Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://new.math.msu.su/department/hydro/papers/Asmolov-Osiptsov-paper-PoF-2009.pdf Äàòà èçìåíåíèÿ: Wed Jun 9 16:28:17 2010 Äàòà èíäåêñèðîâàíèÿ: Sun Apr 10 01:50:34 2016 Êîäèðîâêà:

PHYSICS OF FLUIDS 21, 063301 2009

The inertial lift on a spherical particle settling in a horizontal viscous flow through a vertical slot
Evgeny S. Asmolov1,2,a and Andrei A. Osiptsov1
1 2

Department of Geomechanics, Schlumberger Moscow Research, Moscow 101000, Russia Central Aero-Hydrodynamics Institute, Zhukovsky, Moscow Region 140180, Russia

Received 10 September 2008; accepted 29 April 2009; published online 4 June 2009 The inertial lift force exerted on a small rigid sphere settling due to gravity in a horizontal channel flow between vertical walls is investigated. The method of matched asymptotic expansions is used to obtain solutions for the disturbance flow on the length scales of the particle radius and the channel width inner and outer regions, respectively . The channel Reynolds number is finite, while the particle Reynolds numbers that are based on the slip velocity and the mean shear rate are small. The inner flow is described by the linear Stokes equations. The outer problem is governed by the linear Oseen-like equations with the particle effect approximated by a point force. The outer equations are solved numerically using the two-dimensional Fourier transform of the disturbance velocity field. The lift coefficient is evaluated as a function of governing dimensionless parameters: the particle coordinate across the channel, the channel Reynolds number, and the slip parameter. The particle always migrates away from the walls, with an equilibrium position being on the channel centerline. Close to the walls, the lift coefficient is the same regardless of the slip velocity and the channel Reynolds number. At large channel Reynolds numbers, a local maximum of the migration velocity forms near the channel centerline due to a combined effect of the slip, the linear shear, and the curvature of the undisturbed velocity profile. The results obtained are extended to the case when the drag on a particle has components both parallel and perpendicular to the undisturbed flow. One of primary applications of the results is modeling of the cross-flow migration of settling particles during particle transport in a hydraulic fracture. © 2009 American Institute of Physics. DOI: 10.1063/1.3148277
I. INTRODUCTION

The migration of a single particle settling under gravity at small Reynolds numbers in shear flows of Newtonian fluids was considered in many theoretical studies.1­13 The inertial lift on a sphere was calculated for different flow configurations, either unbounded or wall bounded, either linear shear or channel flows, and a particle slip velocity directed either along the fluid velocity or along the shear gradient. In the present study, we consider the case when a particle settles in a horizontal viscous flow through a vertical slot Fig. 1 . Apparently, the lift force on a particle in the present geometrical configuration has not been studied before. The problem is important in view of possible applications in chemical engineering and petroleum technology. For example, the flow configuration shown in Fig. 1 corresponds to the transport of proppant particles in a hydraulic fracture, where the cross-flow migration is caused by the lateral force exerted on individual settling particles. It is known that lateral migration of particles may have a significant effect on the overall settling rate.14 The settling-induced lift force considered in the present study can be used in a model of particle migration across fluid streamlines in a hydraulic fracture. The crossfracture particle concentration profile may then be incorporated into the models of particle transport, which are implea

Author to whom correspondence should be addressed. Electronic mail: easmolov@slb.com.

mented into existing commercialized simulators of hydraulic fracturing.14 The migration of a particle moving with a nonzero slip velocity in a shear flow of a viscous fluid is due to a small fluid inertia. When the particle Reynolds number, Rs = aUs / , is small, the inertial term in the Navier­Stokes equations is small compared to the viscous one at distances of the order of the particle radius a from the particle center in the inner region . Here, Us is the dimensional magnitude of the slip velocity velocity of the sphere center relative to an undisturbed flow velocity at the point , is the kinematic viscosity of the fluid, and the prime superscript is used hereinafter to denote the dimensional velocities and spatial coordinates. To the leading order in Rs, the disturbance flow in the inner region is governed by the Stokes equations. However, the inertial term decays with the distance slower than the viscous one. These terms are of the same order at sufficiently large distances from the sphere in the outer region . For the uniform undisturbed flow, the length scale of the outer region, where the two terms balance, is the Oseen length LOs = a / Rs = / Us a. Theoretical studies of particle migration due to the inertial effect were based on solving the Navier­Stokes equations using perturbation methods. A regular perturbation technique can be used when the distance of the particle from the wall is small compared to the length scale of the outer region, i.e., it is assumed that the wall lies within the inner region of the flow. Cox and Brenner3 considered the migra© 2009 American Institute of Physics

1070-6631/2009/21 6 /063301/9/$25.00

21, 063301-1

Downloaded 05 Jun 2009 to 134.32.190.77. Redistribution subject to AIP license or copyright; see http://pof.aip.org/pof/copyright.jsp


063301-2

E. S. Asmolov and A. A. Osiptsov

Phys. Fluids 21, 063301 2009

FIG. 1. The sketch of the flow pattern near a particle settling in a viscous flow through a vertical slot.

tion of a non-neutrally buoyant sphere translating in a linearshear flow at distances from the wall d being large compared to the particle radius, a d LOs. They expressed the lift in terms of Green's functions. The lift was evaluated in linear and parabolic wall-bounded flows.4 ­ 6 The regular perturbations were also applied to the case when d a.10 For an unbounded linear-shear flow, another length scale of the outer region, LSa = / G 1/2, based on the shear rate G, was introduced.1 Saffman1 showed that the particle effect in the outer region is equivalent to a point force exerted on the fluid at the center of the sphere. Oseen-like equations governing the outer disturbance flow were solved in terms of Fourier transforms. The disturbance induced by the sphere in the outer region produces a small lateral correction to a uniform flow in the inner region. The migration velocity in the shear direction was calculated for a particle moving parallel to the streamlines of an unbounded flow in the strong-shear limit, when the two small particle Reynolds numbers based on the slip velocity and the shear rate are related as R
s

R

1/2 p

= a G/

1/2

1.

1

is large, the wall effect is significant in thin layers near the walls of a width of the order of lR-1/2.13 In the core of the c flow, excluding near-wall layers, the wall effect is negligible, and the disturbance field in the outer region can be treated as an unbounded one. The effect of the curvature of the velocity profile on the lift is significant. The goal of the present work is to calculate the migration velocity of a non-neutrally buoyant particle settling in a horizontal viscous flow bounded by vertical impermeable walls. One would expect that the wall effect at large Rc is significant only in the thin layers near the slot walls. In the bulk of the flow, the lift is expected to be small for the configuration considered, since it is zero in an unbounded linear flow. However, it will be shown below that a local maximum of the migration velocity appears at distances of the order of R-1/3 from the channel center due to a combined effect of the c slip, the linear shear, and the curvature of the undisturbed velocity profile. The lift coefficient is of the same order over the entire channel width. This means that the scaling we use to nondimensionalize the lift force is applicable everywhere in the flow domain, both near the walls and in the core of the flow. The paper is organized as follows. Governing equations of the viscous flow past a small sphere settling in a horizontal slot flow are presented in Sec. II. Solutions in the inner and outer regions are constructed using the method of matched asymptotic expansions in Secs. III and IV, respectively. The outer solution is presented in terms of a twodimensional Fourier transform. An ordinary differential equation for the Fourier transform of the normal velocity is integrated numerically using an orthonormalization method. Numerical results and a comparison with previous studies are presented in Sec. V. The extension of the approach for arbitrary direction of the slip velocity is presented in Sec. VI. The results are summarized in Sec. VII.

This inequality means that the linear-shear flow dominates the uniform flow in the outer region. In the case when the sphere moves in an arbitrary direction to the shear flow, other components of the migration velocity were also found for the strong-shear case.2,12 The lift is zero in an unbounded linear flow when the slip velocity is perpendicular both to the fluid velocity and its gradient, as for the fracture configuration. Asmolov8 and McLaughlin9 removed Saffman's restriction 1 on the relative sizes of the two Reynolds numbers and considered a general case of an arbitrary ratio of the Reynolds numbers. They evaluated the migration velocity of a sphere translating parallel to the streamlines of an unbounded shear flow as a function of the slip parameter = Rs / R1/2. The velocity is maximum in the strong-shear p limit when 1 and is very small, of the order of -5 ln , 1. in the opposite weak-shear limit, Hogg11 and Asmolov13 studied the inertial migration of a non-neutrally buoyant particle settling in a vertical channel flow. In this case, the migration is due to the following two effects: the Oseen-like inertial migration and the wall effect. The lift coefficient and the particle equilibrium positions were calculated as the functions of the slip parameter and the channel Reynolds number Rc = Uml / , where l and Um are the channel width and the maximum flow velocity. When Rc

II. GOVERNING EQUATIONS

Consider a rigid sphere settling due to gravity in a flow of a viscous incompressible Newtonian fluid through a vertical slot. The sketch of the flow configuration is shown in Fig. 1. The undisturbed flow in a slot may have not only a horizontal but also a vertical component. The origin of a Cartesian coordinate system is at the center of the sphere and translates with the particle velocity U p. Hence, the particle is at rest in this frame of reference. The x-axis is parallel to the slot walls and directed along the undisturbed flow, and y -axis is perpendicular to the walls. The fluid velocity relative to the sphere has generally x-and z-components, Usx and Usz, so 2 that Us = Usx + Usz2. The particle radius a is small compared to the slot width l. The particle Reynolds number and the length scale of the outer region based on the average shear rate, Gav = Um / l, are introduced by R p = U ma 2/ l , L = aR
-1/2 p

=

l/U

m

1/2

= lR

-1/2 c

.

The method of matched asymptotic expansions, based on a small parameter

Downloaded 05 Jun 2009 to 134.32.190.77. Redistribution subject to AIP license or copyright; see http://pof.aip.org/pof/copyright.jsp


063301-3
1/2 p

The inertial lift on a spherical particle

Phys. Fluids 21, 063301 2009
-1/2 c -1/2 c

=R

= a U m/ l

1/2

1,

u =- v + R
p

p

r =- Uszez + R
x

is applied to solve the equations governing the disturbance flow. The ratio of the particle size to the channel width and the particle Reynolds numbers based on the particle slip velocity and the local shear rate are also small, a / l = R p/ R
c 1/2

r - y ex +4 2y 2R-1e c
-1 1/2 Rc

on r =1 , 1- d/l , 4

u=0 u0

on y =-

d/l,

-1 1/2 Rc

=R

-1/2 c

1,

R

s

1.

as r .

The other dimensionless groups, Usx = Usx / Um, Usz = Usz / Um, d / l, Rc are taken to be of the order of unity, where d is the distance of the particle to the nearest wall. Flow disturbances due to drag on the particle dominate those 2 11 . due to shear and particle rotation, when Us 13 The asymptotic analysis follows the study of the inertial migration in a vertical channel for the case when the slip velocity is parallel to the undisturbed flow. The novelty of the present study is that the undisturbed velocity of the fluid in the absence of the sphere has two components, v = vx ex + vz ez ,
vx = Usx + U

III. INNER SOLUTION

We seek the dimensionless disturbance velocity and pressure fields in the form of asymptotic series in , u = u0 + u1 + o , p = p0 + p1 + o .

Substituting these expansions into Eqs. 3 and 4 in the leading-order approximation, we obtain the Stokes equations in an unbounded flow,
2

u0 - p0 =0 ,

5 6

· u0 =0 , y y -4 l l
2 2

m

,

vz = Usz ,

u0 =- Usze u0 0

z

on r =1 ,

7 8

where ex and ez are the unit vectors of the x- and z-axes, respectively, and =4-8d / l is the dimensionless shear rate at the particle center. To the leading order in , the particle translates parallel to the walls, and its migration velocity in the normal direction due to the fluid inertia and the wall effect is small compared to the slip velocity. Hence, the problem can be treated as a quasisteady one. The dimensionless variables are introduced by the formulas r = r /a, u = u / U m,
p

as r .

It should be noted that the boundary condition 7 involves the uniform velocity only. The terms in Eq. 4 corresponding to the shear rate and the particle angular velocity are of order , and the term due to the curvature of the undisturbed velocity profile is of order 2. Hence, these terms should be omitted in the leading-order approximation. The solution of Eqs. 5 ­ 8 is the well-known Stokes solution, u0 =- U
sz

=

p

l / U m,

F p = F p/ aUm ,

e

where u is the disturbance of the fluid velocity and p and F p are the dimensionless rotational velocity and the force on the particle, respectively. The dimensionless undisturbed velocity field is given by v = v /Um = Usx + yR
-1/2 c

z

1 3 + 4r 4r

3

+

3 zr 1 1 - 4 r2 r r3

.

This axisymmetric solution gives the drag on the sphere, F p =6 Uszez, and yields no lateral force or torque. The velocity field decays with distance from the sphere as r-1, u0 u
S

-4 2y 2R

-1 c

ex + Uszez .

2

as r ,
sz

We first consider in detail the case when the particle settles under gravity in a horizontal undisturbed flow, so that the slip velocity is perpendicular to the fluid velocity. Then the x-component of the force on the particle is zero. It follows from Faxen formula that the corresponding slip velocity is Usx = O 2 , and, to the first order, may be neglected. The extension of the analysis to the case Usx 0 is presented in Sec. VI. The dimensionless Navier­Stokes equations governing the disturbance flow are written as R
1/2 c

3 uS =- U 4

ez zr + . r r3

The Stokeslet velocity field uS corresponds to the viscous flow driven by the point force, F f = F f ez =- F p =-6 Uszez . Collecting the terms of order in the asymptotic expansions of Eqs. 3 and 4 and taking into account Eq. 2 , we derive the equations governing the first-order inner solution,
2

u·

u+ v·

u+ u·

v =- p +

2

u, 3

u1 - p1 = R

1/2 c

u0 + Uszez ·

u0 ,

9 10

· u =0 . The boundary conditions are specified as follows. The no-slip condition is prescribed on the particle surface and the channel walls, and it is assumed that the velocity disturbance decays far from the particle,

· u1 =0 , u1 = R
-1/2 c p

r-y e

x

on r =1.

11

Note that even to this order of , the curvature of the undisturbed velocity profile does not enter into the inner-flow

Downloaded 05 Jun 2009 to 134.32.190.77. Redistribution subject to AIP license or copyright; see http://pof.aip.org/pof/copyright.jsp


063301-4

E. S. Asmolov and A. A. Osiptsov

Phys. Fluids 21, 063301 2009
-1/2 2 cY 1/2 c

equations. Since Eqs. 9 ­ 11 are linear, the solution of these equations can be sought as the superposition of three terms,13 u 1 = R 1/2u c
PP

V = Vx,0, Vsz ,

Vx = Y -4R

,

Vsz = UszR

. 14

+ unb + w .

12

The boundary conditions on the particle surface for the three terms are u
PP

= 0,

unb = R on r =1.

-1/2 c

p

r - y ex ,

w=0

The first term in Eq. 12 is the solution of inhomogeneous equations 9 and 10 for a uniform undisturbed flow obtained by Proudman and Pearson,15 u
PP

The slip parameter Vsz characterizes the relative size of the slip velocity and the linear-shear term in the undisturbed velocity. Hence, this parameter is analogous to the ratio of the particle Reynolds numbers = Rs / R1/2 used in the studies p of particle migration in linear-shear flows.8,9 The strongshear limit corresponds to the case 2 Vsz 1. The matching condition with the inner region can be satisfied if the point force F f is introduced into the momentum equation.1 Therefore, the equations governing the outer field are reduced to the linear Oseen-like equations, V
x

3 =U 32

2 sz

3111 2- + 2 - 3 + 4 rr r r z 2r z e - r3 r
z

3z2 r 1- 2 rr .

U +V X

sz

U dVx + U yex + P - Z dY
sz

2

U

=-6 U · U =0 , U =0 U =0 on

R ez ,

312 + 4- + 3 - 4 rr r

Two other terms satisfy homogeneous Stokes equations 5 and 6 . The term unb is the solution for a force- and torque-free neutrally buoyant particle in an unbounded linear-shear flow. It accounts for the effect of particle rotation in a shear flow. The last term, w r , is the solution of Eqs. 5 and 6 , which tends to a uniform velocity w at infinity. The first-order solution u1 does not decay with r, and the boundary condition at infinity has to be replaced by a matching condition with the outer flow. The outer limit of the inner solution is u
1 r

Y =- R1/2d/l, c

R

1/2 c

15 1- d/l ,

as R .

To solve the outer-flow equations, the two-dimensional Fourier transforms U , P of the disturbance field are introduced as U k x, Y , k P k x, Y , k
z

=

1 4
2 - -

z

U P

=u

PP

r/r + w , 13

exp - i kxX + kzZ dXdZ . 3z2 r z 2r z e +2 3 - r2 r r r The velocity and pressure fields are then given by the inverse Fourier transforms, U P =
- -

u

PP

=u

PP

r

=

3 U 16

2 sz

1-

z

.

The first two terms in Eq. 12 give no contribution to the migration velocity because of the symmetry. Hence, the migration is due to the transverse component of the velocity w , which should be found from the matching condition with the outer flow. The migration velocity of an inertialess particle is Vm = wy .
IV. OUTER SOLUTION

U P

exp i kxX + kzZ dkxdkz .

16

The Oseen-like equations 15 are rewritten in terms of the Fourier transforms as follows: ikxVx + ikzVsz U + =- 3 U 2 dVx U ex + dY y P- U

sz

Y ez ,

The outer coordinates are introduced by the scaling R = X , Y , Z = r = r / L. The velocity and pressure in the outer region can be presented as u= U+o , U = U x, U y , U z , p=
2

· U =0 , 17 = ikx, U =0 d , ikz , dY = d2 - k 2, dY 2 R
1/2 c 2 k2 = k2 + kz , x

P+o

2

.

As the disturbances are small compared to the undisturbed flow in the outer region, the inertial terms in the Navier­Stokes equations 3 can be linearized, R
1/2 c

on

Y =- R1/2d/l, c

1- d/l .

u·
2

u+ v· V·

u+ u·

v
3

=

U+ U·

V +O

.

Here, the components of the undisturbed flow velocity are written in terms of the outer coordinates as follows:

Thus we obtain the system of linear ordinary differential equations with respect to Y . Multiplying the x-component of the momentum equation 17 by ikx, the z-component by ikz, adding the resulting equations, and using the equation of continuity, one obtains the value of P as

Downloaded 05 Jun 2009 to 134.32.190.77. Redistribution subject to AIP license or copyright; see http://pof.aip.org/pof/copyright.jsp


063301-5

The inertial lift on a spherical particle

Phys. Fluids 21, 063301 2009

P=

1 k2 +

- ikxVx - ikzV Y .

sz

dUy dVx U + ikx dY dY

y

Vm =

Re
- -

Uy - U

S y

Y 0

dkxdk

z

.

19

3 ikzU 2

sz

Differentiating the last equation with respect to Y , substituting the resulting expression into the y -component of the momentum equation, one can derive a single ordinary differential equation governing the Fourier transform of the lateral velocity,
2

- ikxVx + ikzV

sz

- ikx8R

-1/2 c

U

y

=-

3 dY , ikzUsz dY 2 18 on Y =- R1/2d/l, c R
1/2 c

dUy =0 Uy = dY

1- d/l .

Thus, the problem is reduced to finding Uy kx ,0, kz from the numerical solution of Eq. 18 . The numerical integration of the finite-difference version of Eq. 18 presents a certain difficulty at large Rc. The routine technique fails to converge in this case, since it does not permit to resolve properly all the linearly independent solutions of the ordinary differential equation. Therefore, the technique does not allow one to calculate the lift for channel Reynolds numbers larger than approximately 100.11 The problem is eliminated in the present work similar to the study13 using the orthonormalization method,16 which makes it possible to integrate Eq. 18 up to kmax = 128 for any Rc. For large k, the asymptotic approximation of the Fourier transform similar to that of the work11 can be obtained, Uy kx,0, kz - U =U
sz S y

kx,0, k

z -5

The term on the right-hand side of Eq. 18 is equivalent to a jump condition for the second derivative at the origin of the coordinate system, d 2U y 3 =- ikzUsz , 2 dY 2 where f = f +0 - f -0 is the magnitude of the jump. The function Uy and its first and third derivatives are continuous at the origin. These features of Uy can be proven if one integrates Eq. 18 successively four times over a small interval - Y , Y , Y 0, and solves a linear system for d i Uy / dY i , i =0­3. Matching of the outer field back to the inner flow requires the outer limit of the two-term inner expansion, u0 + u1, to be equal to the outer solution at the origin. As a result one has for the outer limit of the first-order term Eq. 13 ,1 u
1 r

3 k xk z +O k 16 k5

as k .

20

The first term on the right side of Eq. 20 is an odd function of kx and kz, and its integral over kx is zero. For this reason, the contribution of large k to Eq. 19 is small, of the -3 order of kmax. The numerical integration of Eq. 19 is fulfilled using the plane polar coordinates k, = arccos kx / k instead of kx, kz. It can be verified directly from Eq. 18 that Uy kx, Y , kz = Uy - kx, Y ,- kz , where the overbar denotes the complex conjugated value. Hence, it is sufficient to integrate Eq. 19 over the first and second quadrants.

V. NUMERICAL RESULTS

= U-U

S

R0

,

where US is the Stokeslet velocity field written in terms of the outer coordinates, 3 US =- U 4
sz

ez ZR + , R R3

which matches with the outer limit of u0. Small distances R correspond to large k, and the matching condition means that the numerical solution of Eq. 18 should be equal to the two-dimensional Fourier transform of the Stokeslet at large Fourier numbers, Uy = U where U
S y S y

The lift is obtained numerically as a function of the three main dimensionless groups: the particle position across the slot d / l, the slip parameter Vsz, and the channel Reynolds number Rc. For the sake of illustration, we consider wide ranges of variation of Vsz and Rc, which cover the magnitudes relevant to fracturing applications: 0.001 Vsz 4, 0.1 Rc 1000. The results are compared with the predictions of the work5 for a small sphere settling between parallel walls in a stagnant fluid. Vasseur and Cox5 considered the case when the distances from the walls are also much larger than the sphere radius but still smaller than the outer-region length scale, so the walls lie within the inner region. Then the migration velocity is given by5 V
VC m

1+ O k

-1 7

as k

1,

= RsUszc

VC m

.
VC m

is given by

3 ikx UszY exp - k Y . US = y 8k The migration velocity Vm of an inertialess particle is expressed in terms of the Fourier transform of the lateral velocity at the origin as1

The lift coefficient c polynomial c
VC m 9

d / l can be approximated by the
3

= - 0.3125 - 0.0996 + 403.7 - 518
11

+ 14.34

5

- 116.3

7

,

= d/l -1/2.

Downloaded 05 Jun 2009 to 134.32.190.77. Redistribution subject to AIP license or copyright; see http://pof.aip.org/pof/copyright.jsp


063301-6

E. S. Asmolov and A. A. Osiptsov

Phys. Fluids 21, 063301 2009

The lift is directed away from the walls. The maximum VC value of the lift coefficient is attained on the wall: cm d / l 0 =3 / 32. The particle equilibrium position where the lift vanishes is on the channel axis, at d / l = 0.5. For a particle settling in a horizontal channel flow, we found that the migration velocity should be scaled in a similar manner, Vm = R1/2UszVszcm = RsUszcm , p or, in the dimensional form, Usz2a cm d/l, Vsz, Rc . 21

Vm =

The quadratic dependence of the migration velocity on the slip velocity differs from the linear scaling obtained for the case when the slip velocity is parallel to the undisturbed flow.13 This quadratic scaling can be deduced from the behavior of the Fourier transform of the lateral velocity, which, in turn, follows from the solution of Eq. 18 for the strongshear case, Vsz 1. The equation is linear; hence the solution is proportional to the right-hand side, Uy kzUsz. The slip parameter Vsz enters into the term in the round brackets on the left-hand side multiplied by kz. All other terms are even functions of kz as they involve kx and k2 only. For this reason, the solution can be sought in the form of series in Vszkz when the slip parameter is small, U y = k zU
sz 0

kx, k2 + Vszk

z1

k x, k 2 + ¯

as V

sz

1.

The leading-order term in the expansion is linear in kz and, hence, its contribution to the integral 19 is zero. It also follows from the inverse Fourier integral 16 that the corresponding velocity field is an odd function of Z. Hence, the 2 first-order term, kz UszVsz 1, only contributes to Eq. 19 . As a result, Vm UszVsz as Vsz 1, and we obtain the scaling 21 for the migration velocity. The lift coefficient cm d / l , Vsz , Rc is shown in Fig. 2 as a function of the cross-channel particle coordinate for differVC ent Vsz and Rc in comparison with cm d / l .5 The curves are presented only for a half of the channel 0 d / l 0.5 since they are antisymmetric with respect to the channel centerline. The lift coefficient is positive for all values of the governing dimensionless parameters, i.e., the particle always migrates from the walls toward the channel centerline. Near the walls, the lift coefficient coincides with the predictions by Vasseur and Cox5 given by cm d / l 0, Vsz , Rc =3 / 32. The migration velocities, calculated for different small but nonzero values of the slip parameter, 0.001 Vsz 0.1 and scaled using Eq. 21 , tend to the same dependence solid curves in the limit Vsz 0 the strong-shear case . For Rc 1, the undisturbed flow affects the migration only slightly. It can be seen in Fig. 2 a that the dependence of the lift coefficient on the particle position calculated for different slip parameters, 0.001 Vsz 4, is very close to the results for migration of a particle settling in a stagnant fluid.5 The lift coefficient decreases with the distance from the wall when Rc 10 Figs. 2 a and 2 b . At large Rc, the wall

FIG. 2. Lift coefficient vs particle coordinate across the slot and comparison with the predictions by Vasseur and Cox Ref. 5 : a Rc =1, b Rc =10 c Rc = 100, and d Rc = 1000.

effect manifests itself only in the thin layers near the walls at distances of the order of the outer-region length scale, L = lR-1/2. c One would expect that the migration velocity is small at Rc 1 in the core of the flow where the wall effect is negligible since it is zero in an unbounded linear flow.2 However,

Downloaded 05 Jun 2009 to 134.32.190.77. Redistribution subject to AIP license or copyright; see http://pof.aip.org/pof/copyright.jsp


063301-7

The inertial lift on a spherical particle

Phys. Fluids 21, 063301 2009

this is valid only when Vsz 2. At smaller slip parameters and large channel Reynolds numbers, cm is finite in the core of the flow, and an extra maximum forms near the channel centerline see solid and dashed curves in Figs. 2 c and 2 d . The origin of the extra maximum cannot be explained by the wall effect, which vanishes with the increase in the distance from the wall; also, the shear rate is small near the axis. However, the curvature of the undisturbed profile is finite and another length scale of the outer region can be introduced based on the curvature, Lc = lR-1/3, and Eq. 18 c governing the Fourier transform of the lateral velocity is then rewritten as
c2 c - ikcVc + ikz V xx c sz c

-8ikc U x

c y

=-

3 c d Yc ikz Usz , dY c 2 on Y c =
c2

22

Uc = y

dUc y =0 dY c

R1/3/2- , c
2/3 c

Vc =4 2 Y c - Y x

,

c Vsz = UszR

,
-1/6 c

23 Y, k c = R 1/6k . c

= 1/2- d/l R1/3, c

Y c = y /Lc = R

c When the new slip parameter Vsz is fixed, Eq. 22 does not involve Rc. The distances to the walls scaled by Lc are large when Rc is large and the particle is close to the channel center the stretched distance is of order unity . This means that the wall effect is negligible in this region and the outer flow can be treated as an unbounded parabolic flow. As a result, the solution of Eq. 22 and the lift coefficient scarcely depend on Rc. Figure 3 presents cm as a function of c obtained for given Vsz and various channel Reynolds numbers: Rc = 100 , 300 , 1000. The dependences calculated for 1. Thus, the lift different Rc are close to each other when near the axis at large Rc is due to the combination of the slip and the unbounded linear and quadratic shear rates. The mi1. The migration vegration velocity is linear in when 1, i.e., when the locity reaches the maximum value when linear and quadratic terms in the undisturbed profile 23 become comparable.

FIG. 3. Lift coefficient near the channel centerline at large Rc vs dimension1 less distance from the axis = 1 / 2- d / l Rc /3. a Strong-shear limit: c c c Vsz 1, b Vsz = 2, and c Vsz =4.

VI. ARBITRARY DIRECTION OF THE SLIP VELOCITY

We have considered above the inertial migration of a particle settling in a horizontal flow in a vertical slot when the slip velocity is perpendicular to the Poiseuille flow of the carrier fluid. The restriction can be removed and the results obtained can be extended to the case when the drag on a particle has both horizontal and vertical components, F p =- F f = Fxex + Fzez =6
11 13

Usxex + Uszez .

24

Hogg and Asmolov studied the case when the drag on a sphere is directed along the fluid flow, F p = Fxex =6 Usxex. The migration velocity is linear in the slip velocity,
x Vm = R1/2Usxcm d/l, Usx, Rc . p

25

The results obtained numerically in the present work and in the studies11,13 can be combined for the strong-shear case.

Solution in the inner region is constructed in the general case similar to Sec. III. The leading-order term and the first term in the first-order solution 12 are again the solutions obtained by Stokes, u0, and by Proudman and Pearson,15 uPP, respectively, but corresponding to another uniform undisturbed velocity, Usxex + Uszez. The second term in Eq. 12 is the same and accounts for the effect of particle rotation in a shear flow. All the terms give no contribution to the lateral force because of the symmetry, and the migration is again due to the transverse component of the velocity w , which is found from the matching condition with the outer flow. The particle effect on the outer disturbance flow is equivalent to the point drag force on the particle, and Eq. 15 can be rewritten with the right side followed from Eq. 24 : -6 Usxex + Uszez R . Since the Oseen-like equations 15

Downloaded 05 Jun 2009 to 134.32.190.77. Redistribution subject to AIP license or copyright; see http://pof.aip.org/pof/copyright.jsp


063301-8

E. S. Asmolov and A. A. Osiptsov

Phys. Fluids 21, 063301 2009

are linear, the numerical solutions corresponding to each drag component can be constructed separately. However, the slip velocities also enter into the expression for the velocity of the undisturbed flow Eq. 14 , which should be written for the general case as V = Vsx + Y -4R Vsx = UsxR
1/2 c -1/2 2 cY

ex + Vszex ,

.

Hence, the solutions corresponding to each drag component will depend on both Vsx and Vsz when the slip velocities are finite. However, for the strong-shear case, Vsx 1, Vsz 1, the effect of Vsx on the part of the solution proportional to the vertical drag Fz can be neglected and vice versa. Then, combining Eqs. 21 and 25 , we obtain the following formula for the migration velocity: Vm = R
1/2 p x Usxcm + UszVszc z m

as V

sx

1, V

sz

1. 26

It should be noted that the two terms in Eq. 26 can be sizable even for the case when the x-component of the slip velocity is small compared to the z-component Usz 1. The reason is that the first term in Eq. 26 Usx is linear in the slip velocity, while the second one is quadratic. The cross-flow migration of particles in dilutesuspension flows is important in a variety of chemical engineering and oilfield applications, e.g., hydraulic fracturing. The results of the present study can be used in the existing continuum models17­19 of suspension transport through a fracture. Usually, these models are obtained under the assumption that the cross-flow particle concentration profile is uniform. However, it is known that the migration of particles across fluid streamlines can lead to the formation of a central sheet of concentrated suspension with particle-free layers near the walls.14,20 As a result, the rate of particle sedimentation to the fracture bottom increases significantly as compared to that for a uniformly dispersed suspension. In order to model this phenomenon, the lift force obtained in the present work can be now included into a continuum model of particle migration in a dilute-suspension flow through a vertical slot. The inclusion of a nonuniform cross-flow concentration profile resulting from the inertial migration of particles will improve existing models of particle transport in fractures implemented into commercial simulators of hydraulic fracturing.
VII. CONCLUSIONS

which is solved numerically using the orthonormalization method. The migration velocity is calculated as a function of the three governing dimensionless parameters: the particle distance to the closest wall d / l, the channel Reynolds number Rc, and the slip parameter Vsz. The migration velocity scales like Vm = Usz2acm / and is always positive, i.e., the particle migrates away from the walls with an equilibrium position being on the channel centerline. The lift coefficient cm is the same close to the wall regardless of the values of Rc and Vsz. It decreases with the distance from the wall when Rc 10. At larger channel Reynolds numbers, a local maximum of the migration velocity forms at distances of the order of R-1/3 from the channel c centerline, which is due to a combined effect of the slip, the linear shear, and the curvature of the undisturbed velocity profile. Extension of the results is obtained for the case when the drag on a particle has components both parallel and perpendicular to the undisturbed flow. The settling-induced lift force obtained in the present work can be then used in a continuum model of particle migration in a dilute-suspension flow through a vertical slot. A cross-flow particle concentration profile resulting from migration can be further used in models of particle transport through a slot, which are being developed for various oilfield or chemical engineering applications, e.g., for mathematical modeling of hydraulic fracturing.
ACKNOWLEDGMENTS

The authors are grateful to J. R. A. Pearson, P. Hammond, and E. Siebrits for fruitful discussions on the results and many helpful comments; and to the management of Schlumberger for the permission to publish this paper.
1

2

3

4

5

6

7

8

9

10

The inertial migration of a small rigid spherical particle settling under gravity in a horizontal flow of a Newtonian fluid through a vertical slot is investigated. Matched asymptotic expansions are applied to find the solution of the Oseen-like equations governing the outer disturbance flow past a particle on the length scale of the channel width and the solution of the Stokes equations governing the inner flow on the length scale of the particle radius. The problem is reduced to a fourth-order ordinary differential equation for the two-dimensional Fourier transform of the lateral velocity,

11

12

13

14

15

P. G. Saffman, "The lift on a small sphere in a slow shear flow," J. Fluid Mech. 22, 385 1965 . E. Y. Harper and I.-D. Chang, "Maximum dissipation resulting from lift in a slow viscous shear flow," J. Fluid Mech. 33, 209 1968 . R. G. Cox and H. Brenner, "The lateral migration of solid particles in Poiseuille flow. I. Theory," Chem. Eng. Sci. 23, 147 1968 . B. P. Ho and L. G. Leal, "Inertial migration of rigid spheres in twodimensional unidirectional flows," J. Fluid Mech. 65, 365 1974 . P. Vasseur and R. G. Cox, "The lateral migration of a spherical particle in two-dimensional shear flows," J. Fluid Mech. 78, 385 1976 . R. G. Cox and S. K. Hsu, "The lateral migration of solid particles in a laminar flow near a plane," Int. J. Multiphase Flow 3, 201 1977 . P. Vasseur and R. G. Cox, "The lateral migration of spherical particles sedimenting in a stagnant bounded fluid," J. Fluid Mech. 80, 561 1977 . E. S. Asmolov, "Dynamics of a spherical particle in a laminar boundary layer," Fluid Dyn. 25, 886 1990 . J. B. McLaughlin, "Inertial migration of a small sphere in linear shear flows," J. Fluid Mech. 224, 261 1991 . P. Cherukat and J. B. Mclaughlin, "The inertial lift on a rigid sphere in a linear shear flow field near a flat wall," J. Fluid Mech. 263, 1 1994 . A. J. Hogg, "The inertial migration of non-neutrally buoyant spherical particles in two-dimensional shear flows," J. Fluid Mech. 272, 285 1994 . K. Miyazaki, D. Bedeaux, and J. B. Avalos, "Drag on a sphere in slow shear flow," J. Fluid Mech. 296, 373 1995 . E. S. Asmolov, "The inertial lift on a spherical particle in a plane Poiseuille flow at large channel Reynolds number," J. Fluid Mech. 381, 63 1999 . M. J. Economides and K. G. Nolte, Reservoir Stimulation Wiley, New York, 2000 . I. Proudman and J. R. A. Pearson, "Expansions at small Reynolds numbers for the flow past a sphere and a circular cylinder," J. Fluid Mech. 2, 237 1957 .

Downloaded 05 Jun 2009 to 134.32.190.77. Redistribution subject to AIP license or copyright; see http://pof.aip.org/pof/copyright.jsp


063301-9
16

The inertial lift on a spherical particle
19

Phys. Fluids 21, 063301 2009 H. Gu and E. Siebrits, in Proceedings of the Tenth International Conference on Hyperbolic Problems: Theory, Numerics and Applications, Osaka, Japan, 13­17 September 2004, edited by F. Asakura Yokohama Publishers, Yokohama, 2004 . K. G. Nolte, Proceedings of SPE Eastern Regional Meeting, Charleston, West Virginia, USA Society of Petroleum Engineers, Richardson, 1988 , Paper No. 18537, p. 145.

17

18

S. K. Godunov, "On the numerical solution of the boundary-value problem for the system of linear ordinary differential equations," Usp. Mat. Nauk. 16, 171 1961 . J. R. A. Pearson, "On suspension transport in a fracture: framework for a global model," J. Non-Newtonian Fluid Mech. 54, 503 1994 . S. J. McCaffery, L. Elliott, and D. B. Ingham, Topics in Engineering Comput. Mech. Publ., Boston, 1998 , Vol. 32.

20

Downloaded 05 Jun 2009 to 134.32.190.77. Redistribution subject to AIP license or copyright; see http://pof.aip.org/pof/copyright.jsp