PHYSICAL REVIEW E 79, 031401 共2009兲

Short-time motion of Brownian particles in a shear flow
1

Takuya Iwashita1,* and Ryoichi Yamamoto1,2,†

Department of Chemical Engineering, Kyoto University, Kyoto 615-8510, Japan
2
CREST, Japan Science and Technology Agency, Kawaguchi 332-0012, Japan
共Received 16 November 2008; published 4 March 2009兲

The short-time motion of Brownian particles in an incompressible Newtonian fluid under shear, in which thefluid inertia becomes important, was investigated by direct numerical simulation of particulate flows. Threedimensional simulations were performed, wherein external forces were introduced to approximately form
Couette flows throughout the entire system with periodic boundary conditions. In order to examine the validityof the method, the mean-square displacement of a single spherical particle in a simple shear flow was calculated, and these results were compared with a hydrodynamic analytical solution that includes the effects of thefluid inertia. Finally, the dynamical behavior of a monodisperse dispersion composed of repulsive sphericalparticles was examined on short-time scales, and the shear-induced diffusion coefficients were measured forseveral volume fractions up to 0.50.
DOI: 10.1103/PhysRevE.79.031401

PACS number共s兲: 82.70.⫺y, 82.20.Wt, 05.40.⫺a, 83.50.Ax

I. INTRODUCTION

The short-time motion of small particles fluctuating in a
Newtonian fluid is strongly affected by fluid inertia, that is,the vorticity of the host fluid surrounding the dispersed particles. If a dispersed particle accelerates due to Brownianforces, it affects the motion of the host fluid in the neighborhood of the particle, while the vorticity generated by theparticle’s motion then affects the motion of the same particle.
These effects are referred to as memory effects and have an
important role in the dynamical motion of a dispersion on
short-time scales 关1–3兴.
The vorticity diffuses away on a kinematic time scale ␶␯= a2 / ␯, where a is the radius of the spherical particle and ␯ isthe kinematic viscosity of the host fluid 共⬃10−6 s for 1 ␮m
in water兲. When the vorticity diffuses away much faster thanthe particle’s motion, i.e., ␶␯ Ⰶ ␶B = M / 6␲␩a, where M is themass of a Brownian particle and ␩ is the shear viscosity of
the host fluid, the motion of a Brownian particle is wellapproximated by the normal Langevin equation 共LE兲, which
is valid for strong damping 共Reynolds number Re→ 0兲 or
long-time scales; however, this equation, wherein the effectsof the fluid inertia are ignored, is not applicable to a dispersion composed of neutrally buoyant particles since ␶B is
comparable to ␶␯. For a complete understanding of the shorttime motion of a dispersion, the inertias of the particle andthe host fluid cannot be neglected.
One way to account for memory effects is to simultaneously resolve the fluid motion with the particle motion as aboundary condition to be satisfied. We refer to this approachas the direct numerical simulation 共DNS兲 approach. Within
the DNS approach, various numerical methods have beendeveloped 关4–9,15兴, and the power-law decay behavior in thevelocity autocorrelations of a free Brownian particle has
been accurately reproduced. This slow relaxation of the correlation behavior is one of the main features of memory ef-*iwashita@cheme.kyoto-u.ac.jp†

ryoichi@cheme.kyoto-u.ac.jp1539-3755/2009/79共3兲/031401共5兲

fects. Although most of these methods have been applied todispersions composed of free Brownian particles at thermal
equilibrium, dispersions under flows that are far from equilibrium have not been examined in detail, even for the simple
case of a single Brownian particle in a shear flow on a shorttime scale, ␶␯. Furthermore, most numerical methods used
for concentrated dispersions under shear, which are widelyused for measuring rheological properties, are limited to a
Reynolds number of zero, and the short-time motions of thedispersed particles in concentrated dispersions cannot be correctly tracked.
Recently, we have developed a numerical method, known
as the “smoothed profile method 共SPM兲” 关10,11兴, for the
DNS of particulate flows. Its computational accuracy andefficiency have been examined carefully by Lio et al. 关12兴for several flow problems. The SPM has been applied to adispersion composed of free Brownian particles at thermal
equilibrium 关13,14兴. We have also succeeded in reproducing
the power-law decay behavior in the translational and rotational velocity autocorrelations of a Brownian particle, andthese results agree well with hydrodynamic analytical solutions for a free Brownian particle in an infinite fluid that
accounts for memory effects.
In order to simulate dispersions in nonequilibrium conditions on short-time scales in which memory effects become
significant, we have modified the SPM. The primary objective of the present work is to accurately examine the shorttime motion of Brownian particles in a simple shear flow byusing the modified SPM. In this paper, we first present themodified SPM, in which external forces are introduced to
impose a shear flow into the system. In order to validate themethod, we next compare the numerical results for the meansquare displacement 共MSD兲 with a hydrodynamic analyticalsolution of a Brownian particle in simple shear flows that
account for fluid inertia. Furthermore, we apply our method
to a dispersion composed of many spherical particles undershear. The MSD in the vorticity direction is then calculated
for several volume fractions, and the time evolution is discussed. Finally, the shear-induced diffusion coefficients aremeasured from the long-time behavior of the MSD, and the031401-1©2009 The American Physical Society

PHYSICAL REVIEW E 79, 031401 共2009兲

TAKUYA IWASHITA AND RYOICHI YAMAMOTO

dependence of the determined diffusion coefficients on thevolume fraction is examined.
II. SIMULATION METHOD

Let us consider a monodisperse dispersion of repulsive
spherical particles in a Newtonian host fluid. The dispersion
is subjected to shear by an external force. The position of theith particle is Ri, the translational velocity is Vi, and therotational velocity is ⍀i. The velocity and pressure fields ofthe host fluid are v共x , t兲 and p共x , t兲, respectively. These field
quantities are defined on three-dimensional Cartesian grids,x 苸 关0 , Lx兴关−Ly / 2 , Ly / 2兴关0 , Lz兴. In order to distinguish theparticle and fluid domains on the grids, a smoothed function
␾共x , t兲, which is equal to 1 in the particle domains and 0 in
the fluid domains, is introduced. These domains are separated by a thin interfacial domain of thickness ␰. The length
unit is taken to be the lattice spacing 䉭, and the time unit is␳ f ⌬2 / ␩, where ␳ f denotes the density of the host fluid. The
time evolution of the ith dispersed particle with mass M i andmoment of inertia Ii is governed by Newton’s equations of
motion,CV
M iV̇i = FH
i + Fi + Gi ,Ṙi = Vi ,⍀Ii · ⍀̇i = NHi + Gi ,yzFIG. 1. Schematic diagrams of a zigzag velocity flow. The arrows represent the velocity vectors of the host fluid in the flow
direction. The x, y, and z axes represent the flow, velocity-gradient,and vorticity directions, respectively.

共1兲冦␥˙ 共− y − Ly/2兲 共− Ly/2 ⬍ y ⱕ − Ly/4兲
共− Ly/4 ⬍ y ⱕ Ly/4兲
vx共y兲 = ␥˙ y
˙␥共− y + Ly/2兲 共Ly/4 ⬍ y ⱕ Ly/2兲,

共2兲H
where FH
i and Ni are the hydrodynamic forces and torques
exerted by the host fluid on the particle 关10,11兴. FCi is arepulsive force that is employed to prevent particle overlaps,
and a truncated Lennard-Jones potential, V共rij兲 = 4关共␴ / rij兲36− 共␴ / rij兲18 + 1 / 4兴 for rij ⬍ 21/18␴ or V共rij兲 = 0, is adopted inthis work. Here rij = 兩Ri − R j兩 and ␴ = 2a represents the diameter of particles. GVi and G⍀
i are random forces and torques,
respectively, due to thermal fluctuations. These fluctuations
are introduced as white noise with a zero mean and correla⍀
and
具G⍀
tions
具GVi 共t兲GVj 共0兲典 = ␣VI␦共t兲␦ij
i 共t兲G j 共0兲典= ␣⍀I␦共t兲␦ij, where ␣V and ␣⍀ are numerical parameters that
control the translational and rotational particle temperatures,
namely, TV and T⍀ 关13,14兴. The angular brackets denote taking an average over an equilibrium ensemble. The temperatures are determined by the following procedure. First asingle Brownian particle at thermal equilibrium is simulatedwith fixed ␣V and ␣⍀. Then the translational and rotational
long-time diffusion coefficients are obtained from the simulation. By comparing these diffusion coefficients with
Stokes-Einstein diffusion coefficients, DV0 = kBTV / 6␲␩a for⍀3
the translational motion and D⍀
0 = kBT / 8␲␩a for the rotational motion, we finally can determine the temperatures.
Since both ␣V and ␣⍀ are chosen to satisfy kBTV = kBT⍀, thetemperatures are simply written as kBT throughout this paper.
The time evolution of the host fluid is governed by the
Navier-Stokes equations,␳ f 共⳵tv + v · ⵱v兲 = ⵱ · ␴ + ␳ f ␾f p + ␳ f fshear ,共3兲␴ = − pI + ␩兵⵱v + 共⵱v兲T其,共4兲

where the incompressibility condition, ⵱ · v = 0. fshear共x , t兲 isan external force field that is introduced to enforce the following velocity profile:

x

冧

共5兲

where ␥˙ denotes the shear rate of the imposed flow and ydenotes the distance in the velocity-gradient direction. Thisvelocity profile, schematically depicted in Fig. 1, enables us
to solve the motion of the host fluid with periodic boundary
conditions. Couette flow profiles are approximately formed
over a range from y = −Ly / 4 to y = Ly / 4, although the zigzag
profile strictly becomes a Couette flow when Ly → ⬁. ␾f p
represents the body force that ensures the rigidity of particles
and the appropriate nonslip boundary conditions at the fluid/particle interface, which is further elaborated in Refs.
关10–12兴.
III. RESULTS AND DISCUSSION

The computational domain has three dimensions 64⫻ 64⫻ 64 and periodic boundary conditions. The numerical parameters for both the host fluid and the spherical particles are⌬ = 1, ␩ = 1, ␳ f = 1, a = 5, ␰ = 2, and particle density ␳ p = 1. The
imposed shear rate is ␥˙ = 0.005, and the temperature is kBT= 0.07. The temperature was determined by measuring thelong-time diffusion coefficient of a single Brownian particle
at thermal equilibrium before the shear was imposed. Thissystem has dimensionless parameters such that the Peclet
number Pe= 6␲␩a3␥˙ / kBT ⯝ 170 and the particle Reynolds
number Rep = ␳ p␥˙ a2 / ␩ = 0.125. The initial configuration of
the spherical particle is located at the central position of thesystem. The volume fraction of a single particle is ⌽= 0.002.
An important feature of a single Brownian particle in a
simple shear flow is that the MSD in the flow direction varies
asymptotically with time as t3. Although we would expect t
dependence of the MSD in the long-time limit, this nondif-

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SHORT-TIME MOTION OF BROWNIAN PARTICLES IN A …

具Rxi 共t兲2典 = v2x 共y 0兲t2 + 具Rxi 共t兲2典␥˙ + 具Rxi 共t兲2典0 ,共6兲assuming that, at t = 0, the particle is in the initial position
共0 , y 0 , 0兲, wherein the first term on the right-hand side of Eq.
共6兲 is a shear contribution only, and represents the simple
translation of a particle along the streamline in the flow direction. The second term is the coupling term describedabove, which represents the convection induced by diffusion.
The third term, which is in the same form as the MSD of afree Brownian particle, is a thermal contribution only. Similar to that of the x direction, the MSDs in the velocitygradient direction 共y兲 and the vorticity direction 共z兲 are derived theoretically; 具Riy共t兲2典 = 具Riy共t兲2典0 and 具Rzi 共t兲2典 = 具Rzi 共t兲2典0.
These quantities are determined analytically for a single
Brownian particle obeying the normal LE or the generalized
Langevin equation 共GLE兲 with memory effects 关16–18兴.
The time evolution of the MSD in the x direction 共䊊兲 fora single spherical particle under shear is shown in Fig. 2. The
MSD was calculated via 具兩Rxi 共t兲 − Rxi 共0兲 − vx共y 0兲t兩2典 to eliminate the purely shear contribution that corresponds to the firstterm of Eq. 共6兲. The MSD is scaled by kBT. The MSD is inexcellent agreement with the analytical solution for the GLE.
For short times of up to t ⬃ 102, the motion of the Brownian
particle in the x direction is like that of a free Brownian
particle obeying the GLE. For long times, t Ⰷ ␶␯ = 25, theMSD asymptotically approaches 2D0␥˙ 2t3 / 3, where D0= kBT / 6␲␩a. This t3 regime in the MSD is approached in amuch slower manner, t−1/2, than that of the analytical solution for the LE. The transient behavior of the MSD depends
strongly on whether the memory effects are taken into account.
The MSD 具兩Rzi 共t兲 − Rzi 共0兲兩2典 in the z direction 共䊏兲 is alsoplotted in Fig. 2. The numerical results of the MSD in the z
direction agree well with the analytical solution for the GLEof a free Brownian particle where the t regime is approached
in a much slower manner, t−1/2, and the diffusive motion is
attained on time scales of O共104兲. The diffusion time characterizing the diffusive motion is ␶D = a2 / D0 ⯝ 3.4⫻ 104,
which measures the particle diffusion over the particle size.
Another dynamical quantity of interest is the positionalcross-correlation 具Rxi 共t兲Riy共t兲典 of a Brownian particle in a
simple shear flow. This analytical form is derived for boththe LE and GLE. These long-time behaviors show the samedependence on t, which is D0␥˙ t2. The cross correlation was
calculated via 具(Rxi 共t兲 − Rxi 共0兲 − Vxi 共0兲t)(Riy共t兲 − Riy共0兲)典. The numerical results 共䉭兲 are plotted in Fig. 2 and are in good
agreement with the analytical solution for the GLE.
This detailed analysis of the MSD is sufficient to confirm
the validity of our method of incorporating memory effects;
however, within our method, it is assumed that the thermal
fluctuations can be represented by white and Gaussian noise.
This means that correlations between thermal noise at different times are completely ignored, although in the theoreticalframework of the GLE, noises are correlated. The thermal

4

10

t3

GLE : x direction
GLE : z direction
GLE : cross correlation
LE : x direction
LE : cross correlation

103

Mean square displacement

fusional behavior is explained as a result of a coupling between diffusive motion in the velocity-gradient direction andconvective motion due to shear in the flow direction . Theoretical solutions to this problem show that the MSD in theflow direction 共x兲 is composed of three parts,

PHYSICAL REVIEW E 79, 031401 共2009兲

t2
t

102

101

0

10

10-1

-2

10

τν

τB

τD

-3

10

100

101

102

t

103

104

105

FIG. 2. 共Color online兲 Mean square displacement scaled by kBTof a single spherical particle fluctuating in a Newtonian fluid undershear at kBT = 0.07 and ␥˙ = 0.005: 共䊊兲 flow direction, 共䊏兲 vorticity
direction, and 共䉭兲 positional cross correlation. The analytical solution of the MSD in the flow direction 共x兲 for a Brownian particle ina shear flow: the dashed line represents the LE and the solid line
represents the GLE. The dashed-dotted line is the analytical solution of the MSD in the vorticity direction 共z兲 for the GLE, which isthe same form as that for the GLE of a free Brownian particle The
analytical solution of the positional cross correlation for a Brownian
particle in a shear flow: the dotted line represents the LE and thedashed line represents the GLE. The Brownian time ␶B ⯝ 4.5, thekinematic time ␶␯ = 25, and the diffusion time ␶D ⯝ 3.4⫻ 104. The
parameters for both the particle and fluid are a = 5, ␰ = 2, ␳ p = 1, ␳ f= 1, and ␩ = 1. The time unit is ⌬2 / ␯ = 1, and the length unit is ⌬= 1.

noise memory, which we ignore, is considered to affect theparticle’s motion on very fast time scales t ⬍ ␶B ⬃ 4.5. On
these very short-time scales, t ⬍ ␶B, a gap between the hydrodynamic analytical solutions and the numerical results isclearly observed in previous studies 关13,14兴. Although our
method is not applicable to particle motion for t ⬍ ␶B, themethod can be accurately applied to the motion of particles
in a shear flow on time scales t ⬎ ␶B, in which the memory
effects become significant.
Most numerical approaches so far performed for concentrated particle dispersions are based on the Stokesian dynamics 关19兴; thus those simulations are valid only for the zero
Reynolds number limit. Recently, concentrated dispersions at
finite Reynolds number have been simulated using the DNS
methods based on the stochastic rotation dynamics 关20兴 or
the lattice Boltzmann method 关21–23兴, We have applied themodified SPM to concentrated dispersions at a finite Reynolds number Rep = 0.125 in order to examine the short-time
motion of Brownian particles in a shear flow at finite-volume
fractions. Simulations were performed at several volume
fractions from a very dilute case with ⌽ = 0.002 to very dense
cases with ⌽ = 0.4 and 0.5, which are apparently higher thanthe previous DNS simulations.

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PHYSICAL REVIEW E 79, 031401 共2009兲

TAKUYA IWASHITA AND RYOICHI YAMAMOTO102

Φ = 0. 002
0. 10
0. 20
0. 30
0. 35
0. 40
0. 50

1

0

10

0.05
.

10-1

Dz/γa20.04

Dz /γa2t2.

Mean square displacement

10

t

-2

10

0.03
0.02
0.01

10-3

0

-4

10

0.1

0

γτB γτν
.

-2

10

.

-1

10

10

0

1

.

γt

10

0.2

0. 3

Φ
10

0.4

γτD
.

2

3

10

FIG. 3. 共Color online兲 The time evolution of the mean-squaredisplacement in the vorticity direction 共z兲 for several volume fractions ⌽ at kBT = 0.07 and ␥˙ = 0.005. The Peclet number Pe⯝ 170 andthe particle Reynolds number Rep = 0.125. The inset shows the volume fraction dependence of Dz scaled by ␥˙ a2. The parameters forboth the particle and fluid are a = 5, ␰ = 2, ␳ p = 1, ␳ f = 1, and ␩ = 1.

Since Pe⯝ 170, the hydrodynamic shear forces become
dominant over the thermal forces in the particle’s motion.
The initial configurations of the dispersed particles are randomly distributed.
Figure 3 shows the time evolution of the MSD in the z
direction for several volume fractions, up to ⌽ = 0.5. The
time is scaled by 1 / ␥˙ . As the volume fraction is increased,
the MSD grows more rapidly in time due to the hydrodynamic and direct interactions between the particles, resulting
in a increase in the slope of the MSD for a small strain ␥˙ t
⬃ O共10−1兲. For high volume fractions 0.2ⱕ ⌽ ⱕ 0.4, the diffusive behavior is attained at a smaller strain ␥˙ t ⬃ O共1兲 rather
than at ⌽ = 0.002 as for a Brownian particle. The accelerated
Stokesian dynamics 共ASD兲 for non-Brownian particles at⌽ = 0.2 show that the diffusive region is attained at larger
strains than at least 10 关24兴. Compared with these numericalresults, the onset of the diffusive region in the present simulations is much faster. The time at which the diffusive region
is attained shifts to shorter times at higher volume fractions.
For the highest volume fraction ⌽ = 0.5, however, we seethat the motion of Brownian particles is trapped within theeffective cages formed by the surrounding particles. This isbecause the particles start to form stringlike objects in theflow direction, and finally the whole system evolves into a
two-dimensional ordered structure in the plane perpendicular
to the flow, similarly to the flow-induced ordering commonly
observed in experiments 关25–27兴 and by computer simulations 关28–30兴 under shear flow. In the present case with
Rep = 0.125 and Pe⬃ 170, we found that the shear-inducedordering occurs at the high volume fraction between ⌽= 0.4 and 0.5.
The long-time diffusion coefficient Dz was calculated bylinearly fitting the data over the diffusion regions. The inset
of Fig. 3 shows the volume fraction dependence of Dz scaledby ␥˙ a2. The diffusion coefficient increases rapidly up to ⌽= 0.3 and reaches a plateau at a volume fraction beyond ⌽⬎ 0.3. This behavior of Dz is remarkably different from thatat thermal equilibrium, where the diffusion coefficient decreases with increasing volume fraction. The enhancement of
the diffusion coefficient with increasing volume fraction is atypical characteristic of shear-induced diffusion coefficients.
These results exhibit the same qualitative behavior as theexperimental results obtained for non-Brownian particles
关31–33兴.
IV. CONCLUSION

In conclusion, the short-time motion of Brownian particles in an incompressible Newtonian fluid under shear was
examined by using a modified SPM in which external forces
are introduced to approximately form a simple shear flow
throughout the entire system with periodic boundary conditions. The validity of the method was carefully examined bycomparing the present numerical results for the MSD withthe hydrodynamic analytical solution of the GLE of a single
Brownian particle in a simple shear flow. In the present
study, we aimed to modify the original SPM by incorporating
thermal fluctuations so that the modified SPM is valid for␶B ⬍ t, while other computational methods such as Brownian
dynamics and ASD, which are based on the LE that ignores
memory effects due to fluid motions, are valid only for ␶BⰆ ␶D ⬍ t. Simulations were then performed for monodisperse
dispersions of repulsive spherical particles at volume fractions ranging from ⌽ = 0.002 to 0.50. We found that the MSD
in the vorticity direction grows rapidly in time and with increasing particle volume fraction. At a strain ␥˙ t ⬃ O共1兲, thediffusive region is attained for 0.2ⱕ ⌽ ⱕ 0.4. The onset ofthe diffusive region shifts to shorter times at higher volume
fractions. For ⌽ = 0.5, however, the particles are no morediffusive because of the shear-induced ordering. The diffusion coefficient in the vorticity direction was obtained fromthe long-time behavior of the MSD. For volume fractions of
up to 0.3, the diffusion coefficient rises rapidly with increasing volume fraction. It then levels off for volume fractions
beyond 0.3.

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