arXiv subjects

David J. Needham

Publications and source records attributed to David J. Needham.

3 recordsLinked to original sources

arXiv2026

Stationary distribution of active Brownian particles in a bounded domain

Self-propelled particles such as swimming micro-organisms often accumulate at boundaries. Active Brownian particles (ABPs), that is, particles moving at constant speed with an orientation undergoing rotational diffusion, provide a simple model. ABPs have a finite probability to be on the boundary. Their probability density therefore includes both regular bulk and singular boundary components. Exchanges between these represent capture at the boundary and re-entry into the bulk. We examine the distribution of ABPs in the stationary regime. We first obtain a similarity solution that describes the probability density near the turning points, where orientation is tangential to the boundary and particles re-enter the bulk. This shows that the bulk density is singular and the boundary density is non-differentiable. We then add spatial diffusion to the ABP model and investigate the limit $\eps \to 0$ of small spatial diffusivity. We use matched asymptotics to describe the regularising effect of a finite $\eps$, focusing on the vicinity of the turning points. We assess our predictions against numerical simulations using both lattice Markov chain and finite-element discretisations.

cond-mat.soft↗

arXiv2025

The role of mutants in the spatio-temporal progression of inflammatory bowel disease: three classes of permanent form travelling waves

Despite its high prevalence and impact on the lives of those affected, a complete understanding of the cause of inflammatory bowel disease (IBD) is lacking. In this paper, we investigate a novel mechanism which proposes that mutant epithelial cells are significant to the progression of IBD since they promote inflammation and are resistant to death. We develop a simple model encapsulating the propagation of mutant epithelial cells and immune cells which results from interactions with the intestinal barrier and bacteria. Motivated by the slow growth of mutant epithelial cells, and relatively slow response rate of the adaptive immune system, we combine geometric singular perturbation theory with matched asymptotic expansions to determine the one-dimensional slow invariant manifold that characterises the leading order dynamics at all times beyond a passive initial adjustment phase. The dynamics on this manifold are controlled by a bifurcation parameter, $σ$, which depends on the ratio of growth to decay rates of all components except mutants and determines three distinct classes of permanent-form travelling waves that describe the propagation of mutant epithelial and immune cells. These are obtained from scalar reaction-diffusion equations with the reaction being (i) a bistable nonlinearity with a cut-off, (ii) a cubic Fisher nonlinearity and (iii) a KPP or Fisher nonlinearity. Our results suggest that mutant epithelial cells are critical to the progression of IBD. However, their effect on the speed of progression is subdominant.

math.AP↗

arXiv2024

Modelling visibility and surface deformation in particle-fluid flow fields generated by helicopter rotors

As a helicopter descends towards a bed of sand, a high velocity particle laden cloud can form around the helicopter body, a phenomenon known as "brownout", and a consequence of which can potentially be a significant deterioration in visibility for the helicopter pilot. Here we consider a recently developed physically based rational mathematical model for the generation of wind-driven particle flow fields from otherwise static particle beds, one application of which is the scenario considered here. We introduce a directional opacity measure, defined for each observation angle from the helicopter cockpit, and show how visibility may vary in the model as certain parameters are varied. In particular, we demonstrate a counterintuitive result suggesting that, with specific yet potentially realistic parameter choices, pilot visibility may be improved in some viewing directions if the helicopter were hovering at a lower altitude. We also calculate the associated deformation of the upper surface of the particle bed, and show how certain surface deformation features may be sensitive to variation of key parameters.

physics.flu-dyn↗