism behind this threshold is the same at all finite Reynolds numbers but becomes qualitatively different in the limit of infinite Reynolds number (inviscid flow). In addition we show that in the latter case (inviscid flow) the threshold is set solely by the flow near the stagnation point, whereas at finite Reynolds numbers the threshold also depends on the flow far from the stagnation point. The threshold also depends on obstacle geometry and we show that fibers whose cross section is flattened along the flow direction have greater size selectivity than fibers with a circular cross section.

physics.flu-dyn↗

arXiv2021

Localised Radial Patterns on the Free Surface of a Ferrofluid

This paper investigates the existence of localised axisymmetric (radial) patterns on the surface of a ferrofluid in the presence of a uniform vertical magnetic field. We formally investigate all possible small-amplitude solutions which remain bounded close to the pattern's centre (the core region) and decay exponentially away from the pattern's centre (the far-field region). The results are presented for a finite-depth, infinite expanse of ferrofluid equipped with a linear magnetisation law. These patterns bifurcate at the Rosensweig instability, where the applied magnetic field strength reaches a critical threshold. Techniques for finding localised solutions to a non-autonomous PDE system are established; solutions are decomposed onto a basis which is independent of the radius, reducing the problem to an infinite set of nonlinear, non-autonomous ODEs. Using radial centre manifold theory, local manifolds of small-amplitude solutions are constructed in the core and far-field regions, respectively. Finally, using geometric blow-up coordinates, we match the core and far-field manifolds; any solution that lies on this intersection is a localised radial pattern. Three distinct classes of stationary radial solutions are found: spot A and spot B solutions, which are equipped with two different amplitude scaling laws and achieve their maximum amplitudes at the core, and ring solutions, which achieve their maximum amplitudes away from the core. These solutions correspond exactly to the classes of localised radial solutions found for the Swift-Hohenberg equation. Different values of the linear magnetisation and depth of the ferrofluid are investigated and parameter regions in which the various localised radial solutions emerge are identified. The approach taken in this paper outlines a route to rigorously establishing the existence of axisymmetric localised patterns in the future.

math.DS↗

arXiv2016

Convection in a Single Column -- Modelling, Algorithm and Analysis

The group focused on a model problem of idealised moist air convection in a single column of atmosphere. Height, temperature and moisture variables were chosen to simplify the mathematical representation (along the lines of the Boussinesq approximation in a height variable defined in terms of pressure). This allowed exact simple solutions of the numerical and partial differential equation problems to be found. By examining these, we identify column behaviour, stability issues and explore the feasibility of a more general solution process.

physics.ao-ph↗