1 Introduction
Migration of particles towards inertial focusing positions and formation of particle trains are prominent examples of inertial effects in flow of dilute suspensions in conduits (Segre & Silberberg Reference Segre and Silberberg1961; Matas et al.
Reference Matas, Glezer, Guazzelli and Morris2004). Using a characteristic velocity
$U$
, length scale
$L$
and fluid kinematic viscosity
$\unicode[STIX]{x1D708}$
, flow inertia is characterized by Reynolds number defined as
$\mathit{Re}=UL/\unicode[STIX]{x1D708}$
. In confined geometries, such as microchannels, the inertial flow of dilute suspensions has been utilized to self-assemble particles into trains at inertial focusing positions (Di Carlo et al.
Reference Di Carlo, Irimia, Tompkins and Toner2007, Reference Di Carlo, Edd, Humphry, Stone and Toner2009; Hur, Tse & Di Carlo Reference Hur, Tse and Di Carlo2010; Lee et al.
Reference Lee, Amini, Stone and Di Carlo2010). Although the number of trains and the location of focusing positions can be engineered by changing channel dimensions and flow inertia (Chun & Ladd Reference Chun and Ladd2006; Amini, Lee & Di Carlo Reference Amini, Lee and Di Carlo2014), controlling the interparticle spacing necessitates understanding hydrodynamic interaction between particles. Controlling the distance between particles is crucial for applications such as high-speed imaging, flow cytometry and entrapment of live cells in droplets for tissue printing applications, where knowing the arrival time of particles aids in adjusting the imaging or cell entrapment sequence (Edd et al.
Reference Edd, Di Carlo, Humphry, Koster, Irimia, Weitz and Toner2008).
The inertial ordering of particles in trains and interparticle spacings have been experimentally observed in tubular conduits and rectangular microchannels (Matas et al.
Reference Matas, Glezer, Guazzelli and Morris2004; Hur et al.
Reference Hur, Tse and Di Carlo2010). While the underlying mechanism for inertial migration of particles towards focusing positions can be explained by analytical solution of the lateral force on isolated particles (Ho & Leal Reference Ho and Leal1976; Asmolov Reference Asmolov1999; Hood, Lee & Roper Reference Hood, Lee and Roper2015), quantitative and predictive models for the spacing between particles self-assembled in trains are more difficult to ascertain due to the complexity of hydrodynamic interactions between multiple moving particles at finite
$\mathit{Re}$
(Martel & Toner Reference Martel and Toner2014). By studying inertial flow of a dilute suspension in tubes of
$O(10~\text{mm})$
diameter, the underlying mechanism for train formation was attributed to the presence of reversing streamline regions, where fluid parcels approach the particle surface and reverse their paths (Matas et al.
Reference Matas, Glezer, Guazzelli and Morris2004). Numerical studies of suspensions of hard spheres at dilute to intermediate concentrations under finite inertia shear flow show that the reversing motion also exists in relative trajectories of particles with respect to one another (Haddadi & Morris Reference Haddadi and Morris2014, Reference Haddadi and Morris2015). By increasing
$\mathit{Re}$
, the size of the reversing flow region increases, which leads to a shorter distance between particles and a monotonic reduction of the average interparticle spacing in trains (Matas et al.
Reference Matas, Glezer, Guazzelli and Morris2004). Through experimental study of the reversing streamlines in microchannels, it has been demonstrated that, due to an interplay between viscous disturbance flow, which deviates the particles from inertial focusing points, and inertial lift force, which tends to restore particle location, the average spacing between particles changes until it reaches an equilibrium value in the channel downstream (Lee et al.
Reference Lee, Amini, Stone and Di Carlo2010). Although explained in a phenomenological manner, there remains no general predictive model for interparticle spacing. The lateral and axial ordering of particles in microchannels has been studied by changing the concentration of the suspension and the dimensions of the channel cross-section. Increasing the concentration of the suspension leads to formation of multiple trains for certain channel dimensions. A preferred interparticle spacing also forms around
$2.2D$
(Humphry et al.
Reference Humphry, Kulkarni, Weitz, Morris and Stone2010).
In the present work, experiments and numerical simulations have been employed to study the effect of inertia and concentration on the spacing between particles dynamically self-assembled into trains in microchannels. The axial distance between two consecutive particles, denoted as
$d_{x}$
, depends on the conduit Reynolds number
$\mathit{Re}_{c}$
, which is defined using the hydraulic diameter as the characteristic length scale. Inertia can also be defined at the particle length scale by
$\mathit{Re}_{p}$
, determined by the shear rate and diameter of the particle. For a pressure-driven flow in a channel with rectangular cross-section (channel schematic displayed in figure 1), a parabolic velocity profile along with the difference between channel dimensions in
$Y$
and
$Z$
directions leads to a varying shear rate on the particle. For a channel with aspect ratio of 0.58, which is defined as
$\unicode[STIX]{x1D706}=l_{y}/l_{z}$
, particles are expected to equilibrate near the longer walls in the
$Y$
direction (Amini et al.
Reference Amini, Lee and Di Carlo2014). Therefore,
$\mathit{Re}_{p}$
can be defined using the mean shear rate, particle diameter
$D$
and the channel length in the
$Y$
direction (
$l_{y}$
) as
$\mathit{Re}_{p}=(D/l_{y})^{2}\mathit{Re}_{y}$
, where
$\mathit{Re}_{y}=\langle u\rangle l_{y}/\unicode[STIX]{x1D708}$
. Matas et al. (Reference Matas, Glezer, Guazzelli and Morris2004) examined the spacing between particles within trains in
$O(10~\text{mm})$
tube flow and reported a monotonic reduction of the average interparticle spacing. Here, the discussion is focused on the most probable spacing between consecutive particles in trains inside a rectangular microchannel, which is important for microchannel imaging and cell sorting applications. In order to measure the location of the most probable spacing, the probability density function of the interparticle distance has been computed for particles self-assembled into trains, and the modes, which distinctly form in curves of the probability density function, are chosen as a metric of the preferred spacing. The effect of
$\mathit{Re}_{p}$
on the axial spacing between consecutive particles (
$d_{x}$
) has been studied for suspensions of concentration
$\unicode[STIX]{x1D719}\sim O(0.001)$
. It will be shown that the preferred spacing between particles switches from
$5D$
for low
$\mathit{Re}_{p}$
to
$2.5D$
at higher
$\mathit{Re}_{p}$
, without the presence of intermediate values. The results also demonstrate that increasing the concentration of suspensions to
$O(0.01)$
results in the reduction of the distance between particles in trains, and the preferred spacings measured for
$\unicode[STIX]{x1D719}\sim O(0.001)$
suspensions will no longer form. Further increase of
$\unicode[STIX]{x1D719}$
leads to unsteadiness in the spacing and prevents self-assembly of particles into ordered trains.
Schematic of the channel with two inlets utilized for self-assembly of particles into one train. A particle suspension is pumped from one inlet and fluid without particles is infused from the opposite inlet. Dimensions of the channel in the
$X$
(flow),
$Y$
(lateral) and
$Z$
(transverse) directions are 3 cm,
$35~\unicode[STIX]{x03BC}\text{m}$
and
$60~\unicode[STIX]{x03BC}\text{m}$
respectively, which leads to the formation of two inertial focusing positions near the walls in the
$Y$
direction.
2 Experimental procedure
The experimental observations have been made in a dual-inlet polydimethylsiloxane (PDMS) microchannel bonded to a glass slide produced according to standard soft lithography protocols (Duffy et al.
Reference Duffy, Mcdonald, Schueller and Whitesides1998). The dual-inlet channel is utilized in order to self-assemble particles predominantly into one train in order to limit hydrodynamic interactions between particles to a single train. Dimensions of the channel in the axial (
$X$
or flow direction), lateral (
$Y$
) and transverse (
$Z$
) directions are 3 cm,
$35~\unicode[STIX]{x03BC}\text{m}$
and
$60~\unicode[STIX]{x03BC}\text{m}$
respectively. A dilute suspension of polystyrene spherical particles with diameter
$12~\unicode[STIX]{x03BC}\text{m}$
(
$\unicode[STIX]{x1D70C}_{p}=1.05~\text{g}~\text{cm}^{-3}$
) dispersed in a suspending fluid composed of deionized water, 0.002 weight per volume (
$\text{w}/\text{v}$
) triton X-100 and
$0.1~\text{v}/\text{v}$
glycerol is pumped into the channel with a controlled flow rate utilizing a syringe pump from one inlet, accompanied by pumping fluid without particles at an equal flow rate from the second inlet. The volume fraction of particles in the suspension is
$\unicode[STIX]{x1D719}=0.004$
. More concentrated suspensions of
$\unicode[STIX]{x1D719}=0.011{-}0.025$
are used to study disruption of trains. The flow rate ranges from 85.5 to
$342~\unicode[STIX]{x03BC}\text{l min}^{-1}$
, leading to finite inertia at
$\mathit{Re}_{c}=30{-}120$
inside the channel. Lowering the flow rate below
$85.5~\unicode[STIX]{x03BC}\text{l min}^{-1}$
does not generate sufficient inertia for a steady-state interparticle spacing. Increasing the flow rate above
$342~\unicode[STIX]{x03BC}\text{l min}^{-1}$
leads to experimental errors such as delamination of the PDMS from the glass slide, leakage at inlets and alteration of the channel size due to high flow-induced deformation in PDMS microchannels (Dendukuri et al.
Reference Dendukuri, Gu, Pregibon, Hatton and Doyle2007). The channel configuration allows formation of particle trains at two inertial equilibrium points close to the walls in the
$Y$
direction for
$D=12~\unicode[STIX]{x03BC}\text{m}$
particles such that infusing the suspension from one inlet mainly generates a single train close to one wall. The schematics of the channel are displayed in figure 1. The axial distance between two adjacent particles (
$d_{x}$
) has been computed from recorded snapshots captured using high-speed imaging (Phantom V710). The interparticle spacing in multiple sections along the channel has been measured and the values corresponding to the farthest region from the entrance, spanning the last
$500~\unicode[STIX]{x03BC}\text{m}$
section of the channel, have been reported in order to achieve a steady-state spacing.
3 Computational method
In this section, a brief explanation of the governing equations, computational parameters and the lattice Boltzmann method (LBM) that is used to compute particle trajectories in inertial conduit flow is presented. More details about the fundamentals of the LBM are presented in Ladd (Reference Ladd1994a ,Reference Ladd b ) and Nguyen & Ladd (Reference Nguyen and Ladd2002), a review article by Aidun & Clausen (Reference Aidun and Clausen2010) and a complete description of Galilean invariance errors due to the presence of particles in Clausen & Aidun (Reference Clausen and Aidun2009). The non-dimensional forms of the equations governing the fluid phase are
$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}=0, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \mathit{Re}\left(\frac{\unicode[STIX]{x2202}\boldsymbol{u}}{\unicode[STIX]{x2202}t}+\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}\right)=-\unicode[STIX]{x1D735}p+\unicode[STIX]{x1D6FB}^{2}\boldsymbol{u}, & \displaystyle\end{eqnarray}$$
where length has been non-dimensionalized using the hydraulic diameter
$h_{d}$
, velocity by the average fluid velocity at the channel inlet
$\bar{U}$
and the pressure by
$\unicode[STIX]{x1D70C}\bar{U^{2}}$
. The LBM algorithm for suspensions of hard spheres, which has been developed by Ladd (Reference Ladd1994a
,Reference Ladd
b
) with further improvement for moving particles by Aidun, Lu & Ding (Reference Aidun, Lu and Ding1998), is based on the Boltzmann equation for the fluid phase coupled with Newtonian dynamics for solid particles. By computing the force
$\boldsymbol{F}_{i}$
and torque
$\boldsymbol{T}_{i}$
exerted by the fluid on a particle of mass
$m_{i}$
and moment of inertia
$I_{i}$
, the translational velocity
$\boldsymbol{V}_{i}$
and rotational velocity
$\unicode[STIX]{x1D734}_{i}$
of the particle are calculated as
$$\begin{eqnarray}\displaystyle & \displaystyle m_{i}\frac{\text{d}\boldsymbol{V}_{i}}{\text{d}t}=\boldsymbol{F}_{i}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle I_{i}\frac{\text{d}\unicode[STIX]{x1D734}_{i}}{\text{d}t}=\boldsymbol{T}_{i}. & \displaystyle\end{eqnarray}$$
In order to calculate particle trajectories in a confined conduit, a rectangular computational box with
$128\times 36\times 65$
lattice units in
$X$
,
$Y$
and
$Z$
directions is chosen, where
$X$
indicates the direction of the flow. The rectangular box is periodic in the flow direction and is confined by solid walls in the
$Y$
and
$Z$
directions. The specified dimensions of the computational box are chosen in accordance with the dimensions of the microchannel, which leads to observation of two equilibrium points close to the walls in the
$Y$
direction. The resolution of the particles was set at 12.9 lattice units (lu) per diameter. The results of the numerical trajectory analysis were found to be independent of further increase in the resolution by simulating particles with a larger size of
$D=16.5~\text{lu}$
in a
$256\times 72\times 130~\text{lu}^{3}$
channel. In addition, the results have been examined to be independent of the periodicity in the flow direction by iterating some sample trajectories for a
$256\times 36\times 65~\text{lu}^{3}$
box.
4 Results and discussion
In figure 2, the probability density functions (p.d.f.) of
$d_{x}$
for
$\mathit{Re}_{p}=2.8$
, 5.6 and 8.3 are presented. The highest peak in the p.d.f. corresponds to the most probable spacing between two consecutive particles at a certain
$\mathit{Re}_{p}$
. Although the adverse effect of particle sedimentation was minimized by matching the density of particles and the fluid, fluctuation in the number of particles observed in each imaging snapshot is inevitable. Considering that the concentration of suspensions used for inertial self-assembly applications is small (
$\unicode[STIX]{x1D719}<O(0.01)$
), the average length fraction defined as
$\langle L_{f}\rangle =\langle N\rangle D/L$
, where
$\langle N\rangle$
is the average number of particles per frame monitored throughout the experiment and
$L$
is the frame length (
$\simeq 500~\unicode[STIX]{x03BC}\text{m}$
in the present study), can be used as a replacement for volume fraction
$\unicode[STIX]{x1D719}$
(Di Carlo Reference Di Carlo2009). The average length fraction
$\langle L_{f}\rangle$
is a direct measure for proximity between particles, which affects hydrodynamic interactions. In the experimental measurements discussed here,
$d_{x}$
is compared in trains with identical
$\langle L_{f}\rangle$
. This approach assists in differentiating the influence of
$\mathit{Re}_{p}$
on spacing from possible effects of fluctuations in the number of particles in trains. In the probability density functions of figure 2 the value of
$\langle L_{f}\rangle$
is 0.1. It is observed in figure 2 that for
$\mathit{Re}_{p}=2.8$
the preferred spacing, which corresponds to the peak in the p.d.f. curves, forms around
$\simeq 5D$
, and for high
$\mathit{Re}_{p}$
of 8.3, the preferred spacing is located around
$2.5D$
. When varying
$\mathit{Re}_{p}$
, no intermediate preferred spacing is observed and peaks of the p.d.f. are consistently observed around
$2.5D$
and
$5D$
.
The probability density functions (p.d.f.) of axial spacing between particles,
$d_{x}$
, for
$\mathit{Re}_{p}=2.8$
, 5.6 and 8.3 (corresponding to
$\mathit{Re}_{c}=30$
, 60 and 90). The aspect ratio of the channel cross-section is
$\unicode[STIX]{x1D706}=l_{y}/l_{z}=0.58$
and the particle diameter is
$D=12~\unicode[STIX]{x03BC}\text{m}$
. With a slight deviation (
${<}0.4D$
), the formation of most probable spacings is observed around
$2.5D$
and
$5D$
, the modes (highest peaks) in the p.d.f.
Although infusing the suspension from a separate inlet increases the probability of forming one train with axial alignment of particles, i.e. axial ordering, a small number of particles may migrate towards the opposite wall. Sporadic migration of particles towards the opposite wall can start at the channel inlet and during merging. Two consecutive particles that are located on opposite sides of the channel form a lateral ordered pair. The peaks observed in the p.d.f. curves correspond to the spacing between consecutive particles, without distinguishing between axial and lateral ordered pairs. To make a distinction between axial and lateral orientations, the spatial probability distribution of neighbouring particles is presented in figure 3. The spatial probability distribution,
$p(\boldsymbol{r})$
, is obtained by constructing a histogram around each particle which is populated by the location of adjacent particles. The histogram is normalized by the total number of samplings. Considering that the pair orientation vector is used to construct the histograms, the distributions span the entire space around the reference particle, including negative
$X$
and
$Y$
values, although the probability of occupying negative locations is not significant. It is observed in figure 3 that at
$\mathit{Re}_{p}=2.8$
, particle pairs with axial ordering (
$\unicode[STIX]{x0394}Y=0$
) equilibrate close to
$d_{x}\simeq 5D$
. With increasing inertia to
$\mathit{Re}_{p}=8.3$
, the preferred spacing between particles forms around
$2.5D$
. In the spatial representation of
$p(\boldsymbol{r})$
at
$\mathit{Re}_{p}=2.8$
, a secondary zone at
$d_{x}\simeq 2.5D$
is also seen due to the lateral pair configuration (
$\unicode[STIX]{x0394}Y\neq 0$
), which results in the formation of a secondary peak around
$2.5D$
in the probability density functions of figure 2. The spacing between particles with lateral ordering has been predominantly observed around
$2.5D$
for all
$\mathit{Re}_{p}$
studied.
Humphry et al. (Reference Humphry, Kulkarni, Weitz, Morris and Stone2010) measured the axial spacing between particles in single-inlet channels and reported
$d_{x}\simeq 2.2D$
, mainly for suspensions flowing at
$\mathit{Re}_{p}\sim O(1)$
in microchannels. Considering that inertial flow of a dilute suspension in a single-inlet channel leads to the formation of particle trains close to both walls, the probability of observing lateral ordering increases. Therefore, an axial spacing of
$2.2D$
between particles agrees well with our results.
The spatial probability distribution of finding the adjacent particle in the vicinity of a reference particle
$p(\boldsymbol{r})$
at (a)
$\mathit{Re}_{p}=2.8$
and (b)
$\mathit{Re}_{p}=8.3$
. A streak of low probability that is observed at
$\unicode[STIX]{x0394}X\approx 2.5D$
and
$\unicode[STIX]{x0394}Y\approx 1.2D$
corresponds to laterally ordered pairs of particles. Snapshots taken from experiments are included to exhibit the self-assembled trains (the scale bar shows a length of
$60~\unicode[STIX]{x03BC}\text{m}$
).
Experimental observation of interparticle spacing at preferred axial locations of
$d_{x}\simeq 2.5D$
and
$5D$
can be further investigated using numerical simulations of particle relative motions. The classes of relative motion of particles in a dilute suspension can be assembled from trajectories of isolated pairs of particles (Haddadi & Morris Reference Haddadi and Morris2014). Pair trajectories are obtained by calculating the trajectory of one particle with respect to the coordinate frame located at the centre of mass of the second particle. The LBM has been utilized for studying trajectories, and has been proven to be an efficient technique for numerical simulation of inertial suspension flow. The evolution of sample pair trajectories until reaching a steady-state separation at
$\mathit{Re}_{p}=2.8$
and 8.3 is demonstrated in figure 4. To identify the most probable pair separation at each
$\mathit{Re}_{p}$
, the steady-state separation of pair trajectories with various initial configurations has been examined numerically. A large number of initial configurations, composed of 107 pair orientations for each
$\mathit{Re}_{p}$
, have been simulated to construct a thorough representation of the relative motions. The steady-state pair separations are used to populate a spatial histogram that is constructed around the reference particle. The spatial probability densities of pair equilibrium separations for
$\mathit{Re}_{p}=2.8$
and 8.3 are shown in figure 5(a,b) respectively. It is seen that at
$\mathit{Re}_{p}=2.8$
and for all initial configurations, the most probable spacing for an axially ordered pair is located at
$5D$
. For pairs with lateral orientation, the preferred spacing forms at
$d_{x}=2.5D$
. At
$\mathit{Re}_{p}=8.3$
, axially ordered spacings at
$2.5D$
and
$5D$
are more pronounced, where the probability of reaching an equilibrium at
$2.5D$
is significantly higher. In addition, pairs with lateral ordering tend to equilibrate at
$2.5D$
from the reference particle for all
$\mathit{Re}_{p}$
values.
Selected pair trajectories calculated numerically at (a)
$\mathit{Re}_{p}=2.8$
and (b)
$\mathit{Re}_{p}=8.3$
projected on the
$\mathit{XZ}$
plane. Blue triangles correspond to the starting points of the second particle relative to the reference particle shown at the origin, and red circles correspond to the final steady-state separations.
Attractors at preferred spacings and their probabilities, calculated by simulating the motion of pairs starting from various initial configurations at (a)
$\mathit{Re}_{p}=2.8$
and (b)
$\mathit{Re}_{p}=8.3$
. The boundary of the reference particle is also indicated in the figure. The figure comprises attractors forming in axial and lateral directions. Attractors form at
$5D$
and
$2.5D$
, and the most probable axial attractors are located at
$5D$
for
$\mathit{Re}_{p}=2.8$
and at
$2.5D$
for
$\mathit{Re}_{p}=8.3$
. Attractors of lateral orientation are located at
$2.5D$
for both
$\mathit{Re}_{p}$
values.
The match between preferred spacings observed in experiments (figures 2 and 3) and the steady-state separations between pairs in numerical simulations (figure 5) implies that a finite number of attractors contribute to the formation of preferred interparticle spacings in trains. The attractor with highest probability at low
$\mathit{Re}_{p}$
is located at
$5D$
. With increasing
$\mathit{Re}_{p}$
, rather than a gradual shift in the location of the attractors, the probability of the
$5D$
attractor decreases and the
$2.5D$
attractor becomes more probable. Using numerical simulations of the disturbance streamlines around a single force-free particle moving in a rectangular channel, the equilibrium distance between two particles has been related to the formation of closed-streamline regions adjacent to the particle (Humphry et al.
Reference Humphry, Kulkarni, Weitz, Morris and Stone2010). Considering that the disturbance streamlines around a single particle are affected by the presence of other particles, the fundamental description of the flow leading to formation of attractors remains an open question. It should be noted that pair trajectory attractors are not particular to confined inertial flows and have also been observed in the relative motion of particles in Stokes flow of dilute suspensions in two-dimensional microchannels, where linearity of the equations governing Stokes flow allows analytical progress (Uspal & Doyle Reference Uspal and Doyle2012). It should be emphasized that in the present work, the location of the attractors in the pair trajectory space is obtained from numerical sampling of the pair motion. Although a large number of initial pair configurations have been simulated to calculate the location of attractors, drawing a definite conclusion about the total number and location of pair trajectory attractors is not possible. However, the combination of experimental observations and numerical sampling of the pair space corroborates the existence of preferred spacings between adjacent particles at each
$\mathit{Re}_{p}$
. We also note that there is a deviation (between
$0.2D$
and
$0.4D$
for the location of the preferred spacings) from the
$2.5D$
and
$5D$
attractors in experimental measurements. This deviation may be due to unavoidable experimental limitations, including pixel size accuracy in image processing, dispersion in particle size distributions and the presence of multiple particles in a train (more than an isolated pair), which does not exist in simulations of an isolated pair.
The question arises of whether the locations of the attractors depend strongly on the channel and particle size or are more generally observed. The generality has been examined for selected particle sizes and dimensions of the channel cross-section. The locations of the attractors are measured for
$\mathit{Re}_{p}=2.8$
and 8.3, where the preferred spacing for
$D=12~\unicode[STIX]{x03BC}\text{m}$
in
$\unicode[STIX]{x1D706}=0.58$
channels has been previously explained to be respectively around
$5D$
and
$2.5D$
. Similarly, for
$D=4.8$
and
$20~\unicode[STIX]{x03BC}\text{m}$
particles in
$\unicode[STIX]{x1D706}=0.58$
channels the location of the attractors remained unchanged. Identically, for
$D=12$
and
$20~\unicode[STIX]{x03BC}\text{m}$
in a
$\unicode[STIX]{x1D706}=1$
channel (
$l_{y}=l_{z}=60~\unicode[STIX]{x03BC}\text{m}$
), the attractors at
$\mathit{Re}_{p}=2.8$
and 8.3 form around
$5D$
and
$2.5D$
.
For inertial self-assembly of particles in trains, the average length fraction of particles
$\langle L_{f}\rangle$
in trains is an important parameter in addition to
$\mathit{Re}_{p}$
. Attaining a tangible increase in length fraction is achieved by increasing the volume fraction
$\unicode[STIX]{x1D719}$
to
$O(0.01)$
. It has been shown that increasing
$\unicode[STIX]{x1D719}$
leads to formation of multiple trains in microchannels with large aspect ratios of the channel cross-section (Humphry et al.
Reference Humphry, Kulkarni, Weitz, Morris and Stone2010). In order to study the spacing between particles for a more concentrated suspension, experiments have been conducted at
$\unicode[STIX]{x1D719}=0.011{-}0.025$
, which results in
$\langle L_{f}\rangle \simeq 0.3{-}0.6$
. (It should be noted that
$\langle L_{f}\rangle =0.1$
is achieved at
$\unicode[STIX]{x1D719}=0.004$
.) The volume fraction of suspensions is still considered to be in a dilute rheological regime; however, increasing
$\unicode[STIX]{x1D719}$
to
$O(0.01)$
leads to a pronounced increase of
$\langle L_{f}\rangle$
to 0.6, which can alter the dynamic self-assembly of particles into trains. Numerical simulations have also been conducted and the average distance between pairs of consecutive particles versus time has been monitored. Figure 6(a) shows numerical values of
$\langle d_{x}\rangle$
versus dimensionless time
$t^{\ast }=t\langle U\rangle /D$
(
$\langle U\rangle$
is the average inlet velocity) at
$\mathit{Re}_{p}=2.8$
and includes snapshots taken from experimental measurements. For numerical simulations of larger
$\langle L_{f}\rangle$
, the number of particles in the computational box has been increased without changing the box size. The average distances between particles in
$X$
and
$Z$
directions, denoted as
$\langle d_{x}\rangle$
and
$\langle d_{z}\rangle$
respectively, are computed as the average centre-of-mass separations of particle pairs in each time step. It is observed that with increasing
$\langle L_{f}\rangle$
to 0.3 (three times higher than previous results) the preferred axial spacing decreases to
$\simeq 3.2D$
from the previously observed
$5D$
for
$\langle L_{f}\rangle \simeq 0.1$
. Although at
$\langle L_{f}\rangle \simeq 0.3$
particles are still self-assembled into ordered trains and the final
$\langle d_{x}\rangle$
reaches a steady value, with further increase of
$\langle L_{f}\rangle$
to 0.6 interparticle distance starts to fluctuate, which prevents the formation of a self-assembled train, as can be seen in experimental snapshots. The disruption of trains can be viewed as fluctuations of spacing in either
$y$
or
$z$
direction computed numerically. Figure 6(b) exhibits
$\langle d_{z}\rangle$
for
$\langle L_{f}\rangle =0.3$
and 0.6, where apparent temporal fluctuations in spacing and disruption of particle ordering are seen at
$\langle L_{f}\rangle =0.6$
.
Variation of average spacing between particles versus time for larger concentrations of particles: (a) average axial separation
$\langle d_{x}\rangle$
at
$\mathit{Re}_{p}=2.8$
and (b) average transverse separation
$\langle d_{z}\rangle$
at
$\mathit{Re}_{p}=2.8$
. Dimensionless time is defined as
$t^{\ast }=t\langle U\rangle /D$
.
5 Conclusion
The experimental measurements of the distance between consecutive particles self-assembled in trains, which have been validated by numerical simulations, can be used as a predictive model for dynamic self-assembly of particles with precise separations from one another. Depending on the
$\mathit{Re}_{p}$
, the most probable spacings in a dilute regime of
$\unicode[STIX]{x1D719}=O(0.001)$
, corresponding to ‘trajectory attractors’, are located around
$2.5D$
and
$5D$
. The attractors appear to be generally observed for varying ratios of particle size to channel dimensions and channel aspect ratios, although a more exhaustive investigation of the space would be required to confirm this result. By increasing the concentration of suspensions to
$\unicode[STIX]{x1D719}=O(0.01)$
, the spacing between particles decreases until it reaches an unsteady state spacing where focusing of the particles into trains is no longer possible. The results could be helpful in practical applications, suggesting the ability to make use of either of the two attractors if flow conditions are controlled precisely.
Acknowledgements
The support from ONR Young Investigator Program is acknowledged. We would like to thank Professor A. Ladd at University of Florida–Gainsville for providing the initial lattice Boltzmann solver.
