Slip velocity and lift

Transcription

1 J. Fluid Mech.(), vol.,.-. Cambridge University Press DOI:.7/S75 Printed in the United Kingdom Sli velocity and lift By D. D. JOSEPH AND D. OCANDO Deartment of Aerosace Engineering and Mechanics and the Minnesota Suercomuter Institute, University of Minnesota, Minneaolis, MN 5555, USA (Received 5 November and in revised form 3 August ) The lift force on a circular article in lane Poiseuille flow erendicular to gravity is studied by direct numerical simulation. The angular sli velocity Ω s = Ω + γ, γ is the angular velocity of the fluid at a oint where the shear rate is γ where ( ) and Ω is the angular velocity of the article, is always ositive at an equilibrium osition at which the hydrodynamic lift balances the buoyant weight. The article migrates to its equilibrium osition and adjusts Ω so that Ω s > is nearly zero because of Ω γ. No matter where the article is laced it drifts to an equilibrium osition with a unique, slightly ositive equilibrium angular sli velocity. The angular sli velocity discreancy defined as the difference between the angular sli velocity of a migrating article and the angular sli velocity at its equilibrium osition is ositive below the osition of equilibrium and negative above it. This discreancy is the quantity that changes sign above and below the equilibrium osition for neutrally buoyant articles, and also above and below the lower equilibrium osition for heavy articles. The existence and roerties of unstable ositions of equilibrium due to newly identified turning oint transitions and those near the centreline are discussed. The long article model of Choi & Joseh () that gives rise to an exlicit formula for the article velocity and the velocity rofile across the channel through the centreline of the article is modified to include the effect of the rotation of the article. In view of the simlicity of the model, the exlicit formula for U and the velocity rofile are in surrisingly good agreement with simulation values. The value of the Poiseuille flow velocity at the oint at the article's centre when the article is absent is always larger than the article velocity; the sli velocity is ositive at steady flow.. Introduction The exeriments of Segré & Silberberg (96, 96) have influenced the fluid mechanics studies of migration and lift. They studied the migration of dilute susensions of neutrally buoyant sheres in ie flows at Reynolds numbers between and 7. The articles migrate away from the wall and centreline and accumulate at.6 of a ie radius. The lift on heavier than liquid articles is also influenced by the factors that determine the equilibrium osition of neutrally buoyant articles. The heavy articles must reach an equilibrium that balances the hydrodynamic lift and buoyant weight. If the buoyant weight is very small, the equilibrium osition of the articles will be close to the value for the neutrally buoyant case. The effect of increasing the weight is to lower the equilibrium osition whose zero is established for the case of zero buoyant weight. Analysis of the lift of a neutrally buoyant, heavy and light articles may be framed in a uniform way by looking for the equilibrium ositions of articles evolving to steady flow under the action of shear from the Poiseuille flow as ositions for which

2 D. D. Joseh and D. Ocando the buoyant lift, the lift minus the buoyant weight, vanishes. Loosely seaking, these equilibrium ositions can be regarded as generalized Segré-Silberberg radii. Most attemts to exlain the Segré-Silberberg effects have been based on linearized low-reynolds-number hydrodynamics. Possibly the most famous of these attemts is due to Saffman (965), who found that the lift on a shere in a linear shear flow is given by / L = 6.6η a U s γ / v + lower order terms, (.) where U s = U f U is the sli velocity, U f is the fluid velocity and U is the article velocity, a is the shere radius, ν = η ρ f, η is the fluid viscosity, ρ f is the fluid density and γ is the shear rate. The lower order terms are 3 U s a ρ f [ π Ω s ( π ) γ ], (.) 8 where Ω s = Ω + γ (.3) is the angular sli velocity and Ω is the article angular velocity. Sli velocities are imortant because shear gradients define the local environment against which article translations are measured. There are a number of formulas like Saffman's that are in the form of U s multilied by a factor, which can be identified as a density multilied by a circulation as in the famous formula ρu Γ for aerodynamic lift. A review of such formulae can be found in McLaughlin (99). Formulae like Saffman's cannot exlain Segré-Silberberg's observations, which require migration away from both the wall and the centre. There is nothing in these formulae to account for the migration reversal near.6 of a radius. Moreover the sli velocity U s, the angular sli velocity Ω s, the article velocity and the article angular velocity, which are functionals of the solution are rescribed quantities in these formulae. The fluid motion drives the lift on a free body in shear flow; no external forces or torques are alied. If there is no shear, there is no lift. In Poiseuille flow, there is not only a shear, but a shear gradient. Gradients of shear (curvature) roduce lateral forces. At the centreline of a Poiseuille flow the shear vanishes, but the shear gradient does not. To understand the Segré-Silberberg effect, it is necessary to know that the curvature of the velocity rofile at the centre of Poiseuille flow makes the centre of the channel an unstable osition of equilibrium. A article at the centre of the channel or ie will be driven by shear gradients toward the wall; a article near the wall will lag the fluid and be driven away from the wall. An equilibrium radius away from the centre and wall must exist. Ho & Leal (97) were the first to combine these effects in an analysis of the motion of a neutrally buoyant shere rotating freely between lane walls so closely saced that the inertial lift can be obtained by erturbing Stokes flow with inertia. They treated wall effects by a method of reflection and found an equilibrium osition at.69 from the centre. Vasseur & Cox (976) used another method to treat wall effects and their results are close to Ho & Leal s near the centreline but rather different than those of Ho & Leal near the wall. Feng, Hu & Joseh (99) studied the motion of solid circles in lane Poiseuille flow by direct numerical simulation (DNS). The circle migrates to the.6 of a radius equilibrium osition. They comared their two-dimensional results with those of Ho & Leal and Vasseur & Cox. Schonberg & Hinch (989) analyzed the lift on a neutrally buoyant small shere in a lane Poiseuille flow using matched asymtotic methods. The same roblem for neutrally buoyant and non-neutrally buoyant small shere has been studied using asymtotic method by Asmolov (999). The linearized analysis given so far takes the

3 Sli velocity and lift 3 effect of inertia ( u )u into account only in an Oseen linear system; the comarison of the results of these analyses with exeriments is far from erfect. The analysis is heavy and exlicit formulae for lift are not obtained. Recent three-dimensional numerical studies by Kurose & Komori (999) and Cherukat, McLaughlin & Dandy (999) focus on lift and drag on a stationary shere in a linear (not Poiseuille) shear flow. Issues related to equilibrium ositions do not arise in these studies. The recent DNS study of migration of a single liquid dro in Poiseuille flow by Mortazavi & Tryggvason () is relevant to our work. They carried out mainly two-dimensional simulations and a few three-dimensional studies at R = using coarse grids. Though their methods have not been used for solids, the migration to intermediate equilibrium ositions is a shared feature. Their results showed that the equilibrium osition of the dros is close to the two-dimensional redictions, and the two- and three-dimensional results were similar in other resects. This aer aroaches the roblem of migration and lift in a different way. We have used DNS as a diagnostic tool to analyse the role of the sli velocity and the angular sli velocity on migration and lift. In this way, we are able to establish a simle icture of lift and migration that, in articular, clarifies the role of the angular sli velocity, and is not restricted to low Reynolds numbers. We also use DNS to formulate and validate a long article model that gives a very good, comletely exlicit analytical aroximation to the velocity and sli velocity of circular articles. Our analysis is carried out in two dimensions, but should aly in rincile to three dimensions, which is at resent under study.. Mechanism for lift First, we can look at formulae in a fluid without viscosity for which viscous drag is imossible. The most famous formula for lift on a body of arbitrary shae moving forward with velocity U in a otential flow with circulation Γ was given by L = ρuγ (.) where ρ is the fluid density and L is the lift er unit length. The lift on circular cylinder of radius a is of secial interest. A viscous otential flow solution for a stationary cylinder rotating with angular velocity Ω which satisfies the no sli boundary condition is given by u ( r) = eθ Ωa / r, u( a) = eθ Ωa. (.) The circulation for this viscous otential flow is Γ = u dx = π Ωa. (.3) When this rotating cylinder moves forward it generates a lift L er unit length L = πρa UΩ. (.) The direction of the lift can be determined by noting that the velocity due to rotation adds to the forward motion of the cylinder at the to or bottom of the rotating cylinder according to the directions of Ω and U. In figure, the velocity is smaller on the A referee called our attention to a aer by Zhao, Shar & James (998) on transverse lift on a free circular article in steady flow in a channel. The three main results of our aer, (i) the stability of intermediate equilibrium ositions arising from the dynamic resolution of the initial value roblem, (ii) the long body model, and (iii) the exlanation of the Segré- Silberberg effect by identification of the sli quantity, which changes the sign of the lift across the equilibrium osition, are not touched in their aer. Our results and theirs agree where they overla. DDJ//aers/SliVelocity/SV-Lift-REV.doc

4 D. D. Joseh and D. Ocando Low Pressure U Ω High Pressure FIGURE. The lift er unit length L = πρa UΩ on a cylinder of radius a moving forward at seed U and rotating with angular velocity Ω in such a way as to reduce the velocity at the bottom and add at the to. bottom of the cylinder; by the Bernoulli equation, the ressure is greater there and it ushes the cylinder u. Another formula for the lift off on a article in an inviscid fluid in which uniform motion is erturbed by a weak shear was derived by Auton (987) and a more recent satisfying derivation of the same result was given by Drew & Passman (999). They find that in a lane flow L = πa 3ρ Ω f U s e 3 y (.5) where du Ω f = γ =. dy If d u dy >, the shere is lifted against gravity when the sli velocity U s is ositive; if U s is negative, the shere will fall. Particles which lag the fluid migrate to streamlines with faster flow, articles which lead the fluid migrate to streamlines with slower flow. There are striking differences between (.5) and (.); first, (.) deends on the angular velocity of the article, but (.5) deends on the angular velocity of the fluid. Both formulae, leave the sli velocity undetermined, U s aears in (.5) because of the shear, in (.), U f =. The sli velocities have to be rescribed in these theories because the article velocity is not determined by viscous drag. Similarly, the angular velocity of the article cannot arise from torques arising from viscous shears. The effects of article rotation cannot be obtained by the method of Auton (987). The lift formula (.) catures the essence of the mechanism in which the motion of the article relative to the fluid is such as to increase the ressure on the side of the article as it moves forward. The lift on a sherical or circular neutrally buoyant article in a shear flow is different; there is no exterior agent to move and rotate the freely moving article. Instead the article is imelled forward and rotated by the shear flow. Previous theoretical studies and our simulations show that the relevant velocity is the sli velocity and the relevant circulation is roortional to an angular sli velocity discreancy Γ s Ω s Ω se where Ω se is the angular sli velocity in equilibrium (steady flow) and L = CU s Γ s (.6) where C is a to-be-determined function of fluid and flow arameters. This conclusion will be established in the sequel. For now we simly note that in our simulations the angular sli velocity discreancy Ω Ω < when the cylinder is s se

5 Sli velocity and lift 5 Channel height W Poiseuille flow y x Particle of diameter d du γ = w dy wall FIGURE. Migration and lift of a single article in a lane Poiseuille flow. above the equilibrium (Segré-Silberberg) osition and Ω s Ω se > when it is below the equilibrium (figure ). 3. Governing equations, roblem formulation The simulation method used in this aer is described in Joseh (), Choi & Joseh (), and Patankar et al. (). A circular article of diameter d is laced in a lane Poiseuille flow in a channel of height W where it moves forward in the direction x and migrates u or down in the direction y under the action of hydrodynamic forces (figure ). The undisturbed Poiseuille flow (no article) is given by y y d u u( y) = um, γ =, W W d y. (3.) W W um W um = u =, γ = =. w 8η W η Flows are indexed by a Reynolds number w based on the wall shear rate γ. R = γ d v, v =η ρ w f (3.) 3.. Governing equations The fluid-article system is governed by the Navier-Stokes equations for the fluid and Newton's equations for rigid body motions. The fluid satisfies u ρ f + u u = + u + e x t η, (3.3) where is the constant ressure gradient. The article satisfies Newton's equations of motion: The motion U of the mass centre is governed by du ρ = ( ρ ρ f ) ge y + e x + { + ηd[ u] } n d S, (3.) d t V where V = πa is the volume of the article and ( + ηd[ u] ) n is the stress on the surface S of the article that is a circle of radius = d and π () d = a () a S dθ. (3.5) The angular velocity around the mass centre at x = X is governed by dω I = ( x X ) [( + η D[ u] ) n] d S, (3.6) d t DDJ//aers/SliVelocity/SV-Lift-REV.doc

6 6 D. D. Joseh and D. Ocando where I = ρ πa is the moment of inertia. In steady (equilibrium) flow the forward motion of the article is given by π + e x { + ηd[ u] } er dθ =. (3.7) πa The ressure gradient force on the article is balanced by the resultant of the stress traction vector. The long article model discussed by Joseh (), Choi and Joseh () and here in section is a realization of (3.7). 3.. Dimensionless arameters A dimensionless form of these equations was given in Joseh (), Choi & Joseh (), and Patankar et al. (). The arameters at lay are the channel width diameter ratio W d which is fixed at here, the density ratio ρ ρ f and the gravity Reynolds number R ρ ( ρ ρ ) gd 3 G = f f /η. (3.8) Thus, there are four dimensionless arameters at lay; the density ratio arameter ρ ρ f does not enter when the accelerations vanish, reducing the number of arameters to three. The case W d = used here is reresentative of the asymtotic condition W/d > (Patankar et al. ), only R and R G enter. Here, we comute -3 - for η =. oise, ρ =g cm, g = 98 cm s, d =cm with ρ = or f ρ =.. For these two conditions R = or ρ f G ρ f R G = 9.8 (3.9) We are going to work with dimensional variables in CGS units, but we do secify R. -3 Hence, when we comute for ρ =.g cm, we understand that R G = The results at steady flow do not deend on ρ ρ so that lift in steady flow results with ρ ρ f =. are valid when R G = 9. 8, indeendent of ρ Stable and unstable equilibria It may be helful to frame the roblem in terms of the lift-off of a heavier than liquid article. A qualitative descrition of the levitation to equilibrium article can be framed as follows: the article rests on the bottom wall when there is no flow. As the ressure gradient is increased, the circular article starts to slide and roll. At critical seed, the article lifts and finally rises to a height in which the lift balances the buoyant weight. It then moves forward freely under zero net force and torque; the article does not accelerate. After the article lifts off, it rises to the equilibrium height y e where the lift balances the buoyant weight. The sli velocity U f U and angular sli velocity Ω Ω f = Ω + γ can be comuted during migration and at equilibrium. For certain Reynolds numbers, heavier than liquid articles undergo turning oint transitions leading to multile equilibrium solutions with different values of y e, U e and Ω e (see figure 3). These transitions do not occur for neutrally buoyant articles; the conditions under which turning oint transitions occur will be the subject of a future aer. For the resent study, it is essential to maintain a shar distinction between unstable and stable equilibria. The behavior of articles at unstable equilibria is in a sense the oosite of stable equilibria (c.f. figures and 3) and there are more f

7 Sli velocity and lift 7 5 y e (cm) 3 Stable branch Unstable branch R FIGURE 3. (Patankar et al. ). Equilibrium height as a function of shear Reynolds number -3 for ρ =. g cm. This curve was found first in the dynamic simulations of Choi & Joseh (). The turning oints and the unstable branch were determined by the method of constrained simulation (see ). The solutions are unique for R < 3 and R < ; between, there are two stable and one unstable solutions. L ( dyn cm - ) stable solutions unstable solution BW = dyn cm - R = y (cm) Centreline FIGURE. Family of lift curves at steady state for different values of R. The lift curves can be anti-symmetrically extended into 6 cm < y < cm. The equilibrium oints for R G =. 7 on these lift curves are those satisfying (.) for which the hydrodynamic lift balances the buoyant weight. The locus of such equilibrium oints y e vs. R are reresented in the curve shown in figure 5. subtle differences. We will indicate the unstable equilibria by dotted lines as in figures and 5. Neutrally buoyant articles are also in a buoyant-weight / lift balance, but both are zero. Freely moving neutrally buoyant articles also migrate to an equilibrium (Segrè Silberberg) osition, which is determined by wall effects and shear gradients. It is very imortant to frame the analysis of migration and lift relative to the osition of equilibrium, a oint at which the lift must change its sign. DDJ//aers/SliVelocity/SV-Lift-REV.doc

8 8 D. D. Joseh and D. Ocando stable branch L (dyn cm - ) 5 unstable branch turning oint stable branch 5 equilibrium oint turning oint equilibrium oint turning oint 3 unstable branch y (cm) Centreline FIGURE 5. Lift vs. y for R = and R G = 9. 8 for the steady forward motion of a circular article in Poiseuille flow. The equilibrium oints are the laces where the lift balances the buoyant weight for ρ =. ρ f ; there are two stable oints and one unstable. The icture is anti-symmetric with resect to the centreline and the grah shows that a region near the centre is unstable.. Constrained and unconstrained dynamic simulation In numerical exeriments, we can examine the hysical effects one at a time; this cannot be done in real exeriments. In a constrained simulation, we let the motion of a article evolve to steady state while holding some variable fixed; for examle, we could require that the angular velocity of the article is zero and monitor the migration to its equilibrium osition, or we could fix the osition of the article and see how the article velocity, angular velocity and lift L evolve to steady state. The fixed osition will be an equilibrium when the lift L balances the buoyant weight L = ρ ρ V ; that is, when e ( ) g f Le ρ = ρ f +. (.) V g For other values of ρ, the fixed osition simulation will evolve to a different steady state, indeendent of ρ ρ f multilying acceleration, in which L is greater or less than L e. The method of constrained simulation was used by Patankar et al. () to analyse turning-oint transitions. The results of such simulations are shown in figure 3. We start a article from rest at a fixed y and R and let the simulation run to steady flow. At steady flow, we comute L giving the trilet ( R y, L), ; now varying y at a fixed R we get a curve L vs. y at a fixed R. A family of such curves for different R is shown in figure. The straight line intersecting the curves reresents a fixed value of R G =. 7. There is a value ρ given by (.) for which L balances the buoyant weight; for all other values of ρ, the articles will migrate.

9 Sli velocity and lift 9 FIGURE 6. Rise vs. time for R =5.. Comare rise of freely rotating and non-rotating articles. Non-rotating articles rise more. A neutrally buoyant, freely rotating article rises closer to the centreline at y = 6 cm than the article for which Ω = γ ; the non-rotating article rises even more. Models which ignore article rotation overestimate lift. A yet smaller lift is obtained when the angular sli velocity is entirely suressed ( Ω s = ), but the article does rise. The greater the angular sli velocity, the higher the article will rise. In an unconstrained dynamic simulation, we introduce a circular article with a rescribed article velocity U and angular velocity Ω at a oint y at the article centre into a fully develoed Poiseuille flow, (3.). The article will migrate to an equilibrium osition y e and will attain an equilibrium velocity U e and angular velocity Ω e. It is ossible to consider migration for different initial conditions; we are comuting solutions of initial-value roblems. In the study of lift, steady solutions are imortant because of the absence of comlicating effects of article acceleration. In a different kind of constrained simulation, we look at the effect on article migration of controlling the angular velocity of the article. In figure 6, we lotted the rise to equilibrium of a neutrally buoyant article for three different values of the angular sli velocity Ω s = Ω + γ = { γ, Ω, } se, (.) in an unconstrained dynamic simulation. The rise is the greatest when the article angular velocity Ω = and the least when the article angular velocity is equal to the local rate of rotation Ω = γ. The rise of a heavier than liquid ρ ρ f =. circular article is lotted in figure 6 for Reynolds number R = γ d ν = 5. and for R =6. in figure 7. The angular sli velocity Ω se > is the equilibrium value that a free circular article takes in torque-free motion when the angular acceleration vanishes. We call attention to the fact that Ω se > is very small, and at equilibrium γ > Ω e, Ω e γ. (.3) w DDJ//aers/SliVelocity/SV-Lift-REV.doc

10 D. D. Joseh and D. Ocando Y = y/d W s =g /-W W se =g/ Freely rotating < W se <g / W s = Time (second) FIGURE 7. Rise vs. time for a circular article ρ ρ f =. when R = 6.. The rise of the article is an increasing function of the angular sli velocity in the range Ω s γ and is a maximum when the article angular velocity is suressed. The freely rotating article has a small ositive angular sli velocity. 6 5 y (cm) 3 starting at the centreline starting at the wall 6 8 t (s) FIGURE 8. Migration of a neutrally buoyant article in an unconstrained simulation at R =. 5. Sli velocities, angular sli velocities, and lift for neutrally buoyant circular articles Multile equilibrium solutions do not aear at moderate numbers when ρ = ρ, R G = ; the equilibrium solutions are unique. Figures 8, 9 and show the evolution to equilibrium, at R =, of a neutrally buoyant article started at the wall and at the centreline from an initial condition of rest. No matter where the article is released it will migrate to a unique equilibrium solution at y =.8 cm. e f

11 Sli velocity and lift 3 5 U s (cm s - ) 5 starting at the centreline starting at the wall t (s) FIGURE 9. Evolution of the sli velocity of the article whose trajectory is shown in figure 8. The sli velocity evolves to zero through ositive values whether the article is started above or below the equilibrium osition. The sli velocity discreancy U U >, - U se =.8 cm s. s se Ω s (s - ). starting at the centreline starting at the wall. 6 8 t (s) FIGURE. Evolution of the angular sli velocity of a neutrally buoyant article at R = to equilibrium (see figure 8). The angular sli velocity function evolves without crossing the equilibrium value. When the angular sli velocity is below the equilibrium value, the article moves downward. When the angular sli velocity is above the equilibrium value, the article moves uward. In figure 9, we show the evolution of the sli velocity to equilibrium. The sli velocity is ositive and of course the greatest for a article released from rest at the centreline. In figure, we show that the angular sli velocity is smaller than its equilibrium value when the article is above the equilibrium osition and, is larger than its equilibrium value when it is below the equilibrium osition. The angular sli velocity discreancy Ω s Ω se DDJ//aers/SliVelocity/SV-Lift-REV.doc

12 D. D. Joseh and D. Ocando.5 (a) F y (dyn cm - ) fixed osition above equilibrium fixed osition below equilibrium t (s) (b) Ω s (s - ).. fixed osition above equilibrium fixed osition below equilibrium t (s) FIGURE. (a) Lift and (b) angular sli velocity in constrained simulations of a article fixed above and below equilibrium. The sign of the lift correlates erfectly with the sign of the angular sli velocity discreancy. R =, y =. 6 and.9 cm, y =.8 cm. e.6. F y (dyn cm - ). -. fixed osition above centreline fixed osition below centreline t (s) FIGURE. Evolution of the lift on a article at a fixed osition at R = slightly above, y = 6.5 cm and below, y = 5.95 cm, the centreline. The lift ushes the article away from the centreline.

13 Sli velocity and lift 3.6 Ω s (s - ).. -. fixed osition above centreline fixed osition below centreline t (s) FIGURE 3. Evolution of the angular sli velocity in a constrained simulation for articles at y = 5.95 cm and 6.5 cm when R =. The evolution is to a steady state with the following roerties: when the angular sli velocity is below the equilibrium value, the article moves uward. When the angular sli velocity is above the equilibrium value, the article moves downward. This behavior is the oosite of the revious cases, because the revious cases were stable equilibrium ositions, and therefore, the force field around them is the oosite. Ω s/ω s max, U s /U s max, L/L max angular sli velocity sli velocity lift force y (cm) Centreline FIGURE. Sli velocities and lift for neutrally buoyant article at R =. Steady-state relative values for the lift force and the sli velocities. Dotted lines corresond to unstable equilibria. In the region close to the wall, the lift force and the sli velocity have a similar nonlinear behaviour. In the region close to the centreline, the lift force aears to be roortional to the angular sli velocity. where Ω se is the angular sli velocity at equilibrium, changes sign with the lift across the equilibrium. In figure, we carry out a constrained simulation in which the circular article is fixed at a osition slightly above and slightly below the value of equilibrium. This figure shows that the sign of the angular sli velocity discreancy changes with the sign of the lift, which is ositive for articles below and negative for articles above the equilibrium. In figures and 3 are shown the evolution of the lift and angular sli velocity, resectively, from constrained simulations at fixed ositions slightly above and slightly below the channel centreline. Figure 3, shows that the angular sli velocity discreancy is negative when the article is above the centreline and is ositive when it is below the centreline. The discreancy changes sign in both the stable and unstable cases, but the sign of the discreancy is oosite in the two cases. DDJ//aers/SliVelocity/SV-Lift-REV.doc

14 D. D. Joseh and D. Ocando ρ ρ f Starting at centreline y e (cm) Starting close to wall TABLE. Position of equilibrium when R =..3. y (cm)..99 starting below equilibrium starting above equilibrium t (s) FIGURE 5. Particle height about equilibrium for a heavier-than-fluid article ( ρ =., ρ f R = ). The initial condition was the steady state solution from a constrained simulation at the initial height. In figure, we have lotted the resultant constrained dynamical simulation comaring distributions of the normalized sli velocity, the angular sli velocity and lift for a neutrally buoyant article are comuted at each fixed osition y for R =. 6. Sli velocities, angular sli velocities and lift for non-neutrally buoyant circular articles The qualitative results, which were established in 5 hold for heavier and lighter than fluid articles. The sli velocity and angular sli velocity are ositive and the angular sli velocity discreancy changes sign across y e, where y e is the lace where the lift discreancy L = L ( ρ ρ f ) πd g (6.) in two dimensions vanishes. For heavier than liquid articles, the osition of equilibrium moves closer to the bottom wall, and for values of ρ ρ f larger than a critical two equilibrium heights exist (table ).

15 Sli velocity and lift Ω s Ω se (s - ) starting below equilibrium starting above equilibrium 6 8 t (s) Figure 6. Angular sli velocity discreancy about equilibrium for a heavier than fluid article ( ρ =., R = ). The evolution on the angular sli velocity discreancy is consistent ρ f with the evolution on the article height. 6 5 y (cm) 3 starting at the centreline starting at the wall 6 8 t (s) FIGURE 7. Migration from steady flow of a heavy article ρ ρ f =. starting at rest near the wall and centreline at R = ( R G = 9. 8). The article starting at the centreline crosses the equilibrium osition and then moves uward. Figures 5 and 6 are lots of the migration to equilibrium of articles starting above and below but near to equilibrium when R = and R = 9. 8 corresonding ρ ρ f =.. Figure 5 shows that the article migrates to the same equilibrium whether it starts from above or below the osition of equilibrium. Figure 6 is a lot of the angular sli velocity discreancy versus time; it shows that the angular sli velocity discreancy changes sign across the osition of the equilibrium. G DDJ//aers/SliVelocity/SV-Lift-REV.doc

16 6 D. D. Joseh and D. Ocando 5 Ω s (s - ) 3 starting at the centreline starting at the wall 6 8 t (s) FIGURE 8. Evolution of the article angular sli velocity of a heavier than fluid article ρ ρ f =. starting at rest near the wall and centreline at R =, R G = For the article starting at the centreline, the angular sli velocity function crosses the equilibrium value. When the angular sli velocity is below the equilibrium value, the article moves downward. When the angular sli velocity is above the equilibrium value, the article moves uward. A change in the sign of the angular sli velocity discreancy is evident at early times when the article falling from the centreline crosses the equilibrium; after this both articles are below the equilibrium and have essentially the same angular sli velocity.. ( s - ) e Ω s - Ω s starting close to the wall starting at the centreline e y c -y c (cm) FIGURE 9. Analysis on the angular sli velocity discreancy in the case of multile equilibrium ( ρ =., R = ). Case (a) article released close to the wall: ρ f ( y ( t = ) =. cm). Case (b) article released at the centreline: ( ( t = ) = 6. y cm). For this data, the buoyant weight intersects the lift force at three oints, and two of them yield stable solutions (figure 5, table ). For case (a), the article travels from a osition close to the wall to the equilibrium height closer to the wall y =. 33 cm, whereas for case (b), the article e travels from the centreline to the equilibrium height far from the wall y = cm. The discreancy changes sign at the equilibrium weight. y = y where the lift balances the buoyant e e

17 Sli velocity and lift 7 Ω s - Ω se (s - ) starting close to the wall starting at the centreline y -y e (cm) FIGURE. Analysis on the angular sli velocity discreancy in the case of unique equilibrium ( ρ =. 5, R = ). Case (a), article released close to the wall: ( ( t = ) =. cm). ρ f Case (b) article released at the centreline: ( ( t = ) = 6. y cm). For this data the buoyant weight intersects the lift force at only one oint (figure 5, table ). Therefore, no matter where the article started it will reach the same equilibrium height at y = cm. The discreancy is ositive if the local value is greater than the equilibrium value, and it is negative for the oosite condition. The angular sli velocity discreancy changes its sign as the article height discreancy does. Note: the initial condition for all the cases was the following: First, to get a fully develoed velocity rofile, a simulation at the initial height ( y ( t = ) =. cm or y ( t = ) = 6. cm) was erformed using a constrained motion on the vertical direction. Secondly, the vertical motion constrain is released, and therefore the article travels to a referential equilibrium height. Figures 7 and 8 treat the same roblem as in figures 5 and 6 but with a different initial condition. In figure7, the article is started from rest near the wall and the centreline Figure 9 treats the roblem of sli velocity and lift for the case R = and R G = 9.8 ( ρ ρ f =. ) in which there are two stable equilibrium heights (figures 3 and 5). In the dynamical simulation of a article started from steady flow near the centreline, the article migrates downward to the higher osition of equilibrium and the angular velocity discreancy is negative. When the article is started from the wall it migrates uward to the lower osition of equilibrium and the sli angular velocity discreancy is ositive, consistent with our hyothesis about the lift and the sli angular velocity discreancy. In figure, we consider the case of heavier than fluid articles ρ ρ f =. 5 migrating at R =. For this relatively lightweight article the equilibrium solutions are unique at R = and the angular sli velocity discreancy changes with the lift as the article aroaches y = y from above or below. e y e 7. Long article model Joseh () roosed a model roblem for the velocity of a long article in Poiseuille flow (also see Choi and Joseh, ). We relaced the circular article of diameter d with a long rectangle whose short side is d. The rectangle is so long that we may neglect the effects of the ends of the rectangle at sections near the rectangle DDJ//aers/SliVelocity/SV-Lift-REV.doc

18 8 D. D. Joseh and D. Ocando y > u A (y ) h A U A d τ A U B u B (y) y h B τ B x FIGURE. Sketch of flow field under consideration and variables involved in the long article model. W = h + h d is the channel height. A B + centre. In that model the long article was assumed to be rigid but it was noted that a more realistic model could be obtained by letting the long article shear. We could choose this shear to be the same as the shear rate of the circular article in the aroximation in which Ω = γ (figure ). The long article model is meant to reresent the constrained forward motion with y fixed after transients have decayed and steady flow is achieved. The model may be comared with the numerical simulation satisfying the fluid equations (3.3) with u t =, the x-comonent of (3.) with d u d t = e x + e x { + ηd[ y] } ndγ = V and (3.6) with d Ω d t =. The y-comonent of (3.) gives the balance of buoyant weight and lift; the article density enters into this balance through the buoyancy term. It follows that buoyancy and the article density do not enter into the constrained simulations, which determine the steady motion of the fluid and the forward motion of the article, and they do not enter into the long article model, which aroximates the simulation. The model leads to an exlicit exression for the article velocity and sli velocity in which vertical migration is suressed. Since the simulation and the model do not deend on ρ, there is a sense in which the results given here are universal. However, each constrained simulation is realizable for a density given by (.) in which the y-comonent of (3.) is satisfied. The forces acting on the long article are the force due to ressure acting on the sides erendicular to the flow, and the force due to shear stress acting on the sides arallel to the flow (figure 7.). The former force is always ositive, while the latter may be ositive or negative deending if the fluid is faster than the article or vice versa, ( τ A + τ B ) l + ( ) d = (7.) ( ) τ A + τ B + d =, = (7.) l where the shear stresses are defined by du A dub τ A = η ( h A ), τ B = η ( h ) dy B (7.3) dy The velocity rofiles above and below the long article are given by U A y u A ( y ) = y ( ha y ) + (7.) η h A

19 Sli velocity and lift 9 U u + B ( y) = y( hb y) η where different velocities ( ) y B (7.5) hb U A, U B were assumed for the to and bottom walls to take into account the angular seed of the circular article. The relation between them is given by U A U B = γ ( hb + d )d, (7.6) where γ ( y) is the shear rate for the undisturbed flow (without the article), given by du γ ( y) = = ( W y). (7.7) dy η The shear rate on the article s sides arallel to the flow may be evaluated from (7.), (7.5) and (7.6), du ( B ) A U γ h + d B d ( ha ) = ha + + (7.8) dy η h h du B U B ( hb ) = hb +. (7.9) dy η hb Substituting, recursively, (7.8) and (7.9) in (7.3), and then the resultant equation in (7.), we find that at the to and bottom of the long article (diameter d): ( η)( d + ha + hb ) hahb + γ ( hb + d ) had U A = (7.) ( ha + hb ) ( η)( d + ha + hb ) hahb γ ( hb + d ) hbd U B = (7.) ( ha + hb ) The average article velocity is: ( η)( d + ha + hb ) hahb γ ( hb + d ) ( hb ha ) d U = ( U A + U B ) =. (7.) ( ha + hb ) The undisturbed flow field (without the article) can be written as: u( y) = y( W y) (7.3) η At the osition where the centre of the article is located ( y = hb + d), the undisturbed fluid velocity is: ( u hb + d) = ( h d) ( h ) B + A + d (7.) η The article sli velocity can be defined as: U s = u( hb + d) U (7.5) which can be written as: ( )[ ha + hb hb + d ha + d ( d + ha + hb ) hahb ] + ( hb + d ) hb ha d U s = ( η )( )( ) γ ( ) ( ha + hb ) (7.6) The channel height W and the osition h A and h B satisfy the following conditions: ha = W ( hb + d), y = hb + d ha + hb = W d, and the shear rate at the article centre is γ ( hb + d ) = ( W d h B ) η (7.7) Then the sli velocity can be simlified: A A DDJ//aers/SliVelocity/SV-Lift-REV.doc

20 D. D. Joseh and D. Ocando (a) (b) y (cm) 8 6 undisturbed long article model DNS y (cm) 8 6 undisturbed long article model DNS u (cm s - ) (c) u (cm s - ) (d) y (cm) 8 6 undisturbed long article model DNS y (cm) 8 6 undisturbed long article model DNS u (cm s - ) (e) u (cm s - ) (f) y (cm) 8 6 undisturbed long article model DNS y (cm) 8 6 undisturbed long article model DNS u (cm s - ) u (cm s - ) FIGURE. Velocity rofiles through the centre y of a article and the article velocity at R =. The velocity rofiles of the undisturbed flow, of the DNS simulation and the long article model are comared: (a) centreline y = 6. cm, (b) the unique equilibrium osition when ρ =. 5 ( y = cm), (c) the higher equilibrium osition when ρ f ρ =. ( y = cm), (d) y = 3. cm, (e) the lower equilibrium osition when ρ f ρ =. ( y =. 33 cm), (f) y =. 75 cm. The centreline of (a) is unstable. U ρ f s = η [( ha + hb )( hb + d)( h + d) ( d + h + h ) h h + ( h d h )( h h ) d] A A B A B B B A ( h + h ) A = η B ( W d ) [ d( W y ) + d ( W d )] when d, we can get U. s (7.8)

21 Sli velocity and lift (a) (b) y (cm) 8 6 undisturbed long article model DNS y (cm) 8 6 undisturbed long article model DNS u (cm s - ) u (cm s - ) FIGURE 3. Velocity rofiles at steady state on a line through the centre of a article at y =.75 cm, R =. (a) reference size ( d =. cm), (b) small article ( d =. 5 cm). As the article is smaller, the difference between disturbed and undisturbed velocity rofiles is smaller. 6 5 U (cm s - ) 3 DNS long article model y c (cm) Centreline FIGURE. Particle velocity vs. article osition. A comarison of the long article model with DNS for a circular article is given in figure and 3. In these constrained simulations we fix the y osition of the article and comute the dynamic evolution to equilibrium at R =. The diameter of the article in figure is cm and in figure 3 it is.5 cm. The rofiles in the figures are at equilibrium and on a cross-section through the centre of the article. The agreement is rather better than might have been anticiated given the severe assumtions required in the model. The agreement is quite good away from the centreline, even close to the wall. Equations (7.-7.6) can be recommended for an analytical aroximation for the velocity of a circular article in Poiseuille flow. In figure, we comare the article velocity from the simulation with the long article model. In figure 5, we comare the sli velocity, and in Figure 6 we show γ. how nearly the article angular velocity is given by ( ) y DDJ//aers/SliVelocity/SV-Lift-REV.doc

22 D. D. Joseh and D. Ocando U s / u(y c ) DNS long article model y c (cm) Centreline FIGURE 5. Sli velocity/fluid velocity ratio vs. article osition at R =. Sli velocity evaluated using DNS results vs. sli velocity in the long article model. These are relative values of the sli velocity, U, with resect to the fluid velocity on the undisturbed flow at the s article centre u y ). The largest discreancy is at about the distance from the wall to the centreline. ( 8 DNS long article model Ω (s - ) y c (cm) Centreline FIGURE 6. Particle angular velocity, Ω at R =. The angular velocity of the article is aroximated in the long article model as half the value of the shear rate on the undisturbed Ω = γ y. flow evaluated at article s centre osition ( ) LPM 8. Summary and conclusion The lift and migration of neutrally buoyant and heavier-than-liquid circular articles in a lane Poiseuille flow was studied using direct numerical simulation. The study looks at the relation of sli velocity and angular sli velocity to lift and migration. No matter where the neutrally bouyant article is released, it will migrate to a unique equilibrium height and move forward with a unique steady article

23 Sli velocity and lift 3 velocity and rotate with unique steady angular velocity. Neutrally buoyant articles migrate to a radius which can be called the "Segré Silberberg" radius. This radius is a reference; heavier-than-liquid articles also migrate to an equilibrium radius that is close to the Segré-Silberberg radius if the article density is close to the fluid density. The articles migrate to an equilibrium osition y e with shear rate γ e such that the local fluid rotation γ e is slightly greater than the article angular velocity Ω. The angular sli velocity, Ω s = Ω + γ e is always ositive but at equilibrium it is very small; Ω γ e can be roosed as an aroximation. The sli velocity at equilibrium U s = U fe U is always ositive and slowly varying. Since the shear rate and sli velocities are one signed they do not exlain why the lift changes sign across the equilibrium radius. We found that the quantity, which does change sign at y e, is the angular sli velocity discreancy; the angular sli velocity minus the equilibrium angular sli velocity Ω Ω. Ω Ω > when y < y e and Ω s Ω se < when y > ye. The adjustment of the angular velocity of a free article is very critical to lift. One might think of the angular velocity discreancy as a shear flow analogue to the circulation in aerodynamic lift. We derived a shear version of our long article model. The long article model arises when the circular article is relaced with a long rectangle of the same diameter as the circle, but so long that we may neglect end effects. In the shear version we allow the rectangle to shear at the rate γ of the local rotation. Using this model we can find exlicit exressions for the fluid rotation in which the velocity on either side of the long article is matched by the fluid velocity; then we satisfy the articles equation of motion in which the shear stress force balances the ressure dro force. This leads to exlicit exression for the velocity of the article (7.) and the sli velocity (7.6) that is always ositive. The shear version of the long article model is in good agreement with the results of numerical comutation of the motion of a free circular article at oints of stable equilibrium, both with resect to the article velocity and the fluid velocity on the cross-section containing the centre of the circular article. The results given in this aer are for two dimensions. It remains to be seen how such results carry over to three dimensions. We note that the celebrated lift formula (.) is a two-dimensional result. 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 of Basic Energy Sciences; by a grant from the Schlumberger foundation; from STIM-LAB Inc.; and by the Minnesota Suercomuter Institute. s se s se REFERENCES ASMOLOV, E. S. 999 The inertial lift on a sherical article in a lane Poiseuille flow at large channel Reynolds number. J. Fluid Mech. 38, AUTON, T.R. 987 The lift force on a sherical body in a rotational flow, J. Fluid Mech. 83, CHERUKAT, P., MCLAUGHLIN, J. B. & DANDY, D. S. 999 A comutational study of the inertial lift on a shere in a linear shear flow field, Int. J. Multihase Flow, 5, CHOI, H. G. & JOSEPH, D. D. Fluidization by lift of 3 circular articles in lane Poiseuille flow by direct numerical simulation, J. Fluid Mech. 38, July, -8. DREW, D. A. & PASSMAN S. 999 Theory of Multicomonent Fluids, Sringer. DDJ//aers/SliVelocity/SV-Lift-REV.doc

24 D. D. Joseh and D. Ocando FENG, J., HU, H. H. & JOSEPH D. D. 99 Direct simulation of initial values roblems for the motion of solid bodies in a Newtonian fluid. Part : Couette and Poiseuille flows, J. Fluid Mech. 77, 7-3. HO, B. P. & LEAL,, L. G. 97 Inertial migration of rigid sheres in two-dimensional unidirectional flows, J. Fluid Mech. 65, JOSEPH, D. D. Interrogations of Direct Numerical Simulation of Solid-Liquid Flow, to aear, see also htt:// KUROSE, R. & KOMORI, S. 999 Drag and lift forces on a rotating shere in a linear shear flow, J. Fluid Mech. 38, MCLAUGHLIN, J.B. 99 Inertial migration of a small shere in linear shear flows, J. Fluid Mech., 6-7. MORTAZAVI, S. & TRYGGYASON, G. A numerical study of the motion of dros in Poiseuille flow. Part, Lateral migration of one dro, J. Fluid Mech., PATANKAR, N., HUANG, P. Y., KO, T. & JOSEPH D. D. Lift-off of a single article in Newtonian and viscoelastic fluids by direct numerical simulation, J. Fluid Mech. 38, 67-. SAFFMAN, P. G. 965 The lift on a small shere in a slow shear flow, J. Fluid Mech., 385; and Corrigendum, J. Fluid Mech. 3, 6 (968). SCHONBERG, J. A. & HINCH, E. J. 989 Inertial migration of a shere in Poiseuille flow, J. Fluid Mech. 3, SEGRE, G. & SILBERBERG, A. 96 Radial Poiseuille flow of susensions, Nature, 89, 9. SEGRE, G. & SILBERBERG, A. 96 Behavior of macroscoic rigid sheres in Poiseuille flow, Part I, J. Fluid Mech.,, VASSEUR, P. & COX, R.G. 976 The lateral migration of a sherical article in two-dimensional shear flows, J. Fluid Mech. 78, ZHAO, Y. SHARP, M. K. & JAMES, K. 998 Finite-element analysis of transverse lift on circular cylinder in D channel flow. J. of Engineering Mech. 5-6.