Self-similarity in particle accumulation on the advancing meniscus

Abstract

When a mixture of viscous oil and non-colloidal particles displaces air between two parallel plates, the shear-induced migration of particles leads to the gradual accumulation of particles on the advancing oil–air interface. This particle accumulation results in the fingering of an otherwise stable fluid–fluid interface. While previous works have focused on the resultant instability, one unexplored yet striking feature of the experiments is the self-similarity in the concentration profile of the accumulating particles. In this paper, we rationalise this self-similar behaviour by deriving a depth-averaged particle transport equation based on the suspension balance model, following the theoretical framework of Ramachandran ( J. Fluid Mech. , vol. 734, 2013, pp. 219–252). The solutions to the particle transport equation are shown to be self-similar with slight deviations, and in excellent agreement with experimental observations. Our results demonstrate that the combination of the shear-induced migration, the advancing fluid–fluid interface and Taylor dispersion yield the self-similar and gradual accumulation of particles.

Full text

J. Fluid Mech. (2021), vol.925, A10, doi:10.1017/jfm.2021.647
Self-similarity in particle accumulation on the
advancing meniscus
Yun Chen1,RuiLuo
1, Li Wang2and Sungyon Lee1,†
1Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN 55455, USA
2School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA
(Received 2 November 2020; revised 2 June 2021; accepted 15 July 2021)
When a mixture of viscous oil and non-colloidal particles displaces air between two
parallel plates, the shear-induced migration of particles leads to the gradual accumulation
of particles on the advancing oil–air interface. This particle accumulation results in the
fingering of an otherwise stable fluid–fluid interface. While previous works have focused
on the resultant instability, one unexplored yet striking feature of the experiments is the
self-similarity in the concentration profile of the accumulating particles. In this paper,
we rationalise this self-similar behaviour by deriving a depth-averaged particle transport
equation based on the suspension balance model, following the theoretical framework of
Ramachandran (J. Fluid Mech., vol. 734, 2013, pp. 219–252). The solutions to the particle
transport equation are shown to be self-similar with slight deviations, and in excellent
agreement with experimental observations. Our results demonstrate that the combination
of the shear-induced migration, the advancing fluid–fluid interface and Taylor dispersion
yield the self-similar and gradual accumulation of particles.
Key words: lubrication theory, suspensions, thin films
1. Introduction
Particle-laden flows have been studied extensively both theoretically and experimentally,
owing to their relevance in avalanches and mudflows (Savage & Lun 1988; Gray & Ancey
2009; Gray & Kokelaar 2010), three-dimensional printing of complex fluids (Lewis 2006;
Roh et al. 2017)and cell migration in biological systems (Vejlens 1938; Zhou & Chang
2005). In particular, there is much interest in understanding hydrodynamic interactions
of suspended particles that may lead to non-uniform particle distribution or particle
†Email address for correspondence: sungyon@umn.edu
© The Author(s), 2021. Published by Cambridge University Press. This is an Open Access article,
distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.
org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium,
provided the original work is properly cited. 925 A10-1
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Y. Chen, R. Luo, L. Wang and S. Lee
accumulation on a fluid–fluid interface. For instance, Vejlens (1938) experimentally
studied blood flowing through capillary tubes and observed a much stronger red
colour in the region behind an advancing meniscus, which indicates a higher local
concentration of red blood cells. More recently, Zhou & Chang (2005) demonstrated
that the accumulation of red blood cells at the meniscus causes the penetration failure
of blood suspensions into a capillary with a diameter smaller than 100 micrometres,
which is a major obstacle in miniaturising blood diagnostic tools. The observation of
particle accumulation on the fluid–fluid interface is not limited to blood flows and even
extends to geological systems. Bhattacharji & Smith (1964) observed the concentration
gradient of minerals in rock-structure fissures, which is due to the accumulation of solid
particles.
To rationalise the mineral concentration gradient (Bhattacharji & Smith 1964),
Bhattacharji & Savic (1965) hypothesised that the accumulation of solid particles is
caused by a recirculation flow near the interface. They developed a mathematical model
to predict the velocity field of this recirculation flow in the absence of the particles, later
known as a ‘fountain flow’. Following their work, Karnis & Mason (1967) experimentally
investigated the accumulation of particles behind an advancing meniscus by tracking
aluminium trace particles to obtain the fountain flow streamlines. In qualitative agreement
with the theoretical prediction (Bhattacharji & Savic 1965), their results indicate that
particles entering the fountain flow region at the meniscus move radially away from the
axis towards the wall and are directed radially inwards again as they exit the fountain flow
region. At the same time, the particles close to the wall are trapped in the slow-moving
region and remain at the wall, resulting in the accumulation of particles at the meniscus.
In addition, Karnis & Mason (1967) claimed that the particle accumulation must vanish
when the wall effects are eliminated.
Despite the apparent success of the fountain flow model, Chapman (1990) proposed
that shear-induced migration be an alternative mechanism for particle accumulation, the
effect of which does not vanish with wall effects. Shear-induced migration refers to the
tendency of particles to move from high-shear to low-shear regions (Leighton & Acrivos
1987; Phillips et al. 1992; Nott & Brady 1994), which has been considered experimentally
(Hookham 1986;Abottet al. 1991; Koh, Hookham & Leal 1994) and theoretically
(Cook 2008;Wardet al. 2009; Murisic et al. 2011,2013; Lee, Stokes & Bertozzi 2014;
Mavromoustaki & Bertozzi 2014; Wang & Bertozzi 2014; Lee, Wong & Bertozzi 2015;
Snook, Butler & Guazzelli 2016). In particular, a number of recent studies have focused
on building the theoretical framework for particle transport in the axial direction, while
capturing the effects of shear-induced migration normal to the flow direction. Select
examples include modelling of particle dispersion in different configurations (Griffiths
& Stone 2012; Ramachandran 2013; Christov & Stone 2014), while others considered the
effects of particle friction (Lecampion & Garagash 2014) and a yield-stress interstitial fluid
(Hormozi & Frigaard 2017).
Then, connecting shear-induced migration to particle accretion, particles migrate
towards the channel centreline in pressure-driven flow and assume a higher particle
average velocity than that of the fluid upstream of the interface. This velocity differential
results in a net flux of particles towards the meniscus and causes particle accumulation
(Xu, Kim & Lee 2016; Luo, Chen & Lee 2018). Chapman demonstrated that while the
accumulation rate increases with the particle size, there exists a non-zero asymptote
when the particle size approaches zero. More recently, Ramachandran & Leighton (2007)
supported the mechanism proposed by Chapman and confirmed that particle accumulation
is observed even when the wall effects vanish. Therefore, we conjecture that particle
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Self-similarity in particle accumulation
accumulation is governed primarily by the shear-induced migration far upstream, while
the fountain flow near the meniscus may play a secondary role.
One of the consequences of particle accumulation on the fluid–fluid interface is the
emergence of the interfacial instability. For instance, Tang et al. (2000) first observed
viscous fingering, when a mixture of particles and viscous fluid displaces air inside a
Hele-Shaw cell. This surprising phenomenon, or ‘particle-induced viscous fingering’,
contrasts with the stable case of a pure viscous liquid displacing air. Following their work,
Ramachandran & Leighton (2010) observed a similar fingering instability upon squeezing
the particle suspension between two parallel plates. More recently, Xu et al. (2016)
experimentally characterised fingering patterns for varying particle volume fractions.
They also successfully validated the effects of shear-induced migration that leads to the
accumulation of particles on the advancing oil–air interface, which was followed by the
exploration of the role of channel confinement (Kim, Xu & Lee 2017)and linear stability
analysis to predict the critical wavenumber (Hooshanginejad, Druecke & Lee 2019).
Despite the recent development in particle-induced viscous fingering, one of the
fundamental phenomena that leads to fingering remains unexplained. While particle
accumulation on the interface has been attributed to shear-induced migration, Xu et al.
(2016) noted that particles accrete on the advancing interface in such a way that
the depth-averaged particle volume fraction ¯
φis independent of time when scaled
appropriately. In this paper, we focus on understanding this self-similar behaviour of
particle accumulation both experimentally and theoretically. In § 2, we first quantitatively
show the self-similar behaviour by defining an inner radius of the suspension, Rin ,
which separates the quasi-steady region far upstream of the interface from the particle
accumulation region. In § 3, we mathematically model the suspension flow as a continuum
and obtain a depth-averaged particle transport equation, based on the theoretical
framework developed by Ramachandran (2013). The numerical solutions to the transport
equation are also presented in § 4and the paper is concluded in § 5.
2. Experiments
2.1. Materials and methods
We hereby describe the experimental method and data that were previously included in
(Xu et al. 2016;Luoet al. 2018). The experiments are performed by radially displacing
air with a particle suspension inside a Hele-Shaw cell, as depicted in the schematic
in figure 1(a). The cell consists of two plates of plexiglass, each with dimensions of
30.5×30.5×3.8 cm, that are separated by a gap thickness hwith standard shims
(McMaster) secured at four corners. The suspension is prepared by mixing PDMS
silicone oil (United Chemical, viscosity ηl=0.096 Pa ·s, density ρl=0.96 g cm−3)
with fluorescent polyethylene particles (Cospheric, diameter d=125–150 μm, density
ρp=1.00gcm
−3) inside a syringe to an initial volume fraction of φ0. The suspension
is left to sit for a sufficient time to allow entrapped air bubbles to escape with minimal
particle settling.
A complete list of experimental parameters (i.e. hand φ0)is summarised in table 1.
A syringe pump (New Era Pump Inc., model NE-1010) is used to inject the suspension
into the Hele-Shaw cell through a hole drilled at the centre of the bottom plate via a clear
vinyl tube. For all the experiments discussed herein, the volumetric flow rate Qis kept
constant at 150 mL min−1. A light-emitting diode (LED) panel (EnvirOasis, 75 W, 4200
Lumin) is placed under the cell to provide uniform illumination. All the experiments are
recorded with a digital camera (Canon 60D, resolution 1920 ×1080, field of view 64◦)
directly from above, at a rate of 30 frames per second.
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Y. Chen, R. Luo, L. Wang and S. Lee
(b)t
3s
6s
9ss
12 s
15 s
18 s
(a)
Camera
φ0 = 0.2 φ0 = 0.35
2 cm
Hele-Shaw cell
Vp
h
d
Q
Figure 1. (a) A schematic of the experimental set-up. (b) Time-elapsed images of the suspension injected into
a Hele-Shaw cell with gap thickness h=1.4 mm, at φ0=0.2(a)andφ0=0.3(b). Here, the particle diameter
dand the injection flow rate Qcorrespond to d=125 μmandQ=150 ml min−1, respectively.
h(mm)φ
0(×100 %)
1.48,11,14 ,15,17,20–35
1.38,11,14 ,17,20–35
1.214,15,17,20–28,30–35
Table 1. Experimental parameters of the gap thickness, hand particle concentration, φ0.Theφ0range that is
denoted as ‘# – #’ increases by an increment of 1 %.
We process the videos collected from experiments in MATLAB with two specific
goals: to track the suspension–air interface and to measure the depth-averaged particle
concentration, ¯
φ, in each experimental image. To track the interface, we use the built-in
edge detector in MATLAB, which takes the smoothed images from median filter as an
input. The detector utilises the ‘Canny’ method (Canny 1986) to identify changes in local
intensity gradient to find maxima, which are treated as edges. As indicated in figure 2(c),
R(t)corresponds to the radius of the circle fitted from all data points on the suspension–air
interface at a given time t.
Next, to extract ¯
φ, we first systematically subtract the background image from all
the experimental images. This allows us to eliminate any inherent non-uniformity in
lighting and to obtain the light intensity matrix Ifrom the processed images. Then, the
depth-averaged concentration ¯
φis correlated to I, based on the following relationship
(Grasa & Abanades 2001;Xuet al. 2016):
¯
φ(r)=kln I(r)−ln Imin
ln Imax −ln Imin
,(2.1)
where Imin and Imax are the minimum and maximum light intensity values of the given
image, respectively. The empirical parameter, k, is acquired from mass conservation:
¯
φ(r)is integrated from the centre to the suspension–air interface such that 2 R
δ¯
φrdr=
φ0(R2−δ2). Note that the lower bound of the integral, δ, is chosen to be approximately
6din order to account for the injection hole.
Based on the extracted values of ¯
φ(r,t), we calculate the radial particle flux, f(r,t),
across a given cylindrical surface inside the suspension. As illustrated in the schematic in
figure 1(a), we consider a cylindrical control volume whose radius is given by the arbitrary
radial position, r, between the inlet and the interface, R(t). Conservation of particles inside
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Self-similarity in particle accumulation
(a)(b)
(c)
InterfaceSuspension
R(t)
d
d
2s 3s 4s 5s 6s I II III
III
0.40
0.35
0.30
0.25
0.20
t = 1 s
0.35
0.30
0.25
0.20
0.44 0.35
0.30
0.25
0.7 0.8 0.9 1.0
0.20
z
oh
r
u¯p > u¯
0.39
0.34
0.29
0.24
0.19
0.14
2 cm
246 0.2 0.4 0.6
r/R
r/R
0.8 1.0
r (cm)
φ
¯
φ
¯
φ
¯
Figure 2. (a) The depth-averaged concentration profile ¯
φof the suspension with the initial concentration φ0=
0.25 plotted as a function of the radial distance, r, at different times, t.(b) The plots of ¯
φat different times
collapse into a single curve, when ris normalised by the instantaneous interfacial radius, R.Below(b)isa
close-up plot of ¯
φversus r/Rin Region III. (c) The colour map of the measured particle concentration, ¯
φ,fora
suspension at φ0=0.25 at t=3 s. The schematic illustrates three regions of the suspension during injection.
Region I is the transient region near the injection centre where the particles are undergoing shear-induced
migration in the z-direction. Region II corresponds to the region where the suspension has reached a quasi-fully
developed flow with constant ¯
φ, while ¯
φincreases near the interface in Region III.
the control volume yields f(r,t)=φ0Q−dVp/dt, which balances the rate of particle
injection at the inlet with the time rate of change of the total volume of particles inside the
control volume, or Vp=r
δ2π˜r¯
φ(˜r)hd˜r. Hence, to obtain f(r,t), we compute the change
in Vpfor all r(at an increment of δr) between two adjacent times, t1/2=t∓δt/2. Here,
δris a small radial distance, while δtcorresponds to a small temporal deviation from the
reference time t. For the given set of experimental data, we set δr∼3dand δt=1/30 s,
which are limited by the spatial resolution of the image and the video frame rate.
2.2. Experimental observations
Figure 1(b) shows the time-elapsed images from two experimental runs at h=1.4 mm:
φ0=0.2(a)andφ0=0.35 (b). Interfacial deformations associated with viscous fingering
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Y. Chen, R. Luo, L. Wang and S. Lee
are clearly observed at φ0=0.35, while the suspension–air interface remains stable and
circular at φ0=0.2. We further quantify this particle accumulation on the interface by
experimentally measuring ¯
φ,see figure 2(a), which shows an increase with rat φ0=0.25,
where ris the radial coordinate defined from the injection centre. Furthermore, a colour
map of the experimental image at t=3s in figure 2(c) shows a higher colour intensity
close to the interface, corresponding to a local increase in ¯
φat φ0=0.25. Both the plot
of ¯
φand the colour map verify the accumulation of particles on the interface even in the
stable regime (i.e. φ0<0.3), which is the focus of our current study.
Notably, as shown in figure 2(b), all ¯
φ(r,t)profiles approximately collapse on to a single
curve, when ris normalised by the instantaneous position of the suspension interface R(t).
This plot of ¯
φwith r/Rprovides direct evidence that particles accumulate on the advancing
interface in a self-similar manner, despite some deviations in the normalised curve near the
interface. To further analyse this, we presently divide ¯
φ(r/R(t)) into three distinct regions
(I, II and III), as labelled in figure 2(b).
In Region I near the injection point, ¯
φ(r/R(t)) is shown to decrease slightly with
increasing r/R. We conjecture that the suspension is initially uniform upon exiting
the injection hole (near r=0), with the volume fraction of φ0=0.25. Then, as the
suspension flows radially out inside the channel, suspended particles gradually migrate
across streamlines and focus near the centreline (i.e. shear-induced migration). This
gradual transition from uniform to non-uniform particle concentration in the z-direction
causes the average particle velocity to increase, which leads to a drop in ¯
φfrom φ0. Hence,
Region I is characterised by the transient migration of particles across the streamlines
owing to the gradient in shear rate.
In Region II, the suspension flow has reached a quasi-fully developed limit, so that ¯
φis
relatively constant, independent of r/R. We presently refer to this value of ¯
φin Region
II as ¯
φup, where ¯
φup(φ0)<φ
0. Then, in Region III, ¯
φstarts to increase with r/R,as
particles start accumulating near the fluid–fluid interface. As evident in figure 2(a), the
maximum value reached in ¯
φconsistently increases with time. Hence, the close-up plot of
Region III below figure 2(b) reveals that ¯
φdoes not collapse into a single curve, when r
is normalised by R. However, while ¯
φmay not be self-similar over the entire domain of
r/R, the relative sizes of Regions II and III and the general trend in ¯
φstrongly suggest
that particles accumulate in a self-similar way. In addition, following this peak, ¯
φsteeply
decreases towards the outer edge of the interface. While this could partly be attributed
to the curved shape of the meniscus which affects the measurements, the decrease in ¯
φ
covers a distance that is consistently larger than the size of the meniscus, h, and again
follows a self-similar trend. Hence, the gradual increase and a subsequent decrease in ¯
φ
are important features of our experimental results, which will be explored in our model.
Despite our characterisation of each region, the boundary between Regions II and III
is difficult to determine based on the experimental results in figure 2(b), as ¯
φincreases
gradually when approaching the interface. Hence, instead of ¯
φ, we consider the extracted
values of the radial particle flux, f, as a function of r.Figure 3(a) shows the flux profile for
φ0=0.25 over rat t=6 s. Consistent with the plot of ¯
φin figure 2(b), falso maintains a
constant value away from both the inlet and the interface. However, fincreases sharply at
a certain radial position, which we currently define as Rin(t). Then, Rin(t)must correspond
to the boundary between Regions II and III, as it clearly separates the quasi-steady region
and the particle accumulation region downstream of the interface.
Finally, we characterise the temporal evolution of Rin(t)for varying φ0at
h=1.4 mm. As plotted in figure 3(b), Rin(t)increases linearly with t1/2for all values of
φ0considered. From the volume conservation of the mixture (i.e. πR(t)2h=Qt), we have
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Self-similarity in particle accumulation
(a)(b)
1.0 9
8
7
6
5
4
3
0.9
0.8
Rin
φ0 = 0.25
φ0 = 0.2
φ0 = 0.24
φ0 = 0.25
φ0 = 0.27
φ0 = 0.28
t = 6 s
Rin (cm)
0.7
f(cm3 s–1)
0.6
24
r (cm) t1/2 (s1/2)
6 2.0 2.5 3.0 3.5 4.0 4.5
0.5
Figure 3. (a) Particle flux, f, plotted as a function of rwhen φ0=0.25. Here, Rin(t)is defined at the
location that fstarts increasing from a constant value upstream. In (b), Rin(t)increases linearly with t1/2for
varying φ0.
R(t)∝(Q/h)1/2t1/2, where Qand hremain constant in the data plotted. Hence, the linear
scaling of Rin(t)and R(t)with t1/2confirm that the size of the particle accumulation zone
must also grow proportionately with R(t), which is consistent with self-similarity in ¯
φ.
3. Theory
3.1. Theoretical formulation: suspension balance model
In order to rationalise the self-similar behaviour of the ¯
φprofile, we hereby treat the
particle–oil mixture as a continuum and develop a thin-film suspension model of our
system. We consider the injection of a suspension inside a Hele-Shaw cell from the
centre Oand define a z–rcylindrical coordinate system as illustrated in the schematics
of figure 2(c). The present model is based on the suspension balance model by Nott &
Brady (1994), which includes a two-phase formulation expressed for the bulk suspension
and particulate phase.
The mass and momentum conservation equations for the mixture are given as
∇·u=0,(3.1a)
∇·T=0,(3.1b)
where uis the velocity of the suspension and Tis the total stress tensor. Similarly, the
governing equations for the particle phase correspond to
∂φ
∂t+∇·(φup)=0,(3.2a)
∇·Tp+F=0,(3.2b)
where the superscript ‘p’ refers to the particle phase; hence, upand Tprepresent the
particle velocity and stress tensor, respectively.
The inter-phase drag force depends on the relative velocity between the particle and the
bulk suspension:
F=−18 μl
d2
φ
H(φ) (up−u), (3.3)
where the hindrance function corresponds to H(φ) =(1−φ)5(Richardson & Zaki 1954);
μlis the viscosity of the suspending liquid. By combining with (3.1a), we can rewrite
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Y. Chen, R. Luo, L. Wang and S. Lee
(3.2a)as
∂φ
∂t+u·∇φ=−∇·[φ(up−u)].(3.4)
The bulk suspension and particle phase are coupled through the constitutive equations
for stress tensors
T=−pI+Tp+2μlE,(3.5a)
Tp=Tp
n+2(μs(φ) −μl)E,(3.5b)
where Iis the identity matrix, and Eis the bulk suspension rate of strain tensor. We take
the following form of the normal stress of the particulate phase in each direction as
Tp
n=−μn(φ) ˙γ⎛
⎝
10 0
0λ20
00λ3⎞
⎠
(flow)
(velocity gradient)
(vorticity)
,(3.6)
where λ2and λ3are scalar parameters first defined by Bird, Armstrong & Hassager
(1987) independent of φ, which can be determined from rheological measurements of
the suspension (Morris & Brady 1998; Zarraga, Hill & Leighton 2000; Boyer, Guazzelli
& Pouliquen 2011). Note that ˙γ=(2E:E)1/2is the strain rate; μs(φ) and μn(φ) are the
effective shear and normal viscosities of the bulk suspension,respectively.
Amongst different empirical models for the effective viscosities (Krieger 1972;Morris
& Boulay 1999;Zarragaet al. 2000), we presently employ those developed by Morris &
Boulay (1999),written as
μs(φ)
μl=1+2.5φ1−φ
φm−1
+0.1φ
φm21−φ
φm−2
,(3.7a)
μn(φ)
μl=0.75 φ
φm21−φ
φm−2
,(3.7b)
where φm=0.6 is the maximum packing fraction of particles.
Our experiments are performed by injecting the mixture at a constant volumetric flow
rate Q. Hence, the incompressibility of the mixture requires the following constraint for
local volume conservation:
Q=2πrh/2
−h/2
urdz,(3.8)
where the subscript ‘r’ corresponds to the radial component of the velocity field and h
is the channel gap thickness. In addition, the condition of no particle flux at the walls
corresponds to
φup
z|z=±h/2=d2H(φ)
18μl
[∇·Tp]z+φuzz=±h/2=0,(3.9)
where the subscript ‘z’ is the vertical component.
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Self-similarity in particle accumulation
We hereby introduce dimensionless variables (denoted with an asterisk) to simplify the
governing equations:
r∗=r
R0,z∗=z
R0,μ
∗
s/n=μs/n
μl
,p∗=p
μlU0R0/h2,
u∗
r=ur
U0,u∗
z=uz
U0,Tp∗=
Tp
μlU0/h,t∗=t
R0/U0,
⎫
⎪
⎪
⎬
⎪
⎪
⎭
(3.10)
where R0is the characteristic radial distance and ≡h/R0, while the characteristic radial
velocity is defined as U0≡Q/(πR0h). In particular, since R(t)=(Q/(πh))1/2t1/2,the
time-dependent location of the interface can be expressed as R∗=t∗1/2based on the
dimensionless variables defined.
3.2. Model assumptions and multiple-time-scale expansion
Our model rests on a few assumptions based on the experimental set-up and our
observations. First, we reasonably assume =h/R01, as R0∼O(10−1)mand h∼
O(10−3)m based on our experiments. Next, we consider two important time scales that
characterise the current system: τz,the time scale of particle migration across the thin gap
and τr,the time scale of particle advection in the r-direction. While τr∼R0/U0,τzcan be
derived as τz∼(h2/d2)˙γ−1, where ˙γ∼U0/h, by examining (3.4)and(3.3). Alternatively,
τz=h2/D, where Dis the shear-induced diffusivity that scales as d2˙γ(Leshansky, Morris
&Brady2008; Griffiths & Stone 2012). Then, the ratio of the time scales corresponds to
τz/τr=χ, where χ=(h/d)21.
Based on the characteristic experimental parameters, the typical value of χ ranges
from 0.6 to 1. Nonetheless, we consider the limit where χ is a small parameter, such that
χ 1, for the following reasons. First, assuming χ 1 allows us to simplify our
present analysis and to systematically add the effects of Taylor dispersion into our transport
equation, in the manner of Ramachandran (2013). Second, the disparity of the time scales
is evident in the experimental plot of ¯
φ(see figure 2b), in which the size of Region I – the
transient region near the injection centre – is consistently small compared to the overall
radial distance. This implies that the time it takes for particles to migrate across the thin
gap must be small compared to that of particle transport in the radial direction. Hence, it is
reasonable to assume τzτr,orχ 1, as a starting point in the present analysis.
Finally, we neglect the transient region near the injection centre (Region I) and only focus
on Regions II and III in our model.
Given the assumptions above, we first employ the lubrication approximations and reduce
the governing equations based on 1. Second, we follow the multiple-time-scale
expansion by Ramachandran (2013) and break up the time variable into the short (i.e.
t∗
1=t∗) and long (i.e. t∗
2=χt∗) time scales, such that
∂
∂t∗=∂
∂t∗
1+(χ ) ∂
∂t∗
2+O((χ )2). (3.11)
Then, the dependent variables can be expanded in the order of χ.Wedenotethe
leading order term with the superscript ‘(0)’ and the term linear in χ with ‘(1)’ (e.g.
φ=φ(0)+(χ )φ(1)+O((χ)2)). Note that all dependent and independent variables
henceforth are presented in their dimensionless form unless otherwise noted; the asterisk
(∗)is subsequently dropped for brevity.
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Y. Chen, R. Luo, L. Wang and S. Lee
Specifically, we model our current system in two steps. First, we compute the particle
concentration profiles (e.g. φ(0),φ(1)) and radial velocity profiles (e.g. u(0)
r,u(1)
r)inthe
order by order fashion. This yields the expression for the particle flux with respect to
the local depth-averaged particle concentration, up to O(χ ). Second, we utilise the
expression for the particle flux and derive an evolution equation for ¯
φbased on the particle
volume conservation, which is solved numerically with appropriate initial and boundary
conditions. Overall, our model procedure closely follows the derivation of Ramachandran
(2013), which we frequently refer to for more details.
3.3. Leading order solutions
Combining (3.1b)and(3.5a), the momentum conservation equations of the bulk
suspension can be obtained to the leading order in and in χ as
∂rp(0)=∂
∂z[μs(φ(0))∂zu(0)
r],(3.12a)
∂zp(0)=0.(3.12b)
Note that we have assumed the flow to be axisymmetric; hence, the equation in the
θ-direction is neglected. Also, at the leading order, the particle transport equation (3.4),
combined with (3.2b), (3.3), (3.5b)and(3.6), yields
∂
∂zH(φ(0))∂
∂zλ2μn(φ(0))∂u(0)
r
∂z=0,(3.13)
while the no-flux boundary condition (3.9) reduces to
∂
∂zλ2μn(φ(0))∂u(0)
r
∂zz=±h/2=0.(3.14)
By integrating (3.13) subject to (3.14), we obtain
μn(φ(0))∂zu(0)
r=const. (3.15)
Similarly, based on (3.12b) and the condition ∂zur(z=0)=0, we can integrate (3.12a)
with respect to zto obtain
μs(φ(0))∂zu(0)
r=∂rp(0)z.(3.16)
Next, by combining (3.16)and(3.15), we can obtain an implicit expression for φ(0)(z)
as
1
g(φ(0)(z)) =∂rp(0)z
const.=C0z.(3.17)
where g(φ(0))≡μn(φ(0))/μs(φ(0))and C0is a constant independent of zbut may have a
dependence on r. Then, we integrate (3.16) again and apply no-slip boundary conditions
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Self-similarity in particle accumulation
φ
¯
= 0.1
φ
¯
= 0.2
φ
¯
= 0.3
φ
¯
= 0.4 φ
¯
= 0.1
φ
¯
= 0.2
φ
¯
= 0.3
φ
¯
= 0.4
0.4
(a)(b)
0.3
z
0.2
0.1
0.2 0.5 1.00.4
φ(0) ur
(0)/u¯
0.6 1.5
0.8
0
0.50.5
0.4
0.3
0.2
0.1
0
Figure 4. (a) Local particle concentration profile φ(0)and (b) the normalised velocity profile, u(0)
r/¯u,asa
function of zfor varying depth-averaged concentrations ¯
φ.
(i.e. u(0)
r(z=±1/2)=0) to obtain an expression for the mixture velocity,
u(0)
r(z)=z
−1/2
∂rp(0)˜z
μs(φ(0)(˜z)) d˜z,(3.18)
where
∂rp(0)=2r1/2
−1/2z
−1/2˜z
μs(φ(0)(˜z)) d˜zdz−1
(3.19)
can be determined from the volume conservation of the suspension as
1=2r1/2
−1/2
u(0)
rdz.(3.20)
Finally, we compute the resultant local particle concentration profile φ(0)(z)from
(3.17), by treating the depth-averaged particle concentration, ¯
φ=1/2
−1/2φ(0)dz, as a known
parameter. By systematically varying the value of ¯
φ, we obtain a complete set of solutions
for u(0)
r(z)and φ(0)(z), examples of which are plotted in figure 4. The plot of φ(0)(z)in
figure 4(a) shows that there is a higher concentration at the centreline, z=0, owing to
the shear-induced migration of particles. Then, the corresponding velocity profiles, u(0)
r,
are normalised by the local mean velocity, ¯u=1/2
−1/2u(0)
rdz, and plotted for varying ¯
φ.
As shown in figure 4(b), the velocity profile becomes more blunt at z=0 for increasing
¯
φ. This is consistent with the physical picture of particle aggregation near the centreline.
With u(0)
r(z)and φ(0)(z)thus known, we subsequently compute the dimensionless particle
flux, f(0), as a function of ¯
φ:
f(0)=2r1/2
−1/2
u(0)
r(z)φ(0)(z)dz.(3.21)
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Y. Chen, R. Luo, L. Wang and S. Lee
3.4. O(χ ) solutions
Within the lubrication framework, we consider the governing equations at O(χ ) to derive
u(1)
rand φ(1).AtO( χ ), the momentum conservation of the mixture leads to
∂rp(1)=∂
∂z[μs∂zur](1),(3.22a)
∂zp(1)=0,(3.22b)
while (3.4) reduces to
∂φ(0)
∂t1+u(0)
r
∂φ(0)
∂r+u(0)
z
∂φ(0)
∂z=1
18
∂
∂zH(φ(0))∂
∂zλ2μn
∂ur
∂z(1).(3.23)
We then integrate (3.23)fromz=−1/2toz=1/2. With no particle flux condition at
the walls (i.e. (3.9)atO(χ)), the right-hand side of (3.23) vanishes upon integration, such
that
1/2
−1/2
∂φ(0)
∂t1dz+1/2
−1/2u(0)
r
∂φ(0)
∂r+u(0)
z
∂φ(0)
∂zdz=0.(3.24)
On the left-hand side, we apply the chain rule and integration by parts to the second and
third terms, respectively, so that
1/2
−1/2
u(0)
r∂rφ(0)dz=1
r∂rr1/2
−1/2
u(0)
rφ(0)dz−1/2
−1/2
1
r∂r(ru(0)
r)φ(0)dz,(3.25a)
1/2
−1/2
u(0)
z∂zφ(0)dz=(φ(0)u(0)
z)|z=1/2−(φ(0)u(0)
z)|z=−1/2−1/2
−1/2
φ(0)∂zu(0)
zdz.
(3.25b)
Based on the impermeability boundary condition and continuity, we derive the following
particle transport equation at O(χ ):
∂¯
φ
∂t1+1
2r
∂f(0)
∂r=0.(3.26)
By combining (3.23)with(3.26) and employing chain rules (e.g. ∂r=∂r¯
φ∂¯
φ), we
re-write the left-hand side of (3.23)as
−∂φ(0)
∂¯
φ
df(0)
d¯
φ
1
2r
∂¯
φ
∂r+u(0)
r
∂φ(0)
∂¯
φ
∂¯
φ
∂r+u(0)
z
∂φ(0)
∂z,(3.27)
where u(0)
z=r−1∂r1/2
zru(0)
rd˜zbased on continuity. After some algebraic manipulation,
(3.23) becomes
S2(¯
φ,z)1
r
∂¯
φ
∂r=1
18
∂
∂zH(φ(0))∂
∂zλ2μn(φ) 
∂ur
∂z(1),(3.28)
where
S2(¯
φ,z)=∂φ(0)
∂¯
φru(0)
r−1
2
∂f(0)
∂r+∂φ(0)
∂z
∂
∂¯
φ1/2
z
ru(0)
r(¯
φ, ˜z)d˜z.(3.29)
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Self-similarity in particle accumulation
0.4
(a)
0.3
z
0.2
0.1
–1.0 0 0.05–0.05 0.10–0.8 –0.6 –0.4
S7S11
–0.2 0 0.2 0.4
0
0.5
0.4
(b)
0.3
0.2
0.1
0
0.5
φ
¯
= 0.1
φ
¯
= 0.2
φ
¯
= 0.3
φ
¯
= 0.4
φ
¯
= 0.1
φ
¯
= 0.2
φ
¯
= 0.3
φ
¯
= 0.4
Figure 5. Here, (a)S7and (b)S11 plotted as a function of zfor varying depth-averaged concentrations ¯
φ.
Finally, integrating (3.28) and applying appropriate boundary conditions leads to
φ(1)=S7(¯
φ,z)∂¯
φ
∂r,(3.30)
where S7is a function of zfor given ¯
φ. In addition, combining (3.30) with the momentum
equations (3.22a)and(3.22b) yields the expression for the perturbation velocity, u(1)
r:
u(1)
r=S11(¯
φ,z)1
r
∂¯
φ
∂r.(3.31)
The detailed derivations of φ(1)and u(1)
r, as well as the expressions of S7and S11,are
included in the appendix.
Notably, φ(1)and u(1)
rdepend on the gradient of ¯
φin the radial direction and on ¯
φ,
as distinct from the leading order terms. To probe the physical meaning of φ(1)and u(1)
r
more closely, we plot the functions, S7and S11,infigure 5. The plot of S7in figure 5(a)
shows that φ(1)is positive near the walls (z=1/2) and negative near the centreline before
sharply vanishing at z=0; these effects are increasingly less pronounced at higher ¯
φ. This
suggests that when ¯
φincreases with r(i.e. ∂r¯
φ>0), the effects of shear-induced migration
may become mitigated, as the particle concentration is reduced near the centreline and
increases near the walls. Similarly, as shown in the plot of S11 in figure 5(b), u(1)
ris positive
near z=0 and becomes negative for increasing zbefore reaching zero at the walls. While
this trend is consistent for all ¯
φ, it is reduced at larger ¯
φ. This implies that for ∂r¯
φ>0, the
velocity profile becomes less blunt near the centreline, where the corresponding particle
concentration decreases.
This dependence of φ(1)and u(1)
ron ∂r¯
φwill directly impact the dynamics of particle
accumulation through the corresponding perturbation in the particle flux, or f(1).With
φ(0),u(0)
r,φ(1)and u(1)
rknown, we compute f(1)as
f(1)=2r1/2
−1/2
[u(0)
r(z)φ(1)(z)+u(1)
r(z)φ(0)(z)]dz=2S12(¯
φ)∂¯
φ
∂r,(3.32)
where
S12(¯
φ) =1/2
−1/2
[S7(¯
φ,z)ru(0)
r(z)+S11(¯
φ,z)φ(0)(z)]dz.(3.33)
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Y. Chen, R. Luo, L. Wang and S. Lee
As shown in figure 6(b), S12(¯
φ) is negative and its magnitude generally decreases with
¯
φ, while f(0)monotonically increases with ¯
φ. Hence, the particle flux must increase more
slowly owing to this perturbation in the flux, as particles accumulate on the interface, or
as ∂r¯
φ>0. We conjecture that this mitigation in the particle flux may lead to the gradual
accumulation of particles, which will be fully examined in the next section.
4. Particle transport equation and self-similarity
In order to compute ¯
φ, we must consider the depth-averaged particle transport equation
that combines terms up to O((χ)2). Hence, we start by integrating (3.4)overthethin
gap at O((χ)2). By utilising 1/2
−1/2φ(1)dz=0 and taking the same steps as O(χ ) (e.g.
(3.25a)and(3.25b)), we obtain
∂¯
φ
∂t2+1
2r
∂f(1)
∂r=0.(4.1)
Finally, we combine (3.26)and(4.1) to derive the following depth-averaged particle
transport equation:
∂¯
φ
∂t+1
2r
∂f(0)
∂r+(χ ) 1
r
∂
∂rS12(¯
φ)∂¯
φ
∂r=0.(4.2)
To find the similarity solution to (4.2), we seek a solution with the following structure:
¯
φ=1
tαψr
√t:=1
tαψ(ξ), (4.3)
where αand ψcan be found. Substituting it into (4.2), we have
−αξψ (ξ) −1
2ψ(ξ)ξ 2+1
2f(0)(¯
φ)ψ(ξ) +1
tα+1/2S
12(¯
φ)ψ(ξ)2+1
√tS12(¯
φ)ψ(ξ ) =0.
(4.4)
Note from the plot of f(0)in figure 6(a), we may reasonably approximate f(0)as a linear
function in ¯
φ,sothatf(0)(¯
φ) =C1for a constant C1. If we assume S12 (¯
φ) =C2¯
φγfor
some constant C2and γ, and choose αsuch that αγ +1
2=0, then (4.4) reduces to
−αξψ (ξ) −1
2ψ(ξ)ξ 2+C1ψ(ξ ) +γ(ψγ−1ψ(ξ)2+ψγψ(ξ )) =0,(4.5)
which provides a rule to determine ψ. As a result, ¯
φ=t1/(2γ)ψ(r/√t)is a self-similar
solution to (4.2). In figure 6(b), we fit S12(¯
φ) with −0.0022 ׯ
φ−1.399 and show that the
latter approximates the former with a slight deviation. This deviation explains why our
solution is not exactly self-similar. However, as shown in figure 6(b), the deviation is so
small that particles are expected to accumulate in an approximately self-similar manner.
Hence, this approximate self-similar nature of (4.1) is consistent with the experimental
measurements of ¯
φin figure 2(b).
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Self-similarity in particle accumulation
(a)(b)
0.4
0
–0.01
–0.02
–0.03
S12
–0.04
–0.05
–0.06
Numerical solution
Fitted curve
Numerical solution
Fitted curve
0.4
0.3
0.3
f(0)
0.2
0.2
0.1
0.1
φ
¯
0.40.30.20.1
φ
¯
Figure 6. (a) The numerical solution of the leading order flux, f(0), is plotted as a function of depth-averaged
concentration ¯
φ. The solid line indicates the linear fit. (b) The solution of S12 is plotted for varying
depth-averaged concentration, ¯
φ, and is fitted with a function −0.0022 ×φ−1.399.
4.1. Simulation results: comparison with experiments
We solve (4.2) numerically using the upwind finite difference scheme, by applying the
following initial and boundary conditions:
¯
φ(r>0,t=0)=0,(4.6a)
¯
φ(r=0,t)=¯
φup(φ0), (4.6b)
¯
φ(r>R(t), t)=0.(4.6c)
The initial condition (4.6a) indicates that there are no particles inside the domain at
t=0, while (4.6c) imposes zero particle concentration downstream of the immiscible
interface, R(t). The latter simultaneously satisfies the condition of zero particle flux for
r>R(t). In addition, we impose ¯
φat the inlet to be ¯
φup, the constant depth-averaged
particle concentration in Region II. In the experiments, the depth-averaged concentration
is φ0at the inlet and gradually transitions to ¯
φup downstream (Region I). As the current
model considers only Regions II and III, it is reasonable to use ¯
φ(r=0)=¯
φup as our
boundary condition, such that the flux at the inlet is equal to the upstream particle flux
f(¯
φup)=f(0)(¯
φup)+(χ )f(1)(¯
φup).
Hence, we first calculate ¯
φup by revisiting the solution of φ(0)(z)in § 3.3. Specifically,
we compute φ(0)(z)from (3.17), subject to the particle volume conservation in the fully
developed limit (Region II): φ0=2r1/2
−1/2u(0)
rφ(0)dz. Note that the solutions in this
regime reduce to the steady-state results of aone-dimensional channel flows, as ∂r¯
φ=0
ensures that φ(1)=0. Finally, we obtain ¯
φup by integrating the resultant φ(0)(z)across
the thin gap. Figure 7 shows the plot of φ0−¯
φup versus φ0from theory (solid line),
together with the experimental measurements (symbols). The results show a qualitative
match between theory and experiments, although the theoretical prediction of φ0−¯
φup is
persistently larger than the experimental measurements. We subsequently utilise ¯
φup(φ0)
from our model as the boundary condition in (4.6b).
We examine some characteristic solutions to (4.2) for varying values of χ.Notethat
the second term in (4.2) characterises the radial advection of particles, while the last
term corresponds to the radial diffusion, or Taylor dispersion, of particles that acts to
smooth out the changes in ¯
φin the r-direction. Hence, the value of χ determines the
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Y. Chen, R. Luo, L. Wang and S. Lee
0.025
0.020
0.015
Theory
h = 1.4 mm
0.010
0.005
0.1 0.2 0.3
φ0 – φ
¯
up
φ0
Figure 7. The difference between the initial concentration and the depth-averaged particle concentration,
φ0−¯
φup, in Region II is plotted as a function of φ0. The solid line represents the numerical solution from
the simplified suspension balance model with the constraint of the particle volume conservation. The square
markers indicate the experimental data for h=1.4 mm.
0.4
0.5 = 0.4
= 0.6
= 0.8
= 1.0
0.3
0.2
0.1
0.25 0.50 r0.75 1.000
0
φ
¯
Figure 8. The numerical solution of depth-averaged particle concentration, ¯
φup, as function of ris plotted for
various values of χ at t=14 s and φ0=0.2. As χ increases, the increase in ¯
φup becomes less steep.
relative importance of Taylor dispersion versus particle advection in the particle transport
equation. Figure 8 shows the plot of ¯
φ(r)at t=14 s for φ0=0.2withvaluesofχ ranging
from 0.4 to 1. As expected, the increase in ¯
φbecomes less steep, as the value of χ is
systematically increased. Following this increase, ¯
φis shown to rapidly decrease to zero at
the interface, largely independent of the value of χ.
Next, we compare the numerical solutions to (4.2) with our experiments for φ0=0.2
and 0.25, respectively, which fall in the regime of clear particle accumulation and no
viscous fingering. Here we empirically set the values of χ to best fit the experimental
results of ¯
φ, as shown in figure 9(a)and(b). The fitted value corresponds to χ =0.9at
both concentrations. While the actual values of χ used do not strictly meet the assumption
of χ 1, we note that the characteristic size of Taylor diffusion in (4.2) is still an order of
magnitude smaller than that of the advection term, as |S12|∼O(0.01)and f(0)∼O(0.1)
in figure 6. The numerical solutions of ¯
φare plotted as a function of dimensional r(in
solid lines) at six different times, t. The markers indicate the experimental measurements
of the depth-averaged particle concentrations.
The numerical solutions agree well with the experimental observation for both
concentrations except for the over-prediction of ¯
φnear the interface particularly for φ0=
0.25. This over-prediction is expected given that our current model sets ¯
φ(r=0)=¯
φup
and neglects Region I in which ¯
φgradually decreases from φ0at r=0to ¯
φup downstream.
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Self-similarity in particle accumulation
0.4
(a)(b)
(c)(d)
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.3
0.2
0.8 r/R1.0
0.3
0.2
0246 0246810
0246
r (cm)
0246
r (cm)
810
φ
¯
φ
¯5s 7s 9s11 s
14 s
t = 3 s
5s 7s 9s11 s
14 s
t = 3 s
2s 3s 4s 5s 6s
t = 1 s
2s 3s 4s 5s 6s
t = 1 s
1.0
0.8
0.6
f(cm3 s–1)
0.4
1.2
1.0
0.8
0.6
Figure 9. Comparison of the numerical solutions of ¯
φto the experiments for (a)φ0=0.2and(b)φ0=0.25,
both with χ =0.9. Solid lines in the plots indicate the theoretical results, while the solid symbols are
experimental measurements. The inset plots indicate the theoretical results of ¯
φversus r/Rin Region III. The
solutions of the dimensional flux, f, are also compared with the experimental measurements for (c)φ0=0.2
and (d)φ0=0.25, respectively. Note that ris a dimensional radial coordinate.
We have also plotted the numerical solutions of ¯
φas a function of r/Rin Region III in the
inset of figure 9(a). It clearly shows that our theoretical results are also self-similar, despite
small deviations near the interface, which is consistent with the experimental observations
in figure 2(b). In addition, the numerical solutions of the dimensional flux, f, are plotted
and compared with the experimental results in figure 9(c,d)for both concentrations at
varying t. The theoretical results are in qualitative agreement with the experiments at all
times.
Overall, our reduced model has demonstrated the shear-induced migration of the
particles far upstream and the presence of the fluid–fluid interface are barebones physical
ingredients that can lead to particle accretion on the interface in the current geometry.
Then, the gradual accumulation of particles is achieved through the inclusion of the
perturbation in the particle flux that depends on both local ¯
φand the gradient of ¯
φin
the radial direction. This perturbation appears in the particle transport equation as the
axial diffusion term,which mitigates how steeply ¯
φincreases towards the interface. The
resultant particle transport equation yields solutions that are approximately self-similar
and in remarkable agreement with the experimental data.
5. Summary and conclusions
In summary, our work has focused on understanding the self-similar behaviour of particle
accumulation observed in suspension flow, through experiments and theoretical modelling.
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Y. Chen, R. Luo, L. Wang and S. Lee
This particle accumulation at the advancing interface is attributed to the shear-induced
migration of particles far up stream, which causes the particles to assume a faster average
velocity than the mixture. We first carry out the experiments by injecting the suspension
into the Hele-Shaw cell with varying initial particle concentrations, φ0, at a constant
volume flow rate Q. We extract the depth-averaged particle volume fraction ¯
φand the
particle flux f(r)via image processing. The evolution of ¯
φas a function of rcollapses on
to a single curve when ris normalised by the instantaneous position of the interface, R(t).
We divide the collapsed curves of ¯
φ(r/R(t)) into three distinct regions: Regions I, II and
III. In particular, Region II refers to the region where the suspension reaches a quasi-fully
developed flow with constant ¯
φ=¯
φup, while ¯
φincreases towards the interface in Region
III. We experimentally quantify the boundary between Region II and Region III as Rin(t).
The linear scaling of both Rin(t)and R(t)with respect to t1/2also confirms the self-similar
growth of particle accumulation.
We develop a thin-film model to rationalise the self-similarity of ¯
φbased on
the suspension balance model. By combining the lubrication approximations and
multiple-time-scale expansion, we obtain the depth-averaged upstream concentration, ¯
φup,
as well as the particle flux (i.e. f(0)and f(1)) for given local ¯
φ(r)and the gradient of
¯
φ(r). Then, we derive a depth-averaged particle transport equation by considering the
conservation of the particle volume. In particular, the perturbation in the flux appears in
the particle transport equation as the axial diffusion term that mitigates how steeply ¯
φ
increases towards the interface. The resultant particle transport equation yields solutions
that are approximately self-similar, and in remarkable agreement with the experimental
data.
The good match between theory and experiments demonstrates that our reduced model
successfully captures the self-similar behaviour of particle accumulation on the advancing
fluid–fluid interface. However, the model has a number of limitations that need to be
addressed. For instance, the validity of the model for χ ∼O(1)is questionable, despite
the good agreement with the experiments. In addition, the present model results suggest
thatthepresenceofthefountain flow and the shape of the meniscus play a minimal role
in the dynamics of particle accretion, which is not well understood. Finally, resolving
the particle dynamics near the fluid–fluid interface is only the first step in predicting the
threshold particle concentration necessary to trigger particle-induced viscous fingering. In
particular, previous experimental measurements have shown that ¯
φneeds to attain a critical
slope with respect to rbefore miscible fingering is observed (Luo et al. 2018). However,
the physical and mathematical basis for this observation is currently not well understood.
Hence, predicting the onset of fingering requires building on the current model for ¯
φ(r,t)
to derive a fully nonlinear two-dimensional model of the suspension flow inside a thin gap.
Acknowledgements. We acknowledge Dr F. Xu for providing the experimental data featured in this
manuscript.
Funding. This work is partially supported by the National Science Foundation: DMR 2003706 for R.L. and
S.L.; DMS 1846854 for L.W.
Declaration of interests. The authors report no conflict of interest.
Author ORCIDs.
Yun C hen https://orcid.org/0000-0002-5341-5125;
Rui Luo https://orcid.org/0000-0002-5673-4850;
Li Wang https://orcid.org/0000-0002-0593-8175;
Sungyon Lee https://orcid.org/0000-0002-4118-1712.
925 A10-18
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Self-similarity in particle accumulation
Appendix A. Derivations of φ(1)and u(1)
r
Let us first define the following terms:
S1(¯
φ,z)=1/2
z−˜z
μs(φ) d˜z,M(¯
φ) =1/2
−1/2
S1(¯
φ,z)dz(A1a,b)
so that we can rewrite ∂rp(0)and u(0)
ras
∂rp(0)=1
2rM(¯
φ),(A2a)
u(0)
r(¯
φ,r,z)=S1(¯
φ,z)
2rM(¯
φ).(A2b)
We integrate equation (3.28)fromzto 1/2 using the no-particle flux condition (3.9)to
obtain
18
λ2H(φ(0))1/2
z−S2(˜z,¯
φ) d˜z1
r
∂¯
φ
∂r=∂
∂zg(φ)μs(φ ) 
∂ur
∂z(1)
.(A3)
After another integration from 0 to z,(A3) becomes
18
λ2
⎡
⎢
⎢
⎢
⎣−z
0
1/2
˜z−S2(˜z,¯
φ) d˜z
H(φ(0))d˜z1
r
∂¯
φ
∂r+B0(r,t)⎤
⎥
⎥
⎥
⎦=g(φ)μs(φ ) ∂ur
∂z(1)
,(A4)
where B0(r,t)is a constant to be determined later.
We now expand the right-hand side of (A4), so that
g(φ)μs(φ ) ∂ur
∂z(1)
=g(φ0)μs(φ ) ∂ur
∂z(1)
+dg(φ(0))
dφ(0)φ(1)μs(φ) ∂ur
∂z(0)
.(A5)
Combining (A5)with(A4), we obtain an expression for φ(1)as
φ(1)=1
dg(φ(0))
dφ(0)μs(φ) ∂ur
∂z(0)18
λ2S3(z,¯
φ)1
r
∂¯
φ
∂r+B0(r,t)
−g(φ(0))μs(φ ) ∂ur
∂z(1),(A6)
where
S3(z,¯
φ) =z
0
1/2
˜z
S2(˜z,¯
φ) d˜z
H(φ(0))d˜z.(A7)
From (3.12a) and the definition of (A2a), we have
μs(φ) ∂ur
∂z(0)
=z
2rM ,(A8)
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Y. Chen, R. Luo, L. Wang and S. Lee
while the momentum equations (3.22a)and(3.22b) yield
μs(φ) ∂ur
∂z(1)
=∂p(1)
∂rz,(A9)
where ∂rp(1)is independent of z. Therefore, the second term on the right-hand side of (A6)
can be written as
g(φ(0))μs(φ ) ∂ur
∂z(1)
=g(φ(0))∂p(1)
∂rz.(A10)
Since g(φ(0))zis also independent of zaccording to (3.17), the second term on the
right-hand side of (A6), [−g(φ(0))(μs(φ)∂zur)(1)], must be a function of rand tonly,
which will be combined with B0(r,t)as a single constant B(r,t). Then, (A6) can be finally
simplified as
φ(1)=1
dg(φ(0))
dφ(0)
z
2rM(¯
φ)
18
λ2S3(z,¯
φ)1
r
∂¯
φ
∂r+B(r,t).(A11)
The constant B(r,t)can be determined using the constraint 1/2
−1/2φ(1)dz=0, such that
B(r,t)=−S5(¯
φ)
S6(¯
φ)
1
r
∂¯
φ
∂r.(A12)
Note that
S4(¯
φ,z)=dg(φ(0))
dφ(0)
z
2M(¯
φ),(A13a)
S5(¯
φ) =1/2
−1/2
S3(¯
φ,z)
S4(¯
φ,z)dz,(A13b)
S6(¯
φ) =1/2
−1/2
1
S4(¯
φ,z)dz.(A13c)
Therefore, (A11) can be further expressed as
φ(1)=1
S4(¯
φ,z)18
λ2S3(¯
φ,z)−S5(¯
φ)
S6(¯
φ)∂¯
φ
∂r.(A14)
Finally, we simplify (A14)as
φ(1)=S7(¯
φ,z)∂¯
φ
∂r,(A15)
by defining
S7(¯
φ,z)=1
S4(¯
φ,z)18
λ2S3(¯
φ,z)−S5(¯
φ)
S6(¯
φ).(A16)
To compute the velocity profile at O(χ ), we expand the left-hand side of (A9)as
μs
∂ur
∂z(1)
=μs(φ(0))∂u(1)
r
∂z+dμs(φ(0))
dφ(0)φ(1)∂u(0)
r
∂z.(A17)
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Self-similarity in particle accumulation
Substituting equation (A17)into(A9) and rearranging, we obtain
∂u(1)
r
∂z=z
μs(φ(0))
∂p(1)
∂r−1
μs(φ(0))
dμs(φ(0))
dφ(0)φ(1)∂u(0)
r
∂z.(A18)
Now, we integrate (A18)fromzto 1/2 and apply the no-slip boundary condition, which
yields
u(1)
r=S1(¯
φ,z)∂p(1)
∂r−1/2
z−1
μs(φ(0))
dμs(φ(0))
dφ(0)φ(1)∂u(0)
r
∂zd˜z,(A19)
subject to
r1/2
−1/2
u(1)
rdz=0.(A20)
Using the previous constraint and the expression of φ(1), we can write the expression of
∂rp(1)as
∂p(1)
∂r=1/2
−1/21/2
z−1
μs(φ(0))
dμs(φ(0))
dφ(0)
∂u(0)
r
∂zS7(¯
φ, ˜z)d˜zdz
M(¯
φ)
∂¯
φ
∂r.(A21)
Again, from (3.12a) and the definition of (A2a), we have
∂u(0)
r
∂z=1
μs(φ(0))
z
2rM .(A22)
Then, u(1)
rcan be expressed as
u(1)
r=[S10(¯
φ,z)−S8(¯
φ,z)]1
r
∂¯
φ
∂r(A23)
from (A21), by defining
S8(¯
φ,z)=1/2
z−1
μ2
s(φ(0))
dμs(φ(0))
dφ(0)˜z
2M(¯
φ)S7(¯
φ, ˜z)d˜z,(A24a)
S9(¯
φ) =1/2
−1/2
S8(¯
φ,z)
M(¯
φ) dz,(A24b)
S10(¯
φ,z)=S1(¯
φ,z)S9(¯
φ). (A24c)
We finally simplify (A23)as
u(1)
r=S11(¯
φ,z)1
r
∂¯
φ
∂r,(A25)
where
S11(¯
φ,z)=S10(¯
φ,z)−S8(¯
φ,z). (A26)
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Y. Chen, R. Luo, L. Wang and S. Lee
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