THE JOURNAL OF CHEMICAL PHYSICS 138, 184905 (2013)

Velocity relaxation of a particle in a confined compressible fluid
Rei Tatsumia) and Ryoichi Yamamotob)
Department of Chemical Engineering, Kyoto University, Kyoto 615-8510, Japan(Received 19 February 2013; accepted 22 April 2013; published online 13 May 2013)
The velocity relaxation of an impulsively forced spherical particle in a fluid confined by two parallel
plane walls is studied using a direct numerical simulation approach. During the relaxation process,the momentum of the particle is transmitted in the ambient fluid by viscous diffusion and soundwave propagation, and the fluid flow accompanied by each mechanism has a different character andaffects the particle motion differently. Because of the bounding walls, viscous diffusion is hampered,and the accompanying shear flow is gradually diminished. However, the sound wave is repeatedlyreflected and spreads diffusely. As a result, the particle motion is governed by the sound wave andbacktracks differently in a bulk fluid. The time when the backtracking of the particle occurs changes
non-monotonically with respect to the compressibility factor ε = ν/ac and is minimized at the characteristic compressibility factor. This factor depends on the wall spacing, and the dependence isdifferent at small and large wall spacing regions based on the different mechanisms causing the
backtracking. © 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4804186]
I. INTRODUCTION

The dynamics of particles dispersed in a fluid flowing
through a microchannel are important for many chemical engineering processes, such as membrane separation and microfluidics. The dynamics of fluids and dispersed particles
are significantly affected by bounding walls; thus, the transport properties of dispersions in some simple bounding geometries have been studied using analytical methods.1 While
such studies were limited to a steady flow system, recently,
the unsteady dynamics of a dispersed particle in a confined
fluid have been investigated.2–9
In a dispersion system, the momentum of a particle ispropagated in the ambient fluid via two mechanisms: viscous
diffusion and sound propagation. Each of these mechanisms
is accompanied by fluid flow of different character, which affects the particle motion. Here, we consider the velocity relaxation of a particle after adding an impulsive force. In abulk fluid, part of the particle momentum is transported bya sound wave to an infinite distance in time, and finally, theparticle motion is governed by shear flow accompanied by
viscous diffusion, which results in the algebraic decay obeying t−3/2 long-time tail.10, 11 However, in a fluid confined bywalls with stick boundary conditions, both the viscous diffusion and sound propagation are affected by the walls. The viscous diffusion is hampered at the walls, and the accompanying shear flow gradually diminishes; however, the sound wave
is repeatedly reflected at the walls and spreads diffusely.3, 4Consequently, the particle motion is finally governed by flowaccompanied by the spreading sound wave. Especially in a
fluid confined between two parallel plane walls, the particle
velocity relaxation presents a negative t−2 long-time decay
differently than that presented in a bulk fluid, which is derived
by the mode-coupling theory3 and Green’s function method.4a) Electronic mail: tatsumi@cheme.kyoto-u.ac.jpb) Electronic mail: ryoichi@cheme.kyoto-u.ac.jp

0021-9606/2013/138(18)/184905/7/$30.00

In the present study, we investigate the dynamics of a single spherical particle in a fluid confined by two parallel planewalls with stick boundary conditions using a direct numerical simulation approach. We use the smoothed profile method(SPM),12, 13 which is applicable to a compressible fluid.14, 15
In SPM, rigid fixed wall boundaries can be imposed similar to
the representation of rigid particles.16 The accuracy of SPM
for the present system is confirmed by calculating the steadystate mobility of the particle, which is compared with the approximate analytical solutions. We examine the velocity relaxation of the particle after adding an impulsive force in the
direction parallel to the walls. The velocity relaxation function corresponds to the velocity autocorrelation function in afluctuating system. We first consider an incompressible fluid
to investigate the wall effects on the dynamics governed only
by viscous diffusion. We then consider a compressible fluid
and investigate the particle motion affected by the spreading
sound wave arising from reflections at the walls.
II. MODEL

We consider a system in which a single particle is dispersed in a Newtonian fluid confined by two parallel planewalls as described in Fig. 1. The stick boundary conditions
are imposed on the walls. The dynamics of fluid is governedby the following hydrodynamic equations as∂ρ+ ∇ · (ρv) = 0,
∂t

(1)

∂ρv
+ ∇ · (ρvv) = ∇ · σ + K P + K W ,∂t(2)

where ρ(r, t) and v(r, t) are the mass density and velocity
fields of fluid, respectively. The stress tensor is given by2T
σ = −p I + η[∇v + (∇v) ] + ηv − η (∇ · v)I, (3)3138, 184905-1© 2013 AIP Publishing LLC

184905-2

R. Tatsumi and R. Yamamoto

J. Chem. Phys. 138, 184905 (2013)

FIG. 1. Geometry of the present model system. A single spherical particle isin a fluid confined by two parallel plane walls. The particle radius is a andthe wall spacing is H. The particle is located at a height l from the lower
wall. The system is considered to be in a simulation box of size Lx × Ly
× Lz . The periodic boundary conditions are imposed on the end of the box
in all directions. The walls are set on the top and bottom of the box in the z
direction with thickness (Lz − H)/2. The walls generate anisotropy for two
directions: parallel and perpendicular to the walls. In this figure, the x andy directions are the degenerate parallel directions, and the z direction is theperpendicular direction.

where p(r, t) is the pressure, η is the shear viscosity, and ηv
is the bulk viscosity. The body force K P is added so that therigidity of particles is satisfied. The external force field K W
is also introduced in order to impose the stick boundary conditions by two parallel plane walls. Additionally, we assume
a barotropic fluid described by p = p(ρ), with the constant
speed of sound c as
dp
= c2 .
dρ(4)

Equations (1)–(4) are closed for the variables ρ, v, and p without consideration of energy conservation.
The motion of the particle is governed by Newton’s andEuler’s equations of motion asMd
V = FH + FW + FE ,dtd
R = V,dt(5)d = NH + NE,(6)dt
where R, V, and  are the position, translational velocity, androtational velocity of the particle, respectively. The particle
has a mass M and a moment of inertia Ip . A hydrodynamic
force F H and a torque N H are exerted on the particle by thefluid, and a repulsive force F W prevents the particle overlapping with the walls. A force F E and a torque N E are externally applied. According to the momentum conservation between the fluid and the particle, the hydrodynamic force andtorque are evaluated from the stress tensor as
(7)F H = dS · σ ,
Ip ·


N

H

=

(r − R) × (dS · σ ),(8)where the integrations are performed over the surface of theparticle.
We use the SPM for the direct numerical simulations in
the present study. The system is composed of three regions:

fluid, particle, and wall. In SPM, the boundaries between thefluid region and the other regions are expressed by the continuous phase profile function. There are no boundaries between
the particle and wall regions because the particle does not penetrate into the wall. For this purpose, we introduce a smoothed
profile function φX (r, t) ∈ [0, 1], where the index X signifies
the region of particle P or wall W . The function φ X represents the boundary between the regions of fluid and X, suchthat φ X = 1 at the region X and φ X = 0 at the other regions.
By the smoothed profile function, the regions of fluid and X
are smoothly connected through thin interfacial region witha thickness ξ . The detailed mathematical expression of φ X isgiven in the previous paper.12 The body force for the particle rigidity K P is expressed as ρφP f P , which constrains thevelocity field in the particle region to be V +  × (r − R).
The external force field imposing the stick boundary conditions by walls K W is expressed as ρφW f W , which constrains
the velocity field in the wall region to be zero.
III. NUMERICAL RESULTS

Numerical simulations are performed for a threedimensional box with periodic boundary conditions. The
space is divided into meshes of length , which is a unit
length. The units of the other physical quantities are definedby combining η = 1 and ρ 0 = 1 with , where ρ 0 is the fluidmass density at equilibrium. The other parameters are set toa = 4, ξ = 2, ρ p = 1, ηv = 0, and h = 0.05, where ρ p is theparticle mass density and h is the time increment of a single
simulation step. The geometry of the present system is described in Fig. 1. The particle is set on the midway between
two walls as l = H/2. The ratio of the wall spacing to theparticle radius, H∗ = H/a, is set to various values. The system size is Lx × Ly × Lz = 512 × 512 × 32 for H∗ < 8 and
Lz = 64 for H∗ ≥ 8.
We investigate the relaxation of the particle velocity after
exerting an impulsive force at the center of the particle. The
impulsive force is assumed to be sufficiently small such that
the Reynolds and Mach numbers of the flow are sufficiently
low. We set the impulsive force to produce an initial particle
Reynolds number of Rep = 10−3 . In the considered system,
two directions are specified: parallel and perpendicular to thewalls; therefore, assuming a low Reynolds number flow, therelaxation of the particle velocity is described as
V (t) =P
· γ (t),Mγ (t) = γ  (t)(I − ẑ ẑ) + γ ⊥ (t)ẑ ẑ,

(9)
(10)

where P is the impulsive force exerted on the particle att = 0 and ẑ is the unit vector in the z direction. The velocity relaxation tensor γ (t) also depends on the wall spacing H and the distance of the particle from the lower wall l.
In the present study, we focus on the parallel motion of theparticle γ  (t); therefore, in Sec. III, we represent the velocity relaxation function in the parallel direction by γ (t) for
simplicity. For this reason, collisions of the particle against
the wall will not occur, and the direct interactions between aparticle and wall, including the overlap repulsion force, are184905-3

R. Tatsumi and R. Yamamoto

J. Chem. Phys. 138, 184905 (2013)not considered in the present simulations. According to thefluctuation-dissipation theorem, the velocity relaxation function is equivalent to the velocity autocorrelation function in afluctuation system,
γ (t) =MV (0)V (t),
kB T

(11)

where kB is the Boltzmann constant and T is the thermodynamic temperature.
The important time scales in the dynamics of a single
particle are those of viscous diffusion and sound propagation over the length of the particle radius: τ ν = a2 /ν andτ c = a/c, respectively, where ν = η/ρ 0 is the kinematic viscosity. We define the compressibility factor by the ratio of thesetime scales as
ν
τc
=
ε=
,
(12)
τν
ac
which provides a measure of the importance of sound propagation in the dynamics of a single particle. In the present
simulations, we adjust the fluid compressibility by the compressibility factor.
A. Steady-state mobility in an incompressible fluid

First, we estimate the particle mobility μ in steady state
for accuracy testing. In these simulations, a constant force wascontinuously exerted on the particle in the direction parallelto the walls. The mobility was calculated as the ratio of theparticle velocity to the added force after reaching steady state.
The mobilities in such situations have been studied by Faxén
using analytical theories.1, 17 The solution was given as thepower series of the ratio of the wall spacing to particle radius,in which the first few terms were derived as
6π ηaμ(H ) = 1 − 1.004λ + 0.418λ3+ 0.21λ4 − 0.169λ5 + O(λ6 ),

(13)

∗

where λ = 2/H . The simulation results are presented inFig. 2 and are compared with the approximate solutions given
by Eq. (13). The simulations were performed with the various

system side lengths in parallel directions to the walls, Lx and
Ly (Lx = Ly ), to investigate the system size effects due to theperiodic boundary condition, which are diminished to an insignificant level at Lx ≥ 256. However, the simulation results
underestimate the mobility even at Lx = 512; the deviations
from the solutions given by Eq. (13) are less than 2% for H∗≥ 6 and approximately 5% only for H∗ = 4. As for H∗ = 4,
however, because the order of the power λ6 is still 10−2 , atruncation error of a few percent arises in Eq. (13). Therefore,
the simulation error for H∗ = 4 can be less than 5%.
From the present results, the walls with stick boundary
conditions are successfully implemented by SPM. The implementation of confinement by different geometries is alsoavailable in SPM.
B. Velocity relaxation in an incompressible fluid

Here, we consider an incompressible fluid as a solvent
fluid. To treat an incompressible fluid, we assume an infinite speed of sound in Eq. (4) and ignore the deviation of thefluid density. With this assumption and the mass conservation
law Eq. (1), the solenoidal condition for the velocity field isderived as∇ · v = 0.

(14)

Therefore, Eqs. (2), (3), and (14) were solved in combination
as the hydrodynamic equations.
The simulation results of velocity relaxation functions forwall spacings H∗ = 4 and 6 are shown in Fig. 3. The relaxation functions decrease monotonically, and the reduction rateis increased with the decrease in the wall spacing H∗ . In abulk fluid, the relaxation function presents the long-time decay obeying the power law given by11
1 ρp  τν 3/2
γ bulk (t) = √as t → ∞.
(15)
9 π ρ0 t
However, the decays in confined fluids are much faster thanthe power-law decay given by Eq. (15). When a fluid is
0

10
0.9

H* = 4
H* = 6bulk (analytical solution)

-2

10

6

-4

0.7

10
-6

-15

10

-20

10

Lx = 64
Lx = 128
Lx = 256
Lx = 512

0.5

10

-10

10
0.6

-5

10

(t)

(t)

a (H )

0.8

-25

10

-8

10

-30

10

40

60

10
0.4

4

6

8

10

12

-2

10

80

t/

-10
-1

0

10

10

100
1

10

2

10

t/

H*
FIG. 2. Wall spacing dependence of the mobility of a single particle in a
fluid confined by two parallel plane walls. The length of the simulation box
sides parallel to the walls are changed to Lx = Ly = 64, 128, 256, and 512
to investigate the effect of the periodic boundary condition. The broken line
represents the analytical solution given by Eq. (13).

FIG. 3. Velocity relaxation functions of a particle in an incompressible fluid
confined by two parallel plane walls. The wall spacings are H∗ = 4 and 6.
The analytical solution of the corresponding function in a bulk fluid is alsopresented.18 The bold dotted line represents the long-time tail At−3/2 , as given
by Eq. (15). The bold dashed-two dotted lines represent the long-time decay
proportional to t−5/2 exp (−νπ 2 t/H2 ).

R. Tatsumi and R. Yamamoto

J. Chem. Phys. 138, 184905 (2013)0(a) 10

= 0.6

-1

10

-2

10

(t)

half-space bounded by a single wall, the velocity relaxation of
the particle motion parallel to the wall presents a t−5/2 longtime tail,19 instead of t−3/2 . Therefore, the decay in a fluid
confined by two parallel walls is expected to be finally proportional to t−5/2 exp (−νπ 2 t/H2 ), corresponding to a continuous spectrum of viscous modes with wave vector component
perpendicular to the walls equal to π /H. The simulation results shown in Fig. 3 can be fitted by this type of decay function. When the decay function is rewrote as the function of
reduced time τ = t/τ ν , i.e., Cτ −5/2 exp [ − (π /H∗ )2 τ ], the coefficients C are estimated as 20 and 30 for H∗ = 4 and 6,
respectively. The rapid decay of the relaxation function is expected to reflect the loss of fluid momentum at the walls withstick boundary conditions. Therefore, the time when the relaxation function in a confined fluid starts to remarkably falls
below that in a bulk fluid will be related to the time scale when
the fluid flow generated by the particle reaches the walls. In an
incompressible fluid, the temporal evolution of the fluid flow
is only accompanied by viscous diffusion, whose time scale
over the distance between the particle surface and the wall
(H/2 − a) is given by τν† = (H /2 − a)2 /ν = (H ∗ /2 − 1)2 τν .
As for the wall spacings H∗ = 4 and 6, the viscous diffusion
time scales are τν† /τν = 1 and 4, respectively. As shown in
Fig. 3, the starting time of the deviation of the function fromthat in a bulk fluid corresponds to the time scales τν† .

-3

10

-4

10

H* = 4bulk (analytical solution)

-5

10

-6

10

-2

10

-1

0

10

10

1

10

2

10

t/
0

(b) 10

= 1.5

-1

10

-2

10

(t)

184905-4

-3

10

-4

10

H* = 4bulk (analytical solution)

-5

10

-6

10

-2

10

-2.5

0

2.5

-3

5 x10

*

(b) H = 6

-5

-2.5

0

2.5

5 x10-30

2.5

5 x10-3(c) bulk fluid

-5

0

10

1

10

2

10

t/

(a) H* = 4

-5

-1

10

-2.5

FIG. 4. Velocity fields around the particle at t/τ ν = 2.81 in confined fluidswith a wall spacing of (a) H∗ = 4, (b) H∗ = 6, and (c) in a bulk fluid areshown. The walls are set on the upper and lower boundaries of the images
for the confined fluids, and the corresponding region is presented for the bulk
fluid. The cross-sections presented here are perpendicular to the walls andparallel to the impulsive force direction and include the particle center. The
direction of the impulsive force is to the right in the images, and the particle
is represented by a black circle. The vorticity of the velocity field ∇ × v
is described using a color scale, moving from negative (darker) to positive(lighter) vorticity. The vorticity is normalized by the factor of τ ν /Rep .

FIG. 5. Velocity relaxation functions of a particle in a compressible fluid
confined by two parallel plane walls with the wall spacing H∗ = 4. The compressibility factors are (a) ε = 0.6 and (b) ε = 1.5. The analytical solutions of
the corresponding functions in a bulk fluid are also presented.20, 21 The functions are described by solid lines (positive values) and dashed-dotted lines(negative values). The bold dotted lines represent the positive long-time tailAt−3/2 , given by Eq. (15). The bold dashed-two dotted lines represent thenegative long-time tail −Bt−2 , given by Eq. (16).

The velocity fields around the particle with various wall
spacings at the time t/τ ν = 2.81 are displayed in Fig. 4. Compared with a bulk fluid, the attenuation of fluid velocity isclearly observed in a confined fluid with H∗ = 4, corresponding to the decay of the shear flow or the vorticity by the walls.
However, the attenuation of fluid velocity is not observedfor H∗ = 6. This result coincides with the fact that the timet/τ ν = 2.81 is earlier than the viscous diffusion time scalefor H∗ = 6, namely, τν† /τν = 4. The walls restrict the viscous
diffusion of the flow field to form a laminar flow, which corresponds to the extended distribution of the vorticity along thewalls.

C. Velocity relaxation in a compressible fluid

In a compressible fluid, the velocity relaxation function
exhibits essentially different behaviors from those in an incompressible fluid. The simulation results of the relaxation
functions are presented in Fig. 5. Remarkable differences are
the sign inversion and the subsequent negative long-time algebraic decay. The power of the long-time decay is t−2 , and184905-5

R. Tatsumi and R. Yamamoto

J. Chem. Phys. 138, 184905 (2013)

For the compressibility factor ε = 0.6, the temporal evolution of the velocity field in a confined fluid is compared withthat in a bulk fluid in Fig. 6. The flow accompanied by a soundwave is source-sink flow: the flow moves from the source(positive divergence region) to the sink (negative divergenceregion). The divergence of the velocity field corresponds tothe fluid density deviation ast/τν = 0.94

-0.1

-0.05

0

0.05

0.1

∇·v =−

-0.1

-0.05

0

0.05

0.1

t/τν = 2.81

-3

-6

-3

0

3

6 x10

-6

-3

0

3

6 x10

-6

-3

0

3

6 x10

-6

-3

0

3

6 x10

-3

t/τν = 6.25

-4

-4

FIG. 6. Temporal evolution of the velocity fields in a confined fluid with H∗= 4 (upside in each time) and in a bulk fluid (downside in each time). The
compressibility factor is ε = 0.6. The walls are set on the upper and lower
boundaries of the images for the confined fluids, and the corresponding regions are presented for the bulk fluids. The cross-sections presented here areperpendicular to the walls and parallel to the impulsive force direction andinclude the particle center. The direction of the impulsive force is to the rightin the images, and the particle is represented by a black circle. The divergenceof the velocity field ∇ · v is described using a color scale, moving from negative (darker) to positive (lighter) vorticity. The divergence is normalized by
the factor of τ ν /Rep .

the coefficient is analytically derived as3, 4
γ (t) = −3 ρp ε2  τν 2
as t → ∞.
8 ρ0 H ∗ t(16)

In the present simulation results, the asymptotic approaches
to Eq. (16) are accurately reproduced. In a bulk fluid, thelong-time decay is positive and proportional to t−3/2 , given by
Eq. (15) as an incompressible fluid. Although the sign inversion can occur for a large compressibility factor as ε = 1.5,
the relaxation function eventually becomes positive.

ρ
D
ln .
Dt ρ0

(17)

In a bulk fluid, a sound wave propagates away from the particle; therefore, eventually, the effect of the sound wave on
the particle motion disappears, and the particle motion is governed by shear flow accompanied by viscous diffusion. However, in a confined fluid, a sound wave is repeatedly reflected
at the walls and continuously affects the particle motion. Multiple reflections at the walls result in the spreading of soundwave, which is known as the overdamped diffusive mode.3, 4
Because viscous diffusion is prevented by the walls and thecorresponding shear flow gradually disappears, the particle
motion is eventually governed by the sound wave. The spreading sound wave is associated with backward fluid flow andcauses backtracking of the particle, which corresponds to thenegative velocity relaxation function.
The long-time decay of the particle velocity reflects theeventual mechanism transmitting the fluid flow: viscous diffusion or sound propagation. In a bulk fluid, the particle velocity finally decays, obeying Eq. (15), which corresponds
to viscous diffusion of the shear flow. The volume of theviscous diffusion region at time t is proportional to (νt)3/2 .
In a confined fluid, the different long-time decay Eq. (16)appears. This negative decay originates from the spreading
sound wave, whose spreading volume at time t is proportionalto H(ct)2 .
In Fig. 7, the relaxation functions with wall spacingsH∗ = 4 and 6 for various compressibility factors are displayed. Discrepancies of long-time decay with Eq. (16) forsmall compressibility factors are presumed to result fromsystem size effects due to the periodic boundary conditions, which will appear after the time when the sound wave
generated by the particle reaches the end of the system:t = (Lx /2 − a)/c = (Lx /2a − 1)ετ ν . The time when the sign inversion occurs changes non-monotonically with respect to thefluid compressibility. With an increase in the compressibility
factor, the time of the sign inversion occurs earlier when thecompressibility factor is small; however, the sign inversion
occurs later when the compressibility factor is large. In otherwords, there is a compressibility factor at which the time ofthe sign inversion is minimized, which we call the characteristic compressibility factor. The characteristic compressibility factor depends on the wall spacing H∗ ; for example, from
Fig. 7, the characteristic compressibility factors are presumed
to be ε ≈ 0.6 for H∗ = 4 and ε ≈ 1.0 for H∗ = 6. More detailed evaluations of the characteristic compressibility factors
for various wall spacings are provided in Fig. 8.
The sign inversion of the velocity relaxation function,
namely, the backtracking of the particle, is expected tooccur after the sound wave reflected at the wall reaches
the particle. The time scale of this event is estimated by

184905-6

R. Tatsumi and R. Yamamoto

J. Chem. Phys. 138, 184905 (2013)00(b) 10(a) 10H* = 4

-1

10

-2

-2

10

(t)

10

(t)

H* = 4

-1

10

-3

10

-4

10

-4

10

incompressible
= 0.3
= 0.6

-5

10

-3

10

= 0.6
= 1.0
= 1.5

-5

10

-6

-6

10

-2

-1

10

0

10

1

10

10

10

2

-2

10

10

-1

t/

10

1

10

2

10

t/

0

0

(d) 10(c) 10H* = 6-1

10

H* = 6

-1

10

-2

-2

10

(t)

10

(t)

0

10

-3

10

-4

10

-4

10

incompressible
= 0.3
= 0.6

-5

10

-3

10

= 0.6
= 1.0
= 1.5

-5

10

-6

-6

10

-2

-1

10

0

10

1

10

10

2

10

10

-2

10

-1

0

10

t/

10

1

10

2

10

t/

FIG. 7. Velocity relaxation function of a particle in a compressible fluid confined between two flat walls. The wall spacings are ((a) and (b)) H∗ = 4 and ((c)
and (d)) H∗ = 6. The compressibility factor has values of ε = 0.3, 0.6, 1.0, and 1.5. The simulation results are represented by solid lines (positive values) anddashed-dotted lines (negative values). The bold dashed-two dotted lines represent the negative long-time tail −Bt−2 , as given by Eq. (16).
†τc = 2(H /2 − a)/c = (H ∗ − 2)ετν . Moreover, sufficient decay of the shear flow around the particle is required, andthe time scale of the shear flow decay is τν† , as given inSec. III B. Here, we define the confined compressibility factor

1.5

1

0.5

0

4

6

8

10

H*
FIG. 8. Classification of calculated velocity relaxation functions for various
wall spacings and compressibility factors. The time when the sign inversionof the velocity relaxation function occurs non-monotonically changes withrespect to the compressibility factor. For each wall spacing, with the increasein the compressibility factor, the decrease and increase of the time of the sign
inversion are represented by upward and downward triangles, respectively.
The time of the sign inversion is minimized at the characteristic compressibility factor, which is represented by open diamond. The broken line represents
Eq. (18) with ε † = 1. The dashed double-dotted line represents ε = 1.34.

as the ratio of the two time scales:
ε† =

†

τc

†
τν

=

4ε
.
H∗ − 2(18)

When the confined compressibility factor is small, ε† < 1, thespreading of the sound wave caused by reflection at the walls
progresses faster than the decay of the shear flow. For smaller
confined compressibility factors, the sound wave spreads andis weakened more rapidly; therefore, further reduction of theshear flow is required to cause backtracking of the particle. In short, when ε† < 1 is satisfied, backtracking of theparticle occurs at a later time for a smaller confined compressibility factor. However, when the confined compressibility factor is large, ε† > 1, the reflected sound wave reaches
the particle after sufficient decay of the shear flow; therefore, particle backtracking occurs at a later time for a larger
confined compressibility factor. Consequently, the characteristic compressibility factor is expected to satisfy ε† ≈ 1. In
Fig. 8, the line of Eq. (18) with ε† = 1 almost coincides withthe characteristic compressibility factors at H∗ ≤ 6; however,this relation fails at H∗ > 6, where another mechanism of
backtracking must be considered.
For the compressibility factor ε = 1.5, a comparison of
the velocity fields in confined and bulk fluids at time t/τ ν= 2.81 is presented in Fig. 9. Although this time is earlier†
than the time τc , the relaxation function in the confined fluid

184905-7

R. Tatsumi and R. Yamamoto

J. Chem. Phys. 138, 184905 (2013)*(a) H = 4 ε = 1.5

-3

-2

-1

0

-2

1

2

3 x10

1

2

3 x10

(b) bulk fluid ε = 1.5

-3

-2

-1

0

-2

FIG. 9. Velocity fields around the particle at t/τ ν = 2.81 (a) in a confined
fluid with wall spacing H∗ = 4 and (b) in a bulk fluid. The compressibility
factor is ε = 1.5. The walls are set on the upper and lower boundaries ofpicture for the confined fluid, and the corresponding region is exhibited forthe bulk fluid. The cross-sections shown here are perpendicular to the walls
and parallel to the impulsive force direction with including the particle center.
The direction of the impulsive force is to the right in the pictures and theparticle is represented by a black circle. The divergence of the velocity field∇ · v is described by a color scale, going from negative (darker) to positive(lighter) vorticity. The divergence is normalized by the factor of τ ν /Rep .

is negative, as illustrated in Fig. 5(b); therefore, the backtracking is caused by the pressure from the sound wave remaining
in the vicinity of the particle. Backtracking by this mechanism is also observed in a bulk fluid when the compressibility
factor is sufficiently large,21, 22 and the condition for which
the backtracking occurs is estimated as ε ≥ 1.34 from the analytical solution.20, 21 There is only a slight difference in thedivergence of the velocity field between the confined and bulk
fluids; however, in a confined fluid, because shear flow is diminished by the walls, backtracking by a sound wave can occur at an earlier time than in a bulk fluid, as demonstrated
in Fig. 5(b). Considering this mechanism of particle backtracking, the characteristic compressibility factor will eventually converges to ε = 1.34 with an increase in the wall
spacing H∗ . The successive change in the dependence of thecharacteristic compressibility factor on the wall spacing,
namely, from ε† = 1 to ε = 1.34, is observed in Fig. 8. The
wall spacing at which this crossover occurs is estimated asH∗ ≈ 7.4.

differently from the power law t−3/2 observed in a bulk fluid.
The rapid decay reflects the loss of shear flow due to the hindrance of viscous diffusion by the walls. Therefore, the time
when the relaxation function in a confined fluid falls belowthat in a bulk fluid corresponds to the time scale of viscous
diffusion over the distance between the particle surface andthe walls, τν† .
In a compressible fluid, sign inversion and subsequent
negative long-time decay obeying t−2 were observed in thevelocity relaxation functions. A corresponding spreading
sound wave arising from the multiple reflections at the walls
was also observed. The time of the sign inversion changed
non-monotonically with respect to the compressibility factor;
that is, the time of the sign inversion was minimized at thecharacteristic compressibility factor. The characteristic compressibility factor satisfies ε† = 1 when the wall spacing issmall and is given by ε = 1.34 when the wall spacing is large,
where the confined compressibility factor ε† is defined by
Eq. (18). The crossover of the dependence of the characteristic compressibility factor on the wall spacing is estimatedto be H∗ ≈ 7.4. The backtracking of the particle is caused by
the spreading sound wave arising from reflections at the walls
when the characteristic compressibility factor satisfies ε† = 1
and is caused by the remaining sound wave in the vicinity ofthe particle when the characteristic compressibility factor isgiven by ε = 1.34. Backtracking via the latter mechanism canalso occur in a high compressible bulk fluid.
ACKNOWLEDGMENTS

This work was supported by KAKENHI 23244087 and
the JSPS Core-to-Core Program “International research network for non-equilibrium dynamics of soft matter.”
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IV. CONCLUSION

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