A PDE model for unidirectional flows: stationary profiles and asymptotic behaviour

TL;DR

This paper models unidirectional pedestrian flows using PDEs, analyzing stationary profiles and boundary layers influenced by domain shape and boundary conditions, supported by analytical and computational results.

Contribution

It introduces a PDE model linking boundary conditions and domain shape to flow profiles, with analytical and computational analysis of boundary layer formation.

Findings

Boundary layers' location and shape depend on inflow/outflow and domain geometry.

Analytical results are validated through computational experiments.

Abstract

In this paper, we investigate the stationary profiles of a convection-diffusion model for unidirectional pedestrian flows in domains with a single entrance and exit. The inflow and outflow conditions at both the entrance and exit as well as the shape of the domain have a strong influence on the structure of stationary profiles, in particular on the formation of boundary layers. We are able to relate the location and shape of these layers to the inflow and outflow conditions as well as the shape of the domain using geometric singular perturbation theory. Furthermore, we confirm and exemplify our analytical results by means of computational experiments.