Lift forces on a cylindrical particle in plane Poiseuille flow of shear thinning fluids

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1 Lit orces on a cylindrical article in lane Poiseuille low o shear thinning luids J. Wang and D. D. Joseh Deartment o Aerosace Engineering and Mechanics, University o Minnesota, Aril 3 Abstract Lit orces on a cylindrical article in lane Poiseuille low o shear thinning luids are investigated by direct numerical simulation. Previous works on this toic or Newtonian luids show that the D channel can be divided into alternating regions deined by the stability o the article s equilibrium. We observe stability regions with the same attern in lows o shear thinning luids and study the eects o shear thinning roerties on the distribution o the stability regions. Joseh and Ocando, [J. Fluid Mech. 454, 63 ()] analyzed the role o the sli velocity U s =U -U and the angular sli velocity 9 s =9-9 on migration and lit in lane Poiseuille low o Newtonian luids. They concluded that the discreancy 9 s -9 se, where 9 se is the angular sli velocity at equilibrium, changes sign across the equilibrium osition. In this aer we veriy that this conclusion holds in shear thinning luids. Correlations or lit orces may be constructed by analogy with the classical lit ormula L= CU / o aerodynamics and the roer analogs o U and / in the resent context are U s and 9 s -9 se. Using dimensionless arameters, the correlation is a ower law near the wall and a linear relation (which can be taken as a ower law with the ower o one) near the centerline. The correlations are comared to analytical exressions or lit orces in the literature and we believe that the correlations cature the essence o the mechanism o the lit orce. Our correlations or lit orces can be made comletely exlicit rovided that the correlations relating U s and 9 s to rescribed arameters are obtained. 1

2 I. Introduction Dierent analytical exressions or the lit orce on a article in a shear low can be ound in the literature. Auton 1 gave a ormula or the lit on a article in an inviscid luid in which uniorm motion is erturbed by a weak shear. Bretherton ound an exression or the lit er unit length on a cylinder in an unbounded linear shear low at small values o Reynolds number. Saman 3 gave an exression or the lit on a shere in an unbounded linear shear low. Saman s equation is in the orm o the sli velocity multilied by a actor, which can be identiied as a density multilied by a circulation as in the amous ormula H U/ or aerodynamic lit. A number o ormulas like Saman s exist and a review o such ormulas can be ound in McLaughlin 4. Formulas like Saman s cannot exlain the exeriments by Segrè and Silberberg 5,6. They studied the migration o dilute susensions o neutrally buoyant sheres in ie lows and ound the articles migrate away rom both the wall and the centerline and accumulate at a radial osition o about.6 times the ie radius. There is nothing in ormulas like Saman s to account or the migration reversal near.6 o the radius. The eect o the curvature o the undisturbed velocity roile was ound to be imortant to understand the Segrè and Silberberg eect. Ho and Leal 7 analyzed the motion o a neutrally buoyant article in both simle shear lows and lane Poiseuille lows. They ound that or simle shear low, the equilibrium osition is the centerline; whereas or Poiseuille low, it is.6 o the channel hal-width rom the centerline which is in good agreement with Segrè and Silberberg 5,6. Choi and Joseh 8, Patankar, Huang, Ko and Joseh 9 and Joseh and Ocando 1 studied article lit in lane Poiseuille lows by direct numerical simulation. They showed that multile equilibrium states exist or heavy articles in lane Poiseuille lows. These

3 equilibrium states can be stable or unstable and the distinction leads to division o the channel into alternating stability regions in the ollowing order: wall stable unstable stable unstable centerline (see Fig. ). Joseh and Ocando 1 analyzed the role o the sli velocity and the angular sli velocity on migration and lit. The sli velocity is U s =U -U where U is the translational velocity o the article and U is the luid velocity. The angular sli velocity is deined as Ω 9 9 Ω C /, where ( C / ) is the angular velocity o the luid at a oint s where the shear rate is C and 9 is the angular velocity o the article. Both U and 9 are evaluated at the location o the article center in the undisturbed low (without the article). Joseh and Ocando showed that the discreancy 9 s -9 se, where 9 se is the angular sli velocity at equilibrium, is the quantity that changes sign across the equilibrium osition. Thus, this discreancy can be used to account or the migration rom both the wall and the centerline to the equilibrium osition. Power law correlations are requently observed in studies o solid-liquid lows. A amous examle is the Richardson-Zaki correlation 11, which is obtained by rocessing the data o luidization exeriments. The Richardson-Zaki correlation describes the comlicated dynamics o luidization by drag and is widely used or modeling the drag orce on articles in solid-liquid mixtures. Correlations can also be drawn rom numerical data; or examle, ower law correlations or single article lit and or the bed exansion o many articles in slurries were obtained by rocessing simulation data (Patankar et al. 9 ; Choi and Joseh 8 ; Patankar, Ko, Choi, and Joseh 1 ). The rediction o ower laws rom numerical data suggests that the same tye correlations could be obtained rom exerimental data as was done by Patankar, Joseh, Wang, Barree, Conway and Asadi 13 3

4 and Wang, Joseh, Patankar, Conway and Barree 14. The existence o such ower laws is an exression o sel-similarity, which has not yet been redicted rom analysis or hysics. The low o disersed matter aears to obey those sel-similar rules to a large degree (Barenblatt 15 ). Most o studies on migration and lit are or Newtonian luids. However, in many o the alications the luids used are not Newtonian and shear thinning is one o the most imortant non-newtonian roerties. Paers treating migration o articles in shear lows o shear thinning luids were done by Huang, Feng, Hu and Joseh 16, Huang, Hu and Joseh 17 and esecially by Huang and Joseh 18. The numerical methods used by these authors are used in this work and will not be described here. Suice to say that the method is based on unstructured body-itted moving grids (ALE method). All these authors use the Carreau-Bird viscosity unction (1) but only Huang and Joseh 18 study the case when there is shear thinning but no normal stresses. In the resent aer, we extend revious studies o lit on a single cylindrical article in lane Poiseuille lows o Newtonian luids to shear thinning luids. We show that the attern o the stability regions in shear thinning luids is the same as that in Newtonian luids. The eects o shear thinning on the distribution o the stability regions are discussed. We veriy that the angular sli velocity discreancy changes sign across the equilibrium osition or both neutrally buoyant and heavy articles. We derive ower law correlations or the lit orce in terms o the sli velocity and angular sli velocity discreancy and demonstrate that these correlations can be made comletely exlicit. II. Governing equations and numerical methods The D comutational domain is shown in Fig. 1. l and W are the length and width o the channel resectively, and d is the diameter o the article. The simulation is 4

5 erormed with a eriodic boundary condition in the x-direction. The geometric arameters are W/d = 1 and l/d =. The values o these arameters are taken rom Patankar at al. 9 where they justiied that the channel length l is suiciently large so that the solutions are essentially indeendent o l. Channel length l Figure 1:The D rectangular comutational domain. A constant ressure gradient gives rise to the Poiseuille low and the direction o the gravity orce is erendicular to the low direction. In simulations in eriodic domains the luid ressure P is slit as ollows: P H g x e x x P H g e x where e x is the unit vector in x-direction, x is the osition vector o any oint in the domain and g is the gravitational acceleration. is eriodic and solved in simulations. We use the Carreau-Bird model or the shear thinning eects: D D D D 3 n1 [1 ( C ) ] (1) where C is the shear rate deined in terms o the second invariant o the rate o strain tensor D. The shear thinning index n is in the range o 1 and D, D, 3 are rescribed 5

6 arameters. We consider cylindrical articles o diameter d with the mass er unit length m = H Fd /4 and the moment o inertia er unit length I = H Fd 4 /3. A dimensionless descrition o the governing equations can be constructed by introducing scales: the article size d or length, V or velocity, d/v or time, V/d or angular velocity and V / D d or stress and ressure. We choose V W /( 1D ), which is the average velocity o the undisturbed Poiseuille low in Newtonian luids. V can be related to the shear rate at the wall C w W /( D ) : V W C w C wd. () 1 Hat variables are dimensionless in the ollowing art. The dimensionless governing equations are ˆ uˆ u ˆ, ˆ ˆ ˆ ˆ d ˆ ˆ ( ˆ ˆ ˆ T R u u e ˆ x 3 u u ) tˆ W (3) or the velocity û and ressure ˆ o the luid and H H duˆ R dtˆ Ge y d W e x 4 3 ˆ ˆ T ˆ 1 ( uˆ uˆ ) nd/ ˆ, (4) F H H d ˆ R dtˆ Ω ˆ 3 ˆ ˆ T xˆ X ˆ 1 ( uˆ uˆ ) 3 n d/ ˆ F (5) or the velocity Û and angular velocity coordinate Xˆ. In equations (3) (5) we use H Vd R, G D ( H H ) gd Ωˆ o the article whose center o mass has the n1 D D and 3 (1 )1 ( ˆ 3C w ) C DV D D. The no-sli condition is imosed on the article boundaries: 6

7 Following is a list o the dimensionless arameters: uˆ Uˆ Ωˆ xˆ X ˆ. (6) H / H, density ratio; W/d, asect ratio; D /D, viscosity ratio; ), shear rate arameter; ( 3C w n, shear thinning index; R H D Vd H C wd D H Wd D, Reynolds number; ( H H ) gd d ( H H ) g G, gravity arameter. D V W 3 H ( H H ) gd Instead o G, we use the gravity Reynolds number RG R G. D W/d =1 and D /D =.1 are constant throughout our simulations; the arameters or luid roerties and d are also constant and lead to =R in our simulations, so does not rovide more inormation. Thus H / H, R, n and R G are the our dimensionless arameters at lay. The Reynolds number R and shear thinning index n together, characterize an undisturbed Poiseuille low. We deine an average Reynolds number R = H u d/d where u is the average velocity o the undisturbed Poiseuille low. In table 1, we list the average Reynolds numbers R or lows characterized by (n, R) airs. R increases signiicantly with n decreasing at a ixed R. 7

8 n R R Table 1: Average Reynolds numbers R or lows characterized by (n, R) airs. III. Undisturbed low We reer to Poiseuille low without articles as undisturbed low. The dimensionless momentum equation in the x-direction or the undisturbed low is d d duˆ ( 3 ). (7) W dyˆ dyˆ An analytical solution or the Poiseuille low o a Carreau-Bird luid is not known. However, a numerical solution can be achieved by an iterative method. First C ( yˆ ) is assumed to be the shear rate o the Poiseuille low o a Newtonian luid and ˆ 3 ( C ( yˆ )) is obtained. A new shear rate roile C ˆ1 ( yˆ ) is then comuted and the stes are reeated until C ˆ( y ˆ) converges. The velocity u ˆ( yˆ ) is obtained by integrating the shear rate. The velocity roiles o the Poiseuille lows o shear thinning luids are qualitatively similar to the arabolic roiles seen in lows o Newtonian luids. However, the maximum velocity in the channel increases signiicantly as n decreases at a ixed R. The viscosity roiles have their minimums at the wall (corresonding to the maximum C ), and their maximums at the centerline (corresonding to zero C ). ˆ 8

9 IV. Stable and unstable equilibrium regions: An equilibrium is achieved or a reely moving and rotating cylindrical article with a given density in a Poiseuille low when the article migrates to a osition y e o steady rectilinear motion in which the acceleration and angular acceleration vanish and the hydrodynamic lit just balances the buoyant weight. Two tyes o simulations are erormed, unconstrained simulation and constrained simulation. In unconstrained simulations, a article is allowed to move and rotate reely to migrate to its equilibrium osition. The initial translational and angular velocities o the article are rescribed and initial-value roblems are solved to obtain the equilibrium state. In constrained simulations, the osition o the article in the y-direction y is ixed and the article is allowed to move in x-direction and rotate. The solution o the low evolves dynamically to a steady state at which the lit orce er unit length L on the article is comuted. Such a steady state will be an equilibrium at y=y i the density o the article is selected so that L just balances the buoyant weight er unit length, satisying: de Lˆ H L gfd H / 4 H 1 (8) where Lˆ is a dimensionless lit orce and reresents the ratio between the hydrodynamic lit orce L and the buoyant orce H gfd /4. From the steady state values which evolve in constrained simulations, we are able to obtain Lˆ on the article at any osition y/d in the channel. We can divide the curve o Lˆ vs. y/d rom the wall to the centerline into our branches by three turning oints (see Fig. ). The turning oint is deined as the osition where the sloe o the Lˆ vs. y/d curve is zero. On the irst and third branches o steady solutions, the sloe o Lˆ vs. y/d 9

10 curve is negative, and the equilibrium oints on these branches are stable. On the second and ourth branches o steady solutions, the sloe o Lˆ vs. y/d curve is ositive, and the equilibrium oints are unstable. We will indicate the unstable branches by dotted lines in the igures..5.4 Stable branch #1 L.3..1 Unstable branch # Turning oint # Stable branch # H / H =1.1 Unstable branch # Equilibrium oint #1 Turining oint #1 Equilibrium oint # Turning oint #3 -.1 Wall 1 3 y/d Centerline 7 Figure. A lot o Lˆ vs. y/d or a low with n=.8 and R = rom constrained simulations. The stable and unstable branches and three turning oints are illustrated. Unstable branches are indicated by dotted lines. Two stable equilibrium oints or a article with H / ρ = 1.1 are shown. From the Lˆ vs. y/d curve, the equilibrium osition or a article with a certain H can be determined. The lit orce required to balance the buoyant weight o a article can be comuted rom (8). I we draw a line on which Lˆ equals to this required lit orce, the oints o intersections between this line and the Lˆ vs. y/d curve are the equilibrium oints or this article. For heavier-than-luid articles with intermediate densities, there exist multile stable equilibrium ositions rom the wall to the centerline (see Fig. 1

11 where two stable equilibrium oints or a article with H / ρ = 1.1 are shown). However, or a neutrally buoyant article ( Lˆ = ), only one stable equilibrium oint exists rom the wall to the centerline. Ho and Leal 7 studied the equilibrium osition o a neutrally buoyant reely moving and rotating shere between lane bounding walls. They assumed that the walls were so closely saced that the lit could be obtained by erturbing Stokes low with inertia. They calculated dimensionless lateral orce vs. lateral osition curves (equivalent to our Lˆ vs. y/d curve) or simle shear low and D Poiseuille low which are shown in Fig. 3. Comaring the dashed line in Fig. 3 which is or D Poiseuille low and the Lˆ vs. y/d curve in Fig., one can see that both o the two lots imly the centerline is an unstable equilibrium osition. However, the dashed line in Fig. 3 indicates that there are two branches rom the wall to the centerline: wall stable unstable centerline, whereas our branches exist according to Fig.. Ho and Leal only considered neutrally buoyant article and did not include the gravity term in the governing equation used in their calculation. The rame o their work did not enable them to study the multi-equilibrium ositions o heavier-than-luid articles. The results shown in Figs. and 3 are not strictly comarable; Ho and Leal studied 3D sheres between lane walls at indeinitely small R whereas our calculation is or D cylinders at much higher Reynolds numbers. The distribution o the equilibrium branches is aected by the shear thinning eects. The Lˆ vs. y/d curves are comuted or the lows with R =, 4 and 8 and n=.7,.8,.9 and 1. (Newtonian luid). Two grous o tyical curves are lotted in Figs. 4 and 5. 11

12 F* L /ρ V* m a κ Figure 3.Lateral orce as a unction o lateral osition, both in dimensionless orm. -, simle shear low; - - -, D Poiseuille low. (Adated rom Ho and Leal 7 ) n=.7 n=.8 n=.9 n=1. L y/d.5 3 Figure 4. Near-the-wall art o Lˆ vs. y/d curves o the Poiseuille lows with R = and n=.7,.8,.9 and 1. (Newtonian luid). The unstable branches are indicated by dotted lines and their starting and ending oints are marked by airs o short vertical lines. 1

13 With the shear index n decreasing, the stable branch near the wall decreases in size and the unstable branch near the wall moves closer to the wall n=.7 n=.8 n=.9 n=1 L y/d Figure 5. Near-the-centerline art o Lˆ vs. y/d curves o the Poiseuille lows with R = 8 and n=.7,.8,.9 and 1. (Newtonian luid). The unstable branches are indicated by dotted lines and short vertical lines are used to mark the starting oints o these unstable branches. With the shear index n decreasing, the unstable branch near the centerline decreases in size. We ind that when the shear thinning eects become stronger, the stable branch near the wall decreases in size; the unstable branch near the wall moves closer to the wall; the stable branch near the centerline increases in size; the unstable branch at the centerline decreases in size. The shrinkage o the unstable branch at the centerline imlies that a article could be lited to a equilibrium osition closer to the centerline i shear thinning eects are stronger. A closer equilibrium osition to the centerline could also be achieved when the ressure gradient is higher, as shown irst in Patankar et al. 9 and conirmed in our simulations. It seems that a higher ressure gradient and stronger shear thinning both 13

14 lead to stronger inertia eects and could lit a article closer to the centerline. In the range o the Reynolds number and shear thinning index we simulated, the unstable branch at the centerline never vanishes. Patankar et al. 9 reorted that in D Poiseuille lows o an Oldroyd-B luid at high Deborah numbers, the centerline can be a stable equilibrium osition and the Segrè and Silberberg eect does not occur. We did not observe the same henomenon in shear thinning luids. V. Angular sli velocity discreancy and net lit orce: Joseh and Ocando 1 studied sli velocities and article lit in D Poiseuille lows o Newtonian luids. The sli velocity is U s =U -U and the angular sli velocity is Ω s Ω Ω, where U and 9 = C / are the translational velocity and angular velocity o the undisturbed Poiseuille low at the osition o the article and C is the local shear rate. The net lit orce is: H L n = L ( H ) d H F g / 4 ˆ ˆ L n L ( 1). (9) H Joseh and Ocando ound that the angular sli velocity discreancy 9 s -9 se, where 9 se is the angular sli velocity at equilibrium, changes sign across the equilibrium osition. Furthermore, they showed that across a stable equilibrium osition, the net lit orce L n has the same sign as the discreancy 9 s -9 se ; whereas across an unstable equilibrium osition, the net lit orce L n has the oosite sign as the discreancy 9 s -9 se. In this section, we veriy that these conclusions hold in shear thinning luids. We ix a article at ositions slightly above (y > y e ) and below (y < y e ) its equilibrium ositions and comute the steady state lit orce and angular sli velocity 9 s. For a neutrally buoyant article, both stable and unstable equilibrium ositions are 14

15 investigated; or a heavy article, both o its two stable equilibrium ositions are investigated. Table shows the results or a neutrally buoyant article and table 3 shows those or a heavy article. y e /d se /( C ) w ixed y /d L/(H gfd /4) (9 s -9 se )/( C ) w Table. The steady state values o L and 9 s -9 se in dimensionless orm at ixed ositions slightly above (y > y e ) and below (y < y e ) the equilibrium ositions o a neutrally buoyant article in the low with n=.7 and R=. The stable equilibrium osition is y e /d=4.35 with 9 se /( C w ) = For the article ixed below (y /d = 4.33), 9 s - 9 se > and L>; or the article ixed above (y /d= 4.36), 9 s -9 se < and L<. The unstable equilibrium osition is the centerline with y e /d=6. and 9 se /( C w ) =. For the article ixed below (y /d = 5.95), 9 s -9 se > but L<; or the article ixed above (y /d=6.5), 9 s -9 se < but L>. y e /d se /( C ) w ixed y /d L n /(H gfd /4) (9 s -9 se )/( C ) w Table 3. The steady state values o the net lit orce L n and 9 s -9 se in dimensionless orm at ixed ositions above (y > y e ) and below (y < y e ) the equilibrium ositions o a heavy article (H /H =1.4) in the low with n=.9 and R=4. Two stable equilibrium ositions exist: y e /d=.918 with 9 se /( C w ) = and y e /d=.6 with 9 se /( C w ) = For either one o the equilibrium ositions, 9 s -9 se > and L n > when the article is 15

16 ixed below; 9 s -9 se < and L n < when the article is ixed above. Table and 3 veriy the conclusions about the discreancy 9 s -9 se, summarized as ollowing: 9 s -9 se < when y > y e ; 9 s -9 se > when y < y e. With a stable equilibrium as the reerence state, negative 9 s -9 se leads to negative L n, ositive 9 s -9 se leads to ositive L n ; with an unstable equilibrium osition as the reerence state, negative 9 s -9 se leads to ositive L n, ositive 9 s -9 se leads to negative L n. (L n =L in the case o a neutrally buoyant article.) These conclusions are or the steady state values o the lit orce and sli velocity and do not hold generally or a moving article with accelerations. VI. Lit correlations Motivated by the conclusion that 9 s -9 se has the same sign as L n across a stable equilibrium osition, we seek the correlations between L n and 9 s -9 se. Such correlations may be constructed by analogy with the classical lit ormula L= CU / o aerodynamics. The roer analogs o U and / in the resent context are U s and 9 s -9 se as irst rosed in Joseh and Ocando 1. We roceed as ollows to obtain the correlations. First we comute L, U s and 9 s as unctions o y by constrained simulations in a low characterized by (R, n). Then we correlate dimensionless arameters based on L and U s (9 s -9 se ) to ower law ormulas. These stes are reeated or dierent lows identiied by (R, n) airs and lead to correlations or each low. The coeicients in such correlations are unctions o R and n which can be obtained by data itting analyses. Finally we obtain correlations between dimensionless L and U s (9 s -9 se ) with coeicients exressed as unctions o R and n. Figure 6 shows the relative values o L, U s and 9 s obtained rom constraint simulations in the low with R= and n=.9. 16

17 9 s / 9 smax,u s /U smax,l/l max sli velocity angular sli velocity lit orce y/d Figure 6. The relative values o L, U s and 9 s in the low with R= and n=.9. Dimensionless arameters based on local quantities are used to exress the correlations. The local dimensionless net orce is: 4H d L( y) ( H H ) gfd / 4 4H d ( y) L n (y). (1) FD( y) FD( y) Two local Reynolds numbers are based on U s and 9 s -9 se resectively: R U H U s ( y) d ( y), D( y) H [ 9s ( y) 9se ] d R9 ( y). (11) D( y) The roduct o R U and R9 is deined as F: 3 H U s ( y) 9s ( y) 9se d F( y) RU R9. (1) D( y) To comute F(y) rom (1), it is necessary to seciy the equilibrium angular sli velocity 9 se =9 s (y e ) where y e is the osition at which the lit equals the buoyant weight. The Lˆ vs. y/d curve (Fig. ) shows that each and every value o y/d on the stable branches is a ossible equilibrium osition (y=y e ) or some article H. The range o ossible y e may be covered by varying the density o the article. Once y e is selected, 9 se is given as 9 s (y e ). The deendence o 9 se and L n on H makes the correlations between 17

18 (y) and F(y) article-density deendent. However, the steady state values o L do not deend on article density. I we derive the correlations between (y) and F(y) or one H, the lit orce is essentially obtained and can be alied to articles with dierent densities. We resent the correlations with the single equilibrium osition o a neutrally buoyant article as the reerence. There are two advantages o this choice: the comlexity o multi-equilibrium ositions o a heavy article is avoided; the correlations are in simle orms which are a ower law or the stable branch near the wall and a linear relation or the stable branch near the centerline. For a neutrally buoyant article, a single equilibrium osition exists at N y y e (the suerscrit is or neutral ) with ( N N N L y e ) and 9 s ( ye ) 9 se. Thus the dimensionless arameters have the ollowing orm: 4H dl( y) ( y) and FD( y) N 3 H U s ( y) 9s ( y) 9se d F( y). D( y) The correlations are in the ollowing orms, m( R, n) ( R, n, y / d) a( R, n) F( R, n, y / d) on the stable branch near the wall; (13) ( R, n, y / d) k( R, n) F( R, n, y / d) on the stable branch near the centerline. (14) We obtain the correlations or lows with n=.7,.8,.9 and 1. (Newtonian luid). In Fig. 7, the correlations on the stable branch near the wall are lotted or the lows with R=. The ower law correlations along with the correlation coeicients I are shown in the igure. In Fig. 8, two examles o the linear correlation between (y) and F(y) on the stable branch near the centerline are lotted or the lows with (R=, n=.7) and (R=8, n=.8). It can be seen that our correlations describe the data aithully. 18

19 the stable branch near the wall 1 = 37.3F.439 I = 1 1 = 8.49F.46 I =.9965 (y) = 1.589F.44 I = = F.43 I =.9988 n=1(newtonian) n=.9 n=.8 n= F (y) Figure 7. The ower law correlations between (y) and F(y) on the stable branch near the wall or the lows with R= and n=.7,.8,.9 and 1 (Newtonian luid). y) the stable branch near the centerline = F I =.978 = 8.879F I =.999 R=, n=.7 R=8, n= F (y) 19

20 Figure 8. The linear correlation between (y) and F(y) on the stable branch near the centerline or the lows with (R=, n=.7) and (R=8, n=.8). The reactor a, the exonent m and the sloe k in (13) and (14) are unctions o R and n. In table 4, the coeicients a, k and m are listed along with R, n, and the average Reynolds number R which can be viewed roughly as a arameter or the combined eects o R and n. Coeicients a, m and k are also lotted against R in Figs n R R a m k Table 4.The reactor a, the exonent m and the sloe k as unctions o the shear index n and the Reynolds number R. 1 a 1 regime 1 regime R Figure 9. The reactor a vs. the average Reynolds number R.

21 m R regime1 regime Figure 1. The exonent m vs. the average Reynolds number R. 1 k R Figure 11. The sloe k vs. the average Reynolds number R. Figures 9 and 1 reveal that the ower law correlation (13) on the stable branch near the wall has two regimes. Flows o Newtonian luids and weak shear thinning lows all into regime1 where the reactor a increases with R increasing and the exonent m is in the range o.4.5. Regime has three lows (n=.7, R=4), (n=.7, R=8) and (n=.8, R=8) and can be identiied as a strong shear thinning regime where the reactor a decreases with R increasing and the exonent m is in the range o From the values o the exonent m, we can tell that in regime the deendence o the lit orce on 1

22 the roduct o sli velocities is stronger than that in regime1. It is noted that the two lows (n=1., R=16) and (n=.8, R=8) have very close values o R but substantially dierent coeicients a, m and k (see table 4); this indicates that article lit in strong shear thinning lows is dierent with that in lows o Newtonian luids at high Reynolds number. Figure 11 exhibits one regime o the linear correlation (14) where the sloe k decreases with R increasing. Figures 9-11 also suggest that ower law or linear unctions o R could be used to aroximate the reactor a and the exonent m in regime1 and the sloe k. However, the error o such aroximations would be considerable. The reason o such error is that a, k, and m deend on both n and R; one single arameter R cannot ully describe the deendence o the coeicients on the low. We cannot ully determine the coeicients a, m and k as unctions o R and n because o insuicient data. I we ocus on lows o Newtonian luids (n=1), R is the only active arameter and we exect to get satisactory a(r), k(r) and m(r) aroximations by data itting analyses. The coeicients a, k, and m in lows o Newtonian luids are listed as unctions o R in table 5. R a m k Table 5.The reactor a, the exonent m and the sloe k as unctions o the Reynolds number R or lows o Newtonian luids. Data are consistent with those in table 4. Data itting analyses yield:.48 a 5.34R, I =.94; (15) m.7r.386, I =.99; (16)

23 .515 k 3.5R, I =.96. (17) Inserting (15) (17) into the correlations (13) and (14), we obtain correlations which aly to lows o Newtonian luids with a Reynolds number in the range o 16. λ( y) 5.34R λ( y) 3.5R.48 F( y).515 (.7R.386) F( y) on the stable branch near the wall; on the stable branch near the centerline. (18) (19) Relacing (y) and F(y) in (18) and (19) with their dimensional orms and re-arrange, we obtain the equations in the ollowing orm.7r.48.14r.7.14r 1.7 N L 4.R ρ η U (Ω Ω ) s s se on the stable branch near the wall; N L 18.6R ρ U (Ω Ω )d s s se on the stable branch near the centerline. Note that or Newtonian luids, D(y) reduces to D..386 d.1r.159 () (1) Although correlations () and (1) are derived using the equilibrium o a neutrally buoyant article as the reerence, they can be alied to heavy articles. To demonstrate this, we irst obtain U s and 9 s or heavy articles at their equilibrium states rom unconstrained simulations; these values are then inserted into () and (1) to calculate the lit orces which should match the values o the buoyant weight o the heavy articles. Two examles are shown in table 6: a article with H /H =1.16 in a low with R =4 and a article with H /H =1.45 in a low with R =8. In both cases two stable equilibrium ositions exist. The lit orce or y e close to the wall is comuted using () and the lit orce or y e close to the centerline is comuted using (1). It can be seen that the comuted dimensionless lit orces are close to the values o the dimensionless buoyant 3

24 weight (H /H -1) o the articles. In this way we demonstrate that the correlations derived or neutrally buoyant articles can be alied to heavy articles. N R 9 se / ( C w ) H /H -1 y e /d 9 s /( C w ) U s /( C w d) Lˆ Table 6. Comutation o the lit orces on heavy articles using the correlations () and (1). The comuted dimensionless lit orces are close to the values o the dimensionless buoyant weight (H /H 1) o the articles. Correlations () and (1) aly to D motion o a article in a Poiseuille low. They may be comared to well-known lit exressions or a article in a linear shear low with shear rate C. The comarisons are at best tentative because the linear shear neglects the eects o the shear gradient which is a constant in the Poiseuille low and not small; also because the lit exressions in linear shear lows are or indeinitely small Reynolds number erturbing Stokes low on an unbounded domain. Bretherton ound that the lit er unit length on a cylinder at small values o R H C d / D is given by L (.679 ln( 1.16DU R / 4)) s. ().634 Saman 3 derived an exression or the lit on a shere in a linear shear low L 6.46H D U C a lower order terms (3) s where a is the radius o the shere. For a neutrally buoyant article at equilibrium, L = and rom () and (3), U s =. The Bretherton and Saman ormulas thus redict that the sli velocity is zero or a neutrally buoyant article at equilibrium in an unbounded linear shear low. Patankar et 4

25 al. 9 argued that zero sli velocity is always one solution or a neutrally buoyant article reely moving in an unbounded linear shear low, but it may not be the only solution and it can be unstable under certain conditions not yet understood. Feng, Hu and Joseh 19 showed that a neutrally buoyant article migrates to the centerline in a Couette low where U s =. From our simulations or D Poiseuille lows, U s at the equilibrium osition o a neutrally buoyant article (see Fig. 6); whereas 9 s = 9 se at equilibrium gives rise to zero lit. We ind that our exression or the lit on the stable branch near the centerline (1) is similar to the leading term in Saman s exression (3). I we make ollowing changes in equation (1): R H Vd /D R H Cd / D, the ower o R (-.515) (-.5), and use d = a, equation (1) becomes: N L 365.H D U C ( 9 9 a. (4) s s se ) Comaring (4) and the leading term in (3), we note that both exressions are linear in U s ; both have a similar deendence on H, D, and a ater noting that (4) is or the lit orce er unit length. However, the deendence on C and 9 is greatly dierent. N s 9 se Another ormula or the lit on a article in an inviscid luid in which uniorm motion is erturbed by a weak shear was derived by Auton 1 and a more recent satisying derivation o the same result was given by Drew and Passman. In a lane low they ind L Fa HU F H C s9 a U s, (5) 3 3 which is similar to our correlation (1) but diers rom (1) in several ways: (5) has a constant reactor or inviscid luids whereas viscous eects enter into (1) through R; 5

26 the lit orce deends on 9 - sin o the luid in (5) but on the angular velocity discreancy 9 in (1); (5) is or 3D sheres and (1) is or D cylinders. N s 9 se We comare the lit orces comuted rom the direct numerical simulation and rom the lit exressions (1), (), (3) and (5) in igure 1. Our correlation (1) and Bretherton s exression () are or D cylinders and the dimensionless lit Lˆ is ˆ comuted as L L /( H gfd / 4) ; Saman and Auton s exressions (3) and (5) are or ˆ sheres and Lˆ is comuted as L L /( H g Fa ). The sli velocity U s, which is a unctional o the solution, is rescribed in Bretherton, Saman and Auton s exressions and undetermined in their theories. To calculate the lit orces rom these exressions, we use the values o U s obtained rom our DNS. The values o U s, 9 s and N 9 se obtained rom the DNS are used in the calculation o (1) DNS Correlation (1) Saman Bretherton Auton L y/d Figure 1. A comarison o the lit orces comuted rom the direct numerical simulation and rom the lit exressions (1), (), (3) and (5). The lit orces on the stable branch near the centerline in a low o Newtonian luid with R=8 are lotted. We draw the readers attention to the act that the lit exressions (1), (), (3) and (5) 6

27 aly to dierent scenarios and are not strictly comarable. Our correlation (1) is or a reely rotating D cylinder without accelerations in a lane Poiseuille low. Bretherton s exression () and Saman s exression (3) are both or the lit on a article in an unbounded linear shear low with an indeinitely small Reynolds number; the dierence is that the ormer alies to a non-rotating D cylinder while the latter alies to a rotating 3D shere. Auton s exression (5) alies to a ixed 3D shere in an inviscid luid in which uniorm motion is erturbed by a weak shear. Exressions (), (3) and (5) cannot redict the change o sign across the equilibrium osition; whereas our correlation (1) reroduces the DNS results aithully. Our correlations rovide exlicit exressions or the lit orce on a article in terms o the sli velocity U s and the angular sli velocity discreancy 9 s - 9 se. We emhasize that the relative angular motion is characterized by 9 s - 9 se rather than 9 s or 9. By using the discreancy, we are able to account or the Segrè and Silberberg eect. Our correlations cover the whole channel excet the unstable regions. We believe that our correlations cature the essence o the mechanism o the lit orce. Correlations () and (1) are derived or the steady state values o L, U s and 9 s, i.e., they aly to articles with zero acceleration. For a migrating article, correlations () and (1) are not valid, although they might give good aroximations when the acceleration o the article is small. The alication o such correlations is to determine arameters o a article at equilibrium, e.g., the equilibrium osition, translational velocity and angular velocity. For this end, correlations which relate U s and 9 s to rescribed arameters are needed. We will show derivation o such correlations is easible in the next section. 7

28 VII. Correlations or sli velocity and angular sli velocity To make correlations () and (1) comletely exlicit, we need correlations which relate U s and 9 s to R and y/d in steady lows o Newtonian luids. We illustrate the rocedure or 9 s. In Fig. 13, the steady state values o 9 s /( C w ) obtained in constrained simulations are lotted against y/d or ive values o R. I these data are lotted on a loglog lot o 9 s /( C w ) versus R, we obtain straight lines one or each value o y/d rom the wall to the centerline (ive o which are shown in Fig. 14), leading to ower law correlations: Ω y d R C s ( /, ) r ( y / d ) w b( y / d) R RD Ω ( y / d, R) b( y / d) R. (6) r ( y / d ) s H d 9 s/(cw) R= R=4 R=8 R=1 R= Wall y/d Centerline Figure 13. The steady state values o the dimensionless angular sli velocity 9 s /( C w ) in lows o Newtonian luids as a unction o y/d. The reactor b and exonent r in these ower law correlations, which are unctions o y/d, are lotted in Fig. 15. With more data oints, these unctions could be itted to 8

29 slines, making (6) comletely exlicit..1 9 s /( C w ).1 y/d=1. y/d=. y/d=3. y/d=4. y/d= Figure 14. Power law correlations between 9 s /( C w ) and R at ive values o y/d. R b(y/d) Wall y/d Centerline r(y/d) Wall y/d Centerline Figure 15. The reactor b and exonent r in correlation (6) as unctions o y/d. A similar rocedure or U s leads to U s ( y / d, R) q( y / d ) C d w c( y / d) R q y d RD U s ( y / d, R ) c ( y / d ) R ( / ). (7) H d As or b and r in (6), c and q could be it to slines i more data oints were available. Unlike correlation (6) which can be ound at values o y/d rom the wall to the centerline, correlation (7) can only be ound at values o y/d on stable branches o steady solutions. It does not correlate well with the data or the unstable branches; in act or some values o R, U s is slightly negative at some values o y/d on the unstable branch near the wall, which is incomatible with a ower law in the orm (7). 9

30 In addition to (6) and (7), we also need a correlation between N Ω se, the angular sli velocity o a neutrally buoyant article at equilibrium, and R, in order to make () and (1) comletely exlicit. Table 7 shows that Ω N se /( C ) is essentially constant w indeendent o R. Using the average o these values, we obtain: Ω se R C N ( ) 3 w 5.11 RD 9 ( R) (8) N 3 se H d R Ω /( C ) N se w Table 7. The dimensionless angular sli velocity o a neutrally buoyant article at equilibrium is essentially a constant in lows o Newtonian luids with R= 16. I we now insert (6) - (8) into () and (1), we obtain comletely exlicit (assuming suicient data oints or b, r, c and q to be it to slines) correlations or the lit orce:.7 R.386 y y ( ) ( ) y q y r R 3 D d d L 4.R c( ) R [ b( ) R 5.11 ] d d H d on the stable branch near the wall; (9) y y ( ) ( ) y q y r 3 D d d L 18.6R c( ) R [ b( ) R 5.11 ] d d H d on the stable branch near the centerline. (3) These ormulas allow us to calculate L or any value o y/d on the stable branches o the Lˆ vs. y/d curve (Fig. ), obviating the need or urther numerical simulations. The equilibrium osition y e /d o a article o density H can be ound as the value o y/d at which the lit orce equals the buoyant weight: L( y e Fd / d, R) ( H H ) g ; 4 3

31 the sli velocities at equilibrium can then be calculated by inserting y e /d into (6) and (7): Ω se RD r ( ye / d ) Ωs ( ye / d, R) b( ye / d) R ; H d U se q y d RD U s ( ye / d, R ) e c ( ye / d ) R ( / ). H d The corresonding translational velocity U and angular velocity 9 o the article at equilibrium may then be calculated as U = U (y e ) U se and 9 = 9 se - C y ) /. ( e VIII. Conclusions We study liting o a cylindrical article in lane Poiseuille lows o shear thinning luids. It is known that certain regions in a channel are unstable and a article cannot equilibrate in an unstable region. For examle, Ho and Leal 7 ointed out that the centerline is an unstable equilibrium osition in a D Poiseuille low. Our studies show that the domain rom the wall to the centerline in a D Poiseuille low can be divided into our regions with the ollowing order: wall stable unstable stable unstable centerline. The distribution o these regions is aected by shear thinning. Our results show that when shear thinning eects become stronger, the unstable region at the centerline shrinks, indicating that the equilibrium osition o a article could be closer to the centerline. The conclusion that the angular sli velocity discreancy 9 s - 9 se changes sign across an equilibrium osition established by Joseh and Ocando 1 in Newtonian luids is conirmed in shear thinning luids. Across a stable equilibrium osition, 9 s - 9 se has the 31

32 same sign as the net lit orce L n ; across an unstable equilibrium osition, 9 s - 9 se has the oosite sign as the net lit orce L n. Correlations or the lit orce on a article in terms o the sli velocity U s and the angular sli velocity discreancy 9 s - 9 se are derived. The correlations are a ower law near the wall and a linear relation (which can be taken as a ower law with the ower o one) near the centerline. The correlations aly to both neutrally buoyant and heavy articles and cover the whole channel excet the unstable regions. Two regimes, one with no or weak shear thinning eects and the other with strong shear thinning eects, are identiied or the ower law correlation (13) whereas only one regime is ound or the linear correlation (14). It is noted that article lit in strong shear thinning lows is dierent with that in lows o Newtonian luids at high Reynolds number. We are able to obtain correlations between L and U s (9 s - 9 se ) with coeicients exressed as unctions o R; these correlations cover the lows o Newtonian luids with the Reynolds number in the range o The correlation is comared to well known analytical exressions or lit orce in shear lows and similarities between them are revealed. The major dierence between them is that the angular sli velocity discreancy 9 s - 9 se is used in our correlations instead o the shear rate or 9 s. We also demonstrate that correlations which relate U s and 9 s to rescribed arameters can be constructed and will make the correlations or L comletely exlicit. Thus the lit orce in steady lows can be calculated using correlations at any value o y/d on stable branches rom the rescribed arameters; the equilibrium osition o a article with a certain density can then be determined by the balance between the lit orce and its buoyant weight. 3

33 Acknowledgement This work was artially suorted by the National Science Foundation KDI/New Comutational Challenge grant (NSF/CTS ); by the US Army, Mathematics; by the DOE, Deartment o Basic Energy Sciences; by a grant rom the Schlumberger oundation; rom STIM-LAB Inc.; and by the Minnesota Suercomuter Institute. We would like to thank T. Hesla or his hel in rearing the aer. Reerences: 1. T. R. Auton, The lit orce on a sherical body in a rotational low, J. Fluid Mech. 183, 199 (1987).. F. P. Bretherton, Slow viscous motion round a cylinder in a simle shear, J. Fluid Mech. 1, 591 (196). 3. P. G. Saman, The lit on a small shere in a slow shear low, J. Fluid Mech., 385 (1965); and Corrigendum, J. Fluid Mech. 31, 64 (1968). 4. J. B. McLaughlin, Inertial migration o a small shere in linear shear lows, J. Fluid Mech. 4, 61 (1991). 5. G. Segrè and A. Silberberg, Radial Poiseuille low o susensions, Nature, 189, 9 (1961). 6. G. Segrè and A. Silberberg, Behavior o macroscoic rigid sheres in Poiseuille low: Part I, J. Fluid Mech. 14, 115 (196). 7. B. P. Ho and L. G. Leal, Inertial migration o rigid sheres in two-dimensional unidirectional lows, J. Fluid Mech. 65, 365 (1974). 33

34 8. H. G. Choi and D. D. Joseh, Fluidization by lit o 3 circular articles in lane Poiseuille low by direct numerical simulation, J. Fluid Mech. 438, 11 (1). 9. N. A. Patankar, P. Y. Huang, T. Ko, and D. D. Joseh, Lit-o o a single article in Newtonian and viscoelastic luids by direct numerical simulation, J. Fluid Mech. 438, 67 (1). 1. D. D. Joseh, and D. Ocando, Sli Velocity and Lit, J. Fluid Mech. 454, 63 (). 11. J. F. Richardson and W. N. Zaki, Sedimentation and Fluidization: Part I, Trans. Instn. Chem. Engrs. 3, 35 (1954). 1. N. A. Patankar, T. Ko, H. G. Choi and D. D. Joseh A correlation or the lit-o o many articles in lane Poiseuille o Newtonian luids, J. Fluid Mech. 445, 55 (1). 13. N. A. Patankar, D. D. Joseh, J. Wang, R. D. Barree, M. Conway, and M. Asadi, Power law correlations or sediment transort in ressure driven channel lows, Int. J. Multihase Flow, 8, 169 (). 14. J. Wang, D. D. Joseh, N. A. Patankar, M. Conway, and R. D. Barree Bi-ower law correlations or sediment transort in ressure driven channel lows, Int. J. Multihase Flow, in ress (3). 15. G. I. Barenblatt, Scaling, Sel Similarity and Intermediate Asymtotics, Cambridge Univ. Press (1996). 16. P. Y. Huang, J. Feng, H. H. Hu, and D. D. Joseh, Direct simulation o the motion o solid articles in Couette and Poiseuille lows o viscoelastic luids, J. Fluid Mech. 343, 73 (1997). 34

35 17. P. Y. Huang, H. H. Hu, and D. D. Joseh, Direct simulation o the motion o ellitic articles in Oldroyd-B luids, J. Fluid Mech. 36, 97 (1998). 18. P. Y. Huang and D. D. Joseh, Eects o shear thinning on migration o neutrally buoyant articles in ressure driven low o Newtonian and viscoelastic luids, J. Non-Newtonian Fluid Mech. 9, 159 (). 19. J. Feng, H. H. Hu, and D. D. Joseh, Direct simulation o initial values roblems or the motion o solid bodies in a Newtonian luid. Part : Couette and Poiseuille lows, J. Fluid Mech. 77, 71 (1994).. D. A. Drew and S. Passman Theory o Multicomonent Fluids, Sringer (1999). 35