# Unsteady Combustion of the Heptane-in-Water Emulsion Foamed with Hydrogen-Oxygen Mixture
## 2 Problem Setup  
Let us consider the combustion of the foamed emulsion considered recently in [19]. That foam represents a two-phase medium consisting of micro-scale gas bubbles of \(\sim\)200 \(\upmu\)m diameter separated by thin films of an oil-in-water emulsion. The n-heptane is used as an oil, while the hydrogen-oxygen mixtures with varying equivalence ratios are used as foaming gases. The combustion is initiated via a pilot flame at the top opened end of the channel and then propagates downwards through the foam (Figure 1). According to the prior experimental analysis [19; 20], different modes of flame propagation can be observed depending on the foam composition. The flame accelerates and even transits to detonation in the absence of heptane [20] or at its low content in the foam [19]. Vice versa, at high heptane content, the flame quenches, and the foam does not burn out completely in the reactor. One of the main goals of this paper is to understand whether the recently presented mathematical model [27; 36] with an account of the chemical kinetics of heptane/hydrogen oxidation can reproduce the basic features of flame development in the foamed emulsion. As soon as such a model can reproduce the process in a considered combined fuel, one can use the model for further interpretation of the possible modes of foams combustion and develop novel ways of combustible foams application. According to the previous studies of foams combustion, there is the following description of the basic mechanism of flame propagation through the foam. Foam decays into droplets in the pre-flame zone due to the thermal and dynamic impact on the foam from the flame front. Herewith, it should be noted that the foam, by its structure, consists of gas bubbles separated by thin liquid films intersected in the vertices called Plateau borders. When the foam decays, the thin films are corrupted first, while the thicker parts, Plateau borders, maintain their continuity. As a result, the liquid phase is decomposed into coarse droplets of a size equal to the Plateau border one. According to [PERSON]'s law, the bubble diameter \(R_{b}\) is related to the Plateau border size \(L\) as \(R_{b}=\sqrt{2}L\), so the radius of coarse droplets emerging from the foam decay can be estimated as 0.7\(R_{b}\). According to [19], the bubble size in the considered foam is estimated as 200 \(\upmu\)m, and the diameter of the coarse droplets equals 140 \(\upmu\)m. Herewith, the average distance between the droplets is also known and equals 200 \(\upmu\)m. In such a way, the foam decays into the array of droplets suspended in the gas, hydrogen-oxygen mixture in the process of flame propagation. Further, after  
Figure 1: Schematic of the numerical model and problem setup. The flame propagates downwards through the foam that decays into the droplets in the pre-flame zone (for convenience, the channel is shown horizontally. The top end is to the left, the bottom is to the right). The numerical model based on the two-phase gas dynamics includes submodels of momentum and heat exchange in the regions of foam and droplets, combustion kinetics, the structure of the emulsion droplet, and the non-uniform spatial distribution of liquid in the foam. the foam is decayed in the pre-flame zone, the foaming gas is fed into the flame front together with the heptane vapor emitted from the surface of the droplets. The combustion propagates downwards in the form of the flame front. Recently, we have proposed mathematical models suitable to predict both slow [36] and fast [27] combustion processes in foamed emulsions and hydrogen microfoams. Elaborated models provide a good agreement between the calculations and experimental data from [36] and [20]. They are based on the general two-phase hydrodynamic description with an account of:  
* Gas-liquid multiphase interactions,
* Evaporation,
* Foam decay into droplets,
* Droplets fragmentation at high flow rates,
* Combustion. According to the two-phase hydrodynamic model, the medium represents the blend of two continuums.