Dynamic determinants of the upper torso angular velocity ...
Slip Velocity and Liftslip angular velocity discrepancy defined as the difference between the slip...
Transcript of Slip Velocity and Liftslip angular velocity discrepancy defined as the difference between the slip...

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Slip Velocity and Lift
D.D. Joseph, D. Ocando and P.Y. Huang
Department of Aerospace Engineering and Mechanics and the Minnesota Supercomputer Institute,University of Minnesota, Minneapolis, MN 55455, USA
October 2000
AbstractThe lift force on a circular particle in plane Poiseuille flow is studied by direct numericalsimulation and by explicit mathematical analysis of a long particle model. The model leadsto a formula for the particle velocity that is in excellent agreement with simulation valuesfor the circular cylinder. The value of the Poiseuille flow velocity at the point at theparticle's center when the particle is absent is always larger than the particle velocity; theslip velocity is positive at equilibrium (at steady flow). The angular slip velocity
ps Ω−=Ω 2/γ� , where 2/γ� is the angular velocity of the fluid at a point where the shearrate is γ� and pΩ is the angular velocity of the particle, is always positive at equilibrium.The particle migrates to its equilibrium position and adjusts pΩ so that 0>Ω s is nearlyzero because of 2/γ�≈Ω p . No matter where the particle is placed it drifts to anequilibrium position with a unique, slightly positive equilibrium angular slip velocity. Theslip angular velocity discrepancy defined as the difference between the slip angularvelocity of a migrating particle and the slip angular velocity at its equilibrium position ispositive below the position of equilibrium and negative above it. This discrepancy is thequantity that changes sign above and below the equilibrium position for neutrally buoyantparticles, and also above and below the lower equilibrium position for heavy particles.
1. IntroductionThe experiments of Segré and Silberberg (1961, 1962) have had a big influence on
fluid mechanics studies of migration and lift. They studied the migration of dilutesuspensions of neutrally buoyant spheres in pipe flows at Reynolds numbers between 2and 700. The particles migrate away from the wall and centerline and accumulate at 0.6of a pipe radius.
The lift on heavier than liquid particles is also influenced by the factors thatdetermine the equilibrium position of neutrally buoyant particles. The heavy particlesmust reach an equilibrium that balances the hydrodynamic lift and buoyant weight. If thebuoyant weight is very small, the equilibrium position of the particles will be close to thevalue for the neutrally buoyant case. The effect of increasing the weight is to lower theequilibrium position whose zero is established for the case of zero buoyant weight.
Most attempts to explain the SegréSilberberg effects have been based on linearizedlow Reynolds number hydrodynamics. Possibly the most famous of these attempts is dueto Saffman (1965), who found that the lift on a sphere in a linear shear flow is given by
sorder termlower /6.46 L2/1
2 +���
�= vUa s γη (1.1)

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where pfs UUU −= is the slip velocity, Uf is the fluid velocity and Up is the particlevelocity, a is the sphere radius, fρην = , η is the fluid viscosity, ρf is the fluid densityand γ� is the shear rate. The lower order terms are
γ aU sfs�
��
� −−Ω+−8223 ππρ (1.2)
where
psγ Ω−=Ω2
(1.3)
is the angular slip velocity and pΩ is the particle angular velocity. There are a number offormulas like Saffman's that are in the form of Us times a factor, which can be identifiedas a density times a circulation as in the famous formula ρUΓ for aerodynamic lift. Arelatively recent review of such formulas can be found in McLaughlin (1991).
Formulas like Saffman's cannot explain SegréSilberberg's observations, whichrequire migration away from both the wall and the center. There is nothing in theseformulas to account for the migration reversal near 0.6 of a radius. Moreover the slipvelocity Us, the angular slip velocity Ωs, the particle velocity and the particle angularvelocity, which are functionals of the solution are prescribed quantities in these formulas.
The fluid motion drives the lift on a free body in shear flow; no external forces ortorques are applied. If there is no shear there is no lift. In Poiseuille flow there is not onlya shear but a shear gradient. Gradients of shear (curvature) produce lateral forces. At thecenterline of a Poiseuille flow the shear vanishes, but the shear gradient does not. Tounderstand the SegréSilberberg effect it is necessary to know that the curvature of thevelocity profile at the center of Poiseuille flow makes the center of the channel anunstable position of equilibrium. A particle at the center of the channel or pipe will bedriven by shear gradients toward the wall; a particle near the wall will lag the fluid and bedriven away from the wall. An equilibrium radius away from the center and wall mustexist.
Ho and Leal (1974) were the first to combine these effects in an analysis of themotion of a neutrally buoyant sphere rotating freely between plane walls so closelyspaced that the inertial lift can be obtained by perturbing Stokes flow with inertia. Theytreated wall effects by a method of reflection and found an equilibrium position at 0.69from the center. Vasseur and Cox (1976) used another method to treat wall effects andtheir results are close to Ho and Leal’s near the center line but rather different than thoseof Ho and Leal near the wall. Feng, Hu and Joseph (1994) studied the motion of solidcircles in plane Poiseuille flow by DNS. The circle migrates to the 0.6 of a radiusequilibrium position. They compared their 2D results with those of Ho & Leal andVasseur & Cox. Schonberg and Hinch (1989) analyzed the lift on a neutrally buoyantsmall sphere in a plane Poiseuille flow using matched asymptotic methods. The sameproblem for neutrally buoyant and nonneutrally buoyant small sphere has been studiedusing asymptotic method by Asmolov (1999). The linearized analysis given so far takesthe effect of inertia ( )uu ∇• into account only in an Oseen linear system; the comparisonof the results of these analyses with experiments is far from perfect. The analysis is heavyand explicit formulae for lift are not obtained.

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This paper approaches the problem of migration and lift in a different way. Basicallywe have used direct numerical simulation (DNS) to formulate and validate a long particlemodel that gives a very good, completely explicit analytical approximation to the velocityand slip velocity of circular particles. DNS is used here as a diagnostic tool to analyze therole of the slip velocity, and the angular slip velocity on migration and lift. We are able inthis way to establish a rather simple picture of lift and migration that in particularclarifies the role of the angular slip velocity, and is not restricted to low Reynoldsnumbers. Our analysis is carried out in two dimensions but should apply in principle to3D, which is at present under study. †
2. Mechanism for liftFirst we can look on formulas in a fluid without viscosity for which viscous drag is
impossible. The most famous formula for lift on a body of arbitrary shape movingforward with velocity U in a potential flow with circulation Γ was given by
Γ=′ UL ρ (2.1)
where ρ is the fluid velocity and L′ is the lift per unit length.The lift on circular cylinder of radius a is of special interest. A viscous potential flow
solution for a stationary cylinder rotating with angular velocity Ω which satisfies the noslip boundary condition is given by
aarar Ω=Ω= θθ eueu )(,/)(2 . (2.2)
The circulation for this viscous potential flow is22 adx Ω=•=Γ πu . (2.3)
When this rotating cylinder moves forward it generates a lift L′ per unit length
Ω=′ UaL 22πρ . (2.4)
The direction of the lift can be determined by noting that the velocity due to rotation addsto the forward motion of the cylinder at the top or bottom of the rotating cylinderaccording to the directions of Ω and U. In figure 2.1 the velocity is smaller on the bottomof the cylinder; by the Bernoulli equation, the pressure is greater there and it pushes thecylinder up.
† A referee called our attention to an interesting paper by Zhao, Sharp and James 1998 of transverse lift ona free circular particle in steady flow in a channel. The three basic results of our paper, (1) the stability ofintermediate equilibrium positions arising from the dynamic resolution of the initial value problem, (2) thelong body model and (3) the explanation of the SegréSilberberg effect by identification of the slipquantity, which changes the sign of the lift across the equilibrium position, are not touched in their paper.Our results and theirs agree where they overlap.

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LowPressure
HighPressure
U Ω
Figure 2.1. The lift per unit length L′ = 2πρa2UΩ on a cylinder of radius a moving forward atspeed U and rotating with angular velocity Ω in such a way as to reduce the velocity at thebottom and add at the top.
Another formula for the lift off on a particle in an inviscid fluid in which uniformmotion is perturbed by a weak shear was derived by Auton (1987) and a more recentsatisfying derivation of the same result was given by Drew and Passman (1999). Theyfind that in a plane flow
ysf UaL eΩ= ρπ3
34 (2.5)
where
dydu
f ==Ω γ2 .
If du/dy > 0 the sphere is lifted against gravity when the slip velocity Us is positive; if Usis negative the sphere will fall. Particles which lag the fluid migrate to streamlines withfaster flow, particles which lead the fluid migrate to streamlines with slower flow.
There are rather striking differences between (2.5) and (2.4); first (2.4) depends onthe angular velocity of the particle but (2.5) depends on the angular velocity of the fluid.Both formulas leave the slip velocity undetermined, Us appears in (2.5) because of theshear, in (2.4), Uf = 0. The slip velocities have to be prescribed in these theories becausethe particle velocity is not determined by viscous drag. Similarly the angular velocity ofthe particle cannot arise from torques arising from viscous shears. The effects of particlerotation cannot be obtained by the method of Auton (1987).
The lift formula ρUΓ=captures the essence of the mechanism in which the motion ofthe particle relative to the fluid is such as to increase the pressure on the side of theparticle as it moves forward.
The lift on a spherical or circular particle in a shear flow is different; there is noexterior agent to move and rotate the freely moving particle. Instead the particle isimpelled forward and rotated by the shear flow. Previous theoretical studies and oursimulations show that the relevant velocity is the slip velocity and the relevant circulationis proportional to an angular slip velocity discrepancy
Γ=∝=Ωs  Ωse (2.6)
where Ωse is the slip angular velocity in steady flow (equilibrium). This conclusion willbe established in the sequel. For now we simply note that in our simulations the angular

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slip velocity discrepancy Ωs  Ωse < 0 when the cylinder is above the equilibrium (SegréSilberberg) position and Ωs  Ωse > 0 when it is below the equilibrium (figure 5.2).
3. Numerical simulation of migration and liftThe simulation method used in this paper is described in the papers by Patankar,
Huang, Ko and Joseph (2000), Choi and Joseph (2000), and Joseph (2000). A circularparticle of diameter d is placed in a plane Poiseuille flow in a channel of height h where itmoves forward in the direction x and migrates up or down in the direction y under theaction of hydrodynamic forces. The fluid satisfies the NavierStokes equations
xf ppteuuuu +∇+−∇=���
� ∇⋅+∂∂ 2ηρ (3.1)
where p is the constant pressure gradient. The particle satisfies Newton's equations ofmotion. The motion Up of the mass center is governed by
( ) { } Γ⋅+−++−−= d][21d
dnuD1ee
Uηρρρ p
Vpg
t pxyfp
pp (3.2)
where Vp = πa2 is the volume of the particle and (p1 + 2ηD[u])•n is the stress on thesurface Γ of the particle that is a circle of radius a = d/2 and
( ) ( )=Γ π θ20
d•d• a . (3.3)
The angular velocity around the mass center at x = X is governed by
( ) ( )[ ] Γ⋅+−∧−= d][2dd nuD1XxΩ ηp
tI (3.4)
where I = ρpπa4 is the moment of inertia.In steady (equilibrium) flow the forward motion of the particle is given by
{ } 0d•][2•1 20
=+−+ θηπ
π
rx pap euD1e (3.5)
The pressure gradient force on the particle is balanced by the resultant of stress tractionvector. The long particle model discussed by Joseph (2000), Choi and Joseph (2000) andhere in section 4 is a realization of (3.5).
In numerical experiments of solidliquid flows we can examine physical effects one ata time; this cannot be done in real experiments. For the present application we look firstat the effect on particle migration of controlling the angular velocity of the particle. Infigure 3.1 we plotted the rise to equilibrium of a neutrally buoyant particle for threedifferent values of the slip angular velocity
.0,,22
��
�
��
��
�Ω=Ω−=Ω seps
γγ (3.6)

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The rise is the greatest when the particle angular velocity Ωp = 0 and the least when theparticle angular velocity is equal to the local rate of rotation Ωp = 2γ� . The rise of aheavier than liquid ρp/ρf = 1.01 circular particle is plotted in figure 3.1 for Reynoldsnumber == νγ 2dR ww � 5.4 and for Rw = 16.2 in figure 3.2. The angular slip velocity Ωse> 0 is the equilibrium value that a free circular particle takes in torquefree motion whenthe angular acceleration vanishes. We call attention to the fact that Ωse > 0 is very small,and at equilibrium
pepe Ω≈Ω> 2,
2γγ . (3.7)
In another constrained motion we fix the y position of the particle and compute theslip velocities and lift (figure 3.3). A fixed particle with nonzero lift forces will migrateif the constraint is relaxed.
0
1
2
3
4
5
0 50 100 150 200
Y = y/d
Time (second)
Ws=g/2W
pWs=g/2
0

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0.5
1
1.5
2
2.5
3
3.5
4
0 10 20 30 40 50
Freely rotating
60 70 80
Y = y/d
Time (second)
Ws= g /2W p
Ws =g /2
0

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4. Long particle modelJoseph (2000) proposed a model problem for the velocity of a long particle in
Poiseuille flow (also see Choi and Joseph, 2000). We replaced the circular particle ofdiameter d with a long rectangle whose short side is d. The rectangle is so long that wemay neglect the effects of the ends of the rectangle at sections near the rectangle center.In that model the long particle was assumed to be rigid but it was noted that a morerealistic model could be obtained by letting the long particle shear. Ocando has noted thatwe could choose this shear to be the same as the shear rate of the circular particle in theapproximation in which Ωp = 2γ� (figure 4.1). We call this long particle model a shearslip model; the diagram for this model is shown in figure 4.1.
� Shear slip modelThe forces acting on the long particle are the force due to pressure acting on the sides
perpendicular to the flow, and the force due to shear stress acting on the sides parallel tothe flow (figure 4.1). The former force is always positive, while the latter may be positiveor negative depending if the fluid is faster than the particle or vice versa,
( ) ( ) 021 =−++ dpplBA ττ (4.1)
y’
y
uA(y’)
uB(y)
UA
UB
τA
τB
x
p1 p2p1 >p2
hB
hA
d
Figure 4.1. Sketch of flow field under consideration and variables involved in the slip models.
( )l
pppdpBA 210−==++ττ (4.2)
where the shear stresses are defined by
)()( BBBAAA hdyduh
yddu ητητ −=
′−= (4.3)
The velocity profiles above and below the long particle are given by
( ) ( )A
AAA h
yUyhypyu′
+′−′=′η2
(4.4)

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( ) ( )B
BBB h
yUyhypyu +−=η2
(4.5)
where different velocities ( AU , BU ) were assumed for the top and bottom walls to takeinto account the angular speed of the circular particle. The relation between them is givenby
ddhUU BBA 2)2( +=− γ
�(4.6)
where ( )yγ� is the shear rate for the undisturbed flow (without the particle), given by
( )yhpdyduy 2
2)( −==
ηγ� (4.7)
The shear rate on the particle’s sides parallel to the flow may be evaluated from (4.4),(4.5) and (4.6),
( )A
B
A
BAA
A
hddh
hUhph
yddu
2)2(
2+
++−=′
γη
�(4.8)
( )B
BBB
B
hUhph
dydu
+−=η2
. (4.9)
Substituting, recursively, (4.8) and (4.9) in (4.3), and then the resultant equation in (4.2),we find that at the top and bottom of the long particle (diameter d):
( )( )BA
ABBABA
A hh
dhdhhhhhdp
U+
++++=
2
)2(2 γη
�
(4.10)
( )
( )BA
BBBABA
B hh
dhdhhhhhdp
U+
+−++=
2
)2(2 γη
�
(4.11)
The average particle velocity is:
( )( )BA
ABBBABABA
P hh
dhhdhhhhhdpUUU
+
−+−++=+=
22
)()2(2
2
γη
�
(4.12)
The undisturbed flow field (without the particle) can be written as:
( )yhypyu −=η2
)((4.13)
At the position where the center of the particle is located )2
( dhy BC += , the undisturbed
fluid velocity is:

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)2
)(2
(2
)2
( dhdhpdhu ABB ++=+ η (4.14)The particle slip velocity can be defined as:
PBS UdhuU −+= )2
( (4.15)
which can be written as:
( )
( )BA
ABBBABAABBA
S hh
dhhdhhhhhddhdhhhp
U+
−++���� ++−+++
=2
2)()2(2)
2)(
2)(( γ
η�
(4.16)
The channel width h and the position hA and hB satisfy the following conditions:
��
−=++−=dhhhdhhh
BA
BA )( and 2dhy BC +=
and the shear rate at the particle center is
( )BB hdhpdh 2
2)2( −−=+
ηγ� (4.17)
Then the slip velocity can be simplified:
( ) ( )�
��
� −−−+++−++++
=2
))(2(212)
2)(
2)((
2dhhhdhhhhhddhdhhh
hhpU ABBBABAABBA
BAS η
( ) 04)2(222
2 ≥�
��
�+−
−= hdyhd
dhp
Cη(4.18)
when 0→d , we can get 0→SU .
A comparison of the shear slip long particle with direct numerical simulation of acircular particle is given in figures 4.24.5. The agreement is rather better than mighthave been anticipated given the severe assumptions required in the model. The agreementis quite good away from the centerline, even close to the wall. Equations (4.24.6) can berecommended for an analytical approximation for the velocity of a circular particle inPoiseuille flow.

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0
10
20
30
40
50
60
0 2 4 6 8 10 12y [cm]
u [
cm
/s]
UndisturbedDNSShear slip model
WALL WALL
Figure 4.2. FIXED PARTICLE AT THE CENTERLINE: yp = 6.0 [cm], Rw = 20. Axialvelocity profiles at steady state. There is no rotation when the particle is on the centerline.The fluid leads the particle US > 0. The long particle model disturbs the flow less than thecylinder.
0
10
20
30
40
50
60
0 2 4 6 8 10 12y [cm]
u [
cm
/s]
UndisturbedDNSShear slip model
WALL WALL
Figure 4.3. FIXED PARTICLE AT y’p = 3.0 [cm]: Rw = 20. Axial velocity profiles at steadystate. At the middle point between the centerline and the wall, the shear slip model gives aclose approximation to the DNS result. The particle’s angular rotation is approximated ashalf of the shear rate at the particle position on the undisturbed fluid velocity profile.

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0
10
20
30
40
50
60
0 2 4 6 8 10 12y [cm]
u [
cm
/s]
UndisturbedDNSShear slip model
WALL WALL
Figure 4.4. FIXED PARTICLE AT y’p = 0.75 [cm]: Rw = 20. Axial velocity profiles at steadystate. Close to the wall, the shear slip model gives an excellent approximation to the DNSresult.
0
10
20
30
40
50
60
0 2 4 6 8 10 12y [cm]
u
[cm
/s]
UndisturbedDNSShear slip model
WALL WALL
Figure 4.5. SMALL PARTICLE AT y’p = 0.75 [cm]: d = 0.5 [cm], Rw = 20. Axial velocityprofiles at steady state. As the particle is smaller, the difference between disturbed andundisturbed velocity profiles is smaller.

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5. Slip angular velocity discrepancyHere we shall show that the quantity that changes sign when the circular particle is on
one or the other side of the equilibrium (SegréSilberberg) position is the slip angularvelocity discrepancy
Ωs  Ωse
where Ωse is the slip angular velocity at equilibrium. We shall establish this result for aneutrally buoyant particle at a Reynolds number
102
==vd
R wwγ
We call this the benchmark case. Identical results may be obtained for other values of Rwbelow the first bifurcation.
� Unconstrained dynamic simulationIn figure 5.1 we show the trajectory of a neutrally buoyant circular particle that isreleased from the centerline and the wall of the channel. No matter where the particle isreleased it will migrate to a unique position of equilibrium at ye = 4.18 cm.
0
1
2
3
4
5
6
0 20 40 60 80 100 120t [s ]
yp
[c
m]
Starting at the CENTERLINE
Starting at the WALL
Figure 5.1. BENCHMARK CASE: Rw = 10, ρs /ρf = 1. Evolution of the particle’s verticalposition. The particle moves from the starting position to the final position without crossingthe equilibrium position.

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In figure 5.2 we show that the slip angular velocity is negative when the particle isabove the equilibrium position and is positive when it is below the equilibrium position.
0.01
0.1
1
10
0 20 40 60 80 100 120
t [s]
Ωs
[s1
]
Starting at the CENTERLINE
Starting at the WALL
Figure 5.2. BENCHMARK CASE: Rw = 10, ρs /ρf = 1. Evolution of the particle’s ANGULARslip velocity. The angular slip velocity function evolves without crossing the equilibriumvalue. When the angular slip velocity is BELOW the equilibrium value, the particle movesDOWNWARD. When the angular slip velocity is ABOVE the equilibrium value, theparticle moves UPWARD.
In figure 5.3 we show the evolution of the axial slip velocity to equilibrium. The axialslip velocity is positive and of course the greatest for a particle released from rest at thecenterline.
0
5
10
15
20
25
30
0 20 40 60 80 100t [s ]
Us
[cm
/s] Starting at the CENTERLINE
Starting at the WALL
Figure 5.3. BENCHMARK CASE: Rw = 10, ρs /ρf = 1. Evolution of the particle’s AXIAL slipvelocity. There is no clear relation between the particle’s vertical position and the axial slipvelocity.

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� Constrained simulationsIn these simulations we fix the position yp of the particle's center. The lift and
particle's forward and angular velocities are allowed to develop in an otherwiseunconstrained motion. At a fixed position ( )pyγ� is given by the undisturbed Poiseuilleflow.
In figure 5.4 and 5.5 we fix the particle's center at yp = 4.16 cm and 4.19 cm, belowand above the equilibrium position ye = 4.18. The lift force is such as to push the particleto ye (figure 5.4) and the slip angular velocity Ωs  Ωse is positive when yp > ye , andnegative when yp < ye .
0.3
0.25
0.2
0.15
0.1
0.05
0
0.05
10 30 50 70 90 110
t [s]
Fy [
dyn
]
FIXED position ABOVE equilibrium
FIXED position BELOW equilibrium
Figure 5.4. FIXED PARTICLE AROUND EQUILIBRIUM: Rw = 10, yp = 4.16 [cm], yp =4.19 [cm]. Evolution of the vertical force on the particle. The vertical force pulls the particletowards the equilibrium position. This result shows that this position is a STABLEequilibrium position.

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0.01
0.1
1
10 30 50 70 90 110
t [s ]
Ωs
[s1
]
FIXED position ABOVE equilibrium
FIXED position BELOW equilibrium
Figure 5.5. FIXED PARTICLE AROUND EQUILIBRIUM: Rw = 10, yp = 4.16 [cm], yp =4.19 [cm]. Evolution of the particle’s angular slip velocity. At STEADY STATE: When theangular slip velocity is BELOW the equilibrium value, the particle moves DOWNWARD(negative force). When the angular slip velocity is ABOVE the equilibrium value, theparticle moves UPWARD (positive force).
In figure 5.6 we fix the particle below and above the center line at y = 6 cm. Thecenterline is an unstable equilibrium. In figure 5.7 we look at the slip angular velocitynear the centerline. The magnitude of the slip angular velocity decreases as the particleangular velocity increases. The angular velocity Ωs asymptotes at large t to itsequilibrium value Ωse.

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0.06
0.04
0.02
0
0.02
0.04
0.06
0 20 40 60 80 100
t [s ]
Fy [
dyn
]FIXED position ABOVE centerline
FIXED position BELOW centerline
Figure 5.6. FIXED PARTICLE AROUND CENTERLINE: Rw = 10, yp = 5.95 [cm], yp = 6.05[cm]. Evolution of the vertical force on the particle. The vertical force pushes the particleaway from the centerline. This result shows that the centerline is an UNSTABLE equilibriumposition. If the particle starts at the centerline, it will stay there, but the slightest perturbationwill force the particle away.
0.06
0.04
0.02
0
0.02
0.04
0.06
0 20 40 60 80 100
t [s ]
Ωs
[s1
]
FIXED position ABOVE centerline
FIXED position BELOW centerline
Figure 5.7. FIXED PARTICLE AROUND CENTERLINE: Rw = 10, yp = 5.95 [cm], yp = 6.05[cm]. Evolution of the particle’s angular slip velocity. At STEADY STATE: When the angularslip velocity is BELOW the equilibrium value, the particle moves UPWARD. When theangular slip velocity is ABOVE the equilibrium value the, particle moves DOWNWARD. Thisbehavior is the OPPOSITE of the previous cases, because the previous cases wereSTABLE equilibrium positions, and therefore THE FORCE FIELD around them is theOPPOSITE.

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Figures 5.8 and 5.9 give the velocity and angular velocity of the long body from theshear slip model and the velocity and angular velocity of the circular cylinder from theDNS results. We can see that the difference between the shear slip model and the DNS isvery small. From figure 5.10 we compare the relative values of the slip velocity SU withrespect to the fluid velocity on the undisturbed flow at the particle’s center ( )cyu . Theslip velocity from DNS is quite flat for most of the region while the slip velocity fromshear slip model increases when it is closer to the wall, but the magnitude of thedifference is so small that we believe the shear slip model is perfectly in agreement withthe results of numerical computation of the motion of a free circular particle.
0
2
4
6
8
10
6 7 8 9 10 11 12yc [cm]
Ωp [
s1]
DNSShear slip model
CENTERLINE WALL
Figure 5.8.. FIXED PARTICLE: 20=wR . Particle’s angular velocity PΩ is approximated inthe shear slip model as half the value of the shear rate on the undisturbed flow evaluated atparticle’s center position ( ) 2cyγ�

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0
10
20
30
40
50
60
6 7 8 9 10 11 12yc [cm]
Up
[c
m/s
]DNSShear slip model
CENTERLINE WALL
Figure 5.9. FIXED PARTICLE: 20=wR . Particle’s velocity in the shear slip model vs. DNSparticle velocity. The slip model overpredicts in the region close to the centerline andunderpredicts in the region close to the wall.
0
5
10
15
20
25
30
35
6 7 8 9 10 11 12yc [cm]
US
/ u
(yc)
* 10
0 [%
]
DNSShear slip model
WALLCENTERLINE
Figure 5.10. FIXED PARTICLE: 20=wR . Slip velocity evaluated using DNS results vs. slipvelocity in shear slip model. These are relative values of the slip velocity, SU , with respect tothe fluid velocity on the undisturbed flow at particle’s center ( )cyu .
� Heavy particlesA position of equilibrium is a steady state of a free particle in which the lift is
balanced by the buoyant weight. For neutrally buoyant particles we may say that theposition of equilibrium is a "SegréSilberberg" position. In figure 5.11 we plotted the

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trajectory of a heavy particle with ρp=== 1.01 ρf ; no matter where it is released it willmigrate to a unique equilibrium ye = 1.02 cm, which is of course smaller than the value ye= 4.18 cm for a neutrally buoyant particle. In figure 5.12e show that the slip angularvelocity changes sign across the position of equilibrium even when the particle is heavy.
0
1
2
3
4
5
6
0 20 40 60 80 100t [s ]
yP
[c
m]
Starting at the CENTERLINE
Starting at the WALL
Figure 5.11 EAVY PARTICLE: Rw = 10, ρs /ρf = 1.01. Evolution of the particle’s verticalposition. The particle starting at the centerline crosses the equilibrium position and thenmoves upward.
0
1
2
3
4
5
0 20 40 60 80 100t [s ]
Ωs=
[s
1 ]
Starting at the CENTERLINE
Starting at the WALL
Figure 5.12 HEAVY PARTICLE: Rw = 10, ρs /ρf = 1.01. Evolution of the particle’s angularslip velocity. For the particle starting at the centerline, the angular slip velocity functioncrosses the equilibrium value. When the angular slip velocity is BELOW the equilibriumvalue, the particle moves DOWNWARD. When the angular slip velocity is ABOVE theequilibrium value, the particle moves UPWARD.

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6. Forces and torques on a neutrally buoyant accelerating free particleWe calculated the forces on a freely moving neutrally buoyant particle by computing
the integrals of the traction vectors in (3.2) and the moment of thee vectors in (3.4), anddisplayed their values as a function of time for the benchmark case in figure 6.1, 6.2 and6.3. These graphs can be compared with the particle trajectory and slip velocities given insection 5. There is no obvious relation between the forces and torques on the one handand the slip velocities on the other.
0.004
0.003
0.002
0.001
0
0.001
0.002
0 20 40 60 80 100 120t [s ]
Fy [
dyn
],
T
[d
yn.c
m]
Vertical force [dyn]Torque [dyn.cm]
Figure 6.1. BENCHMARK CASE: 10=wR , 1=fs ρρ . Evolution of the vertical force andtorque acting on a particle starting close to the CENTERLINE. There is no clear relationbetween the slip parameters and the vertical force.

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0.03
0.02
0.01
0
0.01
0.02
0.03
10 30 50 70 90t [s ]
Fy [
dyn
],
T
[d
yn.c
m]
Vertical force [dyn]Torque [dyn.cm]
Figure 6.2. BENCHMARK CASE: 10=wR , 1=fs ρρ . Evolution of the vertical force andtorque acting on a particle starting close to the WALL. There is no clear relation between theslip parameters and the vertical force.
0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 20 40 60 80 100 120t [s ]
Fx [
dyn
]
Starting at the CENTERLINE
Starting at the WALL
Figure 6.3. BENCHMARK CASE: 10=wR , 1=fs ρρ . Evolution of the horizontal forceacting the particle. The horizontal force acting on a particle starting close to the WALL isalways positive, or in other words the particle is dragged from the wall to the equilibriumposition. The force for the particle starting from the CENTERLINE is first positive and thennegative, because the particle is moving from a higher fluid velocity region to a lowerfluidvelocity region.

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7. Summary and conclusionThe lift and migration of neutrally buoyant and heavierthanliquid circular particles
in a plane Poiseuille flow was studied using direct numerical simulation. The study looksat the relation of slip velocity and angular slip velocity to lift and migration. No matterwhere the particle is released, it will migrate to a unique equilibrium height and moveforward with a unique steady particle velocity and rotate with unique steady angularvelocity. Neutrally buoyant particles migrate to a radius which can be called the "SegréSilberberg" radius. This radius is a reference; heavierthanliquid particles also migrate toan equilibrium radius that is close to the SegréSilberberg radius if the particle density isclose to the fluid density. The particles migrate to an equilibrium position ye with shearrate eγ� such that the local fluid rotation 2/eγ� is slightly greater than the particle angularvelocity Ωp. The angular slip velocity, pes Ω−=Ω 2/γ� is always positive but atequilibrium it is very small; 2/ep γ�≈Ω can be proposed as an approximation. The slipvelocity at equilibrium pees UyU −= γ� is always positive and slowly varying.
Since the shear rate and slip velocities are one signed they do not explain why the liftchanges sign across the equilibrium radius. We found that the quantity which doeschange sign at ye is the angular velocity discrepancy; the angular velocity minus theequilibrium angular velocity Ωs  Ωse. Ωs  Ωse > 0 when y < ye and Ωs  Ωse < 0 when y >ye. The adjustment of the angular velocity of a free particle is very critical to lift. Onemight think of the angular velocity discrepancy as a shear flow analogue to thecirculation in aerodynamic lift.
We derived a shear slip version of our long particle model. The long particle modelarises when the circular particle is replaced with a long rectangle of the same diameter asthe circle, but so long that we may neglect end effects. In the shear slip model we allowthe rectangle to shear at the rate γ� /2 of the local rotation. Using this model we can findexplicit expressions for the fluid rotation in which the velocity on either side of the longparticle is matched by the fluid velocity; then we satisfy the particles equation of motionin which the shear stress force balances the pressure drop force. This leads to explicitexpression for the velocity of the particle (4.12) and the slip velocity (4.16) that is alwayspositive. The shear slip model is in astonishing agreement with the results of numericalcomputation of the motion of a free circular particle, both with respect to the particlevelocity and the fluid velocity on the cross section containing the center of the circularparticle.
Acknowledgement
This work was partially supported by the National Science Foundation KDI/NewComputational Challenge grant (NSF/CTS9873236); by the US Army, Mathematics; bythe DOE, Department of Basic Energy Sciences; by a grant from the Schlumbergerfoundation; from Stimlab Inc.; and by the Minnesota Supercomputer Institute.

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List of symbols
:d distance between particle’s top and bottom walls.:LF lift force.:xF horizontal force acting on the particle.:yF vertical force acting on the particle.:Ah distance between channel’s top wall and particle’s top wall.:Bh distance between channel’s top wall and particle’s top wall.
:p absolute value of the pressure gradient.:t time.
:Au fluid velocity above particle’s top wall.:Bu fluid velocity below particle’s bottom wall.:cu cylinder’s axial velocity.:fu cylinder’s axial velocity.:su axial slip velocity.:U free stream velocity.
:AU particle’s top wall velocity.:BU particle’s bottom wall velocity.
:y vertical coordinate pointing upward.:y′ vertical coordinate pointing downward.:py particle’s vertical coordinate.
� Greek letters:η dynamic viscosity.:γ� shear rate.
:sΩ angular slip velocity.:Γ circulation around cylinder.
:cΩ cylinder’s angular velocity.:sΩ angular slip velocity.:e
sΩ angular slip velocity at equilibrium.
:Aτ shear stress on particle’s top wall.:Bτ shear stress on particle’s bottom wall.

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