Rheology Suspensions Poiseuille PRE

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Rheology of particulate suspensions in a Poiseuille flow

H. Mansouri,1

N. Tahiri,1,2

H. Ez-Zahraouy,1

A. Benyoussef,1

P. Peyla,2

and C. Misbah2

1Laboratoire de Magntisme et de la Physique des Hautes Energies, Facult des Sciences, Universit Mohammed V,

Avenue Ibn Battouta, Rabat B.P. 1014, Morocco2Laboratoire de Spectromtrie Physique, 140 Avenue de la Physique, Universit Joseph Fourier and CNRS, 38402,

Saint Martin dHres, France

Received 3 December 2009; revised manuscript received 6 June 2010; published 12 August 2010

Particulate dense suspensions behave as complex fluids. They do not lend themselves easily to analytical

solution. We propose an analytical model to mimic this problem. Namely, we consider arrays of long parallel

plates which represent a simplification of arrays of chains of spherical particles. This simplified model can be

solved analytically. The effect of effective rotation of the spherical particles is taken into account by attributing

different velocities on each side of the plate that mimics the fact that particles are subject to shear. This work

is an extension of a previous study where particle rotation was disregarded. The flow rate, the dissipation and

the apparent viscosity are studied as a function of the underlying structure. For a single plate placed out of the

flow center, the viscosity is lower when rotation is taken into account. For two plates, the minimal viscosity

corresponds to the situation where the particles are as close as possible to the center and arranged symmetri-

cally with respect to the center. We compute the rheological properties for arbitrary plate positions, and exploit

them for a periodic arrangement. For N plates, and in a confined geometry, the viscosity is about twice as small

as compared to the situation where rotation is ignored. We have conducted a numerical study of a suspensionof spherical particles, and linear chains of spherical particles. The numerical study is in good qualitative and

semiquantitative agreement with the analytical theory considering long plates. This agreement highlights the

fact that our analytical model captures the essential features of a real suspension. The numerical study is based

on a fluid dynamic particle method where the particles are represented by a scalar field having high viscosity

inside.

DOI: 10.1103/PhysRevE.82.026306 PACS number s : 47.57.E, 47.57.Qk, 47.50.d

I. INTRODUCTION

Complex fluids are abundant in nature. Many complex

fluids consist in suspensions of rigid or soft particles which

are suspended in a simple fluid. Examples include industrialfluids cosmetics, foods, etc , biofluids e.g., blood, mu-cus, cartilage , and so on 1 .

The challenge of understanding complex fluids arises

from an intimate coupling between microscales representedby the suspended entities and the global scale of the flow.

The usual scale separation separation of molecular timescales and the global scale of the flow , used for simple

fluids, cannot be justified in the case of suspensions since the

time scale of the suspension motion is of the same order as

that of the imposed flow. In principle, a derivation of a con-

stitutive law should arise from microscopic considerations

1 . However, only few examples can be handled analyti-

cally: i a dilute suspension of spherical particles 24 , or ii of ellipsoids 5,6 , iii dilute suspensions of quasispheri-cal soft particles, such as droplets 79 , capsules 10 , orvesicles 11,12 . If the particles are confined a question ofmuch current interest for microfluidics and/or for concen-

trated suspensions, only numerical or phenomenological ap-

proaches are available 13 . This field knows nowadays anincreasing progress based on numerical solutions of the full

suspension problem 1416 .In order to capture some of the physical ideas that underly

the behavior of concentrated and/or confined suspensions, it

is interesting to conceive of simple models that allow for

analytical tractability. This should help identifying some fea-

tures encountered for high concentration suspensions, and

may help guiding future numerical studies.

Consider that the suspension is made of spherical par-

ticles. Particles undergo both translation and rotation mo-

tions. In the general case i.e., for any spatial configurationof the particles this task is difficult to handle analytically.

We focus on a simplified model system which consists of an

array of long plates as in Ref. 17 . The rotation of a realsuspension of spheres is modeled in the plate model as fol-

lows. We consider that the upper and lower sides of each

plate moves in opposite directions with a velocity to bedetermined self-consistently as depicted in Fig. 1 in ourprevious study 17 both sides moved in the same directionwith the same velocity; i.e., no rotation . This model is ex-

pected to capture the main effect of rotation. This model

b)

a)

FIG. 1. a The case of plates with opposite upper and lowervelocities mimicking the effect of rotation of an array of spheres

b .

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should be viewed as an idealization of a real suspension of

arrays of spheres Fig. 1 where the notion of rotation iseasier to imagine. Our full numerical simulation will show

that the plate model with an effective rotation Fig. 1 acaptures with a better accuracy the properties of a real sus-

pension of spheres as compared to the case where the plates

had only a pure translation.

Our model regarding effective rotation of the plate plateswith an effective rotation, as in Fig. 1 a is inspired by aprevious work due to Ocando and Joseph 18 . These authorswere able to capture several fundamental features 18 of theSegr-Silberberg 19 effect known for a sphere in a Poi-seuille flow at a nonzero Reynolds number the Segr-Silberberg corresponds to the fact that a sphere which is

placed in a cylindrical tube in an imposed Poiseuille flow

adopts an off-centered position which depends on Reynolds

number .

We shall derive the expression of the flow rate, the appar-

ent viscosity and the dissipation. Several interesting features

emerge. In particular, it is found that rotation significantly

lowers the apparent viscosity in confined geometries. Fur-thermore, the dissipation shows some peculiar behavior as a

function of the concentration.

In order to test our prediction, at least at the qualitative

level, a numerical study is conducted by solving the Stokes

equations in the presence of quasirigid spherical particles.

We consider two situations: i the plate is represented by asingle spherical particle, and ii the plate is represented by achain of spherical particles. We find in both cases a remark-

able qualitative agreement between the numerical calculation

and the analytical theory. The agreement is more satisfactory

with a chain of particles than with a single particle, as could

intuitively be expected. In some examples, we shall see that

the agreement is even almost quantitative.This paper is organized as follows. In Sec. II we present

the model equations and the geometry of the system under

consideration. We shall first treat the simple case of a single

plate. In Sec. III we extend the analytical derivation to the

case of an array of plates with arbitrary configurations. Sec-

tion IV is devoted to the discussion of the analytical results.

The numerical study is presented in Sec. V along with a

comparison with the analytical part. A conclusion is given in

Sec. VI, while some technical details are relegated into the

Appendixes AC.

II. MODEL EQUATIONS AND SOLUTION

FOR A SINGLE PLATE

In the low Reynolds number limit which interests us here,

fluids are described by the Stokes equations 20 .

. v = 0, iij = 0 , 1

ij = Pij + 0vi

xj+

vj

xi 2

leading to

P + 02v = 0 , 3

P is the pressure, v is the fluid velocity, 0 is the viscosity of

the ambient fluid, and ij is the stress tensor.

We consider a plane Poiseuille geometry with a pressure

gradient along the x direction denoted as px= P2 P1 /L

0. The channel width is 2w and L represents the laterallength of the plate see Fig. 2 b . In the absence of particlethe flow field is given by

v0 y =px

20 y2 w2 4

and the flow rate by

Q0 = 2

30pxw

3 . 5

Consider now a long particle of length L and of width 2d

moving horizontally in the fluid Fig. 2 . The particle is as-

sumed to be long enough 18 so that the lateral boundaryeffects can legitimately be neglected. Our first task is to de-termine the flow rate in the presence of the particle. The stillunknown translation velocity of the particle is denoted by v1and the rotation frequency by 1. The continuity condition

of the velocity field at the upper and lower sides of the plate

imposes

v+ = v1 +1d, v = v1 1d, 6

where v refers to the fluid velocity on the upper and lower

sides of the plate, respectively. By solving the Stokes equa-

tion in each fluid domain see Fig. 3 a for notations , and by

using the above boundary conditions, we straightforwardly

find

v01 y =px

20y2 + b01y + c01, v12 y =

px

20y2 + b12y + c12,

7

where

b01 = v1 + d1y1 w + d

px

20 y1 + w + d ,

c01 = px

20w2 wb01,

y

x

L

w

(a) (b)

(c)

FIG. 2. A schematic view of the studied system: a Singlespherical particle SSP , b Rectangular plate RP and c Chainsof spherical particles CSP .

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b12 = v1 d1y1 + w d

px

20 y1 w d ,

c12 = v1 d1 px

20

y1 d2 b12 y1 d . 8

y1 is the vertical position of the plate center see Fig. 3 forthe notations .

The rotation velocity is proportional to the velocity gradi-

ent. As in 18 we take this velocity to be approximately thatgiven by the Faxen law 21

1 = dv y

dy. 9

The value = 1 /2 corresponds to the case of a sphere in an

unbounded flow. is taken here as a phenomenological di-

mensionless parameter. We have checked that for a sphere

with a weak confinement, the value 1/

2 is recovered. Inall comparisons with the numerical solution for SP and CSP,

we take = 1 /2 as an effective value for the plates. In other

words, we do not treat as a fitting parameter. It will turn out

that by setting = 1 /2 we capture the essential features which

follow from numerical simulations.

The two quantities v1 and 1 have to be determined self-

consistently. Their determination requires two conditions.

The first one is given by Eq. 9 , whereas the second onefollows from mechanical balance of the plate. The plate is

subjected to shear forces xy on the upper and lower sides

plus the pressure forces on the lateral sides. The stress bal-

ance condition reads

xy 01 xy

12 2dpx = 0. 10

Equations 9 and 10 allow one to determine the unknownsv1 and 1. Use of the two conditions 9 and 10 afterevaluating the flow field from Eqs. 7 10 yields

1 = px

0y1 11

and

v1 =px

20 w d w + d y1

2 w d 2 2dy1

2 12

III. GENERALIZATION TO N PLATES

We have generalized the results to an arbitrary number of

plates having arbitrary positions Fig. 3 . We have foundrecursive expressions for N plates allowing us to determine

both the rotation and translation velocities of a given plate i.

They take the form

vi =1

yi+1 yi1 + 4d vi+1 + d i +i+1 yi yi1 + 2d

+ vi1 d i +i1 yi+1 yi + 2d

+px yi1 yi+1 yi yi1 + 2d yi+1 yi + 2d

20 , 13

where i is the rotation velocity of the particle i and it obeys

the following recursive formula

i = vi1 vi+1 d i1 +i+1

yi1 yi+1 2d+

px 2yi yi1 yi+1

20 .

14The details of the derivation is given in Appendix A. The

flow field in each domain i.e., for each interval yi+1 + dyyi d; see Fig. 3 is solved for in a consistent manner. For

example, when a second particle is introduced, that particle

will rotate according to the flow fields v01 and v12, and so on

see Appendix A .

IV. RESULTS AND DISCUSSION

A. Single plate

We first exploit our results for a single plate. Having de-

termined the full velocity field we can evaluate the total flowrate including the particle , Q = w

wv y dy. We find

Q = 2pxw

3

30 1 1 1 3 3y1

2 1 , 15

where = d/w is the solid volume fraction and y1 =y1 /w is

the dimensionless position of the middle of the particle.

From the comparison with the flow rate Eq. 5 in the ab-sence of the particle, we may define an effective or appar-ent viscosity as =0Q0 /Q this is the very definition inviscometric devices such as a viscosimeter

=

0 1

1 1 3 3y

12 1 . 16

If the particle is at the center of the channel then y1 =0, and

reduces to 0 / 1 3 . The rotation effect disappears since

at the center of the Poiseuille flow the plate feels no torque.

When the particle is off-centered we see that if 0 and inthe interval 0 1 the effective viscosity is reduced due

to rotation as compared to the free-rotation case. The same

tendency is found for a circular particle in a shear flow: the

rotation reduces the effective viscosity from =0 1 + 3 22 down to =0 1 + 2 , which is the 2D analog of theEinstein result 2 derived by Brady 23 . The fact that rota-tion reduces the viscosity is quite intuitive, since the distur-

P1

Domain 01

Domain 12

P

w

-w

x

y

L

Domain 01

Domain N N+1

P+

w

-w

x

y

L

P

yi+1

yN

yi

yi-1

y1

FIG. 3. A schematic view of the studied system showing the two

possible contours and i used in the calculation of the dissipation.

The contour of integration is also shown with the help of arrows.

We show for sake of clarity separately the case of a single particle,

and that of N particles.

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bance of the flow field around a particle which is free to

rotate is less pronounced than when the particle is immobile.

We show in Figs. 4 a and 4 b the behavior of the particle

velocity compared to the fluid velocity in the absence of the

particle and measured at the particle center. The difference

between these two velocities may be called slip velocity. The

slip velocity increases upon increasing the particle size. On

the other hand, for a given size the slip velocity decreases if

the particle is allowed to rotate.

B. Array of plates

We consider now the case of an array of Nparticles. First,

let us recall a result briefly discussed in 17 according towhich a simple link between the flow rate Q and dissipation

can be made. The temporal change of kinetic energy Ec of

the system reads after several manipulations 20

E c = V

ik kvidV+S

vkniikdS, 17

where ik =0 ivk+ kvi is the viscous part of the stress ten-

sor of the suspending fluid, ni is the ith component of the

normal vector pointing from the liquid toward the solidboundary. V represents the total volume occupied by the liq-

uid, whereas S represents the union of the solid surfaces

which are in contact with the liquid. Using the expression of

ik in terms of the fluid velocity, we can write the first term

on the right hand side of Eq. 17 as we leave apart theminus sign

D 0

2 V ivk + kvi

2dV. 18

This is the hydrodynamic dissipation. If inertia is disregarded

the Stokes limit the left hand side of Eq. 17 vanishes and

dissipation D coincides with the work performed by hydro-

dynamic forces. This result is quite useful. Indeed, if we had

to evaluate D directly, this would be ambiguous since we

have no knowledge about the stress in the solid unless elas-ticity equations in the solid are taken into account . This

difficulty is circumvented by using the fact that Eq. 17vanishes and so that the bulk integral is replaced by a surface

integral evaluated in the liquid region adjacent to the solidboundary. This way of reasoning has been used by Einstein

2 , and later by Jeffery 5 in order to evaluate dissipation ofa sphere or an ellipsoid under shear flow. The same type of

trick is also used in order to evaluate the average stress ten-

sor 3,20 for suspensions.From the above considerations, it follows that the dissipa-

tion assumes also the alternative form

D =S

vkniikdS. 19

For a single particle, for example, the integral is performed see Appendix B over the four faces of the plate, plus theupper and lower bounding plates located at y =w of thewhole systems however, those plates do not contribute, dueto the fact that the velocity vanishes at y =w . The contour

1 of integration is shown on Fig. 3. Alternatively, one can

use the global contour shown with broken lines contour inFig. 3 . The equivalence of the two integrals corresponding

to the two different contours and 1 can easily be proven

by evoking the mechanical balance condition 10 .Let us explicit the calculation in the case of a single plate.

Consider the integration over the contour 1, then the stress

on the lateral left and right sides of the plate see Fig. 3 issimply given by

ik= p

2

ikand p

1

ik, respectively. The

normal is equal to 1 on the left side and +1 on the right one.

Defining the flow rate as Q = Sv . dS = Svi . dSi = Svi . nidS

= SvxdSx, one easily deduces from Eq. 19 the followingrelation:

D = pxLQ . 20

It can easily be checked see Appendix B that the sameresult holds for an arbitrary number of particles; the corre-

sponding contours of integration i i = 1 , 2 . . . ,N are shownon Fig. 3. Thus, the dissipation function provides the same

information as the flow rate. Note that the apparent viscosity

defined as =0Q0/

Q gives the same information as theinverse ofD or Q . Thus, a maximal dissipation correspondsto a minimal viscosity and vice versa.

The above results can now be exploited in order to evalu-

ate the dissipation in the presence of N particles. It is conve-

nient to split the dissipation into two contributions rotationand translation parts . The details of the derivation are given

in Appendix C. The result can be written as

D = Dtr + Drot. 21

Dtr is the contribution arising from plate translation and Drotstems from the effect of rotation. We find

-0.4

-0.2

0.0

-1.0 -0.5 0.0 0.5 1.0

0.01

0.1

d=0.2

=0.2

(a)

y/w

(v1-v0

)/v

c

-0.2

-0.1

0.0

-1.0 -0.5 0.0 0.5 1.0

(b)

=0

=0.5

d=0.1

y/w

(v1-v0

)/v

c

-2

-1

0

-0.2 0.0 0.2

dCSP

=0.4

dRP=0.4

(c)

RPCSP

y/w

(v1-v0

)/v

c

FIG. 4. The difference between v1 normalized by the maximumvelocity of the fluid in the free-particle case , the particle velocity,

and the fluid velocity in the absence of particle for a differentparticle sizes normalized with 2w , b results for different s, and

c the numerical results for CSP and analytical results for RP for=0.5.


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Dtr =2px

2w3

3N20 1 1 N2 3 N2

i=1

N

yi2

+ 3N22ij

N

j=1

N

yi yj2 , 22

Drot =2pxw

2

N2 1 N 1

i=1

N

yii + i=1

N

yij=1

N

j .

23

Let us illustrate the results by restricting ourselves to two

particles. The dissipation takes a relatively simple form,

D =px

2w3

60 4 3 + Dst, 24

where

Dst =

px2w3S1

40 2 y

1

2

+ y

2

2

2

10+ 8

2 y13 y2

3 4 2y1y2 2

4 2 y1 y2 3 2 + 4 y12 + y2

2 y1y2

+ 8y12y2

2 25

and

S= y1 y2 32 12+ 8 + 4y1y2 2

+ 2 1 6 2 4 . 26

We have analyzed the behavior of dissipation as a function of

the dimensionless particles positions y1 and y2. We find the

following results. i The maximum dissipation or minimalviscosity is attained when the two particles are located sym-

metrically with respect to the center and are as close as pos-

sible to the center they form a quasi unique block of width4d located at the center . ii The maximum dissipation as afunction of particle positions does not correspond to a hori-

zontal tangent of the function D y1 ,y2 . There exists an ab-

solute maximum with vanishing slope but it corresponds toan unphysical situation where particles would interpenetrate.

This means that the absolute maximum, defined by D / y1=0 and D / y2 = 0 and with a positive determinant of theHessian , has no physical solution. iii For a given N andgiven channel width, pressure gradient and suspending fluid

viscosity , the position of the particle corresponding to maxi-mum dissipation depends only on the volume fraction , as

can be recognized by inspecting the general expression of D

Eq. 21 . Figure 5 a shows the dissipation for differentvalues of as a function of.

In the general case with N particles, and for the range of

parameters explored so far, maximal dissipation is found to

correspond to a periodic arrangement of the plates. We have

thus focused on this structure in order to analyze some rep-

resentative results. We use the analytical results given by

Eqs. 22 and 23 . It is found that the dissipation increases quasilinearly with the periodicity of the structure Fig.6 a for most values of . For close to one the dissipation

becomes a decreasing function of. The crossover from an

increasing to a decreasing behavior of D is not clearly un-

derstood yet. The critical value of at the crossover depends

on the concentration . Indeed, the critical value of in-

creases when decreases. Another result which is worth of

mention is the significant decrease of the viscosity when al-

lowance is made for rotation. This decrease is quite pro-

nounced when the suspension is sufficiently confined seeFigs. 7 a and 7 b , and may be twice as small as that ob-tained when no allowance is made for rotation.

V. NUMERICAL STUDY

A. Model and method

A numerical study of a suspension of single spherical par-

ticle SSP and chains of spherical particle CSP see Fig.2 a and 2 c , has been performed. These particles of radiusd are free to rotate. Each chain contains four particles, with

the particle-particle distance equal , where is slightly big-

ger than the mesh size = 1 . As an initial condition, the

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0(a)

00.30.5

0.7

=0.99

D/D

0

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0(b)

dSSP

=dCSP

=0.1

dRP=0.1

RP

CSP

SSP

D/D

0

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0(c)

RP

CSP

SSP

D/D

0

FIG. 5. The dissipation as a function of : a for differentvalues of, and for N=8, and b the dissipation as a function of

for =0.5 and d=0.1. is varied by acting on N.

c

for N=1 and

=0.5. is varied by acting on d. The particles RB, SSP, or CSPare the center of the channel.

0.2 0.4 0.6 0.8

0.6

0.8

1.0 (a)

10.9

0.5

0.2

=0

D/D

0

0.88

0.90

0.92

0.94

0.3 0.4 0.5 0.6 0.7 0.8

(b)

SSP

CSP

RP

D/D

0

FIG. 6. The dissipation as a function of the periodicity : a for

different values of , and b the numerical results SSP and CSPare compared with the analytical ones obtained for =0.5. =0.3,

and the periodicity is varied by varying the number of particles bykeeping constant .


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distribution of the spheres in the two case SSP and CSP iswell defined taking care to avoid any overlaps . For numeri-cal reasons see description below we solve the nonsteadyStokes equations. After a certain time, the particle configu-

ration attains a stationary state and thus the solution is

equivalent to that of the pure Stokes flow. The fluid equation

of motion around the spheres is taken to be

tv = . , . v = 0 27

with the fluid density.This is a free-boundary problem, where one would have

to prescribe boundary conditions on the moving spheres.

This is not in principle an easy task. A more convenient way

of handling this problem is to make use of fluid-particle

dynamics FPD originally developed in 24 and extendedto three-dimensional 3D by one of the authors 15 . Notethat other methods such as lattice Boltzmann methods

25,26 or Stokesian dynamics 27,28 could be used as well.The virtue of the FPD is that it avoids particle tracking. In

this method, the particles are defined as high-viscosity re-

gions in comparison to that of the solvent. Therefore, the

flow field is defined in the entire domain and not only out-side the spheres

bounded by the walls. Thus, at each time

step Eq. 27 is solved outside and inside the particles. Theviscosity 0 is replaced by a viscosity field. We briefly sum-

marize the main points of the numerical method, while de-

tails can be found in the paper 24 . The presence of the nthparticle is accounted for via an auxiliary field,

n r = 1 + tanh a r rn / /2, 28

where represents the fluid-particle interface thickness and

rn is the off-lattice center of the particle n. Thus, the radius of

a sphere is R = a +. We choose a = 2and =where is the

mesh size. In other words, the difficulty of the sharp interface

problem the interface between each sphere and the fluid is

circumvented by introducing a diffuse albeit abrupt enoughinterface. The viscosity field is represented by the so-calledcharacteristic, or color function

r = 0 + p 0n=1

N

n, 29

where N is the total number of spheres. This expression guar-antees that far enough from the particle, the local viscosity is

=0 the solvent viscosity whereas inside the particle wehave =p the particle viscosity . The viscosity contrast istaken typically to be p /0 =100. This value is chosen in a

such way that recirculation of the fluid inside the spheres is

small enough that the rigidapproximation be legitimate 24 .Equation 27 is solved on a Mac grid 29 where the pres-sure P and the viscosity are calculated at the center of each

square mesh i of size located at position Xi , Yi of thegrid, while the fluid velocity components are calculated at

the center of each segment of the mesh. This ensures that

discretization of each term involved in Eq. 27 is evaluated

at the same point of the mesh grid 29 .For each time step t=0.001, the pressure is calculated

following a standard projection method that enforces incom-

pressibility of the fluid 29 . The typical simulation box sizesare Lx= 2Ly where Ly = 2w; see Fig. 2 . We have consideredthe case with Ly =100 up to Ly =160. The boundary con-

ditions are such that the fluid velocity vanishes on the upper

and lower boundaries y =w . Instead, periodic boundaryconditions are adopted in the x direction. For each time step,

the off-lattice center rn of bead n is moved as follows rn t

+t = rn t +tvn t , where vn t is the fluid velocity aver-aged on the high-viscosity region surrounding the center of

bead n at time t. Then, the viscosity field is reconstructed

29 considering the new positions of the beads at time t+t.

B. Numerical results and discussion

1. Particles velocities

We first compare the results regarding the behavior of the

particle velocities obtained numerically with the analytical

ones, Fig. 4 c . The numerical computation is performed un-

til a steady configuration is reached, in which case the solved

Eq. 27 becomes equivalent up to numerical uncertaintiesto the Stokes equations. The particles stay at the vertical

position no lift is observed , while their interdistance

evolves in time until a steady configuration is achieved. Notethat the curves of the simulations have a smaller extension as

a function of the volume fraction, since the spheres touch the

external boundaries at smaller values of than the plates

would do.

It is clear in Fig. 4 c that the comparison between the

analytical results of a rectangular plate RP with the numeri-

cal results of a single spherical particle SSP or of a chain of

spherical particles CSP are comparable qualitatively. Thequantitative agreement is better when the particles are close

to the center of the channel. A close inspection of Fig. 4 c

shows that the CSP have a slightly larger velocity than the

plate at the center, while the reverse is observed away from

0.8

0.9

1.0

0.0 0.2 0.4 0.6 0.8 1.0

(a)

0.010.025

d=0.1=0.1

/

0*(1-)

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

(b)d=0.1

0

0.1

0.2

=0.5

/

0*(1-)

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

dSSP

=dCSP

=0.1

dPR

=0.1(c)

SSP

CSP RP

/0

*(1-)

FIG. 7. a The effective viscosity as a function of, for differ-

ent values of d. b : the viscosity as a function of , for differentvalues of and c the numerical results are compared with theanalytical ones obtained for =0.5. For a given d N is varied in

order to vary .


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center. The reason is as follows. At the center there is no

rotation by virtue of symmetry. The CSP has more fluid close

to the center where the velocity is maximal than the plate,while they extend over a thicker region on both sides of the

center the two systems have the same volume fraction .Thus the chain disturbs less the flow than the plate, a fact

which results in a slightly higher flow efficiency. When thechain is out of center, the spherical particle rotates, and in thegap separating the particles there is a counter-rotating motion

of two adjacent spheres that leads to higher fluid dissipation.

This implies that the chain translate less efficiently that the

plate does.

2. Behavior of dissipation as a function of volume fraction

The comparison between the numerical and analytical re-

sults is shown on Fig. 5. Figure 5 c shows the dissipation

for the case where the particles are at the center of the chan-

nel. The agreement is remarkably good both for the SSP and

CSP. When the particles are out of center Fig. 5 b , due toa higher friction in the gaps for CSP and due to the higherlateral extent of the SSP in the channel, the dissipation is

higher. Nevertheless the quantitative agreement is rather

good. Note that the CSP results are closer to the analytical

ones, a fact which could be expected.

3. Behavior of the dissipation as a function of the wavelength

for an array of SSP and CSP

The results of comparison are shown on Fig. 6. Here

again we find a quite good agreement between analytical and

numerical results. The agreement is quite satisfactory even

quantitatively. Here again the CSP system provides better

results than the SSP.

4. Behavior of the effective viscosity

Finally, in Fig. 7 c the behavior of the effective viscosity

as a function of the volume fraction obtained numerically for

the CSP and SSP is shown, for dfixed. The numerical results

capture the essential features obtained for RP.

VI. CONCLUSION

In summary, we have analyzed some rheological proper-

ties of a suspension of long plates in a Poiseuille flow. The

study is fully analytical with only numerical tabulations ofseries in the final results . Several qualitative and quantita-

tive features have emerged from this work. For a periodicstructure the behavior of dissipation as a function of is

increasing at small and becomes decreasing at larger . We

have also seen that allowing for rotation of particles may

significantly shift the values of viscosities, and especially in

confined geometries where the shift may attain a factor two.

It must be emphasized that the model is based on a phenom-

enological law for the rotation. This phenomenological law

is appealing, and has already proven to be useful in extract-

ing some interesting features of the migration of particles

due to inertia, as discussed by Joseph and Ocando 18 . Wehave found that the numerical results obtained for the single

spherical particle and the chains of spherical particles is in a

good qualitative agreement with analytical results for the

rectangular plate. The agreement has proven to be even quite

satisfactory at the quantitative level. These results highlight

the fact that our simplistic model is capable of capturing the

basic features.

The prefactor has been chosen to be phenomenological,

and according to Faxen law, we expect it to be of the order of

unity if the suspensions is not too confined. In a confinedsuspension the Faxen factor can be determined numerically

and it is a function of the position in the channel. It will be

interesting in the future to tabulate it numerically and use its

value in the RP model. We expect then a better quantitative

agreement with the full numerical results. It may be specu-

lated that if hydrodynamic interaction among particles is sig-

nificant, then a strong deviation from = 1 /2 may follow.

Note finally that we have prepared the CSP initially, and

then we have let the system evolve. We have found that if the

density is high enough in the chain, then the array of CSP

remains unaffected in the course of time even after 105

simulation steps . Nowadays, preparing experimentally CSP

can be feasible in microfluidic devices, and it will be inter-esting to check this idea experimentally in the future.

For SSP the situation is quite different, however. Lateral

migration occurs, and the final stage seems to be always an

array of particles setting at the center. Still, in this configu-

ration we obtain finally a single CSP, which is well repre-

sented by a plate, as has been seen here. Here only few tests

have been made with a small enough concentration due tocomputational time , so that all particles have enough space

to evolve naturally toward the center. What would happen for

a higher concentration is still unclear. If the initial configu-

ration is taken to be random with high enough concentra-tion , it will be an interesting task to see whether or not an

ordering will take place in the course of time, or rathershould disorder prevail. It will also be interesting to analyze

the far reaching consequences regarding rheology. We hope

to investigate this matter further in a future paper.

ACKNOWLEDGMENTS

This work has benefitted from a financial support from a

French-Moroccan cooperation program Volubilis . C.M. ac-knowledges financial support from CNES Centre NationaldEtudes Spatiales and from ANR Agence Nationale de laRecherche , project MOSICOB.

APPENDIX A: THE DETAIL OF DERIVATION OF THE

PARTICLE TRANSLATIONAL AND ROTATIONAL

VELOCITY IN THE CASE OF N PARTICLES

On the upper and lower sides of the plate i the following

boundary conditions are used

vi+

= vi +id, vi

= vi id, A1

where vi refers to the fluid velocity on the upper and lower

sides of the plate i. Solving the Navier-Stokes equations in

each fluid domain, and using the above boundary conditions,

one finds see Fig. 3 for definitions of domains


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vi,i1 y =px

20y2 + bi,i1y + ci,i1,

vi+1, i y =px

20y2 + bi+1,iy + ci+1,i. A2

The subscript i , i 1 in v refers to the domain between platei and i 1, and so on.

The translational velocities of the particles i 1, particle i

and particle i +1 at the positions yi1, yi, and yi+1 are obtained

from the velocity continuity condition A1 . These condi-tions take the form

vi1

= vi1 i1d=px

20yi1

2+ bi,i1 yi1 d + ci,i1,

vi+1+

= vi+1 +i+1d=px

20yi+1

2+ bi+1, i yi+1 + d + ci+1,i,

vi = vi id=

px

20yi

2 + bi+1,i yi d + ci+1,i,

vi+

= vi +id=px

20yi

2+ bi,i1 yi + d + ci,i1. A3

The plate i feels shear forces xy on the upper and lower

sides plus the pressure forces on the lateral sides. Mechanical

equilibrium condition reads

xy i,i1 xy

i+1,i 2dpx = 0 , A4

where

xy i,i1 = px yi + d + 0bi,i1, xy

i+1,i = px yi d + 0bi+1,i.

A5

From the system of Eqs. A3 we find that

bi+1,i = vi+1 vi

yi+1 yi + 2d+

d i+1 +i yi+1 yi + 2d

px

20 yi+1 + yi ,

bi,i1 = vi vi1

yi yi1 + 2d+

d i +i1

yi yi1 + 2d

px

20yi + yi1

,

ci,i1 = vi +idpx

20 yi + d

2 bi,i1 yi + d ,

ci+1,i = vi+1 i+1dpx

20 yi+1 + d

2 + bi+1,i yi+1 + d .

A6

Inserting these expressions into Eqs. A4 and A5 one ob-tains

vi =1

yi+1 yi1 + 4d vi+1 + d i +i+1 yi yi1 + 2d

+ vi1 d i +i1 yi+1 yi + 2d

+px yi1 yi+1 yi yi1 + 2d yi+1 yi + 2d

20 A7

Generalization of Eq. 9 reads

i = dvi+1,i1 y

dy, A8

where vi+1,i1 y is the fluid velocity profile between the par-

ticles i +1 and i 1 in the absence of the particle i, and is

given by

vi+1,i1 y =px

20y2 + bi+1,i1y + ci+1,i1 . A9

The velocities of the particles i +1, and i 1 in the absence of

particle i obey the relations

vi+1+ = vi+1 +i+1d=

px

20 yi+1 + d

2 + bi+1,i1 yi+1 + d

+ ci+1,i1 , A10

vi1

= vi1 i1d=px

20 yi1 d

2 + bi+1,i1 yi1 d

+ ci+1,i1 , A11

where

bi+1,i1

= vi+1 vi1

yi+1 yi1 + 2d+

d i+1 +i1

yi+1 yi1 + 2d

px

20 yi+1 + yi1 , A12

ci+1,i1 = vi1 i1dpx

20 yi1 d

2 bi+1i1 yi1 d .

A13

Using Eq. A8 , together with the expression of the velocityfield Eq. A2 , one finds

i

=

px

0y

i+ b

i1,i+1.

A14

Using the expression of bi1,i+1 given above, we easily obtain

i = vi1 vi+1 d i1 +i+1

yi1 yi+1 2d+

px 2yi yi1 yi+1

20 ,

A15

which is Eq. 14 .

APPENDIX B: RELATION BETWEEN DISSIPATION

AND FLOW RATE

The dissipation takes the general form Eq. 19 given by


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D =S

vkniikdS.

Expliciting out the expression for N plates we can write

D = L i=1

N

xy+

xy

v

i pxLi=1

N+1

yi+dyi1d

vi,i1 y dy .

B1

From the mechanical equilibrium condition for each par-

ticle xy+ xy

= 2dpx the expression of the dissipation can be

written as

D = pxL 2di=1

N

vi +i=1

N+1

yi+d

yi1d

vi,i1 y dy . B2

The first term is the contribution due to the particles transla-

tion, while the second one expresses the fluid flow between

particles. It is clear that the sum of the two terms inside thebraces are nothing but the total flow rate Q, so that we can

write

D = pxLQ , B3

which is relation 20 .

APPENDIX C: THE EXPRESSION OF THE DISSIPATION

OF N particles

The total flow rate including the particles is given byQ = w

wv y dy and can be written for N particles as as seen

above

Q =i=1

N+1

yi+d

yi1d

vi,i1 y dy +i=1

N

yid

yi+d

vidy , C1

where vi,i1 y is the fluid velocity profile between i and i

1, and vi represents the particles velocity

Using the expression of the particle velocity vi and that of

vi,i1 y written in the first appendix, and integrating we find

that the dissipation takes the form

D = px

2w i=1

N+1 px60

yi1 d3 yi + d

3

+bii1

2 yi1 d

2 yi + d2 + cii1 yi1 yi 2d

2dpxi=1

N

vi . C2Plugging in the expressions of bi,i1 and ci,i1 we easily

obtain the expression of the dissipation as a function of vi,

i, and yi,

D = px

2w i=1

N+1 px20

yi1 d3 yi + d

3

3

yi1 + y i yi1 yi 2d

2

2

+d i1 +i yi1 yi 2d

2

+ vi1 vi+1 yi1 yi 2d

2 2dpx

i=1

N

vi .

C3

Ifi =0 the particle is in pure translation, and we obtain

the expression of the translation contribution reported in a

recent work 17 ,

Dtr =2px

2w3

3N20 1 1 N2 3 N2

i=1

N

yi2

+ 3N22ij

N

j=1

N

y

i y

j2

, C4The contribution due to rotation can easily be identified from

Eq. C3 and after some elementary algebraic manipulations,

it reads

Drot =2pxw

2

N2 1 N 1

i=1

N

yii + i=1

N

yij=1

N

j ,

C5

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