On thermodynamic closures for two-phase flow with interfacial area concentration transport equation

The present work is a part of a modelling of forest fires fighting by aerial means. In this paper, we study different kind of closures for modelling two-phase flows with an almost “infinite range” of scales. Since theories like homogenization are not, in this case, relevant for obtaining the equivalent medium equations, the averaging method has been preferred. The variables are averaged by convolution with a smooth kernel with compact support, as the equations are non-linear, new quantities are defined in order to obtain the equations satisfied by averaged quantities; the entropy production is determined and closures or phenomenological equations are obtained using the second principle of thermodynamics. Main features of this work are, firstly a derivation in this framework of a balance equation for the interfacial area concentration and secondly, since this introduces a new unclosed variable: the mean velocity of interfaces, extended irreversible thermodynamics is used to obtain the general form of the appropriate closures equations.