Travelling Waves Approach in a Parabolic Coupled System for Modelling the Behaviour of Substances in a Fuel Tank

José Luis Díaz Palencia

doi:10.3390/app11135846

Abstract

The aim of this work was to provide a formulation of a non-linear diffusion model with forced convection in the form of a reaction–absorption system. The model was studied with analytical and numerical approaches in the frame of the parabolic operators theory. In addition, the solutions are applied to a gas interaction phenomenon with the intention of producing an inerted ullage in an Airbus A320 aircraft centre fuel tank. We made use of the travelling wave (TW) solutions approach to study the existence of solutions, stability and the precise evolution of profiles. The application exercise sought to answer a key question for aerospace sciences which can be formulated as the time required to ensure an aircraft fuel tank is safe (inerted) to prevent explosion due to the presence of oxygen in the tank ullage.

Highlights

In the 1930s, Fisher [1], proposed a reaction–diffusion model to describe the interactive dynamic of genes
Any travelling wave (TW) moving with the exponential decaying term γ2 as per (25) is positive in the whole domain
The problem PT proposed with a coupled system of reaction diffusion equations to model an aircraft fuel tank inerting process has been discussed with a mathematical approach stressing aspects related with the existence, uniqueness and behaviour of solutions in the travelling wave domain

Summary

Introduction

In the 1930s, Fisher [1], proposed a reaction–diffusion model to describe the interactive dynamic of genes. Kolmogorov, Petrovskii and Piskunov [2] proposed the same equation in combustion theory. In both cases, the models considered a Gaussian order two-diffusion model with a non-linear reaction of the form f (u) = u(1 − u). Afterwards, the Fisher–KPP model was subjected to remarkable mathematical research to explore further applications (see [3,4,5]). Some analyses [6] have shown new patterns of formation in chemistry and biology (compared to those existing in the current literature, see the remarkable references [7,8])

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