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This paper presents a comprehensive theoretical analysis of the interaction between two quasi-steady burning droplets with differing temperatures, sizes and distances, building upon the mass-flux-potential model and flame-sheet assumption. In contrast to existing research, this study introduces a fresh perspective on droplet interactions by considering the different temperatures of the droplets. Utilizing the bispherical coordinate approach, theoretical solutions for the Stefan flow, scalar fields, droplet evaporation/burning rates, interaction coefficients and flame positions have been derived successfully. A comparison with extensive numerical simulations indicates a good agreement between the analytical and numerical results under a variety of conditions. It is revealed that proximity between the droplets causes non-uniform evaporation rates on their surfaces, and in some cases, leads to condensation on the cooler droplet. Notably, when the temperatures of the two droplets differ, this results in an uneven temperature distribution across the flame surface, and increasing the temperature of one droplet substantially elevates the temperature of the nearby flame. This study also establishes a criterion for the transition between different combustion modes, specifically between group and separated combustion. The findings of this study are crucial in deepening our understanding of evaporation and combustion processes, as well as the dynamics of flame spreading, local ignition, and extinction in systems involving multiple droplets. [ABSTRACT FROM AUTHOR]

Blue whirls are newly discovered blue rotating flames transitioned from fire whirls generated on liquid fuel surfaces. Compared with the traditional fire whirls, blue whirls have a distinct flame structure and are almost soot-free. Studies of the blue whirl are expected to promote the development of clean combustion theory and technology. In this paper, we numerically and theoretically investigate the lifting process of fire whirls and blue whirls, and then discuss their structure and transition mechanisms. In contrast to the explanations in the literature, it is found that blue whirls can be created without the occurrence of bubble-type vortex breakdown in the rotating flow. The blue whirl can be considered as a special form of lifted fire whirl with a considerable liftoff distance and without being blown out. According to the simulation, both blue whirls and fire whirls have the typical triple-flame structure. The unique structure of blue whirls is closely related to the thermal expansion of gas and the restriction of the radial flow by the axis. The theoretical solution of the unified lifting curve for fire whirls and blue whirls is obtained, which is qualitatively in agreement with the numerical result. There are 4 different branches in the lifting curve, two of which are stable, respectively representing fire whirls with small liftoff distances, and blue whirls with larger liftoff distances. The unstable branches between the stable branches lead to non-continuous transitions between fire whirls and blue whirls, as observed in the experiment. The current study explains the existing physical phenomenon and makes valuable predictions that may guide future experiments and simulations. [ABSTRACT FROM AUTHOR]

Stabilization of the flame base at the bottom Ekman boundary layer of the fire whirl under strong circulation is analytically and numerically studied. The problem is relevant for fire safety and the associated understanding of the generation mechanism of the newly discovered whirling flame, the so-called blue whirl. Based on the dynamic balance of the triple flame propagation speed and the local flow velocity at the bottom boundary layer, theoretical solutions of the liftoff position, r f , and the critical blowout limit represented by its Ekman number, Ek Bt , are derived. The theoretical results agree well with the numerical results both qualitatively and quantitatively. It is shown that the response of r f to the system Damköhler number, Da , forms a typical C-shaped curve and there are two solutions of r f for relatively large Da , with the one closer to the edge of the fuel pool being stable while the other being unstable. Explicit solution of the blowout limit shows that E k B t ∝ (S L S) 4 / 3 , where S L S is the stoichiometric planar flame speed. This is because not only the triple flame propagation speed linearly increases as S L S increases, but the flame front can also reach a position closer to the edge of the fuel pool where the flow velocity is lower. The effect of the transversal radial velocity gradient (TVG) at the Ekman boundary layer on the stabilization of the fire whirl is also investigated. The TVG makes the flame harder to liftoff and blowout and the effect of which can be described by a non-dimensional parameter, λ , which represents the ratio of the flame curvature and the TVG. The TVG normally plays a secondary role in the triple flame propagation due to the relatively large flame curvature of the triple flame. [ABSTRACT FROM AUTHOR]