# Squeezing multiple soft particles into a constriction: transition to
clogging

Clément Bielinski  Biomechanics and Bioengineering Laboratory (UMR CNRS 7338),
CNRS, Université de Technologie de Compiègne,
60200 Compiègne, France  Othmane Aouane  Helmholtz Institute Erlangen-Nürnberg
for Renewable Energy,
Forschungszentrum Jülich,
Cauerstr. 1, 91058 Erlangen, Germany  Jens Harting  Helmholtz Institute
Erlangen-Nürnberg for Renewable Energy,
Forschungszentrum Jülich,
Cauerstr. 1, 91058 Erlangen, Germany  Department of Chemical and Biological
Engineering and Department of Physics, Friedrich-Alexander-Universität
Erlangen-Nürnberg, Cauerstr. 1, 91058 Erlangen, Germany  Badr Kaoui
<EMAIL_ADDRESS>Biomechanics and Bioengineering Laboratory (UMR CNRS 7338),
CNRS, Université de Technologie de Compiègne,
60200 Compiègne, France

###### Abstract

We study numerically how multiple deformable capsules squeeze into a
constriction. This situation is largely encountered in microfluidic chips
designed to manipulate living cells, which are soft entities. We use fully
three-dimensional simulations based on the lattice Boltzmann method to compute
the flow of the suspending fluid, and on the immersed boundary method to
achieve the two-way fluid-structure interaction. The mechanics of the capsule
membrane elasticity is computed with the finite element method. We obtain two
main states: continuous passage of the particles, and their blockage that
leads to clogging the constriction. The transition from one state to another
is dictated by the ratio between the size of the capsules and the constriction
width, and by the capsule membrane deformability. This latter is found to
enhance particle passage through narrower constrictions, where rigid particles
with similar diameter are blocked and lead to clogging.

##  I Introduction

The flow of particles is largely encountered in microfluidic devices designed
to manipulate, sort, or characterize micro-sized artificial particles or
living cells. The performance of such devices can dramatically be hindered by
the clogging events that take place at the entrance of channels, especially
when the particle size is of the same order as the microchannels width or when
multiple particles arrive suddenly at narrow passages Dressaire2017 ; Hong2017
; vanZwieten2018 ; Zhang2018 . While there is an increasing need to improve
the capability of microfluidic devices to handle high particle throughput,
this study is carried out to address the scenarios that emerge when multiple
soft particles are pushed to flow through an abrupt 90 degrees microfluidic
constriction, see Fig. 1. The constriction has a basic geometrical shape, but
it is one of the most commonly encountered microfluidic devices engineered
with the soft lithography technique. The flow of soft particles into
constrictions is also encountered in vivo. For example, in blood vessels where
the accumulation of fat on their walls may obstruct the flow of red blood
cells and other particles, such as drug-carrier particles.

Figure 1: The numerical setup used to study the flow of capsules (orange-
colored spheres). The microfluidic constriction forms an angle of $90$ degrees
with the channel walls. The flow direction is from the left-lower pre-
constriction chamber to the right-upper post-constriction chamber. The main
geometrical parameter is the ratio of the channel width $D$ to the capsule
diameter $d$: $D/d$. $D$ is varied while holding $d$ constant.

Recent experimental studies have investigated the clogging phenomenon by rigid
particles Marin2018 ; Souzy2020 , which are found to form a stable arch at the
entrance of the microfluidic constriction, and thus, block the channel. The
neck-to-particle size ratio is found to be the leading parameter that
determines the transition threshold to clogging, with permanent clogs
systematically formed for a neck-to-particle size ratio below 3. For
sufficiently large neck-to-particle size ratio, the particles flow either
continuously or intermittently depending on the solid volume fraction. The
corresponding dynamics has been characterized by stochastic mathematical
models. In the case of soft particles, and to the best of our knowledge, the
existing literature focuses solely on the flow and deformation of single
isolated particles passing through constrictions Rorai2015 ; Kusters2014 ;
LeGoff2017 ; Luo2017 ; Fai2017 ; Lei2019 . In this study, we extend these
previous works and get one step toward mimicking living cells in microfluidic
chips by considering a suspension of soft particles.

Figure 2: Motion of multiple capsules ($N=38$) at a microfluidic constriction
for a given flow strength. The left panel shows the effect of the constriction
aperture $D/d$, and the right panel the effect of particle deformability ${\rm
Ca}$. Easy passage of the particles is achieved at large $D/d$ and ${\rm Ca}$.
Arch structures form at the entrance of the constriction for narrower
constrictions and rigid particles. At large ${\rm Ca}$ and narrow
constrictions, the particles cross the constriction one by one, while forming
a regular train and exiting the constriction as a jet.

We perform fully three-dimensional simulations as used in Ref. Kusters2014  to
study the collective motion of multiple particles through a constriction,
while varying two control parameters: i - the particle deformability to cover
the range of rigid (non-deformable) and soft (deformable) particles, and ii -
the width of the constriction. The details of the numerical setup are shown in
Fig. 1. The deformability is found to affect the transition to clogging, and
serves as an additional key parameter in predicting the mechanism and the
behavior of particles squeezing into a microfluidic constriction. We report a
state-diagram that depicts whether, or not, particles clog the channel
depending on their deformability and the ratio of the constriction width to
the capsule diameter. Moreover, we characterize the particle passage by
analyzing the dynamics of the number of particles passing the constriction.

##  II Setup and method

We consider a microfluidic channel with dimensions $L=37.5d$ (length),
$W=6.25d$ (width) and $H=1.875d$ (height) as shown in Fig. 1, where $d$ is the
diameter of the capsules at rest when they adopt a spherical equilibrium
shape. We set $d=2r=8$. All units are given in lattice units as defined by the
lattice Boltzmann method used to compute the flow. The technical details about
the numerical methods used to obtain the results are given in the Appendix.
Both the suspending and the encapsulated fluids are considered to have
identical kinematic viscosity $\nu$ and mass density $\rho$. The channel inlet
contains initially $38$ non-Brownian neutrally buoyant particles with no
viscosity constrast. All the particles have the same geometrical and
mechanical properties (i.e. a monodisperse suspension). They are randomly
placed at the entrance region of the constriction. The present work is limited
to fixed number of particles, and how this parameter alters the behavior of
the system is left for a future study.

A constriction of length $L_{\rm obst}=6.25d$ and width $D$ is placed at a
distance $L_{\rm in}=12.5d$ from the inlet of the channel having the width
$W$. A body force ${f}_{z}$ is applied in the z-direction to generate a flow
with a parabolic velocity profile, whose mid-plane velocity $u_{\rm max}$ in
absence of particles and the constriction is given by

${u}_{\rm max}=\frac{{f}_{z}\,{W}^{2}}{8\rho\nu}.$ (1)

Here, $f_{z}=2.22\times 10^{-6}$ is set in all simulations.

We study the influence of two dimensionless control parameters: i - the
aperture defined as ratio of the constriction width to the capsule diameter
$D/d$, and ii - the particle deformability quantified by the capillary number,

$\mathrm{Ca}=\frac{\rho\nu r\dot{\gamma}}{\kappa_{\rm s}},$ (2)

where $\dot{\gamma}={4u_{\rm max}}/{W}$ is the measured shear rate at the
wall. We use also ${\rm K}=1/{\rm Ca}$ that expresses the dimensionless
elastic modulus of the membrane, and that characterizes the rigidity of the
particles. $D/d$ is set to desired values by varying $D$, while keeping $d$
constant. The value of ${\rm Ca}$ is varied by varying only the shear elastic
modulus $\kappa_{\rm s}$, while holding all the other parameters constant.

##  III Results

###  III.1 Clog and no clog states

All the simulations are carried out on $40$ CPUs during $60$ hours at the
Reynolds number ${\rm Re}=u_{\rm max}R/\nu=0.1$, as is encountered in
microfluidic flows. Snapshots showing both the effect of varying the
constriction aperture $D/d$ (left panel) and the effect of varying the
particle deformability ${\rm Ca}$ (right panel) are shown in Fig. 2. The
aperture degree is varied from $1.25$ (left snapshot) to $D/d=3.25$ (right
snapshot). The capillary number is set to $\mathrm{Ca}=0.001$ to model rigid
particles because when $\rm{Ca}\ll 1$ the capsules deform weakly, and thus,
behave mechanically as rigid particles, see Ref. SM1 . For the case of
$D/d=1.25$ (narrowest constriction), two particles are found to be sufficient
to clog permanently the constriction entrance. For $D/d=2$, more than two
particles clog the constriction by building a stable arch, while the fluid
still continues to flow. For $D/d=3.25$ (widest constriction), the aperture is
large enough to allow and maintain a continuous flux of particles without
observing any clog formation. In the right panel of Fig. 2, we hold the same
aperture degree $D/d=1.25$, and we vary only the capillary number. For non-
deformable particles, with $\mathrm{Ca}=0.001$, again only two particles are
sufficient to clog the constriction. When increasing ${\rm Ca}$, the particles
deform further, and thus, they can squeeze easily through the constriction,
see Ref. SM2 . No clog event is observed for particles with large deformation
capability under flow. At large ${\rm Ca}$ and narrow constrictions, the
particles cross the constriction one by one, while forming a regular train and
exiting the constriction as a jet. The reported results are observed
independently of the initial random positions of the particles.

###  III.2 State-diagram

For a given number of particles ($N=38$) and a given applied flow strength
($f_{z}=2.22\times 10^{-6}$), we explore the state-diagram that gives the
blockage status as a function of the constriction aperture $D/d$ and the
capillary number ${\rm Ca}$. This latter is taken smaller than $0.1$ to avoid
the limit beyond which the capsules, whose membrane elasticity follows a Neo-
Hookean law, are susceptible to undergo continuous elongation Hu2013 .

Figure 3 reports the state-diagram, where we distinguish clearly two main
regions representing the clog (red symbols) and the no clog states (green
symbols). The border separating the two regions (dashed black line) depends on
the deformability of the capsules and their aspect ratio with respect to the
constriction width. It is also sensitive to the initial spatial arrangement of
the particles. Simulations with three initial conditions are represented with
different symbols: hollow squares, saltires and crosses.

Figure 3: The clog and the no clog states of multiple capsules ($N=38$)
obtained when varying their deformability ${\rm Ca}$ (or rigidity ${\rm K}$)
and the constriction width $D/d$, while holding the same applied flow
strength. The deformability character of the particles reduces the threshold
of the transition to the no clog state below $D/d=3$ measured experimentally
for rigid particles Marin2018 . Simulations with three initial conditions are
represented with different symbols: hollow squares, saltires and crosses.

In the limit of weakly deformable capsules, i.e. when ${\rm Ca}\rightarrow 0$
(${\rm K}\rightarrow\infty$), the transition from clog to no clog states
occurs at a critical value that approaches $D/d=3$, which is measured
experimentally for rigid spherical particles and when using a converging-
diverging constriction Marin2018 . By choosing a basic step-like shaped
constriction here, we intentionally rule out the angle of the constriction to
appear as an additional parameter Lopez-Rodriguez2019 . Capsules with a large
degree of deformability, beyond the threshold $\rm{Ca}=0.005$ (below
$\rm{K}=200$), can pass without clogging the constriction independently of the
aspect ratio, even for particles having almost the same size as the
constriction width ($D/d\rightarrow 1$).

Here, we report the effect of the deformability of multiple fluid-filled
particles (not matrix microgel particles). Soft particles can cross the
constriction easily, and by varying their elasticity the threshold for the
transition to the no clog state drops down. This border is not sharp since it
is sensitive to the initial arrangement of the particles. There, both the clog
and the no clog states may emerge. In contrast, far from this blurry border,
only one of the two states emerges with $100\%$ probability. In our previous
work Ref. Kusters2014 , the state-diagram has been reported only for a single
deformable particle for which a single particle can always cross the
constriction when $D\geq d$. Moreover, other works dealing with soft particles
(e.g. Ref. Harth2020 ) have not considered varying the deformability as a
control parameter.

###  III.3 Dynamics of the particle passage

The number of escapees $N(t)$, defined as the number of particles that have
passed through the constriction at time $t$, is given in Fig. 4 for stiff
particles with $\rm{Ca}=0.001$ (upper panel) and for soft particles with
$\rm{Ca}=0.1$ (lower panel).

Three constriction aperture degrees are considered in each figure: $D/d=1.25$,
$2.25$, and $3.25$. For stiff particles and at small aperture $D/d=1.25$, a
clog forms, as shown in Fig. 2. At $D/d=2.25$ and $3.25$, the number of
escapees over time $N(t)$ evolves in the same way linearly at the early stage
and then non-linearly. This means the flux of the particles across the
constriction $dN(t)/dt$ is almost constant at the beginning of the
simulations. It adopts a greater slope for $D/d=3.25$ for which three
particles can pass at the same time through the constriction, while only two
particles can pass for $D/d=2.25$. In the second stage the number of escapees
increases non-linearly and slowly. This latter represents the passage of the
remaining particles that were trapped in the corners close to the channel
walls that are located just at the constriction entrance, where the flow speed
is lower. Moreover, these particles need to overcome the step of the inlet
constriction, which further increases the time needed for them to pass through
the constriction.

(a)

(b)

Figure 4: Dynamics of the number of escapees through the constriction $N(t)$
for rigid spherical particles with ${\rm Ca}=0.001$ (upper panel) and for soft
particles with ${\rm Ca}=0.1$ (lower panel) at various constriction widths
$D/d$. $N(t)$ increases approximately linearly faster, then it slows down non-
linearly before it adopts a plateau, whose value corresponds to the total
number of particles $N$ for the no clog state and to a smaller value
$N(\infty)<N$ for the clog state. No clogging event takes place for deformable
particles, and the evacuation process is fast.

When the particles are highly deformable, as shown in the lower panel of Fig.
4 for the case of $\rm{Ca}=0.1$, all the particles pass easily through the
constriction, even for the narrower constriction of $D/d=1.25$. One may notice
a plateau at $38$ (which is the initial total number of particles contained in
the channel inlet) for $D/d=2.25$ and $3.25$. This corresponds to the scenario
where all particles have successfully passed through the constriction, and
none of them is left in the inlet compartment. This is the total evacuation,
to which we associate the evacuation time $T_{\rm evac}$ that measures the
time needed to evacuate all the particles, and which is found to increase when
narrowing the constriction width. $T_{\rm evac}$ is similar to the time needed
to empty totally one compartment of a sand-clock made of granular particles,
while in this study the particles are soft and are suspended in a viscous
fluid. It is reported in Fig. 5 as a function of $D/d$ and ${\rm Ca}$. $T_{\rm
evac}$ clearly decreases as the aperture $D/d$ and the capillary number ${\rm
Ca}$ increase because particle passage is easier when the constriction is
wider and the particles are more deformable. The dependency of $T_{\rm evac}$
on both $D/d$ and ${\rm Ca}$ is non-linear. The derivation of a scaling law
$T_{\rm evac}=f(D/d,{\rm Ca})$ would be practical in designing microfluidic
constrictions with desired throughput of soft particles. However, in absence
of a theoretical model to guide the scaling, we were only able to extract the
exponents that give the dependency of $T_{\rm evac}$ on $D/d$ and ${\rm Ca}$
for the range of parameters available to our simulations,

$T_{\rm evac}\propto\left(\frac{d}{D}\right)^{0.827}\quad\text{and}\quad
T_{\rm evac}\propto{\rm Ca}^{-0.108}$ (3)

using the data set plotted on log-log scale in both panels of Fig. 5. The
exponent related to ${\rm Ca}$ is low, here, that means the constriction
aperture $D/d$ is the leading parameter. However, its exponent of the
proportionality $0.827$ is larger than $1/2$ known for the evacuation time of
granular dry particles through a pore due to the contribution of the particle
deformation and the presence of a suspending fluid in the present study.

(a)

(b)

Figure 5: Evacuation time $T_{\rm evac}$ needed for $38$ capsules to pass the
constriction as a function of the aperture $D/d$ (the upper panel) and the
capillary number ${\rm Ca}$ (the lower panel). $T_{\rm evac}$ is a decreasing
non-linear function of both ${\rm Ca}$ and $D/d$.

##  IV Discussion and conclusions

Multiple deformable capsules passing through a microfluidic constriction
exhibit similar dynamics as observed for a single capsule reported in Ref.
Kusters2014 , with slight differences due to the collective motion that is
rendered cooperative due to the particle deformation. The capsules either pass
the constriction or they get stuck by building an arch at the constriction
entrance. Here, the flow is not blocked since the fluid can flow around and
through the blocked capsules. This study has examined both the effect of the
constriction confinement and the particle deformability degree for a fixed
number of particles and for a given applied flow strength. The suspension is
dilute when compared to the overall volume of the channel and the
constriction, but it becomes dense at the constriction’s entrance. For the
same flow strength, the transition from the blockage state to the passage is
controlled by the size aspect ratio and the capsule deformability. Capsules
with extremely weak deformability show an almost similar threshold, as
measured experimentally by Marin et al. Marin2018  for rigid particles
$D/d=3$. The capsules have revealed the presence of two types of the blockage
states: permanent and transient, which lead to a non-sharp border in the
state-diagram. The dynamics of the number of particles crossing the
constriction evolves linearly in the beginning of all the simulations, then
non-linearly before it adopts a plateau when all the particles have been
evacuated. The complete passage of the capsule through the constriction is
carefully analyzed, and it may be used when designing microfluidic devices for
handling high throughput of soft particle suspensions. At large ${\rm Ca}$ and
small aperture ($D/d\rightarrow 1$), the particles cross the constriction one
by one by forming a regular train, which could be exploited for diagnosis in
the case of living cells. This study has examined the role of both the
constriction confinement and the particle deformability for a fixed number of
particles, while further exploration of the parameter space is left for a
future work, where we plan to vary the geometry of the constriction and the
number of particles. The small number of particles used in the present study
does not lead to strong stochastic behavior as observed for a large number of
particles. For example, when flowing through a silo Harth2020 , which is
described by statistics and probabilistic mathematical models. Here, faraway
from the border of the state-diagram only one of the clog and no clog states
occurs. At the vicinity of the border these two states may emerge with a
probability due to the weak, but non-negligible effect of the initial
positions of the particles.

## Appendix: Numerical method

###  IV.1 Fluid flow solver

We use the lattice Boltzmann method (LBM) as a mesoscopic numerical method
which allows to recover solutions of the Navier-Stokes equations. We introduce
very briefly the method, while the readers are invited to check out existing
textbooks for more details Succi2001 ; Kruger2016 . The spirit of the LBM
consists of streaming a distribution function $f_{i}$ that gives the
probability to find a number of fluid particles on a lattice node
$\mathbf{r}$, at a discrete time step $t$, and with a discrete velocity
$\mathbf{e}_{i}$. The dynamics of $f_{i}$ is given by

$f_{i}(\mathbf{r}+\mathbf{e}_{i},t+1)-f_{i}(\mathbf{r},t)=\Omega_{i}(\mathbf{r},t)+F_{i}(\mathbf{r},t).$
(4)

$\Omega_{i}(r,t)$ on the right hand side is the collision operator. The time
and space steps are both taken to be unity. Here, we use the Bhatnagar-Gross-
Krook (BGK) collision operator Bhatnagar1954 ,
$\Omega_{i}=-\frac{1}{\tau}[f_{i}(\mathbf{r},t)-f_{i}^{\rm
eq}(\mathbf{r},t)]$, that expresses the relaxation of $f_{i}$ toward its
equilibrium $f_{i}^{\rm eq}$ within the relaxation time $\tau$. $f_{i}^{\rm
eq}$ is given as a truncated expansion of the Maxwell-Boltzmann distribution
for the velocities in an ideal gas. External applied forces, including the
membrane forces, are incorporated in Eq. 4 through the source term $F_{i}$
such that

$F_{i}(\mathbf{r},t)=\omega_{i}\left(1-\frac{1}{2\tau}\right)\left(\frac{\mathbf{e}_{i}-\mathbf{u}}{c_{\rm
s}^{2}}+\frac{\mathbf{e}_{i}\cdot\mathbf{u}}{c_{\rm
s}^{4}}\mathbf{e}_{i}\right)\cdot\mathbf{F}(\mathbf{r},t),$ (5)

where $\mathbf{F}(\mathbf{r},t)$ accounts for either the body force or the
membrane forces. $c_{\rm s}=1/\sqrt{3}$ is the lattice speed of sound and
$\omega_{i}$ are the lattice weights which, for the three-dimensional lattice
with 19 velocities (D3Q19) used here, read as $1/3$, $1/18$ and $1/36$ for
$i=1$, $i=2\dots 7$, and $i=8\dots 19$, respectively. We impose no-slip
boundary conditions on the channel walls using mid-grid bounce-back boundary
conditions.

###  IV.2 Mechanics of particle deformation

We consider strain-softening capsules with zero-thickness membranes that
exhibit Neo-Hookean 2D hyperelastic mechanical behavior with the energy
Barthes-Biesel2016

$E_{\rm s}=\frac{\kappa_{\rm
s}}{2}\int_{A}\left[I_{1}-1+\frac{1}{I_{2}+1}\right]{\rm d}A,$ (6)

where $\kappa_{\rm s}$ is the shear elastic modulus,
$I_{1}=\lambda_{1}^{2}+\lambda_{2}^{2}-2$ and
$I_{2}=\lambda_{1}^{2}\lambda_{2}^{2}-1$ are the two deformation invariants,
$\lambda_{1}$ and $\lambda_{2}$ are the principal stretching ratios, $A$ is
the surface of each capsule, and ${\rm d}A$ the surface element. In addition,
we enforce the constraint of the capsule volume conservation by using an
energy that gives the cost of any deviation of the actual volume of the
capsule $V$ from its original value $V_{0}$ that is the volume of the
spherical capsule at rest,

$E_{\rm v}=\frac{\kappa_{\rm v}}{2}\frac{(V-V_{0})^{2}}{V_{0}},$ (7)

where $\kappa_{\rm v}$ is a numerical parameter whose value is set large
enough to fullfill the volume conservation constraint. Non-physical wrinkles
may also emerge at the surface of the capsules, and are consequently avoided
by applying a bending force $\mathbf{F}_{\rm b}$, which is derived as a
functional derivative of the Helfrich energy originally proposed for lipid
membranes Helfrich1973 ,

$\mathbf{F}_{\rm b}(\mathbf{x}_{i})=2\kappa_{\rm b}[2H(H^{2}-K)+\Delta_{\rm
s}H]\mathbf{n},$ (8)

where $\kappa_{\rm b}$ is the bending modulus,
$H=\frac{1}{2}\sum_{i=1}^{2}c_{i}$ is the mean curvature,
$K=\prod_{i=1}^{2}c_{i}$ is the Gaussian curvature, $c_{i}$ is the principal
curvature, $\Delta_{\rm s}$ is the Laplace-Beltrami operator and $\mathbf{n}$
the normal vector pointing outward from the membrane. The bending modulus is
chosen such as the dimensionless number $B=\kappa_{\rm b}/(\kappa_{\rm
s}r^{2})$, that quantifies the relative importance of the bending rigidity
with respect to the shear elasticity, is small. In this way the bending force
mitigates wrinkle formation, while it does not influence globally the dynamics
and deformation of the capsules. $H$, $K$ and $\Delta_{\rm s}$ are computed
following a discrete differential geometry operators approach Guckenberger2017
.

The membrane of each capsule is discretized into $1280$ triangular elements,
and the force on each membrane node $\mathbf{x}_{i}$, with $i$ refering to the
index of the node, is evaluated following the principle of virtual work such
that

$\mathbf{F}_{\alpha}(\mathbf{x}_{i})=-\frac{\partial
E_{\alpha}}{\partial\mathbf{x}_{i}}.$ (9)

The subscript $\alpha$ denotes either $\\{{\rm s}\\}$ for the strain energy,
$\\{{\rm v}\\}$ for the volume energy or $\\{{\rm b}\\}$ for the bending
energy. The derivatives needed to evaluate the membrane forces are computed
numerically using the finite element method Ref. Kruger2012 . A short-range
repulsive force is implemented to mimic the hydrodynamic lubrication force,
and to avoid overlap between particles or between particles and walls,

$\mathbf{F}_{rep}=\left\\{\begin{array}[]{rcr}&\bar{\epsilon}[(\frac{1}{d_{ij}})^{2}-(\frac{1}{\delta_{0}})^{2}]\frac{\mathbf{d}_{ij}}{d_{ij}}\quad\text{if}\quad
d_{ij}<\delta_{0}\\\ &\mathbf{0}\quad\text{if}\quad
d_{ij}\geq\delta_{0}\end{array}\right.,$ (10)

with $\bar{\epsilon}$ being the strength of the force, and $d_{ij}$ is the
surface-to-surface distance between particles $i$ and $j$ or the distance
between particle $i$ and the solid node $j$ on the wall. The repulsive force
vanishes when $d_{ij}$ is larger than the cutoff distance $\delta_{0}=1$.

###  IV.3 Fluid-structure interaction

The two-way coupling between the fluid flow and the capsule dynamics is
realized using the immersed boundary method (IBM), which is a front-tracking
method developed originally by Peskin to study blood flow in the heart
Peskin1977 . The IBM consists of coupling a moving Lagrangian mesh
$\delta\Omega$ representing the capsule membrane and a stationary Eulerian
grid $\Omega$ Peskin2002 , where the flow is computed with the LBM. The method
has two main steps:

Advection - The flow advects all the capsules’ mesh nodes as if they are
massless pointwise particles. Once the fluid velocity field ${\bf u}(x,y,z,t)$
is computed by the LBM on the Eulerian mesh, the velocity of each membrane
node ${\bf u}(s_{1},s_{2},t)$, where $(s_{1},s_{2})$ are curvilinear
coordinates, is estimated by interpolation of the velocities of its
neighboring fluid nodes using a function $\delta$,

${\bf u}(s_{1},s_{2},t)=\int_{\Omega}\delta({\bf r}(x,y,z,t),{\bf
r}(s_{1},s_{2},t)){\bf u}(x,y,z,t){\rm d}{\bf r},$ (11)

with ${\bf r}(x,y,z,t)\in\Omega$, ${\bf r}(s_{1},s_{2},t)\in\delta\Omega$, and

$\delta({\bf r}_{1},{\bf
r}_{2})=\phi(x_{1},x_{2})\phi(y_{1},y_{2})\phi(z_{1},z_{2}),$ (12)

where

$\phi(x_{1},x_{2})=\frac{1}{4}\left(1+\cos\frac{\pi(x_{1}-x_{2})}{2}\right)$
(13)

if $|x_{1}-x_{2}|\leq 2$, $|y_{1}-y_{2}|\leq 2$ and $|z_{1}-z_{2}|\leq 2$,
otherwise $\delta({\bf r}_{1},{\bf r}_{2})=0$. All the membrane nodes are then
advected using the explicit Euler scheme:

${\bf r}(s_{1},s_{2},t+1)={\bf r}(s_{1},s_{2},t)+{\bf u}(s_{1},s_{2},t)$ (14)

Reaction -  When all the membrane nodes are advected, the overall capsule
deforms into a new shape that is not necessarily its equilibrium shape, and
thus, it tries to relax back to its lowest energy configuration. By doing so
it exerts a force back upon its surrounding fluid. The forces exerted by the
membrane in the Lagrangian mesh ${\bf F}(s_{1},s_{2},t)$ are computed with the
finite difference method and are extrapolated to the fluid nodes using
$\delta$ again as a weight in order to have a force field in the Eulerian grid
${\bf F}(x,y,z,t)$,

${\bf F}(x,y,z,t)=\int_{\delta\Omega}\delta({\bf r}(x,y,z,t),{\bf
r}(s_{1},s_{2},t)){\bf F}(s_{1},s_{2},t){\rm d}A.$ (15)

This force is plugged into the right-hand side of the LBM equation.

## Acknowledgements

The authors acknowledge financial support by the Deutsche
Forschungsgemeinschaft (DFG) within the research unit FOR2688 ‘Instabilities,
Bifurcations and Migration in Pulsatile Flows’ (grant number HA4382/8-1). CB
and BK acknowledge the Ministère de l’Enseignement Supérieur, de la Recherche
et de l’Innovation (MESRI) and the Biomechanics and Bioengineering Laboratory
(BMBI) for financial support.

## References

  * (1) E. Dressaire, A. Sauret, Clogging of microfluidic systems, Soft Matter 13, 37-48 (2017).
  * (2) X. Hong, M. Kohne, M. Morrell, H. Wang, E. R. Weeks, Clogging of soft particles in two-dimensional hoppers Physical Review E 96, 062605 (2017)
  * (3) Z. Zhang, J. Xu, C. Drapaca, Particle squeezing in narrow confinements, Microfluidics and Nanofluidics 22, 120 (2018)
  * (4) R. van Zwieten, T. van de Laar, J. Sprakel, K. Schroen, From cooperative to uncorrelated clogging in cross-flow microfluidic membranes, Scientific Reports 8, 5687 (2018)
  * (5) A. Marin, H. Lhuissier, M. Rossi, C. J. Kähler Clogging in constricted suspension flows, Physical Review E 97, 021102(R) (2018)
  * (6) M. Souzy, I. Zuriguel, A. Marin, Transition from clogging to continuous flow in constricted particle suspensions, Physical Review E 101, 060901(R) (2020)
  * (7) R. Jäger, M. Mendoza, and H. J. Herrmann, Clogging at pore scale and pressure-induced erosion, Physical Review Fluids 3, 074302 (2018)
  * (8) C. Rorai, A. Touchard, L. Zhu, L. Brandt, Motion of an elastic capsule in a constricted microchannel, The European Physical Journal E 38, 49 (2015)
  * (9) R. Kusters, T. Heijden, B. Kaoui, J. Harting, C. Storm Forced transport of deformable containers through narrow constrictions, Physical Review E 90, 033006 (2014)
  * (10) A. Le Goff, B. Kaoui, G. Kurzawa, B. Haszon, A.-V. Salsac, Squeezing bio-capsules into a constriction: deformation till break-up, Soft Matter 13 (41), 7644-7648 (2017)
  * (11) Z. Y. Luo, B. F. Bai, Off-center motion of a trapped elastic capsule in a microfluidic channel with a narrow constriction, Soft Matter 13, 8281-8292 (2017)
  * (12) T. G. Fai, R. Kusters, J. Harting, C. Rycroft, L. Mahadevan, Active elastohydrodynamics of vesicles in narrow, blind constrictions, Physical Review Fluids 2, 113601 (2017)
  * (13) W. Lei, C. Xie, T. Wu, X. Wu, M. Wang, Transport mechanism of deformable micro-gel particle through micropores with mechanical properties characterized by AFM, Scientific Reports 9,1453 (2019)
  * (14) See Supplemental Material at [URL will be inserted by publisher] for a video showing clearly how capsules do not deform for Ca = 0.001
  * (15) See Supplemental Material at [URL will be inserted by publisher] for a video showing clearly how capsules deform a lot for Ca = 0.1.
  * (16) S. Succi, The lattice Boltzmann equation for fluid dynamics and beyond (Oxford University Press, Oxford, UK, 2001)
  * (17) T. Krüger, H. Kusumaatmaja, A. Kuzmin, O. Shardt, G. Silva, E. M. Viggen, The lattice Boltzmann method - Principles and practice (Springer, Switzerland, 2016)
  * (18) P. L. Bhatnagar, E. P. Gross, M. Krook, A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems, Physical Review 94, 511 (1954)
  * (19) D. Barthès-Biesel, Motion and deformation of elastic capsules and vesicles in flow Annual Review of Fluid Mechanics 48, 25-52 (2016)
  * (20) W. Helfrich, Elastic properties of lipid bilayers: Theory and possible experiments, Z. Naturforsch. C 28, 693 (1973).
  * (21) A. Guckenberger, S. Gekle, Theory and algorithms to compute Helfrich bending forces: A review, J. Phys. Cond. Mat. 29, 203001 (2017)
  * (22) T. Krüger. Computer simulation study of collective phenomena in dense suspensions of red blood cells under shear. Wiesbaden: Springer Spektrum, (2012). ISBN 978-3834823755.
  * (23) C. S. Peskin, Numerical analysis of blood flow in the heart, J. Comp. Phys. 25, 220–252 (1977)
  * (24) C. S. Peskin, The immersed boundary method, Acta Numerica 11, 479–517 (2002)
  * (25) D. Lopez-Rodriguez, D. Gella, K. To, D. Maza, A. Garcimartin, I. Zuriguel, Effect of hopper angle on granular clogging, Physical Review E 99, 032901 (2019)
  * (26) K. Harth, J. Wang, T. Borzsonyi and R. Stannarius, Intermittent flow and transient congestions of soft spheres passing narrow orifices, Soft Matter 16, 8013-8023 (2020)
  * (27) X.-Q. Hu, B. Sévénié, A.-V. Salsac, E. Leclerc, and D. Barthès-Biesel, Characterizing the membrane properties of capsules flowing in a square-section microfluidic channel: Effects of the membrane constitutive law, Physical Review E f 8̱7, 063008 (2013)