Jiachen Liu,Xiaoping Guan,Ning Yang,*

1 State Key Laboratory of Multiphase Complex Systems,Institute of Process Engineering,Chinese Academy of Sciences,Beijing 100190,China

2 School of Chemical Engineering,University of Chinese Academy of Sciences,Beijing 100049,China

Keywords:Bubble column Flow regimes Mesoscale Noncooperative game theory

ABSTRACT 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 multiobjective variational problem into a single-objective problem.The mathematical 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.

1.Introduction

Gas-solid flow in fluidization and gas-liquid flow in bubble columns represent two typical types of multiphase flow in chemical and process engineering.In fast fluidization,particles tend to aggregate into clusters when interacting with gas [1-4].In bubble columns,gas continuously fed into the system through a distributor at the column bottom,and evolves into two representative groups of bubbles,i.e.,small bubbles and large bubbles,through breakage and coalescence [5].The aforementioned structures frequently emerge at mesoscales,i.e.,between a single particle/bubble and the whole vessel,and are critical to mass and heat transfer as well as chemical reactions.Understanding the physical mechanisms relevant to the mesoscale structures is of significance to reactor design and optimization.

To characterize the mesoscale structure in gas-solid fluidization,the energy-minimization multiscale (EMMS) approach was proposed [6-10].The heterogeneous structure was resolved into a particle-rich dense cluster phase and a fluid-rich dilute phase,and described by eight parameters and six simplified conservation equations.Since the number of unknown variables is more than the number of equations,a stability condition was put forward to close the conservation equations.The stability condition physically reflects the compromise of two dominant mechanisms,i.e.,the gas seeking the least resistance to penetrate through particle-bed layer during the upward movement,and the solid particles tending to minimize their own potential energy.Here the former was represented by the minimization of suspending and transporting particles per unit volume(Wst→min),and the latter was formulated by the minimization of bed voidage (ε →min).The joint effect,or in other words,the compromise of these two dominant mechanisms,leads to a multi-objective variational problem.To solve the problem more efficiently,the two extremum tendencies were merged and transformed into a single-objective variational problemNst=-Wst/[(1-ε)ρp]=min.The stability condition essentially differentiates the movement tendencies of gas and of solid particles to explore the underlying mechanisms of cluster formation.

Analogously,the EMMS approach was extended to the gas-liquid flow in bubble columns in which a stability condition was proposed as the compromise of two dominant mechanisms [11-15].One is formulated asNsurf→min;and the other asNturb→min.HereNsurfstands for the energy dissipation due to bubble surface instability and oscillation,andNturbdenotes the energy dissipated through the energy cascade process of liquid turbulence.Nsurf→min reflects a liquid-dominant regime,and in this case larger bubbles are likely to break into smaller bubbles and hence smaller bubbles prevail in the system.Nturb→min represents a gasdominant regime to favor the existence of larger bubbles.It should be noted that the stability condition here was formulated in terms of the morphology of gas,i.e.,small bubbles and large bubbles,rather than the movement tendencies of gas and liquid,as bubbles,unlike solid particles,may change their morphology to response the interaction with liquid and partially realize their movement tendencies.This is achieved by the dynamic interfacial phenomena,i.e.,bubble coalescence or breakage.

Since gas-solid systems and gas-liquid systems bear some analogy,it is natural to look for++a generalized approach to identify the dominant mechanisms and the mathematical formulation.In gas-solid systems,the two physical quantityWstand ε was lumped intoNstby a division operator,andNsttherefore physically represents the energy suspending and transporting particles per unit mass.In gas-liquid systems,the two physical quantityNsurfandNturbwas added together,as the two terms are of the same dimension and denote the two different portions of energy dissipation out of the total.Both of the two stability conditions transform the original multi-objective variational problem into a singleobjective variational problem.Recently,a noncooperative game model based the concept of the generalized Nash equilibrium(GNE) was developed to directly solve the multi-objective variational problem[16,17].The overall trend of flow regime transition agrees with the original EMMS model and experimental results,although the GNE calculation shows a single-bubble dominant mechanism.

The motivation and roadmap of this study,as summarized in Fig.1,follows two routes.Firstly,with the given formulations for each dominant mechanism,we compare the different combination of dominant mechanisms,i.e.,single extremum tendencies,the single-objective variational problem,and the multi-objective variational problem.The last is solved by the noncooperative game theory or noninferior solutions.Secondly,we apply the two different approaches to formulate the dominant mechanisms,i.e.,the difference in movement tendencies of different phases,and the difference in morphology of the dispersed phase.A new integrated multi-objective variational problem is proposed and solved by the noncooperative game theory to simultaneously consider the two approaches.It should be pointed out that this study is exploratory and heuristic,and does not focus on more rigorous derivation or discussion on the stability conditions due to the physical complexity of the problem.Interested readers are referred to[6,10,13,14,18-23] for previous studies and more details.Rather,we aim to explore the pathways for different dominant mechanisms to compromise with each other in this study.

2.Model Description

Fig.1.Different pathways of compromise of dominant mechanisms in bubbly flow.

In our previous study,based on experimental observation [5],six parameters were used to characterize the two-phase flow structure in bubble columns,including bubble diameter (dS,dL),gas volume fraction (fS,fL) and superficial gas velocities (Ug,S,Ug,L)for small or large bubbles.Here the subscripts denote small or large bubbles respectively.Then three conservation equations can be formulated,as listed in Table 1.These include a continuity equation for gas,and two force balance equations for small or large bubbles.The three conservation equations are not adequate to obtain the six unknown parameters.A stability condition was then proposed to close the equations.The total energy fed into the system per unit mass of liquidNTarises from the gas input from the column bottom,as given in Eq.(4).NTis consumed in three different routes.In addition toNsurfandNturbmentioned earlier,the third portionNbreakapplies due to bubble breakage and coalescence.According to the theory of bubble or drop breakage in turbulent dispersion [24],energy can be extracted from liquid bulk to break bubbles through the interaction of turbulent eddies and bubbles.This portion is stored temporarily in the form of newlygenerated surface in bubble breakage,and finally dissipated during bubble coalescence.Here we assume bubble breakage and coalescence reaches a dynamic balance at steady state.Hence the three portions constitute the total energy consumptionNT,with the first two directly dissipated at microscale and the last presented at mesoscale.The stability condition was established by analyzing the two dominant mechanisms in bubble columns,namely,Nsurf→min andNturb→min.

Table 1 Conservation equations of DBS model

Then the next question is how to formulate the stability condition.A combined formulationNsurf+Nturb=min was applied in our previous work.In this way,the three conservation equations in Table 1 can be closed.With the given total superficial gas velocityUgand physical properties,the six parameters can be obtained by solving a single objective problem.It should be pointed out that there is no artificial setting for the six parameters for small bubbles or large bubbles.In the three conservation equations,i.e.,Eqs.(1)-(3),the solution spaces for the parameters of the two bubble classes are completely symmetrical,and the model equations for large bubbles or small bubbles are exactly the same.The bubble diameters are only determined by the conservation equations and the stability condition.A bubble class is denoted as ‘‘small bubbles”or‘‘large bubbles”only in terms of the model calculation of bubble sizes.In addition,Wanget al.[25] extended the DBS model to triple-bubble-size (TBS) and multiple-bubble-size (MBS)models by introducing three or more bubble classes.They found that the predictions of TBS and MBS model were reduced to that of DBS model,which suggests that the gas-liquid flow in bubble columns are essentially dominated by two bubble classes.

Regime transition to fully-developed heterogeneous regime can be reflected in this model through a jump change on the curve of total gas holdup as a function of superficial gas velocity[13].Based on this zero-dimensional model,Yanget al.[26] proposed a new drag model and gave a correlation forCd/dbas a function of superficial gas velocityUg.The drag model was further improved[27,28]and recently correlated with the local hydrodynamics of CFD cells[29].Submodels to calculate the drag coefficients in conservation equations and definitions of different energy terms are given in Table 2.Generally,the superficial liquid velocity of bubble column is zero (semi-batch).Guan and Yang [23]extended the DBS model to the co-current and counter-current modes of superficial liquid velocity.

Table 2 Submodels in the DBS model

Table 3 Stability conditions evaluated in this paper

3.Formulation of Stability Condition

3.1.Single-objective method

As shown in Fig.1,the original stability condition is formulated asNsurf+Nturb=min,transforming the original multi-objective variational problem into a single-objective variational problem.In this paper,the original stability condition is termed HDM (hybrid of dominant mechanisms).To elucidate the mechanism onthe compromise of the two dominant mechanisms,we also test the minimization of single dominant mechanism,i.e.,Nsurf=min andNturb=min,termed SDM (single dominant mechanism).The algorithm for solving the models with the two SDMs or the HDM is the same.By scanning the global domain of the free variables meeting the conservation equations,the grid nodes with the minimum of objective functions (Nsurf=min,Nturb=min,Nsurf+Nturb=min)can be found.

3.2.Noninferior solution

A multi-objective optimization problem (MOP),involves more than one objective function to be optimized.In most cases,the possibility is quite low for all the objective functions to simultaneously reach their optimal values.This implies that the absolute optimal solution may not exist.However,there may be a set of relative optimal solutions,and the values of each objective function would not be improved without sacrificing other objective functions.The set of relative optimal solutions is called noninferior solution or Paretooptimal solution [30],an important concept in solving an MOP.According to its definition,for an MOP withnobjective functions,minf(x)=[fi(x)],i=1,2,···,n,x*∈X.If there is noxinXwhich makesfi(x)＜fi(x*),thenx*is a noninferior solution.Moet al.[31]analyzed the noninferior solution of the EMMS model for gas-solid fluidization,and correlated the MOP solutions with flow regime transitions.In this paper,we utilize the noninferior solution for gas-liquid flow in bubble columns,as shown in the bottom-right in Fig.1.

There are several approaches to find the noninferior solutions,such as K-T condition,weighting method,constraint method and so forth.The weighting method is used in this paper.TakeNsurfandNturbas the objective functions,the MOP problem can be formulated as:

The noninferior solution can be obtained by finding the minimum off.

where ω denotes a weight.In this work,it is divided into 10,000 sub-intervals.

3.3.Noncooperative game theory

Game theory was originally proposed by Neumann and Morgenstern[32].It now generally refers to the study of a noncooperative game which investigates how players make decisions,given that everyone’s payoff is influenced by the actions of all the others[33].For the noncooperative game models of the multi-objective variational problem,as shown in Fig.1,there are three basic elements: the players Pi(i=1,2,...,n),the pure strategy sifor each player Piselected from a strategy space Si,and the objective functionJifor each player Pi.The multi-objective variational problem is formulated as a noncooperative game between different objective functions.Each player aims to minimize its individual objective function.The system would finally reach a generalized Nash equilibrium (GNE) at which each individual objective function is optimal with consideration of its rival.When the objective function aim to take the minimum value,the policy tuple S*=(s1*,s2*,...,sn*) satisfiesJi(si*,s-i*) ≤Ji(si,s-i*),?si∈Si,?i,where s-i=(s1,...,si-1,si+1,...,sn),the game reaches a Nash equilibrium.

3.3.1.Game between two dominant mechanisms

Yuanet al.[17]applied the noncooperative game theory to solve the DBS model with several basic assumptions:the players are finite,rational,and have complete information,which means that all the players know the structure of the game and know that they all know this.In the DBS model,there are two representative bubble classes for gas-liquid flow in bubble columns:large bubbles and small bubbles.Yuanet al.[17]chosesmallandlarge bubblesasthe twoplayers,andNsurf=min for the objective function of player S(small bubbles),andNturb=min for the objective function of player L(large bubbles).It should be noted that this method needs to specify the objective functions for each player,in contrast to the noninferior solutions and the combined single-objective problem.

The algorithm of game theory is relatively complex.As mentioned above,there are three conservation equations and six unknown parameters in the DBS model.Three of the unknown parameters are free variables (dS,dL,Ug,S).For a game between two dominant mechanisms,the free variables can be divided into two groups,i.e.,(dS,Ug,S) and (dL),corresponding to small bubbles and large bubbles respectively.For each group (dS,Ug,S),the game requires seeking adL*which corresponds to the minimum ofNturb,and records them as RL={dS,Ug,S,dL*}.For each (dL),the game needs a set(dS*,Ug,S*)which corresponds to the minimum ofNsurf,and records them as RS={dS*,Ug,S*,dL}.The intersection of the two sets is the result of the game,called the generalized Nash equilibrium (GNE).The flowchart is illustrated in Fig.2.

3.3.2.Game between three dominant mechanisms

The original EMMS model for gas-solid fluidization distinguished the different dominant mechanisms based on the observation of movement tendencies of gas and particle,i.e.,the particledominated with minimal potential energy of particles,and the gas-dominated with the least resistance for gas to penetrate through the particle bed.Intrinsically,gas-solid system and gasliquid system should bear some analogy.To reach an equilibrium state,gas-liquid flow may also follow the extremum tendencies analogous to gas-solid flow,in addition to the morphology change in the form of small and large bubbles,as shown below:

(1) Morphology of the dispersed phase:Nsurf→ minvs.Nturb→min.

(2) Movement tendencies of different phases:Wst→minvs.εg→min.

According to the original EMMS model,Wsthere denotes the volume specific energy consumption for suspension and transportation of particles.It can be calculated from the work done by the drag force per unit volume.At steady state,the drag force balances with the buoyancy per unit volumefi(ρl-ρg)g.ThenWstcan be formulated as:

Fig.2.Flowchart of solving the noncooperative game of two dominant mechanisms.

The noncooperative game needs to specify the correspondence of each player and objective functions.For the four objective functions,the game needs at least four free variables to act as four players.Unfortunately,there are only three free variables in the DBS model(dS,dL,Ug,S).One solution to this issue is to reduce the number of objective functions to three.For example,the two dominant mechanisms relevant to the movement tendencies of different phases can be merged into a single-objective function:

A new noncooperative game,as shown in the bottom-left in Fig.1,is then formulated.Small bubbles (dS),as the first player,aim to minimizeNsurfand in this case larger bubbles are not stable and likely to break into smaller bubbles.Large bubbles (dL),as the second player,aim to minimizeNturbwhich favor the existence of large bubbles.Here we should notice that the position ofUg,Sis quite different from that ofdSanddLin the three free variables.Ug,Sactually serves as an equivalent toUg,Laccording to Eq.(1),reflecting to some extent the compromising state of small and large bubbles.Hence we may tentatively infer thatUg,SorUg,L,as the third player,aiming to minimizeWst/((1-εg)ρl).All the stability conditions are tabulated in Table 3.

Table 4 The characteristics of stability conditions

For the game between three dominant mechanisms,the free variables are divided into three groups,i.e.,(dS,Ug,S),(dL,Ug,S) and(dS,dL).For each group (dS,Ug,S),the game needs to search thedL* which corresponds to the minimum ofNturb,and records them as RL={dS,Ug,S,dL*}.For each group (dL,Ug,S),the game needs to search thedS* which leads to the minimum ofNsurf,and records them as RS={dS*,Ug,S,dL}.For each group(dS,dL),the game needs to find theUg,S* which reaches to the minimum ofWst/[(1-εg)ρl],and records them as RU={dS,Ug,S*,dL}.The intersection of these three sets is the solution,as shown in Fig.3.

4.Results and Discussion

4.1.Single dominant mechanism

Fig.3.Flowchart of solving the noncooperative game of three dominant mechanisms.

Fig.4.Bubble diameter and total gas holdup of the solution of single dominant mechanism (SDM).

First,the calculation by the SDM stability condition is illustrated in Fig.4.Basically the different energy dissipation terms are a function of the three free variables(dS,dL,Ug,S).For the given superficial gas velocitiesUg,Fig.4 depicts the points of minimum of different SDMs.Obviously,the single dominant mechanismNsurf=min generates very smallerdSanddL,andNturb=min gives largerdSanddLof the same value.This suggests that SDM leads to an extremum case of gas structure,and the system is dominated either by small bubbles or large bubbles.Also the calculation gives the correspondence of each single dominant mechanism and gas morphology structure:Nsurf=min for small bubbles andNturb=min for large bubbles.This can support the assumption in the model of noncooperative game theory.The total gas holdup increases monotonically with the increase ofUg.

4.2.Noninferior solution

WithNsurfandNturbas two objective functions,a number of noninferior solutions can be obtained for different superficial gas velocities,as shown in Fig.5.Obviously,there are multiple solutions for each gas velocity.Most of noninferior solutions are regularly distributed in the solution space and condensed in the two planes ofdS=1.42 mm ordS=3.0 mm.The spatial distribution can be divided into two branches.This region is amplified,as shown in Fig.6.At lowerUg,most of points are in the plane ofdS=1.42 mm.With increasingUg,more points are turned to the second plane.To some extent,this jump change probably signifies the flow regime transition in bubble columns.In the HDM with stability conditionNsurf+Nturb=min,a jump change was also reported and related to regime transition from homogeneous to heterogeneous regimes [13,14,34].There were also two branches in the solution space,and the difference lies in that there was only one solution for eachUgin the HDM,as shown in Fig.7.With increasingUg,the point withNsurf+Nturb=min shifted from one branch to the other.Both the HDM and the noninferior solutions can reflect the change of structure parameters and the regime transition.

Fig.5.Noninferior solutions at different Ug.

Fig.6.Amplification of the solution space.

Fig.7.The solutions of HDM at different Ug.

4.3.Noncooperative game theory

Fig.8.Bubble diameters and gas holdup of large bubbles and small bubbles obtained by three types of stability condition.

Fig.8(a)shows that the HDM(Nsurf+Nturb=min),GNE(Nsurfvs.Nturb) and GNE (Nsurfvs.Nturbvs.Wst/(1-εg)ρl) all can catch the jump change ofdSwith the increase ofUg.The points in the area near the jump are increased,and there is a transition region for GNE (Nsurfvs.Nturbvs.Wst/(1-εg)ρl).At this region,there will be two solutions at the given gas velocity,corresponding to the situation before and after the jump,as shown in the green dotted box of Fig.8(a).The jump change of bubble diameter leads to the jump change of other structure parameters in the model,and finally the jump change of total gas holdup and regime transition.In Fig.8(b),the HDM withNsurf+Nturb=min gives a large bubble diametersdLwithin the interval 10-40 mm.By contrast,the GNE(Nsurfvs.Nturb) and GNE (Nsurfvs.Nturbvs.Wst/(1-εg)ρl) give much largerdLat the boundary of the interval.In fact,this also suggests that the stability conditionviathe noncooperative game amplifies the difference between the two characteristic structure parameters by establishing the relationship between each player and its corresponding objective function.

Fig.9.Comparison between calculation from the HDM and GNE(Nsurf vs.Nturb)and GNE (Nsurf vs. Nturb vs. Wst/(1-εg)ρl) as well as experiments of Camarasa et al. [36]:variation of total gas holdup with superficial gas velocity.

However,the gas holdup of large bubblesfLobtained from the GNE (Nsurfvs.Nturb) tends to zero,implying that the two-bubble structure disintegrates and reduces to the single-bubble structure.This is consistent with the calculation of Yuanet al.[17],that the single-bubble structure of small bubbles at lowerUgand large bubbles at higherUg.In practice,this is similar to the uniform gas aeration at column inlet under some ideal conditions.In this case,very uniform small bubbles can be generated through a tailor-made needle sparger,and gas holdup can be kept much higher [35].Although the GNE(Nsurfvs.Nturb)gives similar single-bubble structure to SDM at each givenUg,the GNE can predict the jump change and therefore the regime transition,which is more reasonable than SDM in this respect.Despite the single-bubble structure at a givenUg,dSjump with increasingUg,the GNE may to some extent reflect the two-bubble structure within the whole spectrum of flow regime.Small bubbles appear at lowerUg,and becomes a larger one at a critical velocity.

As mentioned earlier,the dominant mechanisms may also be physically differentiated through the movement tendencies of gas and liquid,in addition to the bubble morphology including small bubbles and large bubbles.As a preliminary attempt,this could be formulated as a GNE between three dominant mechanisms.Fig.8 shows that compared to the GNE between two dominant mechanisms,this formulation can indeed reproduce the twobubble-class structure.The volume fraction of large bubblesfLrecovers to a normal value,as shown in Fig.8(d).All the three types of stability conditions can predict the regime transition,but the critical velocities for the transition are different,as shown in Fig.9.For the GNE(Nsurfvs.Nturb),only small bubbles exist at lowerUg,the system with more small bubbles are more stable and delay the regime transition.But the GNE applies too much constraint to the two players,or produces a system of involution,and therefore enlarges the difference of small bubbles and large bubbles by the predefined objective function.In other words,the GNE (Nsurfvs.Nturb) intensifies the objective of each player.In the three-players game,however,the calculation indicates that the GNE (Nsurfvs.Nturbvs.Wst/(1-εg)ρl) contains the preference of each player,though the presence of the third dominant mechanism relevant to movement tendencies apparently complicates the game.The total gas holdup of the GNE (Nsurfvs.Nturbvs.Wst/(1-εg)ρl) lies in the mid of the other methods due to the recovery offL,but the order of the jump change is still in turn HDM,GNE(Nsurfvs.Nturb),GNE (Nsurfvs.Nturbvs.Wst/(1-εg)ρl) from the left to the right,reflecting the extent of complexity of the game.

Fig.10.Relationship of solutions with different stability conditions.

It should be noted that the model calculation in Fig.9 does not quite agree with the experimental data at higherUg,as the DBS model,as a zero-dimensional conceptual model,does not consider the effects of reactor diameter,distributor and height.When the DBS model is integrated into CFD simulation through the drag force,the prediction is close to the experimental data,as given by Xiaoet al.[28].

Finally,the relationship of these different stability conditions is summarized in Fig.10.The rectangular light blue area represents the solution space of the three conservation equations.The large green circle represents the noninferior solutions set,the remaining small circles represent the solutions of SDM,HDM and GNE respectively.It is clear that the solutions of SDM,HDM falls into the noninferior solutions set.Actually,SDM and HDM represent the special cases of noninferior solution sets with different weights(0 or 1 for SDM,and 0.5 for HDM).However,the solution of noncooperative game theory is not included in the noninferior solutions set.This is because noncooperative game and multi-objective optimization are two different concepts.Players in noncooperative games pursue optimal strategy and reach equilibrium under certain rules,and the result may be not optimal or relatively optimal,the prisoner’s dilemma is a classic case.The game assumes that small bubbles correspond toNsurf→min and large bubbles corresponds toNturb→min is added to the noncooperative game,although there is evidence for this assumption.The characteristics of all stability conditions in this paper are summarized in Table 4.

5.Conclusions

This study explores the formulation of different stability conditions for gas-liquid flow in bubble columns.First,the dominant mechanisms of morphology are studied.Single dominant mechanism cannot give reasonable results,but it reveals the correspondence between small bubbles andNsurf,large bubbles andNturb,which supports the strategy of the player’s objective function in noncooperative game theory.The GNE (Nsurfvs.Nturb) can predict the flow regime transition like the HDM.However,the twobubble structure degenerates and reduces to the single-bubble structure.With the increase of gas velocity,there is a jump change in the distribution of noninferior solutions in the solution space,which reflect the flow regime transition.Then,we combine the dominant mechanisms of movement tendencies of different phases and the morphology of the dispersed phase.The GNE(Nsurfvs.Nturbvs.Wst/(1-εg)ρl)can also predict the flow regime transition.Meanwhile,the addition of the dominant mechanism of movement tendencies enhances the existence of large bubbles,the single-bubble structure reverts to the two-bubble structure.In fact,we have also explored using only the dominant mechanisms of the movement tendencies of different phases as the stability condition,the twobubble structure cannot be distinguished and the flow regime transition cannot be identified,which reflects the complexity of the gas-liquid system.The dominant mechanisms of gas-liquid system are then investigated by considering both the movement tendencies of different phases and the morphology of the dispersed phase.Only when the three-players game for reflecting such relationship is established,can the dominant mechanisms of movement tendencies start to play a role and can give reasonable results;otherwise,the original HDM without considering the movement tendencies can also give reasonable prediction.It should be pointed out that this study does not intend to recommend a better stability condition for the real-world flow systems.Instead,we try to explore the path to formulate the stability condition,which may help to understand the roles and interactions of different dominant mechanisms in the origin of complexity in multiphase flow systems.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The authors acknowledge the long-term support from National Natural Science Foundation of China (21925805,22178354,91834303) and the ‘‘Transformational Technologies for Clean Energy and Demonstration” Strategic Priority Research Program of the Chinese Academy of Sciences Grant No.XDA21000000.We thank Yifen Mu and Zhixin Liu of Academy of Mathematics and Systems Science,Chinese Academy of Sciences for helpful discussion on game theory.