Document [original]

DOI 10.1140/epje/i2020-11975-6

Regular Article

Eur. Phys. J. E (2020) 43:50 THE EUROPEAN

PHYSICAL JOURNAL E

Particle pairs and trains in inertial microﬂuidics

Christian Schaafaand Holger Stark

Technische Universit¨at Berlin, Institut f¨ur Theoretische Physik, Straße des 17. Juni 135, 10623 Berlin, Germany

Received 22 May 2020 and Received in ﬁnal form 26 June 2020

Published online: 4 August 2020

The Author(s) 2020. This article is published with open access at Springerlink.com

Abstract. Staggered and linear multi-particle trains constitute characteristic structures in inertial mi-

croﬂuidics. Using lattice-Boltzmann simulations, we investigate their properties and stability, when ﬂow-

ing through microﬂuidic channels. We conﬁrm the stability of cross-streamline pairs by showing how they

contract or expand to their equilibrium axial distance. In contrast, same-streamline pairs quickly expand

to a characteristic separation but even at long times slowly drift apart. We reproduce the distribution of

particle distances with its characteristic peak as measured in experiments. Staggered multi-particle trains

initialized with an axial particle spacing larger than the equilibrium distance contract non-uniformly due

to collective drag reduction. Linear particle trains, similar to pairs, rapidly expand toward a value about

twice the equilibrium distance of staggered trains and then very slowly drift apart non-uniformly. Again,

we reproduce the statistics of particle distances and the characteristic peak observed in experiments. Fi-

nally, we thoroughly analyze the damped displacement pulse traveling as a microﬂuidic phonon through a

staggered train and show how a defect strongly damps its propagation.

1 Introduction

Since the discovery of inertial focusing by Segr´e and Sil-

berberg [1], inertial microﬂuidics has evolved into a ma-

ture research ﬁeld with immense potential for biomedi-

cal applications [2–4]. At higher densities particles do not

only move to an equilibrium position in the channel cross

section but also form regular trains along the channel

axis [5–7]. This feature of inertial microﬂuidics is particu-

lar useful for counting [8–10], sorting [11–14], or manipu-

lating cells [15,16]. While trains of particles were already

observed by Segr´e and Silberberg [1], the ﬁrst systematic

analysis was done by Matas et al. about 40 years later [17].

This triggered further research on particle trains, which

we describe below, in order to understand their occurrence

more thoroughly. In this paper we contribute with our sim-

ulation study to the understanding of staggered and linear

multi-particle trains including particle pairs (see ﬁg. 1).

1.1 Staggered and linear multi-particle trains

The ﬁrst experiments used cylindrical tubes, where parti-

cles focus onto an annulus. In ref. [17] Matas et al. studied

particles with a low conﬁnement ratio (ratio of particle to

cylinder radius between 0.03 and 0.05) and observed par-

ticle trains above the Reynolds number Re =2wumax/ν ≈

Supplementary material in the form of eight .mpg ﬁles

available from the Journal web page at

https://doi.org/10.1140/epje/i2020-11975-6

ae-mail: [email protected]

100. Here, 2wis the channel width, umax the maximum

ﬂow velocity, and νthe kinematic viscosity. With increas-

ing Reynolds number more and more particles assembled

into trains. For channels with quadratic or rectangular

cross sections, the particles focus on one of the four or

two possible equilibrium positions depending on the as-

pect ratio of the cross section [18–20]. For such channels,

typically, a mixture of staggered and linear particle trains

occurs [21]. In staggered trains the particle locations alter-

nate between both channel halves, whereas in linear trains

all particles are located on one side (see ﬁg. 1). For both

structures experiments and simulations report a distinct

axial spacing, where the spacing for linear trains is about

twice the spacing of staggered trains [5,6,22,23].

Lee et al. explained the formation of particle trains by

the combination of two eﬀects: i) inertial lift forces focus

the particles onto their single-particle equilibrium posi-

tions and ii) the viscous disturbance ﬂow together with

the imposed channel ﬂow determine their axial separa-

tion [5]. In ref. [24] the well-deﬁned axial spacing of two

particles in a cross-streamline pair is explained by the ﬂow

ﬁeld around a single particle viewed in its center-of-mass

frame. There one observes two inward spiraling vortices on

the opposite side of the channel, located around 4 particle

radii ahead and behind the particle. The second particle

then follows the streamlines created by the ﬁrst particle

and spirals in damped oscillations toward its equilibrium

position [25]. This idea is also conﬁrmed by analytical

calculations [26]. As shown in ref. [25], the stable cross-

streamline pair also corresponds to ﬁxed points in the lift

Page 2 of 13 Eur. Phys. J. E (2020) 43:50

c) staggered train

Δz(1)

Δz(2)

u

d) linear train

a) cross-streamline pair

Δz

u

b) same-streamline pair

xeq

u

e) microfluidic phonon

Fig. 1. Illustrations of the multi-particle structures discussed

in this work. Two particles either form cross-streamline (a)

or same-streamline (b) pairs. Several particles form the corre-

sponding staggered (c) or linear trains (d). xeq gives the equi-

librium distance from the channel centerline, Δz the distance

between two neighboring particles, and the arrows with u indi-

cate the ﬂow direction. (e) A microﬂuidic phonon in a staggered

particle train is triggered by moving one particle toward the

channel center (pink particle in the upper channel). It moves

faster than the rest of the train and thereby a displacement

pulse travels through the train. In the middle and bottom chan-

nel the maximal displacements of the gray and yellow particles

are shown. The arrow indicates increasing time.

force proﬁles of both particles. Finally, two-dimensional

simulations in ref. [27] indicate that above Re = 80 parti-

cles in a staggered train perform stable oscillations about

these equilibrium positions.

The stability of linear trains is less clear. Lattice-Boltz-

mann simulations performed by Kahkeshani et al. indicate

that same-streamline pairs are stable [6], for which ref. [26]

provided an explanation based on the minimization of the

kinetic energy of the ﬂuid. However, early experiments [5]

and most recent 2D simulations [23] report an increase

of particle spacing over time in agreement with our own

ﬁndings [25]. Some experiments report that for increas-

ing Reynolds number the axial spacing between the parti-

cles decreases [22,28] even to the value of cross-streamline

pairs [6]. However, other experiments report an increase of

the spacing in linear trains with increasing Re, while the

spacing decreases in staggered trains [29]. In the range of

Re ≈1 to 4 recent experiments observe that the spacing

is independent of the ﬂow velocity and that the small-

est channel dimension determines the axial spacing be-

tween pairs [30]. Finally, simulations using a force coupling

method report that trains are only stable up to lengths of

2 to 4 particles depending on the conﬁnement ratio and

the particle Reynolds number [28].

A comparison of the diﬀerent experimental results is

hampered since they use very diﬀerent parameters, such

as the channel Reynolds number and conﬁnement ratio

a/w, where ais the particle radius and wis the half

width of the channel. While experiments with smaller par-

ticles can go to higher channel Reynolds numbers with

Re >100 [17, 22], experiments with larger particles typ-

ically operate at channel Reynolds numbers between 1

and 20 [5, 24, 30]. As the particle Reynolds number is

Rep=(a/w)2Re, both experiments and simulations can

operate at the same particle Reynolds number although

the other parameters are very diﬀerent. This is especially

relevant for Re ≈100, where the secondary ﬂow around a

single larger particle (a/w =0.4) becomes so strong that

it inﬂuences the particle-wall interactions [29]. A detailed

review of the eﬀect of particle size suggests that for a small

conﬁnement ratio a/w 0.1 the particles move closer to-

gether with increasing Reynolds number [17,22,28] while

for larger particles the separation seems to increase [5,29].

Experiments are often limited by the channel length

L. Typical length-to-width ratios are L/2w≈1000. To

overcome the limitation of the channel length, Dietsche

et al. used an oscillatory ﬂow device which switched the

direction of the ﬂow such that the particles stayed in the

channel and were not aﬀected by the switching [30]. They

conﬁrmed the stability of cross-streamline pairs with an

equilibrium axial distance of Δz/a =3.8. For the same-

streamline pair they identiﬁed a range of separations,

7.4<Δz/a<14, where the particles moved together

at low speed. However, the error bars were much larger

than the measured speed values.

1.2 Microﬂuidic phonons

Toward the end of the paper we will also analyze how

perturbations of the regular structure move through a

staggered particle train. An initial particle displacement

triggers a displacement pulse, which travels through the

staggered train while being damped. Due to the resem-

blance with acoustic phonons, such excitations are called

microﬂuidic phonons [31]. So far, these phonons were

only analyzed for ﬂowing droplets squeezed between two

parallel plates at vanishing Reynolds numbers [32, 33].

They provide an interesting model system with non-

linear and non-equilibrium behavior [34]. In this quasi–

two-dimensional geometry the interactions between the

droplets are determined by dipolar interactions [31]. Stud-

ies on such phonons in inertial microﬂuidics do not ex-

ist. Due to the strong inertial damping such microﬂuidic

phonons have to be triggered from outside. Staggered par-

ticle trains are desired for applications since due to the

regular and dense order they enhance particle throughput

and at the same time facilitate particle sorting. Analyzing

how these staggered trains react on perturbations might

help to obtain a better understanding how they can be

crystallized.

Eur. Phys. J. E (2020) 43:50 Page 3 of 13

1.3 Summary of results

In this paper we study the stability of staggered and linear

particle trains using lattice-Boltzmann simulations. First,

we focus on a pair of ﬂowing particles and analyze their

dynamics when both particles are already initialized on

their lateral equilibrium positions. We conﬁrm the sta-

bility of cross-streamline pairs and also show how they

contract or expand to their equilibrium axial distance. In

contrast, same-streamline pairs quickly expand to a char-

acteristic separation but even at long times slowly drift

apart. However, we are able to reproduce the distribution

of particle distances measured in experiments [5]. Then,

we extend our analysis to particle trains. We thoroughly

analyze how a staggered train initialized with an axial par-

ticle spacing larger than the equilibrium distance contracts

non-uniformly. We speculate about a possible realization

and consider a defect in the train. In contrast, particles

in linear trains slowly drift apart non-uniformly, and we

are able to reproduce the statistics of particle distances

observed in experiments [6]. Finally, the damped displace-

ment pulse traveling through a staggered train is presented

and how a defect strongly damps its propagation.

The paper is organized as follows. In sect. 2 we explain

the microﬂuidic setup of our system, describe the imple-

mentation of the lattice-Boltzmann method, and how we

couple the particles to the ﬂuid. In sect. 3, we present the

results for the stability of staggered and linear particle

trains including particle pairs and discuss the phononic

displacement pulse as well as the inﬂuence of defects. We

summarize and close with ﬁnal remarks in sect. 4.

2 System and methods

2.1 Microﬂuidic setup in the simulations

The microﬂuidic setup consists of multiple particles im-

mersed in a Newtonian ﬂuid with density ρand kine-

matic viscosity ν, which ﬂows through a microchannel (see

ﬁg. 1). The channel is of length Land has a rectangular

cross section with width 2wand height 2h. To ensure that

the particle dynamics is limited to the x-zplane, spanned

by the short axis xof the cross section and the channel

direction z, we set the aspect ratio to w/h =0.5. Thus,

only two equilibrium positions along the short axis ex-

ist [18, 19]. To realize the Poiseuille ﬂow through a rect-

angular channel [35] in our lattice-Boltzmann simulations

(sect. 2.2), we drive the ﬂuid with a constant body force,

which stands for the pressure gradient.

We quantify the inﬂuence of ﬂuid inertia by the chan-

nel Reynolds number Re =2wumax/ν, where umax is the

maximum ﬂow velocity in the channel center. In sects. 3.1

and 3.2 we restrict ourselves to the Reynolds number

Re = 20. This corresponds to a typical experimental setup

with a channel width of 2w=25μm, ν=1×10−6m2/s

for water, and umax =0.8m/s [24]. In sect. 3.3 we vary

the Reynolds number between 5 and 100. In the following,

we distinguish between the axial direction along the ﬂow

(z-axis) and the lateral direction perpendicular to the ﬂow

(x-axis).

We place Nneutrally buoyant particle with radius a

in the channel ﬂow. The particles are initialized either in

a staggered or a linear particle train (ﬁg. 1). If not stated

otherwise, the particles are initialized on the equilibrium

position of single particles on the x-axis. The initial axial

spacing Δz0of two neighboring particles in a staggered

conﬁguration diﬀers from the observed equilibrium value

and the initial spacing in a same-streamline conﬁguration

is always chosen smaller than the limiting distance, we

observe at long times. To analyze the stability of parti-

cle pairs and trains, the channel length was always chosen

suﬃciently long (at least L≈(N+1)·10 a) to ensure

that hydrodynamic interactions with images from the pe-

riodic boundary conditions along the channel were not

relevant. When analyzing the microﬂuidic phonons, the

channel length always was L=N·Δz,sothatwesimu-

lated an inﬁnitely long particle train using periodic bound-

ary conditions. Finally, in all our simulations the particle

radius was ﬁxed to a/w =0.4 as in ref. [25].

2.2 Simulation method

To solve the Navier-Stokes equations, we performed simu-

lations in 3D with the lattice-Boltzmann method (LBM)

using 19 diﬀerent velocities vectors (D3Q19) [36] and the

Bhatnagar-Gross-Krook (BGK) collision operator [37].

We relied on the same simulation code used in our pre-

vious publication [18, 25]. The LBM determines the one-

particle probability distribution fi(x, t) on a cubic lattice

with lattice spacing Δx = 1. In addition to space and time

discretization, the LBM also discretizes the possible veloc-

ity vectors, which are indicated by the index iin fi(x, t).

In the case of the velocity set D3Q19, 19 velocity vectors

are implemented. During time Δt the particle distribu-

tion function fi(x, t) evolves according to two alternating

steps:

collision: f∗

i(x, t)=fi(x, t)+ 1

τfeq

i(x, t)−f(x, t),(1)

streaming: fi(x +ciΔt, t +Δt)=f∗

i(x, t),(2)

where feq

iis a second-order expansion of the Maxwell-

Boltzmann distribution in the mean velocity and τis the

relaxation time of the BGK model. For details on this

method we refer the interested reader to ref. [38].

One important feature of the LBM is that the viscosity

is related to the collision relaxation time τ[39],

ν=c2

sΔt τ−1

2,(3)

where c2

s=1/3 is the speed of sound measured in LBM

units (Δx =1,Δt = 1). The lattice-Boltzmann method

does not strictly conserve volume. To ensure incompress-

ibility of the ﬂuid, simulations have to be performed at

small Mach numbers Ma = umax/cs. For all our simula-

tions the Mach number is set to Ma ≤0.1, which corre-

sponds to a maximum possible density variation of 1%.

The exact procedure is explained in ref. [25].

Page 4 of 13 Eur. Phys. J. E (2020) 43:50

To drive the channel ﬂow, we apply a constant body

force according to the Guo-force scheme [40]. At the chan-

nel walls we use regularized boundary conditions [41]. The

main part of our simulation code is provided by the Pala-

bos project [42], which we extended by implementing the

ﬂowing particles. We use a resolution of 60 lattice cells

along the short axis.

For simulating the suspended particles, we rely on

the immersed-boundary method (IBM) proposed by In-

amuro [43]. This scheme belongs to the class of immersed-

boundary methods with direct forcing and was shown to

capture the correct behavior of solid particles in laminar

ﬂow [44]. The immersed-boundary method is based on a

Lagrangian grid, which exists in between the ﬁxed Eule-

rian grid for the ﬂuid. The velocity of the ﬂuid is inter-

polated to the particle surface, where a penalty force en-

sures the no-slip boundary condition. The resulting force

is interpolated back to the ﬂuid. This process is done it-

eratively, where we use the recommended ﬁve iterations.

For further details on these methods we refer the reader to

the original publications [43] and our previous work [25].

3Results

We now present our results starting with the stability of

cross-streamline and same-streamline pairs, which we then

extend to the same types of particle trains. Finally, we ad-

dress microﬂuidic phonons concentrating on the damped

propagation of a displacement pulse.

3.1 Stability of particle pairs

Before we consider multi-particle trains, we ﬁrst analyze

the long-time behavior and stability of a pair of rigid par-

ticles where both particles are initialized at their lateral

single-particle equilibrium positions. We focus on the ax-

ial distance and look for stable axial conﬁgurations. We

already analyzed the dynamics of a pair of particles in-

teracting by inertial lift forces and brieﬂy summarize here

the results relevant for this work [25].

Analyzing two-particle trajectories with diﬀerent ini-

tial positions, we only found one class of bound trajecto-

ries, where two particles on opposing sides of the channel

(cross-streamline conﬁguration) performed damped oscil-

lations toward their equilibrium positions. All other types

of trajectories were unbound; the axial particle distance

increased while the two particles moved individually to-

ward their single-particle equilibrium positions. In ad-

dition, an analysis of the two-particle lift force proﬁles

showed that cross-streamline conﬁgurations are stable and

thereby conﬁrmed observations by other groups [6,24,26].

For the same-streamline conﬁguration we could not iden-

tify a ﬁxed point with a small axial particle distance,

which is reported in experiments at larger Reynolds num-

bers Re = 60 and 90 [6]. Instead, we observed that the

particles moved apart until their interaction was vanish-

ingly small close to the experimentally reported larger

spacing [6,30]. Hence, we concluded that same-streamline

Δzpair

eq /a

0510 15 20 25

t/(w2/ν)

Δz/a

072 144 216 288 360

Traveled distance z(t)/2w

Fig. 2. Axial distance Δz/a as a function of time for a par-

ticle pair in cross-streamline conﬁguration. Both particles are

initialized at single-particle equilibrium positions, x=±xsingle

eq ,

and diﬀerent initial axial distances Δz0are chosen. The dashed

line indicates the equilibrium axial distance Δzpair

eq /a ≈4.2.

The upper axis indicates the traveled distance of the center of

mass of the pair along the channel for Δz0/a =8.

pairs are only one-sided stable: the particles repel after be-

ing pushed together, while, when moved apart, they keep

their distance due to the missing attraction. A similar ob-

servation was reported in one of the earliest works on par-

ticle trains in inertial microﬂuidics [5].

In the following, we focus on the case where both par-

ticles already occupy their equilibrium lateral positions.

Thus, they stream with the same velocity.

3.1.1 Cross-streamline particle pairs

In our previous work we found that particles in all an-

alyzed bound trajectories reach the same value for their

axial distance, Δzpair

eq =4.2a, independent of the initial

positions [25]. In order to observe the damped oscillation,

leading and lagging particles in ﬂow were initialized with

a similar lateral position xlag ≈−xlead. Here, we analyze

the situation where both particles are initialized on the

single-particle equilibrium positions at ±xeq/w ≈±0.4

but with a distance Δz0=Δzpair

eq . In ﬁg. 2 we vary the ini-

tial distance Δz0from 2.5ato 9 aand plot the respective

time course of the axial distance. In all cases the particles

reach Δzpair

eq ≈4.2, even when the initial axial distance is

as large as 9 a. In the graph we also added the traveled dis-

tance of the particle pair for Δz0/a = 8. This shows that

the particle pair relaxes to its equilibrium conﬁguration

on distances much shorter than typical channel lengths of

the order of L/2w≈1000 [5,22].

In ﬁg. 3 we show the time course of the lateral coordi-

nates of the leading and lagging particles. When the initial

axial distance Δz0is larger than the equilibrium value

(ﬁg. 3(a)), the lateral positions hardly change. But this

is suﬃcient to let them approach each other. Only when

the particles are initialized closer together (ﬁg. 3(b)), do

the lateral position change noticeably by ca. 10%. This

then initiates the initial rapid increase of the axial spac-

ing due to the diﬀerent ﬂow velocities (blue line in ﬁg. 2

Eur. Phys. J. E (2020) 43:50 Page 5 of 13

Δz0/a=6

0510

0.39

0.4

0.41

t/(w2/ν)

|x|/w

lagging leading

Δz0/a=2.5

0510 15

0.35

0.4

0.45

t/(w2/ν)

|x|/w

Fig. 3. Leading and lagging particles in a cross-streamline

conﬁguration. Distance from the channel centerline |x|/w as a

function of time. Both particles are initialized at x=±xsingle

eq .

(a) Initial distance Δz0=6a>Δz

pair

eq and (b) Δz0=2.5a<

Δzpair

eq .

with Δz0/a =2.5) followed by a slow relaxation back to

the equilibrium value. Thus, our study also shows how

particle pairs relax toward their preferred distance even

when starting at their lateral equilibrium positions. We

note that in the ﬁnal conﬁguration the leading particle

is located slightly closer toward the channel center but

moves with the same velocity. However, the diﬀerence is

smaller than a lattice unit.

3.1.2 Same-streamline particle pairs

In ref. [25] we already observed that particles initialized on

the same streamline with an initial distance of Δz0/a =5

slowly drift apart, even when positioned on the single-

particle equilibrium position. We explained this behavior

with an asymmetry in the two corresponding lift force

proﬁles: the leading particle is pushed toward the channel

center while the lagging particle toward the walls such

that they move apart. This behavior did not change for

larger Δz0. Hence, our previous simulations indicate that

same-streamline pairs are not stable.

We now analyze this behavior in more detail in ﬁg. 4.

Again, we initialize the particles at the lateral equilib-

rium position and with an axial spacing equal to the axial

distance Δzpair

eq of the cross-streamline pairs. The lead-

ing particle is noticeably and rapidly pushed toward the

channel center while the lagging particle moves outwards

(ﬁg. 4(a)). This drives the particles apart, which is visi-

ble by the rapid increase of the axial distance in ﬁg. 4(b).

Beyond Δz/a ≈5, the particles slowly relax toward their

equilibrium lateral position. However, even at large times,

where both particles should move with the same speed,

we still observe a slow but steady increase of the particle

spacing (ﬁg. 4 (inset)). In experiments such a drift might

be diﬃcult to measure. According to our simulations, be-

0.39

0.414

0.44

x/w

lagging

leading

012345

t/(w2/ν)

Δz/a

5 15253545

9.5

10.5

012 26 40 55 70

Traveled distance z(t)/2w

Fig. 4. (a) Lateral positions and (b) axial distance as a func-

tion of time for a same-streamline particle pair. The dashed line

localizes the maximum lateral displacement. Inset: at larger

times still a slow but steady increase of Δz is observable. The

upper axis indicates the traveled distance of the center of mass

of the pair along the channel. At t=45w2/ν the particles have

moved a distance of z/2w= 650.

yond t=5w2/ν the distance Δz increases by only 6%

while the particle pair travels a distance of 600×2w. This

is of the order of channel lengths used in experiments.

Taking typical values of L=5cmand2w=50μm, we

obtain L/(2w)≈1000 [22].

Experiments typically report a distance of about twice

the equilibrium spacing of cross-streamline pairs, which

would be Δz/a ≈8.4 in our case [6]. Our simulations in-

dicate that the particles go to a somewhat larger spacing.

In experiments the starting conditions are not as well de-

ﬁned as in our case. To reproduce the experimental statis-

tics for particle distances observed in ref. [5], we initialize

76 diﬀerent pair conﬁgurations, where both particles are

randomly placed in the upper channel half with an initial

distance 5a<Δz

0<L/2. In ﬁg. 5 we plot the distribu-

tion of particle distances after the particles have traveled

a distance of 1000 ×2w, which is a typical value in exper-

iments as mentioned above. The simulated distribution

matches well with experiments for the same particle size

with a peak at Δz/a =9.8 [5]. The authors do not specify

the channel Reynolds number explicitly, but the particle

Reynolds number Rep=Re(a/w)2=2.4 is similar to the

value of 3.2 used for this work.

To summarize, while we obtain good agreement with

early experiments [5], we could neither identify an ad-

ditional stable equilibrium distance for higher Reynolds

numbers [6] nor reproduce an attractive interaction of par-

ticles in same-streamline pairs [30].

3.2 Stability of particle trains

Based on the insights we gathered from the behavior of

particle pairs, we continue by analyzing multi-particle

trains and begin with staggered particle trains.

Page 6 of 13 Eur. Phys. J. E (2020) 43:50

CS-pairs

04812

16 20 24

0.1

0.2

0.3

Δz/a

pdf (a.u.)

Fig. 5. Histogram of particles distances for randomly initial-

ized pairs after the center of mass has traveled a distance of

z/2w= 1000. The small peak at Δz/a = 4 corresponds to

cross-streamline pairs, which formed despite the fact that all

particles were initialized on the same channel side. The red

line shows data from experiments for pairs of particles with

the same conﬁnement ratio(a/w =0.4) and the same traveled

distance (2.5 cm) [5]. We rescaled the experimental data by a

factor of 0.4 to match the height of the peak.

t=0

t=3.2

t=4.4

t=7.2

t=16.9

t=32.2

u

Fig. 6. Snapshots of the contraction process of a staggered par-

ticle train at diﬀerent times given in units of w2/ν.Att=0

all particles are initialized with a nearest-neighbor distance

Δz0=6.5a=1.5Δzpair

eq . The color of the trailing pairs corre-

sponds to the lines in ﬁg. 7(b).

3.2.1 Staggered particle trains

When we start the simulations with arbitrarily placed

particles, we obtain staggered particle trains with defects

in it. To concentrate ﬁrst on the ideal case, in the fol-

lowing we analyze how an expanded staggered particle

train contracts toward its equilibrium conﬁguration. For

this we consider 11 particles, which we initialize on their

single-particle position at ±xeq with an axial distance of

Δz0=6.5a≈1.5Δzpair

eq .

As expected from the analysis of the cross-streamline

pairs, the axial distances between the particles decrease in

time. However, as ﬁgs. 6, 7(a), and video 1 in the electronic

supplementary material (ESM) demonstrate, the contrac-

tion does not occur uniformly but rather through the for-

mation of particle pairs. The contraction starts in the front

and back of the train. We observe that initially only the

leading and trailing pairs contract, whereby mainly the

leading particle of the pair moves backwards toward the

lagging particle (ﬁg. 6 (t=3.2)). While the pairs contract,

they slow down. The leading pair stays connected to the

staggered train but the last pair separates from the rest of

the train due to its reduced velocity (see below). This trig-

back of the crystal

front of the crystal

0 10203040

−40

−20

t/(w2/ν)

z/a

02468

t/(w2/ν)

Δz/a

Δz0,1(t)

Δz2,3(t−2)

Δz4,5(t−4)

Fig. 7. (a) Axial positions of all the particles in a stag-

gered particle train plotted versus time. The positions are

given in the center-of-mass frame of the train. (b) Axial par-

ticle distances of the trailing particle pairs as a function of

time. The curves of the second and third pair are shifted such

that they fall onto each other. The initial axial distance is

Δz0=6.5a≈1.5Δzpair

eq .

gers the contraction of the next pair and then a third pair

so that at t=7.2 three individual pairs in the back of the

train exist. The contraction of these pairs always occurs

in the same manner. In ﬁg. 7(b) we plot their particle dis-

tances versus time and have shifted the curves by the time

the previous pair needed to contract and separate. Then,

all three curves fall onto each other. The particle pair in

the front of the staggered train also slows down. The next

particle in line catches up so that a three-particle cluster

exists at t=4.4. This cluster slows down further and the

next two particles can catch up. Ultimately, at t=7.2

a contracted ﬁve-particle cluster exists followed by the 3

trailing pairs. The larger cluster slows down (see below)

so that the three pairs can catch up one by one (t=16.9)

and, ﬁnally, at t=32.2 the staggered train has reached

its equilibrium conﬁguration.

For the contraction of the particle train, two mecha-

nisms are relevant. They are related to viscous drag reduc-

tion of clusters of particles compared to a single particle

and when the clusters are more compact [34,45,46]. In our

case, this means the resistance to an imposed Poiseuille

ﬂow is reduced and therefore the clusters slow down rela-

tive to the ﬂow. Thus, a pair of particles slows down when

the axial distance decreases and the center-of-mass veloc-

ity also decreases for larger staggered trains. We discuss

this in detail in ﬁg. 8. In graph (a) we plot the center-of-

mass velocity of a cross-streamline pair as a function of the

axial particle distance. Although the decrease of the ve-

locity with Δz is very small, it quantitatively agrees with

the reported results of simulations for a pair of rigid par-

ticles (a/w =0.8) moving on the centerline of a conﬁned

ﬂow at Re 1 [46]. In plot (b) we observe for staggered

Eur. Phys. J. E (2020) 43:50 Page 7 of 13

45678

0.85

0.86

0.87

Δz/a

vcom

z/uflow(x)

1510 15 20

0.96

0.98

Nparticles

vcom

z/vsingle

Fig. 8. (a) Axial center-of-mass velocity for a cross-streamline

pair as a function of the particle distance. The velocity is plot-

ted in units of the ﬂuid ﬂow velocity at the lateral position of

the particle pair. The data were extracted from the gray curve

in ﬁg. 2, which starts at Δz0=9a.(b)Axialcenter-of-mass

velocity for a staggered particle train as function of the num-

ber of particles in the train. The velocity is plotted in units of

the single-particle velocity. The gray dashed line is a linear ﬁt

for range with N>5.

02468101214

4.2

4.4

4.6

4.8

pair index

Δzeq/a

3 5 7

9 12 14

back front

Fig. 9. Axial equilibrium distance of neighboring particles in a

staggered train as a function of the pair index, which increases

from the back to the front of the train. The dashed line indi-

cates the particle distance of a single pair, Δzeqpair /a =4.2.

particle trains that the center-of-mass velocity monoton-

ically decreases with increasing number of particles. For

N>5 this decrease is linear. The diﬀerence in velocities

of a cluster consisting of 20 particles and a single particle

is about 5%. A very similar dependence on the particle

number was reported for simulations of a chain of par-

ticles driven by an applied force along a ring in a bulk

ﬂuid [45]. However, in this situation the variation of the

velocity is more pronounced (around 50%). The same type

of collective drag reduction was also reported by Beatus

et al. [31] for a linear chain of droplet disks in a quasi-2D

ﬂow. The authors named this observation the peloton ef-

fect, in analogy to the reduced drag of a group of closely

riding cyclists.

We ﬁnish with a ﬁnal observation. For cross-streamline

pairs we found that all particles relax toward the same

value Δz/a =4.2 for the axial distance. In contrast, as

ﬁg. 9 shows the axial spacing of neighboring particles in

a staggered train is non-uniform. It increases when one

moves forward in the train. For trains with more than

nine particles the axial distance saturates at a value of

Δz/a =4.8, which is about 15% larger than the distance

of a single pair. Finally, we observe that the distance of

the leading pair is slightly reduced for trains consisting of

seven particles or more.

3.2.2 Linear particle trains

Linear particle trains ﬂowing on the same streamline have

been observed both in experiments [6, 22] and simula-

tions [28]. Typically, they have an axial particle spacing

about twice the distance measured for staggered trains. So

far, in sect. 3.1 we found that the axial distance of a same-

streamline particle pair steadily increases in time. In the

following we analyze the stability of longer particle trains

and check if multi-particle interactions can stabilize them.

We initialize the particles in a linear train at the lateral

equilibrium positions of single particles and choose an ini-

tial axial distance of Δz0/a = 4 between neighboring par-

ticles. Figure 10(a) shows that the mean axial distance for

linear trains of diﬀerent sizes increases monotonically in

Δz/a

2 3 5 11

N=5front

back

02040

60 80

t/(w2/ν)

Δz/a

1 2 3 4

pair index

0 285 570 860 1150

Traveled distance z(t)/2w

Fig. 10. (a) Mean axial particle distance as a function of time

for linear trains with diﬀerent numbers of particles. (b) Axial

distance of neighboring particles within a linear train consisting

of 5 particles. From the back to the front the pairs are indexed

by 1 to 4.

Page 8 of 13 Eur. Phys. J. E (2020) 43:50

t=0

t=10

t=20

t=40

t=80

u

Fig. 11. Snapshots of an expanding linear particle train at dif-

ferent times given in units of w2/ν.Att= 0 all particles are ini-

tialized with the axial equilibrium distance of cross-streamline

pairs. The dashed lines correspond to Δz plotted in ﬁg. 10(b).

time. However, while for N= 2 the axial distance hardly

changes after reaching a distance of Δz/a ≈10, the mean

distance of trains with N>2 increases visibly and the

increase is slower for longer trains.

The reason is that the expansion of the linear train is

non-uniform as we show in ﬁg. 10(b), where we plot the

axial distance of neighboring particles for a train with ﬁve

particles. Snapshots of the expanding train are presented

in ﬁg. 11. The expansion for ﬁve and 11-particle trains is

also visualized in videos 2 and 3 of the ESM. After an

initial fast expansion to Δz/a ≈8, which is about twice

the distance in a cross-streamline pair, always the lead-

ing particle is carried away by the imposed ﬂow while the

particle train behind it moves more slowly. This creates a

particle train where the particle distance at one instant in

time increases from the back (pair index 1) to the front

(index 4). In our simulations all particles have traveled

a distance of more than 1000was the upper axis of the

plot shows. Thus, the steady increase of Δz is very slow

and it might not be possible to observe this in a typical

experiment. However, our data show that the ﬁnal conﬁg-

uration in the simulations is not a stable conﬁguration.

The separation of the leading particle from the rest

of the train was also reported in simulations by Gupta

et al. [28]. They also mention stable trains up to a cer-

tain cluster size, an eﬀect the authors name conditional

stability. Their critical cluster size depends on the con-

ﬁnement ratio a/w and the particle Reynolds number. In

their analysis the authors focused on conﬁnement ratios

a/w =0.08–0.14, which is much smaller than in our work.

However, their results indicate that for larger particles the

critical cluster size reduces to N=2.

Finally, we calculate the statistics of the particle spac-

ing in linear trains (ﬁg. 12). For this we randomly place

4, 6, or 11 particles in the upper channel half and en-

sure that there is no overlap between the particles. The

channel length is chosen such that the volume fraction is

ﬁxed at ϕ=0.004 as in ref. [6]. After the particles trav-

eled at least a distance of 160 w, we measure the distance

to the nearest-neighbor particles. The recorded statistics

shows good agreement with experimental data measured

for slightly smaller particles with a/w =0.34 compared to

our particles with a/w =0.4 (ﬁg. 12). Again, we observe a

small peak at Δz/a =4.2 which, in our case, corresponds

0 1020304050

5·10−2

0.1

0.15

0.2

Δz/a

pdf (a.u.)

Fig. 12. Histogram of particles distances for 4, 6, or 11 parti-

cles randomly initialized in the upper channel half after they

have traveled at least a distance of 160walong the channel.

The small peak at Δz/a = 4 corresponds to cross-streamline

pairs, which formed despite the fact that all particles were ini-

tialized on the same channel side. The red line shows data from

experiments by Kahkeshani et al. [6] for slightly smaller parti-

cles with a/w =0.34. We rescaled the experimental data by a

factor of 0.7 to match the height of the peak.

to particles which move to the lower channel half. This

ﬁrst peak is much smaller than in the experiments. This

is most probably due to how the linear trains are set up.

In the experiment the same-streamline trains are created

with two inlets which form a Y-shaped channel, where the

particles enter via one inlet. It sounds plausible that with

this method cross-streamline particle pairs are more easily

formed compared to our simulations, where we initialize

the particles already on the same streamline.

3.2.3 Staggered particle train with defect

Besides the pure cross-streamline and same-streamline

particle trains we also initialized a staggered train with

a single defect. The results for the temporal evolution of

the particle distances and the ﬁnal conﬁguration are pre-

sented in ﬁg. 13.

To create a defect, the fourth and ﬁfth particles are

placed on the same channel side so that two staggered

trains exist, which consist of four and ﬁve particles, re-

spectively. The trains ﬁrst contract individually while they

drift apart from each other as the increasing distance of

particle 4 and 5 indicates (red line in ﬁg. 13, see also video

4 in the ESM). Only after the two trains have reached

their equilibrium conﬁguration, we observe that the lag-

ging four-particle train catches up to the slightly slower

train with ﬁve particles (see ﬁg. 8(b)). In the ﬁnal state

the particle distance of the defect is about twice the equi-

librium particle distance of cross-streamline pairs similar

to observations in [15, 21]. We note that this distance is

not governed by any attractive interaction between the

two particles but solely due to the fact that the larger

leading train moves slower than the smaller trailing train.

Indeed, when we swap the two trains such that the smaller

one is leading, the two trains slowly drift apart in time.

Eur. Phys. J. E (2020) 43:50 Page 9 of 13

2Δzpair

100101102

t/(w2/ν)

Δz/a

t=0

t=5

t=19

t=154

u

Fig. 13. Staggered particle train with defect. Top: snapshots

at diﬀerent times tof the non-uniform contraction from the

initial (t= 0) toward the ﬁnal (t= 154) conﬁguration. Bot-

tom: distances between neighboring particles plotted versus

time. The numbers indicate the pair index increasing from the

back to the front. The ﬁnal particle distance of the defect is

2Δzpair

eq as indicated by the dashed line.

3.3 Microﬂuidic phonons

Regular structures such as the staggered particle trains

can be perturbed and thereby show propagating phononic

excitations or microﬂuidic phonons. To study them in

more detail, we analyze how a cross-streamline train reacts

to a perturbation of a single particle position. Video 5 in

the ESM shows the resulting damped pulse propagation.

We ﬁt a train of 12 particles into a channel and adjust its

length such that periodic boundary conditions generate

an inﬁnitely extended staggered train.

In ﬁg. 14(a) we show the staggered train, where we ini-

tialized the 12 particles with an axial spacing of Δz/a =

4.2, which corresponds to the equilibrium distance of an

isolated particle pair, and with lateral equilibrium posi-

tions at ±xeq/w =±0.4. To perturb the system, we move

one particle inwards toward the channel center as indi-

cated. Thus, it moves faster than the train and approaches

the neighboring particle in front. Figure 14(b) quantiﬁes

the reaction of all the particles by plotting their displace-

ments Δ|x(t)|=|x(t)|−xeq from the equilibrium position

where Δ|x|<0 means motion toward the channel center

and Δ|x|>0 toward the channel wall. While approaching

the neighboring particle, the ﬁrst particle is pushed back

to its equilibrium position and thereby pulls the neigh-

boring particle from the opposite channel side toward the

center. Thus, the whole process repeats such that a dis-

placement pulse travels through the staggered train as il-

lustrated by ﬁg. 14(b). The particle motion is strongly

damped since the inertial lift force pushes the particles

back toward their equilibrium positions. Thus, the initially

displaced particle (pink curve) overshoots only by a small

distance and then relaxes toward its equilibrium position.

Also, the propagating displacement pulse is exponentially

∝exp(−γt)

01234

−0.2

−0.1

t/(w2/ν)

Δ|x|/w

−xeq

+xeq

Δx

Fig. 14. (a) Staggered particle train with an initial displace-

ment of the seventh particle counted from the end (pink) and

equilibrium axial particle distance is Δz/a =4.2. (b) Lateral

particle displacement from the equilibrium position, Δ|x(t)|=

|x(t)|−xeq, plotted versus time for all the particles. The color

coding is the same as in (a). Here Δ|x|<0meansmotion

toward the channel center and Δ|x|>0 toward the channel

wall. The exponential decay of the pulse height with time is

indicated (dashed line) and γis damping rate. The Reynolds

number is Re = 25.

urel

Fig. 15. The swapping mechanism from ref. [25] explains how

the displacement pulse is passed from the lagging to the lead-

ing particle. The curved arrows indicate the particle trajecto-

ries during swapping. The lateral equilibrium position of the

particles and the channel center are marked by the dashed and

dotted lines, respectively. The swapping mechanism is also vi-

sualized in video 6 of the ESM.

damped (dashed line in ﬁg. 14) with a damping rate γ,

which we discuss in more detail further below. We observe

that individual particles in a train return much faster to

their equilibrium positions compared to isolated particles

due to the coupling to neighboring particles, while the re-

laxation time of the whole pulse, γ−1, is roughly twice as

big. Below, we will also discuss in more detail the velocity

of the displacement pulse.

The mechanism for the propagating displacement

pulse can be explained as a sequence of swapping tra-

jectories similar to the one we discussed in a previous

work [25]. As indicated in ﬁg. 15, the displaced particle

approaches the next particle in line, and they swap their

lateral positions such that xafter

lead =−xbefore

lag and vice versa.

In principle, a propagating displacement pulse of the same

type but without damping should also be possible in low-

Reynolds-number ﬂow, as swapping trajectories exist in

this regime as well [47].

We also mention that if the displacement brings the

ﬁrst particle close to or across the channel centerline such

that Δ|x|/w ≤−0.4, it becomes too fast and can no longer

Page 10 of 13 Eur. Phys. J. E (2020) 43:50

−1

x/w

−1

z/w

x/w

Fig. 16. Top: example of a passing trajectory in a staggered

train when the initial lateral displacement from the equilibrium

position is too large. Bottom: snapshot after the particle has

passed two of its neighbors. See also video 7 in the ESM.

20 40 60 80 100

1.14

1.16

1.18

upulse/utrain

Re2

101102

10−1

100

101

γ/(ν/w2)

phonon pair oscillation

Fig. 17. Ratio of pulse to train velocity (main plot) and damp-

ing rate γ(inset) of the displacement pulse as a function of

the Reynolds number. In all cases the initial axial spacing is

Δz0/a ≈4.2 and the initial displacement is Δ|x|0/w =−0.2

toward the channel center. For comparison, we also plot γfor

an oscillating particle pair from [25].

swap its position with the next particle. Instead, it moves

through the staggered train (see ﬁg. 16) and leaves be-

hind a defect, where two neighboring particles move on

the same streamline. This is reminiscent of the passing

trajectory for a particle pair identiﬁed in ref. [25].

3.3.1 Quantitative analysis of the displacement pulse

In ﬁg. 17 we plot the ratio of pulse velocity to train velocity

upulse/utrain (main plot) and the damping rate γ(inset)

of the displacement pulse as a function of the Reynolds

number Re. This is similar to our analysis of the damped

oscillations for a pair of rigid particles [25]. The velocity

ratio is roughly constant in Re so that we identify a lin-

ear dependence upulse ∝utrain ∝Re, which makes sense

since the ﬂuid ﬂow directly determines how fast displaced

particles approach their neighbors. A similar scaling was

observed for the oscillating frequency of a pair of rigid

particles [25]. The pulse velocity is also larger than the

train velocity since the displacement pulse is propagated

by faster moving particles. For the damping rate γof the

displacement pulse in the inset we ﬁnd good agreement

with the damping rate of the oscillating particle pair in the

regime of Re ≤20. The γvalues for the propagating pulse

are slightly lower. The damping rate scales quadratically

with Re, only for higher Reynolds numbers the scaling de-

viates slightly from Re2. Thus, damping of the propagating

33.544.555.56

0.2

0.4

0.6

Δz/a

γ/(ν/w2)

33.544.555.5

1.16

1.17

Δz/a

upulse/utrain

Fig. 18. Damping rate γ(main plot) and ratio of pulse to

train velocity (inset) of the displacement pulse as a function of

axial particle distance Δz. The other parameters are Re =20

and Δ|x|0/w =−0.2. The dashed lines indicate a region with

almost constant γ.

pulse is a clear inertial eﬀect due to the acting inertial lift

force and therefore the scaling with Re2is expected.

In our setting using periodic boundary conditions

along the channel axis, we can compress or expand the

inﬁnitely extended particle train by changing the channel

length. This allows to study pulse propagation under ten-

sion. A possible experimental realization are expanding

or contracting channels, where the channel width changes

abruptly. For example, in an expanding channel the par-

ticle spacing relaxes slowly to its larger distance [5,9]. In

ﬁg. 18 we show ratio of pulse to train velocity upulse/utrain

(inset) and the damping rate γ(main plot) as a function of

the axial particle distance Δz. The velocity ratio again is

nearly constant with varying Δz, while the train velocity

increases linearly with Δz due to the increased friction

(not shown). For the damping rate we ﬁnd three diﬀer-

ent regimes. When the staggered train is strongly com-

pressed, the damping rate is strongly reduced, which is

due to the strong repulsive interactions between particles.

Additionally, we observe that the displacement pulse no

longer travels in one direction only, rather it propagates

in both directions at the same time as we demonstrate in

video 8 in the ESM. For axial distances around the equi-

librium value (3.5<Δz/a<4.5) the damping rate is

almost constant and it slightly increases for Δz/a > 4.5.

Here, the regular train is not stable. Instead, the distances

between two particles contract leaving larger gaps between

particle pairs. Since then the particles relax toward their

equilibrium positions more like an individual particle, the

overall damping rate increases.

Finally, we also varied the initial displacement Δ|x|0

from the equilibrium position and plot in ﬁg. 19 the ratio

of the pulse to train velocity upulse/utrain (inset) and the

damping rate γ(main plot). Positive Δ|x|0means that

the particle is moved toward the channel wall. Since it

thereby slows down relative to the staggered train, it ap-

proaches the particle behind and the displacement pulse

moves against the staggered train. For the velocity ratio

in the inset this means, while for positive and negative

Δ|x|0the pulse velocity is relatively constant, the ratio

upulse/utrain jumps from a value larger than one to a value

smaller than one when Δ|x|0becomes positive. The depen-

Eur. Phys. J. E (2020) 43:50 Page 11 of 13

−0.3−0.2−0.10 0.10.2

Δ|x|0/w

γ/(ν/w2)

−0.3−0.2−0.10 0.10.2

0.8

1.2

Δ|x|0/w

upulse/utrain

Fig. 19. Damping rate γ(main plot) and ratio of pulse to train

velocity (inset) of the displacement pulse as a function of the

initial displacement Δ|x|0. The other parameters are Re =20

and Δz/a =4.2.

0123456

−0.2

−0.1

0.1

t/(w2/ν)

|Δx|/w

Fig. 20. (a) Staggered particle train with a defect and an

initial displacement of the ﬁrst particle counted from the end

(pink). The equilibrium axial particle distance is Δz/a =4.2

and Δz/a =8.4 between the defect particles. (b) Lateral par-

ticle displacement from the equilibrium position, Δ|x(t)|=

|x(t)|−xeq, plotted versus time for all the particles in the

upper and lower channel half. The color coding is the same as

in (a). The arrow indicates the strongly damped displacement

of the ﬁrst particle after the defect has been passed.

dence of the damping rate is less intuitive. It is constant

for large negative Δ|x|0, goes through a minimum and

then increases linearly from Δ|x|0=−0.1 for increasing

Δ|x|0.

3.3.2 Inﬂuence of a defect on the pulse propagation

At the end we study how a defect in the staggered particle

train strongly dampens the propagating pulse and show

our results in ﬁg. 20. We reduce the number of particles to

11 and introduce a defect with two neighboring particles

on the same streamline, as already investigated above. As

illustrated in ﬁg. 20, the pulse is initiated at the pink

particle at the left end of the channel, which is the ﬁfth

particle to the left of the defect. Up to the defect (black

particle) the pulse propagates as before. However, having

passed the defect (blue particle) it is strongly damped such

that the pulse vanishes almost completely.

4 Conclusions

The axial alignment of particles in an inertial microchan-

nel is an important feature for the counting, manipulation,

and sorting of cells. Therefore, the stability of trains is

a crucial ingredient for designing and understanding lab-

on-a-chip devices. As the literature on this topic does not

always agree with their ﬁndings, we focused on a detailed

analysis for one speciﬁc set of parameters.

The stability of cross-streamline pairs is accepted in

the literature. We show that particles in such a pair at-

tract each other over large distances while their lateral

positions hardly change. A cross-streamline pair always

contracts or expands to its equilibrium axial distance. For

same-streamline pairs we thoroughly analyze and thereby

conﬁrm the result of our previous work that the same-

streamline conﬁguration does not have an equilibrium ax-

ial spacing [25]. However, from smaller distances it quickly

expands to a characteristic separation but even at long

times very slowly drifts apart. Their dynamics is domi-

nated by a repulsive interaction, which rapidly decays with

increasing axial distance. With our simulations we also

reproduce the distribution of axial distances for a same-

streamline pair measured in experiments [5]. In particular,

it has a well-deﬁned peak at about twice the distance of

cross-streamline pairs.

Then, we extended our analysis to particle trains with

more than two particles and ﬁrst analyzed how staggered

trains relax to their equilibrium conﬁgurations. In par-

ticular, a staggered train initialized with an axial parti-

cle spacing larger than the equilibrium distance contracts

non-uniformly. The non-uniform contraction is related to

two eﬀects of collective drag reduction: i) when two par-

ticles in a cross-streamline conﬁguration approach each

other they slow down since they exhibit less resistance to

the driving Poiseuille ﬂow and ii) the center-of-mass ve-

locity of staggered trains decreases the more particles are

in the train (peloton eﬀect). These two eﬀects drive the

non-uniform contraction in the front and back of a stag-

gered train. While in the front the leading pair slows down

and collects more and more particles, in the back trailing

pairs of particles separate from the rest of the train. Only

with time the pairs catch up with the particle train in

front of them and form one large train. Finally, we ﬁnd

a master curve for the axial spacing within a staggered

train. The spacing between the particles increases from

the back to the front of the train and ultimately saturates

for suﬃciently long trains. So a staggered train is slightly

expanded at the front relative to the back. In experiments

a particle train slowly expands when it enters a channel

with a suddenly expanding cross section [5,9]. This could

be a means to observe the scenario outlined here.

For linear trains we ﬁnd a very similar behavior as for

same-streamline pairs. Starting from a particle distance

similar to the staggered train, the spacing relaxes in the

beginning phase to a value close to experimental results [6]

about twice the distance of cross-streamline pairs. Then,

the particles continue to slowly drift apart non-uniformly.

The leading particle moves the fastest and separates from

the rest of the train. This conﬁrms a similar observation

Page 12 of 13 Eur. Phys. J. E (2020) 43:50

reported by Gupta et al. for a smaller conﬁnement ra-

tio [28]. In addition, we are able to reproduce the statistics

of particle distances observed in experiments [6].

Finally, we analyzed how a displacement pulse mi-

grates as inertial microﬂuidic phonon through a staggered

train. The displacement is transported from one particle

to another via swapping trajectories, where the inertial

lift forces damps the amplitude of the displacement pulse.

When the initial displacement is too large, the displaced

particle itself moves through the staggered train and leaves

behind a defect. The ratio of pulse to train velocities is

almost constant for varying Reynolds number and axial

particle distance, whereas it is by ca. 30% smaller for ini-

tial displacement toward the wall compared to perturba-

tions toward the channel center. For the damping rate of

the displacement pulse we conﬁrm the quadratic scaling

with the Reynolds number, which identiﬁes damping as

an inertial eﬀect. The damping rate is almost constant for

varying axial distance between the particles or changing

line density. Only when the particles are close together

the damping rate is reduced. This is especially interesting

when the line tension of staggered trains changes rapidly

when entering sections of the channel with expanding or

contracting cross sections [5,9] since then the trains want

to expand or contract.

We hope that with our careful numerical study we

initiate further research on staggered and linear particle

trains to clarify some of the still open questions and also to

provide guidance how microﬂuidic phonons in the inertial

regime behave.

Open Access funding provided by Projekt DEAL. We ap-

preciate helpful discussions with Timm Kr¨uger. We acknowl-

edge support from the Deutsche Forschungsgemeinschaft in the

framework of the Collaborative Research Center SFB 910.

Author contribution statement

All the authors were involved in the preparation of the

manuscript. All the authors have read and approved the

ﬁnal manuscript.

Open Access This is an open access article distributed

under the terms of the Creative Commons Attribution

License (http://creativecommons.org/licenses/by/4.0), which

permits unrestricted use, distribution, and reproduction in any

medium, provided the original work is properly cited.

Publisher’s Note The EPJ Publishers remain neutral with

regard to jurisdictional claims in published maps and institu-

tional aﬃliations.

References

1. G. Segr´e, A. Silberberg, J. Fluid Mech. 14, 115 (1962).

2. J. Zhang, S. Yan, D. Yuan, G. Alici, N.T. Nguyen, M.E.

Warkiani, W. Li, Lab. Chip 16, 10 (2015).

3. H. Amini, W. Lee, D. Di Carlo, Lab. Chip 14, 2739 (2014).

4. D. Stoecklein, D. Di Carlo, Anal. Chem. 91, 296 (2019).

5. W. Lee, H. Amini, H.A. Stone, D. Di Carlo, Proc. Natl.

Acad. Sci. U.S.A. 107, 22413 (2010).

6. S. Kahkeshani, H. Haddadi, D. Di Carlo, J. Fluid Mech.

786, R3 (2016).

7. S.R. Bazaz, A. Mashhadian, A. Ehsani, S.C. Saha, T.

Kr¨uger, M. Ebrahimi Warkiani, Lab. Chip 20, 1023 (2020).

8. A.A.S. Bhagat, S.S. Kuntaegowdanahalli, N. Kaval, C.J.

Seliskar, I. Papautsky, Biomed Microdev. 12, 187 (2010).

9. Y. Deng, S.P. Davis, F. Yang, K.S. Paulsen, M. Kumar, R.

Sinnott DeVaux, X. Wang, D.S. Conklin, A. Oberai, J.I.

Herschkowitz, A.J. Chung, Small 13, 1700705 (2017).

10. W. Tang, D. Tang, Z. Ni, N. Xiang, H. Yi, Anal. Chem.

89, 3154 (2017).

11. A.A.S. Bhagat, S.S. Kuntaegowdanahalli, I. Papautsky,

Lab. Chip 8, 1906 (2008).

12.A.J.Mach,D.DiCarlo,Biotechnol.Bioeng.107, 302

(2010).

13. M.Li,M.vanZee,K.Goda,D.DiCarlo,Lab.Chip18,

2575 (2018).

14. H. Haddadi, H. Naghsh-Nilchi, D. Di Carlo, Biomicroﬂu-

idics 12, 014112 (2018).

15. J.F. Edd, D. Di Carlo, K.J. Humphry, S. K¨oster, D. Irimia,

D.A. Weitz, M. Toner, Lab. Chip 8, 1262 (2008).

16. M. Dhar, J.N. Lam, T. Walser, S.M. Dubinett, M.B. Ret-

tig, D. Di Carlo, Proc. Natl. Acad. Sci. U.S.A. 115, 9986

(2018).

17. J.P. Matas, V. Glezer, ´

E. Guazzelli, J.F. Morris, Phys.

Fluids 16, 4192 (2004).

18. C. Prohm, H. Stark, Lab. Chip 14, 2115 (2014).

19. J. Zhou, I. Papautsky, Lab. Chip 13, 1121 (2013).

20. D. Di Carlo, Lab. Chip 9, 3038 (2009).

21. S. Claire Hur, H.T. Kwong Tse, D.D. Carlo, Lab. Chip 10,

274 (2010).

22. Y. Gao, P. Magaud, L. Baldas, C. Laﬀorgue, M. Abbas, S.

Colin, Microﬂuid. Nanoﬂuid. 21, 154 (2017).

23. X. Hu, J. Lin, X. Ku, Phys. Fluids 31, 073306 (2019).

24. K.J. Humphry, P.M. Kulkarni, D.A. Weitz, J.F. Morris,

H.A. Stone, Phys. Fluids 22, 081703 (2010).

25. C. Schaaf, F. R¨uhle, H. Stark, Soft Matter 15, 1988 (2019).

26. K.Hood,M.Roper,Phys.Rev.Fluids3, 094201 (2018).

27. X. Hu, J. Lin, D. Chen, X. Ku, Microﬂuid. Nanoﬂuid. 24,

25 (2020).

28. A. Gupta, P. Magaud, C. Laﬀorgue, M. Abbas, Phys. Rev.

Fluids 3, 114302 (2018).

29. Z. Pan, R. Zhang, C. Yuan, H. Wu, Phys. Fluids 30,

102005 (2018).

30. C. Dietsche, B.R. Mutlu, J.F. Edd, P. Koumoutsakos, M.

Toner, Microﬂuid. Nanoﬂuid. 23, 83 (2019).

31. T. Beatus, T. Tlusty, R. Bar-Ziv, Nat. Phys. 2, 743 (2006).

32. U.D. Schiller, J.B. Fleury, R. Seemann, G. Gompper, Soft

Matter 11, 5850 (2015).

33. J.B. Fleury, U.D. Schiller, S. Thutupalli, G. Gompper, R.

Seemann, New J. Phys. 16, 063029 (2014).

34. T. Beatus, I. Shani, R.H. Bar-Ziv, T. Tlusty, Chem. Soc.

Rev. 46, 5620 (2017).

35. H. Bruus, Theoretical Microﬂuidics,Oxford Master Series

in Physics, No. 18 (Oxford University Press, Oxford, New

York, 2008).

36. S. Succi, The Lattice Boltzmann Equation for Fluid Dy-

namics and Beyond,Numerical Mathematics and Scientiﬁc

Computation (Clarendon Press, Oxford University Press,

Oxford, New York, 2001).

Eur. Phys. J. E (2020) 43:50 Page 13 of 13

37. P.L. Bhatnagar, E.P. Gross, M. Krook, Phys. Rev. 94, 511

(1954).

38. T. Kr¨uger, H. Kusumaatmaja, A. Kuzmin, O. Shardt, G.

Silva, E.M. Viggen, The Lattice Boltzmann Method: Prin-

ciples and Practice,Graduate Texts in Physics (Springer

International Publishing, 2017).

39. B. D¨unweg, A.J. Ladd, in Advances in Polymer Science

(Springer Berlin Heidelberg, 2008) pp. 1–78.

40. Z. Guo, C. Zheng, B. Shi, Phys. Rev. E 65, 046308 (2002).

41. J. Latt, B. Chopard, O. Malaspinas, M. Deville, A. Mich-

ler, Phys. Rev. E 77, 056703 (2008).

42. The Palabos project, http://www.palabos.org/ (2013).

43. T. Inamuro, Fluid Dyn. Res. 44, 024001 (2012).

44. A. De Rosis, S. Ubertini, F. Ubertini, J. Sci. Comput. 61,

477 (2014).

45. M. Reichert, H. Stark, J. Phys.: Condens. Matter 16,

S4085 (2004).

46. P.J.A. Janssen, M.D. Baron, P.D. Anderson, J.

Blawzdziewicz, M. Loewenberg, E. Wajnryb, Soft

Matter 8, 7495 (2012).

47. S. Reddig, H. Stark, J. Chem. Phys. 138, 234902 (2013).