Gas–solid two-phase flow is ubiquitous in nature and many engineering fields, such as chemical engineering, energy, and mining. The closure of its hydrodynamic model is difficult owing to the complex multiscale structure of such flow. To address this problem, the energy-minimization multi-scale (EMMS) model introduces a stability condition that presents a compromise of the different dominant mechanisms involved in the systems, each expressed as an extremum tendency. However, in the physical system, each dominant mechanism should be expressed to a certain extent, and this has been formulated as a multiobjective optimization problem according to the EMMS principle generalized from the EMMS model. The mathematical properties and physical meanings of this multiobjective optimization problem have not yet been explored. This paper presents a numerical solution of this multiobjective optimization problem and discusses the correspondence between the solution characteristics and flow regimes in gas‒solid fluidization. This suggests that, while the most probable flow structures may correspond to the stable states predicted by the EMMS model, the noninferior solutions are in qualitative agreement with the observable flow structures under corresponding conditions. This demonstrates that both the dominant mechanisms and stability condition proposed for the EMMS model are physically reasonable and consistent, suggesting a general approach of describing complex systems with multiple dominant mechanisms.

气固两相流在自然界和化学工程，能源和采矿等许多工程领域中无处不在。由于这种流动的复杂的多尺度结构，很难关闭其流体动力学模型。为了解决这个问题，能量最小化多尺度（EMMS）模型引入了一个稳定性条件，该条件提出了系统中涉及的不同主导机制的折衷，每种主导机制均表示为极值趋势。然而，在物理系统中，每个主导机制都应在一定程度上表达出来，并根据从EMMS模型中得出的EMMS原理将其表述为多目标优化问题。这个多目标优化问题的数学性质和物理意义尚未探索。本文给出了该多目标优化问题的数值解，并讨论了气固流化过程中的解特征与流态之间的对应关系。这表明，尽管最可能的流动结构可能对应于EMMS模型预测的稳定状态，但在相应条件下非劣解与可观察到的流动结构在质量上是一致的。这表明为EMMS模型提出的主导机制和稳定性条件在物理上都是合理且一致的，这表明了描述具有多种主导机制的复杂系统的通用方法。这表明，尽管最可能的流动结构可能对应于由EMMS模型预测的稳定状态，但在相应条件下，非劣解与可观察到的流动结构在质量上是一致的。这表明为EMMS模型提出的主导机制和稳定性条件在物理上都是合理且一致的，这表明了描述具有多种主导机制的复杂系统的通用方法。这表明，尽管最可能的流动结构可能对应于由EMMS模型预测的稳定状态，但在相应条件下，非劣解与可观察到的流动结构在质量上是一致的。这表明为EMMS模型提出的主导机制和稳定性条件在物理上都是合理且一致的，这表明了描述具有多种主导机制的复杂系统的通用方法。