Xiao Hu,Jianzhong Lin,Dongmei Chen,Xiaoke Ku

Department of Mechanics,State Key Laboratory of Fluid Power and Mechatronic Systems,Zhejiang University,Hangzhou 310027,China

Keywords: Particle non-Newtonian fluids Inertial migration Channel flow Numerical simulation

ABSTRACT The formation of self-organizing single-line particle train in a channel flow of a power-law fluid is studied using the lattice Boltzmann method with power-law index 0.6 ≤n ≤1.2,particle volume concentration 0.8% ≤Φ ≤6.4%,Reynolds number 10 ≤Re ≤100,and blockage ratio 0.2 ≤k ≤0.4.The numerical method is validated by comparing the present results with the previous ones.The effect n,Φ, Re and k on the interparticle spacing and parallelism of particle train is discussed.The results showed that the randomly distributed particles would migrate towards the vicinity of the equilibrium position and form the ordered particle train in the power-law fluid.The equilibrium position of particles is closer to the channel centerline in the shear-thickening fluid than that in the Newtonian fluid and shear-thinning fluid.The particles are not perfectly parallel in the equilibrium position,hence IH is used to describe the inclination of the line linking the equilibrium position of each particle.When self-organizing single-line particle train is formed,the particle train has a better parallelism and hence benefit for particle focusing in the shearthickening fluid at high Φ,low Re and small k.Meanwhile,the interparticle spacing is the largest and hence benefit for particle separation in the shear-thinning fluid at low Φ,low Re and small k.

1.Introduction

Precise control the position of particles in a particle-laden flow is important in chemical engineering such as particle sorting,separation,focusing and sintering [1–3].Conventionally,the external force (e.g.,electric,magnetic and optical fields) or sheath flow are introduced to control the particle position,which will increase the cost and complicate the fabrication process.The inertial fluidic device is a young but growing field of active research to fast focus the randomly distributed particles towards the equilibrium positions and form the ordered particle train without external force,and some conclusions in the Newtonian fluid have been drawn.For example,the neutrally buoyant particles in a circular channel of Newtonian fluid would migrate towards a radial equilibrium position and form the Segré-Silberberg annulus [4]; the migration of initially random distributed particles towards equilibrium positions was a combination of shear gradient and wall effect lift,then the particle trains were formed simultaneously in the flow direction; the rigid particles and cells formed the ordered single-line particle trains and staggered particle trains simultaneously[5];most of the trains started with a large particle and ended with a small one,and the interparticle spacing was decreased under higherRe[6]; the preferred interparticle spacing of single-line particle train was around 5 times of particle diameter at lowRe[7],but was 3.7–7 times of particle diameter under different traveled distances [8],the interparticle spacing decreased with increasingRe[7],and was almost constant in different downstream positions[9];the particle trains were unstable whenRe>105 orRe< 21[10];the single-line particle train would shift a larger spacing for further downstream position,while the spacing of staggered particle train was stable whenRewas below a critical value [11,12]; the stable single-line particle trains were formed by limited number of particles,the leading particle would leave the train in the downstream position [13]; the percentage of particles in trains reached a maximum value asReincreased,and then decreased [14].

The studies mentioned above have been focused on the particle migration in the Newtonian fluid.However,many fluids in practical applications are the non-Newtonian fluid [15]with shear-thinning,shear-thickening,and viscoelasticity properties which would affect the particle migration,equilibrium position and characteristics of particle train.Del Giudiceet al.[16]experimentally found that the particles in a square channel of viscoelastic liquid with strongly shear-thinning effect were driven from the edge to the center when the inertia effect can be ignored,and the shear-thinning effect could separate particles and cells with different shapes and sizes.Xianget al.[17]indicated that the particles suspended in viscoelastic fluids with constant viscosity would focus on a single stream,and the particle focusing was significantly promoted with increasing particle concentration.Del Giudiceet al.[18]showed a single-line particle train in a square channel of weak shear-thinning viscoelastic fluid under ignorable inertial effect,and found that the particle train was not observed in a near-constant viscosity fluid,indicating that the shearthinning effect was responsible for the particle train.Li and Xuan[19]observed that the particles were focused on the multiequilibrium positions along the long wall of a channel in xanthan solution with a strong shear-thinning effect,and they realized a potentially high efficient separation of particles by size.D’Avinoet al.[20]found that the interparticle spacing of three aligned particles in strong shear-thinning viscoelastic liquids would increase to a stable value,and the particles flowed like isolated objects.For the multi-particles system,D’Avinoet al.[21]found that a particle train aligned at the channel centerline of viscoelastic liquid with shear- thinning property when the inertia effect can be ignored,and the ordered particle trains and strings of nearlytouched particles were formed with increasing particle concentration.Nieet al.[22]indicated that three particles in the shear flow of a power-law fluid formed a particle train with equal spacing,regardless of power-law index.Firouzniaet al.[23]found that the shear-thinning property would affect the interactions and trajectories of the particles.

As summarized above and as far as our knowledge,the study on the equilibrium position of particles and formation of ordered particle train with considering the inertial effect in the power- law fluid has not been reported.Therefore,the aim of this study is to assess the effect of power-law index,particle concentration,Reynolds number and blockage ratio on the interparticle spacing and parallelism of particle train.

2.Numerical Method

2.1.Lattice Boltzmann scheme

Lattice Boltzmann method(LBM)is an efficient direct numerical simulation method where the fluid is replaced by the fractious particles.LBM can recover the continuity and momentum equation via Chapman-Enskog expression[24]and is widely used while dealing with the complex geometry and multiphase flow.In the present study,the two-dimensional D2Q9 single-relaxation-time LBGK model with external force is used for its efficiency and high precision [25]:

wherefi(x,t)is the distribution function at position x and timet;Δtis the unit time step;τ is the relaxation time;(r,t)is the equilibrium distribution function[24];wiis a weight factor withw0=4/9,w1,˙˙˙,4=1/9,andw5,˙˙˙,8=1/36;eiis the discrete velocity vectors;Fpis the external force [26],f is the driving force;csis the speed of sound;cis the lattice speed.

The fluid density and velocity are determined by:

Laddet al.[27],Aidunet al.[28]used the momentum exchange method and a modified bounce-back rule to deal with a moving solid particle,the particle boundary node on the links connecting the intra-particle and extra-particle nodes was given by:

wherei′andiare the reflected and incident directions,respectively;t+is the post-collision time;Bi=3ρωi/c2;ub=u0+Ω×xband u0is the translational velocity of the solid particle,Ω is the angular velocity; xb=x + ei/2 - x0with x0being the position of the solid particle.

The hydrodynamic force and torque exerted by fluid at xbare given by:

The added force and torque when the solid particle covered or uncovered the fluid region is:

Therefore,the total force and torque on the solid particle are given by:

Then the Newton’s second law is used to update the motion of the particle.

2.2.Power-law fluid

The viscosity of a power-law fluid can be calculated by:

wheremandnare the power-law consistency and the power-law index,respectively;n=1,n< 1,andn> 1 correspond to the Newtonian,shear-thinning and shear-thickening fluids,respectively;|| can be expressed as:

The fluid velocityuand v in Eq.(13) is determined with Eqs.(2)–(4) in the LBM.The fourth-order finite-difference scheme is used to calculate the local derivative of the fluid velocity in the main stream region:

In the boundary region,the second-order finite-difference scheme is used:

The instantaneous and local relaxation time for all lattices is τ=3μ/(ρc2Δt) + 0.5 which appears in Eq.(1).

2.3.Lubrication force model

In order to avoid the unphysical contact between the particles and between particle and walls,the lubrication force is used when the spacing of two particle surfaces or the distance between particle surface to wall is less than 2Δx[29]:

whereCmis the characteristic forceCm=MU2/awithM,Uandabeing the particle mass,velocity and radius,respectively;ε=10-4is a constant value;dand erare the spacing and direction from the center of one particle to another particle or to the wall,respectively;dmin=2aand Δr=2Δx(Δx=1 is the unit lattice space)stand for the distance less than which the lubrication force will work.

2.4.Problem definition

Migration of the neutrally buoyant circular particles in a twodimensional channel is shown in Fig.1 where the particles are randomly distributed below the centerline,so that only one inertial equilibrium position is accessible.The periodic boundary condition is applied to the flow direction,and no-slip boundary condition is applied on the channel walls.The particles are released when the channel flow is fully developed.The blockage ratiokis defined ask=D/H.The Reynolds number is defined asRe=ρU02-nHn/mwhere ρ andU0are the fluid density and the mean velocity at the inlet,and the channel height isH=140Δx.The migrations of single and two particles are simulated when the length of periodic boundarylpis changed from 2500Δxto 3500Δx(i.e.,17.9H–25H).The numerical results are insensitive tolpwhenlpis changed from 3000Δxto 3500Δx,solp=3000Δxis adopted.For multi-particle system,a large number of particles are realized by increasing the number of particles in the computational domain without changing the computational box size.

Fig.2.Trajectory of particle in a power-law fluid.

3.Validation

3.1.Particle trajectory in a shear flow

In order to verify the feasibility of the method,we simulated the trajectory of a particle near a rotated particle fixed on the centerline in a confined shear flow of power-law fluid withn=0.75,1.0 and 1.25,resprespectively. The numerical results are shown in Fig. 2 where the previous numerical results simulated with different different method by Nie et al. [22] also are presented. The excellent agreements demonstrate that the present numerical method is reliable.

3.2.Inertial migration of single particle

The numerical results of particle inertial migration in a Newtonian fluid with blockage ratiok=0.25,0.35 are shown in Fig.3 where the numerical results simulated with different method[30]also are given.We can see that no matter where the particle is initially released,it always migrates toward the same equilibrium position,which is consistent with the Segré–Silberberg effect[4].

3.3.Comparison with experimental results

Further validation is conducted by comparing our numerical results of single-line particle train with experimental result [7]as shown in Fig.4 whereuxis the velocity inxdirection.The probability density function (PDF) of particle spacing isdp/D=4.4 ± 1.2 for the experimental result[7]as shown in Fig.4(a),the theoretical prediction of particle spacing isdp/D=4.17[31],and the mean particle spacing of our numerical result isdp/D=4.05 as shown in Fig.4(b) and 4.04 as shown in Fig.4(c),respectively,under the same conditions.

Fig.3.Migration and equilibrium position of particle.

4.Results and Discussion

In order to clearly describe the formation process of a singleline particle train,the particles in the train located from right to left are numbered asP1,P2,˙˙˙; and the axial spacings between two neighboring particles are also numbered asd1-2,d2-3,˙˙˙from right to left,respectively.When the leading particle catches up with the lagging particle at the first time,the spacing is calculated by(xlag+L-xlead)/D,wherexlagandxleadare the traveled distance from the channel inlet.

4.1.Formation process of a single-line particle train

The characteristics of a self-organizing single-line particle train withn=1.0,Re=32,Φ=4.0% andk=0.33 are shown in Fig.5 where the colors of curves in Fig.5(a) and (b) correspond to the colors of particles in Fig.5(c).We can see from Fig.5(a)and(c)that the randomly distributed particles first migrate towards the vicinity of the equilibrium position aroundx/H=60,and then start to form an ordered train under the particle–particle ineraction and inertial effect.As shown in Fig.5(a),particles are not perfectly parallel in the equilibrium position.Therefore,a dimensionless parameter IH is used to describe the inclination of the line linking the equilibrium position of each particle(IH=(Hlead-Hlag)/H,whereHleadandHlagare the positions inydirection of leading and lagging particles,respectively).The higher the IH value,the higher the inclination,and the worse parallelism.The leading particle migrates towards the equilibrium position firstly,and the second particle will not migrate towards the equilibrium position untild1-2is large enough and the influence of the leading particle is ignorable.The interparticle spacing increases along the flow direction.The new particle train is self-organized when particle 1 (P1)and 3(P3)catch up with particle 10(P10)as shown in Fig.5(c)with red dotted box.The formation of a single-line particle train is a dynamic process where some particles downstream leave the train and new particles come in simultaneously,and the formed particle train will be kept for a long traveled distance.Fig.5(d) shows the changes of IH and interparticle spacing along the flow direction.It can be seen that the values of IH remain uncahnged forx/H≤300,and decrease obviously with increasingx/Hat 300 ≤x/H≤750 when the particle at the most downstream leaves the equilibrium position.Forx/H≥750,a new particle train is selforganized and the values of IH remain uncahnged again.For the interparticle spacing,the values increase first slowly forx/H≤400,then quickly at 400 ≤x/H≤650,and finally remain uncahnged forx/H≥650.The interparticle spacing and IH remain constant downstream.

4.2.Effect of power-law index n

In order to explore the effect of power-law index on the formation of particle train withRe=32,Φ=4.0%andk=0.33,the characteristics of a particle train in the power-law fluid withn=0.6,0.8 and 1.2 are shown in Fig.6 where the new formed particle train is marked with a red dotted square.We can see that the particles migrate towards the equilibrium position firstly,i.e.,the Segré–Silberberg effect is also valid for the migration of multi-particles in a power-law fluid,and then a single-line particle train is formed in the vicinity of walls.As shown in Fig.6(a),(c),(e),the distance from the equilibrium position(red dotted line)to the channel centerline is reduced with the increase ofn,i.e.,the equilibrium position of particles is closer to the channel centerline in the shearthickening fluid than that in the Newtonian fluid and shearthinning fluid.The particles upstream migrate laterally towards the equilibrium position and the interparticle spacing increases along the flow direction forn=0.6.However,forn=0.8 and 1.2,the particles upstream first move a distance on the horizontal line and then migrate quickly towards the equilibrium position,and the same trend is also observed for the interparticle spacing.

The effect ofnon the inclination IH at differentx/His shown in Fig.7.Although the values of IH decrease along the flow direction,the dependence of IH onnis different at upstream and downstream.At a more upstream location(x/H≤300)where the particle train has not been formed completely,the values of IH decrease with increasingn.However,at the location downstream (x/H=800) where a stable particle train has formed,the values of IH first increase and then decrease with increasingn,and the values of IH are the largest for Newtonian fluid(n=1)and the least for shear-thickening fluid,respectively,which indictaes that the selforganizing single-line particle train has a better parallelism in the shear-thickening fluid.This conclusion has reference value for particle focusing.

Fig.8 shows the effect ofnon the interparticle spacingdp/Dat differentx/H.The interparticle spacing increases along the flow direction,and increases withnat upstream (x/H≤150),but first decrease and then increase withnatx/H≥200.At the location downstream (x/H=800),the interparticle spacing is the largest for the stonger shear-thinning fluid (n=0.6) and the least for the Newtonian fluid (n=1),respectively.This conclusion can provide a reference for the separation of particles.

Particle concentration Φ is an important factor affecting the formation of a single-line particle train.In the computation,we take the particle numberNas 2,6,10,14 and 16(corresponding Φ=N-×π×(D×0.5)2/(L×H)%=0.8%,2.4%,4.0%,5.6% and 6.4%),respectively,in the same computation domain.The effects ofnon the interparticle spacingdp/Dand inclination IH for a pair of particles are shown in Fig.9.The interparticle spacing increases first quickly and then slowly along the flow direction,and increases with decreasingn.Withinx/H=200–700,the growth rate of interparticle spacing is 25.6% for shear-thinning fluid and 11.0% for shearthickening fluid,respectively.The values of IH first increase quickly,then decrease,and finally tend to zero along the flow direction.

The effect ofnon the interparticle spacingdp/Dfor 6 particles is shown in Fig.10.We can see that the values ofdp/Dfirst show oscillation and then tend to a stable value along the flow direction.The larger the power-law index,the faster the values ofdp/Dapproach to a stable value,and the more concentrated the stable values.

Fig.4.Comparison of single-line particle train between the experimental and numerical results (Re=30, k=0.34).(a) experimental result [7],(b) numerical result with 10 particles,(c) numerical result with 16 particles.

Fig.5.Characteristics of self-organizing single-line particle train: (a) trajectories; (b) interparticle spacing; (c) typical distribution of particle train,(d) changes of IH and interparticle spacing along the flow direction.

Fig.6.Characteristics of particle train for different n: (a) interparticle spacing and trajectory, n=0.6; (b) distribution of particle train, n=0.6; (c) interparticle spacing and trajectory, n=0.8; (d) distribution of particle train, n=0.8; (e) interparticle spacing and trajectory, n=1.2; (f) distribution of particle train, n=1.2.

The distributions of 6 particles for differentnalong the flow direction are shown in Fig.11,and corresponding interparticle spacingdp/Dand IH are shown in Fig.12,which is different from the case for a pair of particles as shown in Fig.9.The interparticle spacing increases continuously along the flow direction for shearthinning fluid (n=0.6),but first increases and then tends to be a sable value for Newtonian fluid and shear-thickening fluid.In contrast,the values of IH decrease continuously along the flow direction for shear-thinning fluid,but first decrease and then tend to be a sable value for Newtonian fluid and shear-thickening fluid.The large IH corresponds to the small interparticle spacing.

Fig.7.Effect of n on the values of IH.

Fig.8.Effect of n on the interparticle spacing.

Fig.9.Effect of n on (a) interparticle spacing and (b) the values of IH (N=2, Re=32, k=0.33).

Fig.10.Effect of n on the interparticle spacing: (a) n=0.6,(b) n=1.0,(c) n=1.2 (N=6, Re=32, k=0.33).

Fig.11.Distribution of particle train (N=6, Re=32, k=0.33).

Fig.12.Effect of n on dp/D and IH.

4.3.Effect of particle concentration Φ

The characteristics of a self-organizing single-line particle train consisting of 14 and 16 are shown in Figs.13–17,respectively.The phenomenon of particle train becomes obvious with the increase of the particl number.The particles upstream will keep a relatively stable interparticle spacing for a long traveled distance.The effect ofnon the interparticle spacingdp/Dand IH is different for different particle number.For the case of 14 particles,the interparticle spacing reaches quickly a stable value along the flow direction for shear-thinning fluid as shown in Fig.13.Comparing Fig.14 with 13,we can see that the interparticle spacing reaches a stable value faster,along the flow direction,for the case of 16 particles than that of 14 particles.In other words,the particle train with equal interparticle spacing is formed earlier in the shear-thinning fluid at higher particle concentration Φ.Besides,a staggered particle pair due to the interaction between particles under low inertial effect can be observed in the shear-thickening fluid as shown in Fig.15,and the interparticle spacing of staggered particle pair is stable and smaller than that of the single-line particle train.

Fig.13.Effect of n on the interparticle spacing: (a) n=0.6,(b) n=1.0,(c) n=1.2 (N=14, Re=32, k=0.33).

Fig.14.Effect of n on the interparticle spacing: (a) n=0.6,(b) n=1.0,(c) n=1.2 (N=16, Re=32, k=0.33).

Fig.15.Distribution of particle train (N=14 and 16, Re=32, k=0.33).

Fig.16.Effect of Φ on IH.

Fig.17.Effect of Φ on the interparticle spacing: (a) n=0.6,(b) n=1.0,(c) n=1.2 (N=16, Re=32, k=0.33).

The effect of Φ on the inclination IH for differentnatx/H=150 is shown in Fig.16 where the values ofIHfirst increases rapidly when Φ is increased from 0.8% (N=2) to 4.0% (N=10),and then start to decrease.Therefore,the self-organizing single-line particle train has a better parallelism when the particle concentration(particle number) is increased under the condition ofN> 10,which is consistent with the result in viscoelastic fluids with constant viscosity [17].

For the case of 16 particles,the effect of Φ on the interparticle spacing for differentnat differentx/His shown in Fig.17.It can be seen that the interparticle spacing decreases with increasing Φ for differentn,although the decreasing rate is the largest for shear-thinning fluid,then followed for Newtonian fluid and finally for shear-thickening fluid.Along the flow direction,the interparticle spacing increases rapidly at lower Φ,especially for shearthinning fluid; while slowly at higher Φ,especially for Newtonian fluid and shear-thickening fluid.

4.4.Effect of Reynolds number

The distributions of 10 particles withk=0.33 for differentnandReatx/H=200 are shown in Fig.18,and corresponding IH is shown in Fig.19.It can be seen that,as the Reynolds number increases,the values of IH do not change much and keep relatively small values for Newtonian fluid and shear-thickening fluid,but increase significantly and reach relatively large values for shear- thinning fluid.This means that the self-organizing single-line particle train has a better parallelism in a shear-thickening fluid even at high Reynolds numbers.

Fig.18.Distribution of particle train (N=10, k=0.33).

Fig.19.Effect of Re on IH.

Changes of interparticle spacing along the flow direction at higher particle concentration (16 particles) and higherReare shown in Fig.20.For shear-thinning fluid(n=0.6),the interparticle spacing of all particles reaches a stable value along the flow direction after a period of oscillation upstream.For shear-thickening fluid(n=1.2),the interparticle spacing of some particles first oscillates for a long traveled distance and then reaches a stable value.The change of interparticle spacing for Newtonian fluid is similar to that for shear-thickening fluid,but the amplitude of oscillation is smaller,which is consistent with the result in reference [6,7].

The effect ofReon the interparticle spacingdp/Dfor differentnis shown in Fig.21 where solid and dotted lines represent the case of 10 and 16 particles,respectively.There are obvious difference in the values ofdp/Dbetween the case of 10 and 16 particles.In shearthinning fluid,as the Reynolds number increases,the interparticle spacing first decreases and then increases for the case of 10 particles,while continuously decreases for the case of 16 particles.In Newtonian fluid and shear-thickening fluid,the interparticle spacing continuously decreases with increasingRe.The interparticle spacing increases along the flow direction for all cases,and reaches a stable value atRe=100 in shear-thickening fluid.

4.5.Effect of blockage ratio k

The distributions of 10 particles for differentnandkatx/H=200 are shown in Fig.22,and corresponding IH is shown in Fig.23 where the values of IH increase with increasingk,are larger in shear-thinning fluid than that in Newtonian fluid and shearthickening fluid.The self- organizing single-line particle train has a better parallelism in a shear-thickening fluid with smaller blockage ratio.

The effect ofkon the interparticle spacingdp/Dfor differentnis shown in Fig.24.The interparticle spacing decreases with increasingk,which is due to the confinement of the wall.The larger the blocking ratio,the stronger the wall effect.Along the flow direction,the interparticle spacing increases a lot in shear-thinning fluid,while increases a little in shear- thickening fluid.

Fig.20.Changes of interparticle spacing along the flow direction: (a) n=0.6,(b) n=1.0,(c) n=1.2 (N=16, Re=100, k=0.33).

Fig.21.Effect of Re on the interparticle spacing: (a) n=0.6,(b) n=1.0,(c) n=1.2.(k=0.33).

Fig.22.Distribution of particle train (N=10, Re=32).

Fig.23.Effect of Re on IH.

5.Conclusions

The formation of self-organizing single-line particle train in a channel flow of a power-law fluid is studied using the lattice Boltzmann method.The numerical method is validated by comparing the present results with the previous ones.The effect of powerlaw index,particle concentration,Reynolds number and blockage ratio on the interparticle spacing and parallelism of particle train is discussed.The main conclusions are summarized as follow:

The randomly distributed particles would migrate towards the vicinity of the equilibrium position and form the ordered particle train in the power-law fluid.The equilibrium position of particles is closer to the channel centerline in the shear-thickening fluid than that in the Newtonian fluid and shear-thinning fluid.The particles are not perfectly parallel in the equilibrium position,hence IH is used to describe the inclination of the line linking the equilibrium position of each particle.When self-organizing single-line particle train is formed,the particle train has a better parallelism and hence benefit for particle focusing in the shear-thickening fluid at high particle concentration,low Reynolds number and small blockage ratio.Meanwhile,the interparticle spacing is the largest and hence benefit for particle separation in the shear-thinning fluid at low particle concentration,low Reynolds number and small blockage ratio.

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.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (91852102 and 11632016).

Fig.24.Effect of Re on the interparticle spacing: (a) n=0.6,(b) n=1.0,(c) n=1.2.(N=10, Re=32).

Nomenclature

Cmcharacteristic force,N·m-3

c,cslattice velocity and sound speed,m·s-1

Dparticle diameter,m

d,erthe spacing and direction from the one particle to another particle or to the wall

eidiscrete velocities in directioni,m·s-1

Fc,Futhe added force due to the covered and uncovered fluid region,N·m-3

Fhhydrodynamic force,N·m-3

Fpsplit force,kg·m-3·s-1

Frlubrication force,N·m-3

f body force,N·m-3

fidensity distribution function in directioni,kg·m-3

fieqequilibrium distribution function in directioni,kg·m-3

H,Lchannel height and length,m

idirectioniin a lattice

i′reflected of directioniin a lattice

kblockage ratio

mpower-law consistency

npower-law index

ReReynolds number

Tc,Tuthe added torque due to the covered and uncovered fluid region,N·m

Thhydrodynamic torque,N·m

t+post-collision time,s

Δtunit time step,s

U0maximum velocity,m·s-1

u fluid velocity,m·s-1

ubparticle boundary velocity,m·s-1

u0translational velocity of the particle,m·s-1

ui,j,ui+1,j,ui+2,j,ui-1,j,ui,j+1,ui,j+2,ui,j-1fluid velocity in fluid region,m·s-1

?u/?x,?u/?ylocal derivative of the velocity inxandydirection

wiweight factor in directioni

xb,x0particle boundary and mass center position,m

Δxunit lattice space,m

ε constant value

μ dynamic viscosity,Pa·s

ρ fluid density,kg·m-3

τ dimensionless relaxation time

Φ particle concentration

Ω particle angular velocity,rad·s-1