The thermodynamic nonequilibrium (TNE) effects in a coalescence process of two initially static bubbles under thermal conditions are investigated by a discrete Boltzmann model. The spatial distributions of the typical nonequilibrium quantity, i.e., nonorganized momentum fluxes (NOMFs), during evolutions are investigated in detail. The density-weighted statistical method is used to highlight the relationship between the TNE effects and the morphological and kinetics characteristics of bubble coalescence. The results show that the $xx$ component and $yy$ component of NOMFs are antisymmetrical; the $xy$ component changes from an antisymmetric internal and external double quadrupole structure to an outer octupole structure during the coalescence process. Moreover, the evolution of the averaged $xx$ component of NOMFs provides two characteristic instants, which divide the nonequilibrium process into three stages. The first instant, when the averaged $xx$ component of the NOMFs reaches its first local minimum, corresponds to the moment when the mean coalescence speed gets the maximum, and at this time the ratio of minor and major axes is about $1/2$. The second instant, when the averaged $xx$ component of the NOMFs gets its second local maximum, corresponds to the moment when the ratio of minor and major axes becomes 1 for the first time. It is interesting to find that the three quantities, TNE intensity, acceleration of coalescence, and the slope of boundary length, show a high degree of correlation and attain their maxima simultaneously. The surface tension and the heat conduction accelerate the process of bubble coalescence, while the viscosity delays it. Both the surface tension and the viscosity enhance the global nonequilibrium intensity, whereas the heat conduction restrains it. These TNE features and findings present some insights into the kinetics of bubble coalescence.

中文翻译：

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气泡聚结中的热力学非平衡效应：离散玻尔兹曼研究

通过离散玻尔兹曼模型研究了在热条件下两个初始静态气泡的聚结过程中的热力学非平衡 (TNE) 效应。详细研究了演化过程中典型的非平衡量，即无组织动量通量（NOMFs）的空间分布。密度加权统计方法用于突出 TNE 效应与气泡聚结的形态和动力学特征之间的关系。结果表明，$XX$组件和$\mathrm{是的}\mathrm{是的}$NOMF 的分量是反对称的；这$X\mathrm{是的}$在聚结过程中，组件从反对称的内外双四极结构变为外八极结构。此外，平均数的演变$XX$NOMFs的组成部分提供了两个特征时刻，将非平衡过程分为三个阶段。第一个瞬间，当平均$XX$NOMFs的分量达到其第一个局部最小值，对应于平均聚结速度达到最大值的时刻，此时短轴和长轴的比率约为$1/2$. 第二个瞬间，当平均$XX$NOMFs 的分量获得第二个局部最大值，对应于短轴和长轴之比第一次变为 1 的时刻。有趣的是，TNE 强度、聚结加速度和边界长度斜率这三个量显示出高度相关性并同时达到最大值。表面张力和热传导加速了气泡聚结的过程，而粘度延迟了它。表面张力和粘度都增强了整体非平衡强度，而热传导抑制了它。这些 TNE 特征和发现对气泡聚结的动力学提出了一些见解。