The energy-minimization multiscale (EMMS) model

originally proposed for gas-solid fluidization

features a stability condition to close the simplified conservation equations. It was put forward to physically reflect the compromise of two dominant mechanisms

i.e.

the particle-dominated with minimal potential energy of particles

and the gas-dominated with the least resistance for gas to penetrate through the particle bed. The stability condition was then formulated as the minimization of the ratio of these two physical quantities. Analogously

the EMMS approach was later extended to the gas-liquid flow in bubble columns

termed dual-bubble-size model. It considers the compromise of two dominant mechanisms

i.e

.

the liquid-dominated regime with small bubbles

and the gas-dominated regime with large bubbles. The stability condition was then formulated as the minimization of the sum of these two physical quantities. Obviously

the two stability conditions were expressed in different manner

though gas-solid and gas-liquid systems bear some analogy. In addition

both the conditions transform the original multi-objective variational problem into a single-objective problem. The math

ematical formulation of stability condition remains therefore an open question. This study utilizes noncooperative game theory and noninferior solutions to directly solve the multi-objective variational problem

aiming to explore the different pathways of compromise of dominant mechanisms. The results show that only keeping the single dominant mechanism cannot capture the jump change of gas holdup

which is associated with flow regime transition. Hybrid of dominant mechanisms

noninferior solutions and noncooperative game theory can predict the flow regime transition. However

the game between the two mechanisms makes the two-bubble structure degenerate and reduce to the single-bubble structure. The game of the three mechanisms restores the two-bubble structure. The exploration on the formulation of stability conditions may help to understand the roles and interactions of different dominant mechanisms in the origin of complexity in multiphase flow systems.