Modeling of an aircraft fire extinguishing process with a porous medium equation

Abstract

The aim of this work is to provide a formulation of a non-linear diffusion model in the form of a Porous Medium Equation (PME) with application to a fire extinguishing process in an aircraft engine nacelle. The work starts by describing some relevant publications currently related to fire suppression modelling methods with emphasis in the aerospace sector. The PME is then introduced highlighting some key relevant features (particularly the finite speed of propagation) compared to the classical Heat Equation (HE). We will refer as u to the extinguisher or suppressor concentration in the media, which is postulated to be governed by a PME equation of the form: 1 $$\begin{aligned} u_{t}=\Delta u^m + \vert x \vert ^\sigma u^p, \end{aligned}$$ 2 $$\begin{aligned} u\left( x,0\right) = u_0(x) \in \mathbb {L}_{loc}^\infty (\mathbb {R^N}), \end{aligned}$$ where 3 $$\begin{aligned} m>1, \sigma >0, p<1, \end{aligned}$$ 4 $$\begin{aligned} (x,t) \in Q_T = \mathbb {R^N} \times (0,T) \end{aligned}$$ Without losing generality and in virtue of the mass transfer application, we will consider that any solution is $$u \ge 0$$ . The set of equation and conditions expressed from (1) to (4) will be referred as problem P. From a mathematical perspective, the main areas of analysis are related to the existence of solutions, the obtaining of particular solutions as asymptotic approach and the application or particularization to a representative aircraft engine nacelle domain where a fire may happen.