PHYSICAL REVIEW E 98, 042603 (2018)

Rheological behavior of colloidal suspension with long-range interactions

S. ArietaleanizKIRO GRIFOLS S.L. Polo de Innovación Garaia, Goiru Kalea 1, Edificio B, Planta 2, 20500 Arrasate, Spainand Departamento de Ingeniera Biomédica (TECNUN), Universidad de Navarra, 20009 San Sebastian, Spain

P. Malgaretti*

Max-Planck-Institut für Intelligente Systeme, Heisenbergstr. 3, 70569 Stuttgart, Germanyand IV. Institut für Theoretische Physik, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany

I. PagonabarragaDepartament de Física de la Materia Condensada and Institute of Complex Systems (UBICS), Universitat de Barcelona,

Barcelona 08028, Spainand CECAM Centre Européen de Calcul Atomique et Moléculaire, École Polytechnique Fédérale de Lausanne (EPFL),

Lausanne CH-1015, Switzerland

R. C. Hidalgo†

Departamento de Física y Matemática Aplicada. Facultad de Ciencias, Universidad de Navarra, 31080 Pamplona, Spain

(Received 5 May 2018; published 10 October 2018)

In this work, we study the constitutive behavior of interacting colloidal suspensions for moderate and highconcentrations. Specifically, using a lattice Boltzmann solver, we numerically examine suspensions flowingthrough narrow channels, and explore the significance of the interaction potential strength on the system’smacroscopic response. When only a short-range interaction potential is considered, a Newtonian behavior isalways recovered and the system’s effective viscosity mostly depends on the suspension concentration. However,when using a Lennard-Jones potential we identify two rheological responses depending on the interactionstrength, the volume fraction, and the pressure drop. Exploiting a model proposed in the literature we rationalizethe simulation data and propose scaling relations to identify the relevant energy scales involved in these transportprocesses. Moreover, we find that the spatial distribution of colloids in layers parallel to the flow direction doesnot correlate with changes in the system macroscopic response; but, interestingly, the rheology changes docorrelate with the spatial distribution of colloids within individual layers. Namely, suspensions characterizedby a Newtonian response display a cubiclike structure of the colloids within individual layers, whereas forsuspensions with non-Newtonian response colloids organize in a hexagonal structure.

DOI: 10.1103/PhysRevE.98.042603

I. INTRODUCTION

Transport of particles in confined geometries appears inmany technological and biological processes [1,2]. Examplesspan from engineering applications such as the injection offuel in engines, energy harvesting devices [3] and microflu-idics [4] to physiological processes such as blood flow throughvessels [5–8] and ionic pumping transport in channels [9]. In-terestingly, the rheology of all these systems is quite complexwhen compared to that of Newtonian fluids due to the stressesinduced by the interactions among suspended particles. Ac-cordingly, these systems are usually grouped under the lemmaof soft glassy materials that includes concentrated emulsions,foams, colloids, and even macroscopic granular assemblies[1,2].

In the last decade, a number of experimental techniqueswere introduced, allowing us to capture the rheology of soft

*malgaretti@is.mpg.de†raulcruz@unav.es

glassy materials such as x-ray tomography [10–12], dynamiclight scattering [13], magnetic resonance imaging [14,15],and high-frequency ultrasonic speckle velocimetry [16]. Inparticular, previous outcomes [13–22] demonstrate that therheological response of colloidal suspensions is very sensitiveto the confining conditions, as well as, strongly dependingon the system composition. On one hand, for low volumefractions of the dispersed phase the Newtonian response isgenerally recovered. On the contrary, upon increasing volumefraction the interactions among particles begin to play a role.As a result, particles rearrangements occur, which leads tononlinear macroscopic response and enhancing of the yieldstress. Several research groups have proposed to rationalizethese complex response, assuming that the macroscopic flowis produced by a succession of local elastic deformationsand irreversible plastic rearrangements [23–31]. Thus, theseevents cause long-range stress fluctuations over the system,which create localized fragile zones where the system flows.Moreover, correlations between those plastic events have beendetected [32]. In particular, Goyon et al. [14,15] characterized

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S. ARIETALEANIZ et al. PHYSICAL REVIEW E 98, 042603 (2018)

the rheology of soft glassy materials confined between parallelplates by introducing a nonuniform effective field namednonlocal fluidity.

In this paper, we go one step further exploring the flowof soft glassy materials in a confined geometry. We numeri-cally examine three-dimensional (3D) suspensions of colloidsflowing through narrow channels at low Reynolds number,focusing on the relevance of the interaction potential amongsuspended colloids on the overall rheology of the system.Our results show that the interactions among colloids stronglyaffect the rheology of the system and lead to a non-Newtonianresponse even at smaller volume fractions, compared to sys-tems of noninteracting colloids. Interestingly, we find thatthe onset of the non-Newtonian response is controlled bythe ratio between the magnitude of the applied pressure dropand the magnitude of the colloid-colloid interaction potential.When the attractive interaction potential is strong, for lowvalues of the pressure drop the system shows a non-Newtonianresponse whereas upon increasing the pressure drop it even-tually displays a Newtonian behavior. Finally, in order tocapture the velocity profile in the non-Newtonian regime weexploit a previous mentioned model [14,15] that account forinhomogeneous transport coefficients of the fluid. It is worthmentioning that a similar effort was carried out in two dimen-sions, exploring the rheological response of droplets [33].

II. COLLOIDAL SUSPENSION FLOWING THROUGH ANARROW CHANNEL (LATTICE BOLTZMANN SCHEME)

The system under study is composed by N colloidal parti-cles of radius R, suspended in a Newtonian fluid (see Fig. 1).The system is confined by solid walls along the z directionand periodic boundary conditions are applied along x andy. A lattice Boltzmann (LB) approach is used to model thefluid. This method recovers the solution of the Navier-Stokesequations, and it has provided significant results examiningthe rheological response of complex liquids [34–39].

FIG. 1. Illustration of the colloidal suspension, which movesdriven by a pressure drop �p

Lacting on the Y direction. The system

is confined in the Z direction by solid walls and periodic boundariescondition are imposed on X and Y directions.

According to the LB scheme the fluid dynamics is readout from the single-particle distribution function f (�r, t ) [35].Hence, at each lattice node, the discretized distribution func-tion fi (�r, t ) evolves at discrete time steps �t as,

fi (�r + �ci�t ; t + �t ) = fi (�r; t ) + �ij

(f

eq

j (�r; t ) − fj (�r; t )).

(1)

Note, this evolution rule accounts for the linear momentumstreaming to the neighbors nodes j due to the liquid advectionmotion of velocity �ci . Moreover, a collision operator �ij rulesthe relaxation process toward an equilibrium state, character-ized by f

eq

j (�r; t ). In computing, we use LUDWIG [35,40] alattice Boltzmann implementation, which is able to reproducethe behavior of complex fluids, and include a multirelaxationcollision operator �ij [40]. The system geometry is a 3Dcubic lattice with 19 allowed velocities �ci known as D3Q19

scheme [35]. We use units such that the mass of the nodes, thelattice spacing, and the time step are one and the kinematicviscosity is ν. Thermal fluctuations kBT �= 0 are incorporatedin the lattice Boltzmann model adding an additional noiseterm ξi (kBT ) in Eq. (1) [41]. This additional contributionintroduces fluctuations in the populations in each phase-spacecell. The stochastic properties of this term are chosen toensure that the fluctuation dissipation theorem in equilibriumis satisfied [41,42].

As mentioned above, the system is confined by solid wallson the z direction. To compare with experimental condi-tions, appropriate boundary conditions at the walls are needed[40,43]. In LUDWIG walls are implemented by applying so-called stick boundary conditions [40]. Thus, during propaga-tion, the component of the distribution function that wouldpropagate into the wall node is bounced back and ends up backat the fluid node but pointing in the opposite direction. Thisprocedure produces stick boundary conditions at the middlepoint of the vector joining the wall and fluid nodes.

The rheological response of the colloidal suspensions isrealistically modeled, assuming that the solid particles inter-act with the surrounding fluid also through bounce-back onthe links. Hence, the total force and torque acting on thecolloid are determined using the mechanical constraint thatthe momentum exchange between the solid particle and thenodes from bounce-back vanishes. The interaction betweenthe colloids is defined by two interaction potentials. A short-term soft potential that reads as,

vSP ={

εSP

(σSP

r

)νor < R + rc

0 else, (2)

where εSP accounts for the strength of the contact interaction,σ is the length scale and ν0 characterize the range of theparticle-particle interaction. The numerical implementationof the interaction potential Eq. (2) ensures colloids do notoverlap at the hard-core radius R when introducing a so-called hydrodynamic radius Rh [44]. In general, simulatingmultiparticle suspension all distance calculations are based onthe hydrodynamic radius Rh, and to obtain accurate resultsit is then essential to use a calibrated value [45], whichis typically larger than the physical radius R. Besides this,vSP is truncated at a cutoff distance of rc = 0.25 (in latticeunits). Additionally, a long range of interaction Lennard-Jones

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RHEOLOGICAL BEHAVIOR OF COLLOIDAL SUSPENSION … PHYSICAL REVIEW E 98, 042603 (2018)

potential is also used,

vLJ = 4εLJ

((σLJ

r

)12

−(

σLJ

r

)6), (3)

where εLP accounts for the strength of the contact interaction.Accordingly, a pair of forces �Fij = − �Fji with equal magni-tudes |∇vLJ |, is applied to interacting colloids at a distancer , but vLJ is truncated at a cutoff distance of rc = 5R. In allcases presented here the values of σSP = 1.0, ν0 = 1.0, σLJ =2Rh, R = 5.0, and Rh = 5.2 (in lattice units) are fixed. Notethat using those parameter values the interaction potentialsEq. (2) and Eq. (3) are mainly locally repulsive and long-rangeattractive, respectively.

In the simulations N = 1000 colloidal particles are ini-tially distributed in random positions inside the simulationbox whose dimensions along x and y (w = wx = wy) areadjusted in order to attain the desired volume fraction, φ ∈[0.30, 0.52], whereas for the confining direction we fixedh = 104 nodes. Fluid parameters are chosen to ensure a lowReynolds number Re = Vmh/ν where ν is the fluid kinematicviscosity and Vm is the maximum velocity, which is measuredat the center of the channel. The fluid is subject to a pressuregradient along the y direction, �p

L, which results in a uniform

force parallel to the walls, mimicking a pressure driven flowthat pushes the colloidal particles through the channel. Theforce induces the flow of the suspension, which reaches asteady state after a short transient. In the following, all thevalues of pressure gradient �p

Lare in lattice units (ρkBT /l

being l the lattice unit and ρ the number density of the fluid).In the following we consider a pressure gradient �p

L= 5 ×

10−5ρ kBTl

that corresponds to a pressure gradient of �p

L�

6 × 103Pa/m. We remark that this value of the pressure dropis well within the typical experimental range (see Ref. [46]).

III. NONUNIFORM RHEOLOGICAL FORMULATION

In general, the rheological response of a liquid with negli-gible yield stress reads as,

σ (z) = μ(γ (z)) γ (z), (4)

where the shear strain γ (z) is defined as

γ = ∂vy (z)

∂z(5)

and μ(γ (z)) is the effective viscosity.Goyon et al. [14,15] examined the rheological behavior

of complex flows introducing a novel nonuniform flow rule.According to Refs. [14,15], the fluidity f (z) is defined as theratio between shear rate and shear stress

f (z) = γ (z)

σ (z)= 1

μ(γ (z))(6)

having the dimension of inverse viscosity. Moreover, themodel postulates that the fluidity f (z) is the solution of:

f (z) = fbulk + ξ 2 ∂2f (z)

∂2z(7)

and it depends parametrically on ξ (the so-called fluiditylength) and fbulk. We remark that in our case the limit ξ → 0corresponds to a Newtonian fluid. Here, we study numerically

a pressure driven flow in a rectangular channel with height h.In that geometry, the shear stress varies spatially according to

σ (z) = −�p

Lz, (8)

where �p

Lis the constant pressure gradient along the channel.

Note that two boundary conditions are needed to obtain thesolution of the fluidity, Eq. (7) and one boundary conditionfor the velocity profiles vy (z). Using the numerical data, wededuce the wall fluidity estimating the velocity gradient atthe wall γ (±h/2) = γw, finding f (−h/2) = f (h/2) = fw =σ (±h/2)

γw. Thus, using Eq. (8) the solution of Eq. (7) reads

f (z) = fw − fbulk

cosh(

h2ξ

) cosh

(z

ξ

)+ fbulk. (9)

Finally the velocity profiles vy (z) can be analytically de-duced form Eq. (4), using the momentum balance Eq. (8), andobtaining

dvy (z)

dz= �p

Lz

(fw − fbulk

cosh(

h2ξ

) cosh

(z

ξ

)+ fbulk

). (10)

Each particular solution of Eq. (10) involves an integrationconstant that we fix by imposing that the velocity at thecenter of the channel Vm = vy (0) matches the one obtainedfrom the numerical simulations. We remark that followingthis procedure the velocity profiles within the channel vy (z)is totally determined by the values of fluidity length ξ andbulk viscosity fbulk, which will be used as fitting parameterswhen comparing the LB numerical results with Eqs. (5)–(9).

IV. RESULTS AND DISCUSSION

In the simulations, the colloidal suspension is subject to aconstant pressure gradient �p

L, which induces the flow. In gen-

eral, we obtain that the suspension yield stress is practicallyzero and after a short time, a steady state is reached.

First, we characterize the rheology of the system in theabsence of long-range interactions (εLJ = 0). For such aregime, the system shows a Newtonian response regardlessthe strength of the short-range interaction εSP . As shown inFig. 2, the velocity profiles obtained for different volumefractions all collapse on a parabola, i.e., the system behavesas a Newtonian fluid. Interestingly, the data indicate that theeffective viscosity ηeff = h2

8Vm

�p

Lincreases monotonically with

the volume fraction φ. As expected, for small values of φ thedependence of ηeff (φ)/ηo is compatible with Batchelor’s for-mula [47] whereas for larger values of φ it is better capturedby [48]

ηeff = η0

[1 − φ

1 − cφ

]− 52

, (11)

where c = 1−φc

φcand φc � 0.74, which is the maximum pack-

ing fraction of hard spheres in three dimensions. For compar-ison, the expressions proposed by Quemada [49] and Krieger[50] are also included. Our results show the colloids are nothomogeneously distributed across the channel, rather theyaccumulate at specific locations [see the points location inFig. 2(a)]. Indeed, for the largest value of φ that we explore,

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S. ARIETALEANIZ et al. PHYSICAL REVIEW E 98, 042603 (2018)

-1 -0.5 0 0.5 12z/h

0

0.2

0.4

0.6

0.8

1

v y(z

)/V

m

(a)

φ - 0.30φ - 0.40φ - 0.52Newtonian

0 0.2 0.4 0.6φ

0

0.2

0.4

0.6

0.8

1

1.2

Vm

(φ)/

Vm

(0)

(b)

0 0.2 0.4 0.6φ

0

5

10

15

ηef

f/η

o

LJ = 0

BatchelorMendozaQuemadaKrieger

FIG. 2. (a) Velocity profiles obtained numerically. The datacorrespond to colloidal suspensions without long-range interaction(εSP = 0.5 and εLJ = 0), �p

L= 5 × 10−5, εSD = 0.5. Outcomes for

several volume fractions φ are shown. The velocity values arerescaled with the maximum velocity Vm and compared with a typicalNewtonian response. (b) The maximum velocity Vm(φ) as a functionof φ is illustrated, the values are rescaled with Vm(0) = h2

8η0

�p

L. In

the inset the values of effective viscosity ηeff/ηo are compatible withthe Batchelor’s analytic prediction for non-Brownian suspension. ηo

is the viscosity of the liquid, i.e., ηo = ηeff (0).

the interparticle distance along the z direction is smaller thanthe particles diameter 2R, suggesting that the particles arespatially distributed in a honeycomblike structure. Such anobservation justifies the value of φc = 0.74 used in Eq. (11).We argue the system linear response is expected because thevolume fraction φ is still low. We remark that non-Newtonianresponse was found experimentally for systems with φ > 0.62[14,15].

Next, we turn on the long-range Lennard-Jones interactionpotential with strength εLJ [see Eq. (3)]. Figure 3 summarizesa systematic study, exploring the macroscopic response ofthe system varying εLJ , while keeping constant the pressuregradient �p

L= 5 × 10−5 and the volume fraction φ = 0.52.

Interestingly, the strength of long-range interaction potentialεLJ has a significant impact on the macroscopic rheologicalresponse, as shown in Fig. 3(a), or alternatively in Fig. 3(b).In contrast, the maximum velocity is not sensitive to themagnitude of the short-range interaction εSD [see insets ofFig. 3(a) and Fig. 3(b)]. In particular, Fig. 3(a) shows that

0 0.1 0.2 0.3 0.4 0.5LJ

0

0.2

0.4

0.6

0.8

1

1.2

Vm

(L

J)/

Vm

(L

J=

0)

(a)

0.2 0.4 0.6 0.8SP

0.9

0.95

1

1.05

1.1

Vm/V

m(

SP

=0.

3)

0 0.1 0.2 0.3 0.4 0.5LJ

0.8

1

1.2

1.4

1.6

1.8

2

ηef

f/η

eff(

LJ

=0)

(b)

0.2 0.4 0.6 0.8SP

0.9

0.95

1

1.05

1.1

η eff/η

eff(

SP

=0.

3)

FIG. 3. Maximum velocity Vm (a) in the channel as a functionon εLJ , keeping εSD = 0.5, the values are rescaled with the maxi-mum velocity obtained for εLJ = 0. (b) The corresponding effectiveviscosity ηeff , the values are rescaled with ηeff (εLJ = 0). The insetsillustrates the same but as a function of εSD keeping εLJ = 0.

the maximum velocity decreases and effective viscosity [seeFig. 3(b)] grows upon increasing εLJ . More interesting, anonuniform rheological response is typically found whenεLJ �= 0.

In order to rationalize the simulations data, we use themodel presented in Sec. III. Figure 4(a) illustrates the velocityprofiles obtained numerically, for different values of εLJ

at constant volume fraction φ = 0.52 and pressure gradient�p

L= 5 × 10−5. For comparison, in each case the analytic

solution of Eq. (10) is also included. Interestingly, uponincreasing εLJ the velocity profiles depart from the Poiseuille-like profile typical of Newtonian fluids and a plateau sets atthe center of the channel. Moreover, it correlates with theincreasing of the shear rate close to the walls. We fit thevelocity profiles using Eqs. (5)–(9), while fbulk and ξ arefitting parameters. Figure 4(b) shows the bulk fluidity profilesf (z), i.e., Eq. (9), normalized by f ∗

bulk(φ) = 1ηeff (φ) where

ηeff(φ) is calculated using Eq. (11). Interestingly, Fig. 4(b)shows that for larger values of εLJ the value of the fluidity

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RHEOLOGICAL BEHAVIOR OF COLLOIDAL SUSPENSION … PHYSICAL REVIEW E 98, 042603 (2018)

0 0.5 12z/h

0

0.01

0.02

v y(z

)

(a)

0 0.2 0.4 0.6 0.8 12z/h

0

1

2

3

f(z

)/f∗ bu

lk

(b)T - 47 - 0.0

T - 469 - 0.2

T - 939 - 0.3

T - 1408 - 0.5

T - 2347 - 0.8

FIG. 4. System response obtained varying the strengths of long-range interaction εLJ , at constant φ = 0.52 and �p

L= 5 × 10−5.

(a) Velocity profiles numerical (symbols) compared with the analyticpredictions (lines) solution Eq. (10), see (b) for the color legend. In(b) the corresponding fluidity f (z) profiles Eq. (7) are shown.

is quite reduced and its profile becomes nonuniform all alongthe channel transverse direction. The dependence of the fittingparameters, namely fbulk and ξ on εLJ is referred to further inthe text.

Exploring the dependence of the onset of non-Newtonianregime on the suspension volume fraction φ, we performsimulations varying φ, for εLJ �= 0. Figure 5(a) shows thevelocity profiles obtained numerically for different values ofφ, keeping constant the strength of Lennard-Jones potentialεLJ = 0.2 and the pressure gradient �p

L= 5 × 10−5. For

comparison, in each case the analytic solution of Eq. (10)is also included. As expected, the velocity profiles [seeFig. 4(a)] deviate from the Newtonian regime upon increasingthe volume fraction φ. The corresponding fluidity profiles [seeFig. 4(b)] are nonuniform along the transverse section of thechannel, highlighting the onset of the non-Newtonian regime.Interestingly, when εLJ �= 0 the non-Newtonian response isdetected for volume fractions notably lower than φ = 0.62,which is the critical value obtained experimentally, in theabsence of long-ranged interactions [14,15].

Complementarily, we simulate cases varying the pressuredrop �p

Lat constant φ = 0.52 and εLJ = 0.2. The velocity

and fluidity profiles are presented in Fig. 6(a) and Fig. 6(b),respectively. Interestingly, the non-Newtonian response isattained for smaller values of �p

L, suggesting that in that

regime the particle-particle interaction potential is dominant.Whereas the linear response is recovered for large values

0 0.5 12z/h

0

0.01

0.02

0.03

v y(z

)

(a)

0 0.5 12z/h

0

0.5

1

1.5

2

f(z

)/f∗ bu

lk

(b)φ - 0.42φ - 0.45φ - 0.46φ - 0.48φ - 0.50φ - 0.52

FIG. 5. System response obtained varying the colloidal volumefractions φ, at constant εLJ = 0.2 and �p

L= 5 × 10−5. (a) Velocity

profiles numerical (symbols) compared with the analytic predictions(lines) solution of Eq. (10), see (b) for the color legend. In (b) thecorresponding fluidity f (z) profiles Eq. (7) are shown.

0 0.5 12z/h

0

0.01

0.02

0.03

0.04

v y(z

)

(a)

0 0.5 12z/h

0

0.5

1

1.5

2

f(z

)/f∗ bu

lk

(b)E/

LJ- 1.55

E/LJ

- 3.68

E/LJ

- 6.13

E/LJ

- 8.10

FIG. 6. System response obtained varying the pressure drop �p

L

at constant φ = 0.52 and εLJ = 0.2. (a) Velocity profiles numerical(symbols) compared with the analytic predictions (lines) solutionEq. (10), see (b) for the color legend. In (b) the corresponding fluidityf (z) profiles Eq. (7) are shown.

of �p

L, where the liquid-mediated hydrodynamic interactions

play the most significant role.

A. Onset of the nonuniform rheological behavior

Next, we identify the onset of nonuniform rheologicalresponse, as well as the interrelationship between the externalforcing, �p

L, Lennard-Jones potential, εLJ , and volume frac-

tion, φ. As a first step, we clarify the significance of the threeenergy scales of the system, namely the thermal energy, kBT ,the long-ranged interaction strength εLJ , and the energy, E,associated to the viscous forces. Indeed the latter can be easilyexpressed via its time derivative, namely the dissipated power:

E = fd〈V 〉 × 〈V 〉 = 6πηlR〈V 〉2, (12)

where ηl is viscosity of the fluid phase ηl = η0 and 〈V 〉 is themean velocity of the fluid, which can be estimated in terms ofthe effective viscosity of the suspension ηeff (φ), and reads as

〈V 〉 = 2

3

(h2

8ηeff (φ)

�p

L

), (13)

the average speed of a Newtonian fluid in a rectangular chan-nel. In order to obtain an energy scale, we multiply Eq. (12)by the characteristic advective time τ = rm−σLJ

〈V 〉 , where rm ≈1.122σLJ is the distance at which the long range potentialreaches its minimum (maximum attraction). As a result weobtain,

E = 6πη0(rm − σLJ )R〈V 〉. (14)

For the cases under scrutiny, both E and εLJ are much largerthen kBT , hence, the latter can be disregarded. Accordingly,we are left with two energy scales, namely the dissipative en-ergy, E and the strength of long-range interaction εLJ . WhenεLJ � E we expect that the local inhomogeneity promotedby the long-range interactions will be suppressed due to theaction of the external force and hence the linear responseshould be recovered. In contrast, when εLJ � E we expect theonset of nonlinear response due to the local rearrangements inthe colloidal density induced by the long-range interactions.Accordingly, substituting Eq. (13) into Eq. (14) and equatingit to εLJ provides an estimate for the onset of pressure for thenon-Newtonian response,

�p∗

L= 2

ηeff (φ)εLJ

πη0(rm − σLJ )Rh2. (15)

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S. ARIETALEANIZ et al. PHYSICAL REVIEW E 98, 042603 (2018)

0 0.2 0.4 0.6 0.8 1LJ/ ∗

LJ

0

1

2

3

4

5

ξ/2R

(a)

0 0.5 1/

0

0.5

1

1.5

/f

0.5 0.6 0.7 0.8φ/φ∗

0

1

2

3

ξ/2R

(b)

0.7 0.8φ/φ

0

0.5

1

f/f

1 2 3 4 5 6ΔpL /Δp

L

∗0

1

2

3

4

ξ/2R

(c)

2 4 60

0.5

1

f/f

FIG. 7. Values of ξ in unit of 2R, and fbulk, used when fittingEq. (9) and Eq. (10) to the numerical data presented in Fig. 4(a),in Fig. 5(b), in Fig. 5(c). In all cases, the error bars representconfidence interval for the parameters reproducing the data with error

� =√∑

(vi−vp )2

nless than 5%, where vi and vp are the numerical

results and the model prediction, respectively. The values of ε∗LJ ,

�p∗L

, and f ∗bulk used in each case are explained in the text.

Whereas for fixed εLJ and �p

Lthe critical value of volume

fraction reads:

φ∗ = χ25 − 1

χ25 − c + χ

25 c

(16)

with

χ = π (rm − σLJ )Rh2

2εLJ

�p

L. (17)

While for fixed �p

Land φ [using Eq. (11)] the critical values

of the Lennard-Jones potential:

ε∗LJ = E =πη0(rm − σLJ )Rh2

2ηeff (φ)

�p

L. (18)

For clarity reasons, Fig. 7 summarizes, the values of thefluidity length ξ in units of 2R, and bulk fluidity fbulk, usedto fit the nonuniform model [Eq. (9) and Eq. (10)] to thenumerical data of Fig. 4, Fig. 5, and Fig. 6.

In particular, Fig. 7(a) corresponds to the data shown in inFig. 4, i.e., when changing εLJ with constant φ = 0.52 and�p

L= 5 × 10−5. In this case, the values of εLJ are rescaled

by the characteristic energy ε∗LJ [see Eq. (18)] and fbulk is

normalized by the bulk fluidity calculated using Eq. (11),f ∗

bulk = 1/ηeff (φ). It is noticeable that introducing the long-range interactions potential leads to the increasing of thefluidity length ξ and the reduction of the bulk fluidity fbulk.Interestingly, we found that there is a threshold value εLJ ≈0.4ε∗

LJ (βεLJ � 500) above which the fluidity length, ξ , isof the order of a few particle diameters whereas for εLJ �0.4ε∗

LJ we have that ξ becomes vanishing small. Moreover,for εLJ < 0.4ε∗

LJ we have fbulk/f∗bulk � 1 whereas for εLJ >

0.4ε∗LJ the bulk fluidity strongly diminishes [see inset of

Fig. 7(a)].

Figure 7(b) shows how the onset of nonuniform rheologydepends on the suspension volume fraction, φ, for fixedεLJ = 0.2 and �p

L= 5 × 10−5 (corresponding to Fig. 5). The

values of fbulk are rescaled with f ∗bulk = 1/ηeff (φ), where

φ = 0.42, i.e., the lower explored value. Moreover, the valuesof volume fraction are rescaled with φ∗, which is estimatedusing Eq. (18). Figure 7(b) shows that high values of φ, whichrepresent the most confined situation, leads to lower valuesof bulk fluidity fbulk and larger effective viscosity. Moreover,an abrupt change in the nonuniform fluidity length ξ is de-tected for an specific value of volume fraction. These resultssuggest that the onset of the non-Newtonian regime is dueto a nontrivial coupling between the long-range interactionsamong colloids and the confinement. In particular, for �p

L=

5 × 10−5 and εLJ = 0.2, we find non-Newtonian rheologysets for values of φ above the threshold value 0.75φ∗.

Finally, Fig. 7(c) summarizes the results varying the forc-ing, �p

L(corresponding to Fig. 6). As we pointed out earlier,

for weak values of �p

Lthe system shows a non-Newtonian

response, switching to a Newtonian one upon increasing�p

L. Interestingly, Fig. 7(c) shows that the onset of the non-

Newtonian rheology �p

L≈ 2 �p∗

Lwhere we evaluate εLJ =

0.2 and φ = 0.52 in the scaling function �p∗L

Eq. (15). Theseresults validate the consistency of our analysis.

B. Suspension morphology

The balance between the energy input and internal dis-sipation always leads to steady-state regimes, but revealingdiverse flow morphologies. Thus, we examine the one-pointcorrelation function gz(z), i.e., the probability that a colloidlies at a given distance z from the wall averaged over the x-yplane. Note that gz(z) is proportional to an average volumefraction profile. Moreover, we quantify the degree of localordering on the x-y plane, by means of the radial distributionfunction gxy (r ), which accounts for the probability of findingtwo colloids separated by distance r =

√�x2 + �y2, regions

of width �z = 2R. Complementarily, we describe the localstructural order in the x-y plane, by using the hexatic orderparameter �6, defined as,

�6 =∣∣∣∣∣∣

1

N

N∑j=1

ϕ6(�rj )

∣∣∣∣∣∣ (19)

with

ϕ6(�rj ) = 1

nj

nj∑k=1

exp (i6θjk ). (20)

In the above equation, θjk is the angle between the neighborsk of each particle j . Specifically, the sum runs over the nj

neighbors at a distance r < 1.1 × D, where D = 2R. Thus,�6 is obtained averaging the result of the N particles of thelayer.

First, we discuss the case without long-range interaction(εLJ = 0). Figure 8(a) illustrates the correlation functionprofiles gz(z) obtained for different volume fractions. Forcomparison, we also show the gz(z) obtained initially, whenthe colloids are distributed in random positions inside thesimulation box. We find that for systems with high volume

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RHEOLOGICAL BEHAVIOR OF COLLOIDAL SUSPENSION … PHYSICAL REVIEW E 98, 042603 (2018)

0 0.2 0.4 0.6 0.8 12z/h

0

0.02

0.04

0.06

0.08

0.1

0.12g z

(z)

(a)

randomφ - 0.30φ - 0.42φ - 0.52

0 1 2 3 4r/2R

0

0.2

0.4

0.6

0.8

1

g xy(r

)

(b)wall Ψ

6 = 0.64

core Ψ6 = 0.65

0 1 2 3 4r/2R

0

0.2

0.4

0.6

0.8

1

g xy(r

)

(c)wall Ψ

6 = 0.75

core Ψ6 = 0.67

FIG. 8. Correlation functions profiles, obtained in the absence oflong-range interactions (εLJ = 0). (a) One-point correlation functiongz(z) profiles for φ = [0.30; 0.34; 0.42; 0.52]. Radial distributionfunction gxy (r ) profiles, obtained for (b) φ = 0.3 and (c) φ = 0.52.

fraction, the profiles of gz(z) show local peaks, which denotesa pronounced layerlike ordering and the strong degree of cor-relation between colloid movement and the walls. Figure 8(a)also shows that for higher values of the volume fractionthe distance among the layers �h ≈ 9.45 is slightly smallerthan 2R, which suggests that the colloids are ordered as ina honeycomb lattice. Interestingly, the high of the peaks ofgz(z) is not uniform and the peaks close to the walls becomelarger than those close to the channel axis upon increasing thevolume fraction. Moreover, as the volume fraction decreases(by increasing the channel width W with constant number ofparticles), the amplitude of the peaks of gz(z) reduces, whichindicates loss of layering. Thus, at the smallest channel width(φ = 0.30), the layering across the system practically breaksdown and the walls no longer induce well-ordered layeringin agreement with previously reported numerical [18,51–53]and experimental [10,54–57] studies. Complementarily, inFig. 8(b) and Fig. 8(c) the degree of local ordering on the x-yplane is studied using radial distribution function gxy (r ), fortwo extreme values of volume fraction. Note that the maxi-mum of the curves are always located at values of 2R, whichare multiples of the particle diameter, suggesting that a cubicphase is dominant in the horizontal structure. As expected, itis found that the changes in the two-point correlation functioncorrelates with the differences in the hexatic order parameter,�6 (see legends in Fig. 8).

In previous studies, this particle layering has been re-lated with an anomalous behavior of the suspension viscositywhen changing the volume fraction [18,52,58]. However, forεLJ = 0 we obtain a Newtonian response and a monotonousincrease of the viscosity with the volume fraction. Thus, our

0 0.2 0.4 0.6 0.8 12z/h

0

0.02

0.04

0.06

0.08

0.1

0.12

g z(z

)

(a)random

LJ/k

BT - 0

LJ/k

BT - 2347

0 1 2 3 4r/2R

0

0.2

0.4

0.6

0.8

1

g xy(r

)

(b)wall Ψ

6 = 0.75

core Ψ6 = 0.67

0 1 2 3 4r/2R

0

0.2

0.4

0.6

0.8

1

g xy(r

)

(c)wall Ψ = 0.95core Ψ = 0.90

FIG. 9. Correlation functions profiles, obtained varying the long-range interaction potential εLJ . (a) One-point correlation functionsgz(z). The results correspond to �p

L= 5.0 × 10−5, εSD = 0.5, φ =

0.52. Radial distribution function gxy (r ) profiles, obtained for twovalues of εLJ , (b) below and (c) above the transition

results suggest that the shear-induced hydrodynamic forcesdominate over the long-range wall effects, for the range ofexplored volume fractions.

Next, in Fig. 9(a), we explore the changes in the morphol-ogy of the suspension, which are detected varying the strengthof the long-range interaction potential εLJ at fixed volumefraction, φ = 0.52, and external force, �p

L= 5 × 10−5. In

contrast to what we have shown for εLJ = 0, for very largevalues of the long-range interaction potential εLJ we findthat the wall-induced layering spreads homogeneously acrossthe whole channel, denoting the large-range correlations areenhanced. Complementarily, Fig. 9(b) and Fig. 9(c) illustratethe radial distribution function gxy (r ) obtained for two ex-treme values of long-range interaction potential εLJ . Inter-estingly, while for εLJ = 0 a cubic phase results dominantin the horizontal structure, for εLJ /ε∗ ≈ 0.8 (or equivalentlyεLJ = 2347 kBT ) the position of the maximum indicates thata hexagonal phase is dominant. This change in the g(r )is in agreement with the values of the mean hexatic orderparameter �6 (see legends in Fig. 9). It is noticeable that thelong-range interaction potential induces a significant changein the morphology of the suspension, which correlates withthe change in the rheological response (see Fig. 4).

Finally, varying the pressure drop �p

Land keeping

εLJ /ε∗ ≈ 0.8 (or equivalently εLJ = 2347 kBT ) and φ =0.52 [see Fig. 10(a)], we obtain that the nature of wall-induced layering is not altered. Note, that the hight of themaximums is the practically the same in the whole channeland for both cases. Moreover, in this case both the radialdistribution function gxy (r ) and the hexatic order parameter

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S. ARIETALEANIZ et al. PHYSICAL REVIEW E 98, 042603 (2018)

0 0.2 0.4 0.6 0.8 12z/h

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14g z

(z)

(a)randomE/

LJ- 1.55

E/LJ

- 3.68

E/LJ

- 6.13

E/LJ

- 8.10

0 1 2 3 4r/2R

0

0.2

0.4

0.6

0.8

1

g xy(r

)

(b)wall Ψ

6 = 0.85

core Ψ6 = 0.87

0 1 2 3 4r/2R

0

0.2

0.4

0.6

0.8

1

g xy(r

)

(c)wall Ψ

6 = 0.95

core Ψ6 = 0.92

FIG. 10. Correlation functions profiles, obtained varying thepressure drop �p

L. The results correspond to εLJ = 0.2, εSD = 0.5,

φ = 0.52. (a) One-point correlation function gz(z). Radial distribu-tion function gxy (r ) profiles, obtained for two pressure drops �p

L

(b) �p

L= 10−4 and (c) �p

L= 2.5 × 10−5, which is below the

transition.

presented in Fig. 10(b) and Fig. 10(c), indicates a mild changein the horizontal plane upon increasing �p

L. Interestingly, both

indicators show a weakening of the hexagonal order uponincreasing �p

L. This indicates that larger values of �p

L(i.e.,

larger than those we could attain) may destroy the hexagonalorder.

V. CONCLUSIONS

We examine numerically in three dimensions, the rheolog-ical response of colloidal suspensions when flowing betweenparallel plates under the action of a constant pressure gradient.When the colloids interact solely via a short-range potentialthe colloidal suspension displays a Newtonian response, forthe studied volume fraction. Moreover, the colloids order inlayers parallel to the streamlines. For the highest value of thevolume fraction (φ = 0.52), we obtain the distance amonglayers is smaller that the particle diameter, hence, pinpoint-ing that colloids are organized in a honeycomblike struc-ture. When introducing a long-range Lennard-Jones potentialthe macroscopic response of the suspensions changes, fromNewtonian to non-Newtonian, depending on the interactionstrength, the volume fraction and the pressure drop. Exploit-ing a theoretical framework proposed by Goyon et al. [14,15]we rationalize the simulations data and identify the relevantenergy scales involved in this transport process. The latterallows us to propose some scaling relations, which are in goodagreement with the numerical data. Finally, we find that thedistribution of colloids among layers does not correlate withchanges in the macroscopic response. However, the changes inthe macroscopic response partially correlates with the type ofstructure within individual layers. Suspensions characterizedby a Newtonian response display a more cubiclike structureof colloids within individual layers, whereas suspensionswith non-Newtonian response colloids organize in a morehexagonal-like structure.

ACKNOWLEDGMENTS

The Spanish MINECO (Projects FIS2017-84631-P) andPIUNA University of Navarra supported this work. The au-thors thank I. Zuriguel for fruitful discussions and F. Alarconfor his technical support.

[1] P. Coussot, Rheometry of Pastes, Suspensions and GranularMaterials-Application in Industry and Environment (Wiley,New York, 2005).

[2] P. Coussot, Rheophysics of pastes: a review of microscopicmodelling approaches, Soft Matter 3, 528 (2007).

[3] D. Brogioli, Extracting Renewable Energy from a SalinityDifference Using a Capacitor, Phys. Rev. Lett. 103, 058501(2009).

[4] J. Guzowski, K. Gizynski, J. Gorecki, and P. Garstecki, Mi-crofluidic platform for reproducible self-assembly of chemi-cally communicating droplet networks with predesigned num-ber and type of the communicating compartments, Lab Chip 16,764 (2016).

[5] A. M. Forsyth, J. Wan, P. D. Owrutsky, M. Abkarian, and H.A. Stone, Multiscale approach to link red blood cell dynamics,shear viscosity, and atp release, PNAS 108, 10986 (2011).

[6] G. R. Lazaro, A. Hernandez-Machado, and I. Pagonabarraga,Rheology of red blood cells under flow in highly confinedmicrochannels. ii. effect of focusing and confinement, SoftMatter 10, 7207 (2014).

[7] M. Thiébaud, Z. Shen, J. Harting, and C. Misbah, Predictionof Anomalous Blood Viscosity in Confined Shear Flow, Phys.Rev. Lett. 112, 238304 (2014).

[8] M. Brust, O. Aouane, M. Thiébaud, D. Flormann, C. Verdier,L. Kaestner, M. W. Laschke, H. Selmi, A. Benyoussef, T.Podgorski, G. Coupier, C. Misbah, and C. Wagner, The plasmaprotein fibrinogen stabilizes clusters of red blood cells in micro-capillary flows, Sci. Rep 4, 4348 (2014).

[9] P. Malgaretti, I. Pagonabarraga, and J. M. Rubi, Entropic Elec-trokinetics, Phys. Rev. Lett 113, 128301 (2014).

[10] O. Bunk, A. Diaz, F. Pfeiffer, C. David, C. Padeste, H.Keymeulen, P. R. Willmott, B. D. Patterson, B. Schmitt,D. K. Satapathy, J. F. van der Veen, H. Guo, and G.H. Wegdam, Confinement-induced liquid ordering investi-gated by x-ray phase retrieval, Phys. Rev. E 75, 021501(2007).

[11] T. Börzsönyi, B. Szabó, G. Törös, S. Wegner, J. Török, E.Somfai, T. Bien, and R. Stannarius, Orientational Order andAlignment of Elongated Particles Induced by Shear, Phys. Rev.Lett. 108, 228302 (2012).

042603-8

RHEOLOGICAL BEHAVIOR OF COLLOIDAL SUSPENSION … PHYSICAL REVIEW E 98, 042603 (2018)

[12] S. Wegner, R. Stannarius, A. Boese, G. Rose, B. Szabo, E.Somfai, and T. Borzsonyi, Effects of grain shape on packing anddilatancy of sheared granular materials, Soft Matter 10, 5157(2014).

[13] J.-B. Salmon, L. Bécu, S. Manneville, and A. Colin, Towardslocal rheology of emulsions under couette flow using dynamiclight scattering, Eur. Phys. J E Soft 10, 209 (2003).

[14] J. Goyon, A. Colin, G. Ovarlez, A. Ajdari, and L. Bocquet,Spatial cooperativity in soft glassy flows, Nature (London) 454,84 (2008).

[15] J. Goyon, A. Colin, and L. Bocquet, How does a soft glassymaterial flow: Finite size effects, non local rheology, and flowcooperativity, Soft Matter 6, 2668 (2010).

[16] S. Manneville, L. Bécu, and A. Colin, High-frequency ultra-sonic speckle velocimetry in sheared complex fluids, Eur. Phys.J. Appl. Phys. 28, 361 (2004).

[17] S. Mueller, E. W. Llewellin, and H. M. Mader, The rheology ofsuspensions of solid particles, Proc. R. Soc. A 466, 1201 (2010).

[18] S. Gallier, E. Lemaire, L. Lobry, and F. Peters, Effect of con-finement in wall-bounded non-colloidal suspensions, J. FluidMech. 799, 100 (2016).

[19] V. Doyeux, S. Priem, L. Jibuti, A. Farutin, M. Ismail, andP. Peyla, Effective viscosity of two-dimensional suspensions:Confinement effects, Phys. Rev. Fluids 1, 043301 (2016).

[20] M. Foglino, A. N. Morozov, O. Henrich, and D. Marenduzzo,Flow of Deformable Droplets: Discontinuous Shear Thinningand Velocity Oscillations, Phys. Rev. Lett. 119, 208002 (2017).

[21] M. E. Rosti, L. Brandt, and D. Mitra, Rheology of suspensionsof viscoelastic spheres: Deformability as an effective volumefraction, Phys. Rev. Fluids 3, 012301 (2018).

[22] K. Suzuki and H. Hayakawa, arXiv:1711.08855.[23] K. Nichol, A. Zanin, R. Bastien, E. Wandersman, and M.

van Hecke, Flow-Induced Agitations Create a Granular Fluid,Phys. Rev. Lett. 104, 078302 (2010).

[24] K. A. Reddy, Y. Forterre, and O. Pouliquen, Evidence of Me-chanically Activated Processes in Slow Granular Flows, Phys.Rev. Lett. 106, 108301 (2011).

[25] M. Bouzid, M. Trulsson, P. Claudin, E. Clément, and B.Andreotti, Nonlocal Rheology of Granular Flows Across YieldConditions, Phys. Rev. Lett. 111, 238301 (2013).

[26] D. L. Henann and K. Kamrin, Continuum Modeling of Sec-ondary Rheology in Dense Granular Materials, Phys. Rev. Lett.113, 178001 (2014).

[27] D. Chen, K. W. Desmond, and E. R. Weeks, Experimen-tal observation of local rearrangements in dense quasi-two-dimensional emulsion flow, Phys. Rev. E 91, 062306 (2015).

[28] M. Bouzid, A. Izzet, M. Trulsson, E. Clément, P. Claudin, andB. Andreotti, Non-local rheology in dense granular flows, Eur.Phys. J E Soft 38, 125 (2015).

[29] M. Bouzid, M. Trulsson, A. Izzet, A. F. de Coulomb, P. Claudin,E. Clément, and B. Andreotti, Non-local rheology of densegranular flows, EPJ Web Conf. 140, 11013 (2017).

[30] T. Gueudre, J. Lin, A. Rosso, and M. Wyart, Scaling descriptionof non-local rheology, Soft Matter 13, 3794 (2017).

[31] B. Géraud, L. Jørgensen, C. Ybert, H. Delanoë-Ayari, and C.Barentin, Structural and cooperative length scales in polymergels, Eur. Phys. J E Soft 40, 5 (2017).

[32] P. Jop, V. Mansard, P. Chaudhuri, L. Bocquet, and A. Colin,Microscale Rheology of a Soft Glassy Material Close to Yield-ing, Phys. Rev. Lett. 108, 148301 (2012).

[33] M. Sbragaglia, R. Benzi, M. Bernaschi, and S. Succi, Theemergence of supramolecular forces from lattice kinetic modelsof non-ideal fluids: applications to the rheology of soft glassymaterials, Soft Matter 8, 10773 (2012).

[34] S. Succi, The lattice Boltzmann equation: for fluid dynamics andbeyond (Oxford University Press, Oxford, 2001).

[35] M. E. Cates, K. Stratford, R. Adhikari, P. Stansell, J. C. Desplat,I. Pagonabarraga, and A. J. Wagner, Simulating colloid hydro-dynamics with lattice boltzmann methods, J. Phys.: Condens.Matter 16, S3903 (2004).

[36] J. Harting, J. Chin, M. Venturoli, and P. V. Coveney, Large-scalelattice boltzmann simulations of complex fluids: Advancesthrough the advent of computational grids, Philos. Trans. RoyalSoc. A 363, 1895 (2005).

[37] J. Harting, C. Kunert, and J. Hyväluoma, Lattice boltz-mann simulations in microfluidics: Probing the no-slipboundary condition in hydrophobic, rough, and surfacenanobubble laden microchannels, Microfluid Nanofluid 8, 1(2009).

[38] N. Rivas, S. Frijters, I. Pagonabarraga, and J. Harting, Meso-scopic electrohydrodynamic simulations of binary colloidalsuspensions, J. Chem. Phys. 148, 144101 (2018).

[39] E. S. Asmolov, A. L. Dubov, T. V. Nizkaya, J. Harting, andO. I. Vinogradova, Inertial focusing of finite-size particles inmicrochannels, J. Fluid Mech. 840, 613 (2018).

[40] J. C. Desplat, I. Pagonabarraga, and P. Bladon, Ludwig: Aparallel lattice-boltzmann code for complex fluids, Comput.Phys. Commun 134, 273 (2001).

[41] R. Adhikari, K. Stratford, M. E. Cates, and A. J. Wag-ner, Fluctuating lattice boltzmann, Europhys. Lett. 71, 473(2005).

[42] S. P. Thampi, I. Pagonabarraga, and R. Adhikari, Lattice-boltzmann-langevin simulations of binary mixtures, Phys. Rev.E 84, 046709 (2011).

[43] G. Kähler, F. Bonelli, G. Gonnella, and A. Lamura, Cavitationinception of a van der waals fluid at a sack-wall obstacle, Phys.Fluids 27, 123307 (2015).

[44] G. Bryant, S. R. Williams, L. Qian, I. K. Snook, E. Perez, and F.Pincet, How hard is a colloidal “hard-sphere” interaction? Phys.Rev. E 66, 060501 (2002).

[45] N.-Q. Nguyen and A. J. C. Ladd, Lubrication corrections forlattice-boltzmann simulations of particle suspensions, Phys.Rev. E 66, 046708 (2002).

[46] M. J. Fuerstman, A. Lai, M. E. Thurlow, S. S. Shevkoplyas,H. A. Stone, and G. M. Whitesides, The pressure drop alongrectangular microchannels containing bubbles, Lab Chip 7,1479 (2007).

[47] G. K. Batchelor and J. T. Green, Brownian diffusion of par-ticles with hydrodynamic interaction, J. Fluid. Mech 56, 401(1972).

[48] C. I. Mendoza and I. Santamaría-Holek, The rheology ofhard sphere suspensions at arbitrary volume fractions: An im-proved differential viscosity model, J. Chem. Phys. 130, 044904(2009).

[49] D. Quemada, Rheology of concentrated disperse systems andminimum energy dissipation principle, Rheol. Acta 16, 82(1977).

[50] I. M. Krieger and T. J. Dougherty, A mechanism for nonnew-tonian flow in suspensions of rigid spheres, J. Rheol. 3, 137(1959).

042603-9

S. ARIETALEANIZ et al. PHYSICAL REVIEW E 98, 042603 (2018)

[51] A. Komnik, J. Harting, and H. J. Herrmann, Transport phe-nomena and structuring in shear flow of suspensions near solidwalls, J. Stat. Mech. (2004) P12003.

[52] K. Yeo and M. R. Maxey, Ordering transition of non-browniansuspensions in confined steady shear flow, Phys. Rev. E 81,051502 (2010).

[53] R. Hartkamp, A. Ghosh, T. Weinhart, and S. Luding, A study ofthe anisotropy of stress in a fluid confined in a nanochannel, J.Chem. Phys. 137, 044711 (2012).

[54] X. Cheng, J. H. McCoy, J. N. Israelachvili, and I. Cohen, Imag-ing the microscopic structure of shear thinning and thickeningcolloidal suspensions, Science 333, 1276 (2011).

[55] M. Zurita-Gotor, J. Bławzdziewicz, and E. Wajnryb, LayeringInstability in a Confined Suspension Flow, Phys. Rev. Lett. 108,068301 (2012).

[56] B. Snook, J. E. Butler, and E. Guazzelli, Dynamics of shear-induced migration of spherical particles in oscillatory pipe flow,J. Fluid Mech. 786, 128 (2016).

[57] S. Pieper and H. J. Schmid, Layer-formation of non-colloidalsuspensions in a parallel plate rheometer under steady shear, J.Nonnewton Fluid Mech 234, 1 (2016).

[58] S. D. Kulkarni and J. F. Morris, Ordering transition and struc-tural evolution under shear in brownian suspensions, J. Rheol.53, 417 (2009).

042603-10