# Motion and deformation of capsules flowing through a corner in the inertial
and non-inertial regimes

Damien P. Huet  Department of Mathematics, University of British Columbia,
1984 Mathematics Road, Vancouver BC V6T 1Z2, Canada  Antoine Morente
Department of Mathematics, University of British Columbia, 1984 Mathematics
Road, Vancouver BC V6T 1Z2, Canada  Guodong Gai  Department of Mathematics,
University of British Columbia, 1984 Mathematics Road, Vancouver BC V6T 1Z2,
Canada  Anthony Wachs<EMAIL_ADDRESS>Department of Mathematics, University
of British Columbia, 1984 Mathematics Road, Vancouver BC V6T 1Z2, Canada
Department of Chemical and Biological Engineering, University of British
Columbia, 2360 E Mall, Vancouver BC V6T 1Z3, Canada

###### Abstract

We investigate the inertial and non-inertial dynamics of three-dimensional
elastic capsules flowing through a square channel presenting a sharp corner.
Our study analyzes the trajectory, surface area, velocity and membrane stress
of the capsules in the case of a single capsule, a system of two interacting
capsules and a train of ten capsules released upstream of the corner. The
channel Reynolds number $Re$ ranges from 0.01 to 50 and the Capillary number
$Ca$, which measures the ratio of the viscous and elastic stresses, ranges
from 0.075 to 0.35. We find that in the inertial regime, the membrane stretch
and stress increase dramatically as compared to the non-inertial case, and
that the velocity overshoot inside the corner is also enhanced. The maximum
capsule deformation is observed to depend nearly linearly on $Ca$ and $Re$.
Additionally, we report a repelling mechanism between two confined capsules
when their initial interspacing distance $d$ is smaller than a critical value
$d_{c}$. The deformation of the leading capsule is found to be mitigated by
the presence of the following capsule. In the case of multiple capsules
flowing through the corner, we observe that the increase in the maximum
surface area of the trailing capsules eventually saturates at the tail of the
train. Moreover, we find that the corner tends to separate the capsules
regardless of their upstream interspacing distances $d$. This study
contributes to the elaboration of practical guidelines for controlling capsule
breakup and predicting throughput in both inertial and non-inertial
microfluidic experiments.

##  I Introduction

Membrane-enclosed fluid objects, or capsules, are everywhere in natural and
industrial processes, from red blood cells (RBCs), circulating tumor cells
(CTCs) or flowing eggs in biology to encapsulated substances in the
pharmaceutical, cosmetic and food industries [1]. The study of microcapsules
in particular is of primary importance in a variety of biological
applications, such as sorting and enriching solutions of biological
microcapsules, e.g. to segregate RBCs or CTCs, as well as efficiently
manufacturing capsules enclosing an active substance in the field of targeted
drug delivery [2, 3]. In the past decade, microfluidic devices have been shown
to accomplish a variety of tasks including cell segregation based on size and
deformability [4, 5, 6, 7], concentration enrichment [8, 9, 10] and cell
characterization [11, 12, 13]. Moreover, the increase in computing power has
recently allowed numerical studies to contribute to the design of microfluidic
devices. For example, Zhu et al. [4] numerically investigated an original
microchannel geometry consisting of a semi-circular pillar located at the
center of a microchannel: their study showed that this design can efficiently
segregate cells based on membrane deformability. Recently, experiments were
conducted using their microfluidic design and concluded that it can indeed
sort cells based solely on membrane stiffness, with relatively high efficacy
[5]. With regards to cell characterization, Gubspun et al. [11] proposed a
method to determine capsule properties such as the membrane shear modulus by
comparing the experimental and numerical “parachute” shape of capsules in a
straight microchannel. While the majority of microfluidic investigations
operate in Stokes conditions, in recent years the design and study of inertial
microfluidic devices has risen due to their ability to accurately segregate
capsules by size and to extract them from their solvant [7, 14]. Inertial
focusing in microfluidic devices typically relies on a spiral-shaped channel
concentrating heavier capsules to the outer, lower-curvature edge of the
channel, while lighter capsules concentrate closer to the inner, higher-
curvature edge. A smooth geometry such as a spiral-shaped channel usually does
not induce a high strain nor stress on a suspended capsule even in inertial
regimes, however little is known about the strains and stresses induced by
commonly encountered sharp geometries such as forks and corners on a capsule
flowing in the presence of inertia. Moreover, the effect of such sharp
geometries on the hydrodynamic interactions of a train of several capsules in
inertial regimes is also an open question. More insight in these directions is
of practical interest in the design and operation of inertial microfluidic
devices because (i) the devices should not compromise the mechanical integrity
of the capsules, i.e. it is critical to avoid capsule breakup, and (ii) cell-
sorting processes typically operate in very dilute regimes to avoid capsule
interactions, while a better understanding of such interactions would allow to
operate these devices at a moderate to high concentration optimizing efficacy
and throughput.

In the past four decades, a significant research effort has been invested into
the modeling and the study of capsule deformations in non-inertial regimes,
primarily because this regime is encountered in microcirculation such as
capillary vessels and in traditional microfluidic devices. Using formalism
from the thin-shell theory [15], Barthès-Biesel & Rallison first published an
analytical solution for the time-dependant deformation of an elastic capsule
in an unbounded, creeping shear flow in the limit of small deformations [16].
Over a decade later, Pozrikidis was able to go beyond the assumption of small
deformations using a Boundary Integral Method (BIM) [17]. The same method was
used to consider finite deformations of sheared capsules which inner and outer
fluid viscosities differ [18], as well as to study the contribution of bending
stresses [19], allowing to consider RBCs suspended in an unbounded shear flow
[20]. Besides unbounded geometries, Zhao et al. [21] simulated RBCs in
straight and constricted channels using a spectral BIM. A similar method was
later used by Hu et al. [22] to consider an initially spherical capsule
flowing through a square channel of width similar to the capsule diameter: the
originality of their work is that they performed experiments and showed
remarkable agreement between the measured and the computed capsule shape.
Concomitantly, Park and Dimitrakopoulos [23] studied the deformation of a
capsule with non-unity viscosity ratio flowing through a sharp constriction.
More recently, Balogh & Bagchi [24, 25, 26] used a Front-Tracking Method (FTM)
to analyze the motion and deformation of RBCs through complex geometries
resembling capillary vessels found in human microcirculation: their
simulations exhibited in particular the well-known cell-free layer observed
experimentally between the RBCs and the vessel walls [27, 28].

Regarding the study of flowing capsules in the presence of inertia, the
aforementioned analytical theory for small deformations as well as the popular
BIM both fall short of accounting for the convective term in the fluid
momentum equation. Doddi & Bagchi [29] first studied inertial capsules in the
context of two interacting capsules in a shear flow using the FTM. They showed
in particular that the two capsules engage in spiralling motions at
sufficiently high inertia. The inertial motion of a deformable capsule was
then studied in straight microchannels [30, 31], where several equilibrium
positions are found away from the channel centerline, along the cross-section
diagonals. With regards to curved channels, Ebrahimi & Bagchi [32] recently
investigated the migration of a single capsule over an impressive amount of
varying parameters: the channel Reynolds number, the capsule deformability, as
well as the aspect ratio and curvature of the channel were all varied
independently. Their study shows that for sufficiently high inertia, exactly
two focusing locations appear near the centers of the vortices of the
secondary flow, known as Dean’s vortices. However no mention of the membrane
internal strains and stresses is found in their work, as their goal was not to
investigate the capsule integrity in such flows.

While straight and curved microchannels are essential components of
microfluidic devices, such simple geometries do not account for the numerous
junctions, corners and coils commonly found in these devices. To bridge this
gap, Zhu & Brandt [33] investigated the non-inertial motion and the
deformation of a single elastic capsule in a sharp corner. They showed that
the capsule follows the streamlines of the undisturbed flow regardless of
membrane deformability. Due to lubrication forces, the capsule velocity
decreases when approaching the corner, reaches a minimum along the corner
diagonal, and rises back to its steady state with an overshoot increasing with
deformability. Similarly, the surface area of the capsule reaches a maximum
inside the corner and reaches its steady value with an undershoot more
pronounced as deformability is increased. Also reported in their study is the
maximum stress in the capsule membrane, which can be used to assess mechanical
integrity and characterize the cell mechanical properties. They find that the
maximum stress deviation increases and shifts from the front to the top of the
capsule with increasing deformability. Wang et al. [8, 9] later considered the
inertial and non-inertial path selection of a single capsule through Y- and
T-junctions, both typically encountered in microfluidic geometries. They
observe that at high inertia, the capsule does not necessarily favor the
daughter branch with the largest flow rate, and that this effect is more
pronounced for stiff membranes (corresponding to a low capillary number).
Recently, Lu et al. [10] investigated the interaction and path selection of
capsules in a T-junction at moderate inertia, with the goal of enriching
capsule solutions. When considering a pair of capsules, they show that the
leading capsule is weakly affected by the presence of a trailing capsule, but
that the reverse is not true. They find that the trailing capsule enters a
different branch depending on the initial interspacing distance and on the
flow rate split ratio between the two daughter branches of the T-junction.
They then consider a train of capsules and find two distinct regimes: (i) the
interspacing distance is low and the capsule interaction is high, resulting in
an unsteady regime and affecting the trajectories of the capsules, and (ii)
the interspacing distance is large and the capsule interaction is low, leaving
the capsule trajectories identical to that of a single capsule. Interestingly,
they report that the critical interspacing distance between two capsules
plotted against the flow rate split ratio of the daughter branches results in
a master curve independent of membrane deformability, capsule size, and
Reynolds number.

In this study, we investigate the inertial and non-inertial motion and the
interaction of deformable capsules flowing through a sharp corner, which is a
very common geometry in microfluidic devices. As the efficiency of these
devices is defined in terms of the capsules throughput, which can be optimized
by increasing the flow rate as well as the concentration of capsules, our
objective is two-fold: first, we aim to quantify the effet of inertia on the
deformation of a single capsule in a microfluidic-relevant geometry, second,
we seek to describe the hydrodynamic interactions and deformation differences
between leading and trailing capsules when a pair and a train of capsules are
considered. The rest of this paper is organized as follows. In Section II, we
describe the governing equations as well as the flow configuration and the
considered parameter space. In Section III, we give an overview of our
numerical method and we investigate the impact of the inlet length. We analyze
the motion of a single capsule in Section IV, both in the non-inertial and in
the inertial regimes. Section V is devoted to the analysis of binary
interactions of a pair of capsules, where the influence of the initial
interspacing distance is investigated. In Section VI, we consider a train of
ten capsules flowing through the corner and we discuss the velocity and
deformation discrepancies between the leading and trailing capsules. Finally,
we conclude in Section VII.

The documented source code allowing to reproduce all of the simulations and
figures presented in this study is freely available online [34].

##  II Governing equations and problem statement

The capsule membrane $\Gamma$ is assumed infinitely thin and is surrounded by
an incompressible, Newtonian fluid of constant viscosity and density. In all
of this study, the capsule inner and outer fluids are assumed identical: in
particular their viscosity ratio is unity. The fluid is described by the mass
and momentum conservation equations:

$\nabla\cdot\bm{\tilde{u}}=0$ (1)
$\frac{\partial\bm{\tilde{u}}}{\partial\tilde{t}}+\bm{\tilde{u}}\cdot\nabla\bm{\tilde{u}}=\frac{1}{\tilde{\rho}}\nabla\tilde{p}+\tilde{\nu}\Delta\bm{\tilde{u}}+\frac{1}{\tilde{\rho}}\bm{\tilde{f_{b}}}$
(2)

where $\bm{\tilde{u}}$ is the velocity field, $\tilde{p}$ is the pressure
field, $\tilde{\rho}$ is the density, $\tilde{\nu}=\tilde{\mu}/\tilde{\rho}$
is the kinematic viscosity, $\tilde{\mu}$ is the dynamic viscosity and
$\bm{\tilde{f_{b}}}$ is a body term accounting for the action of the membrane
on its surrounding fluid. The dimensional quantities are denoted by the $\sim$
symbol. The membrane exhibits elasticity and bending resistance, and its
action on the fluid is local, resulting in the following expression for
$\bm{\tilde{f_{b}}}$:

$\bm{\tilde{f_{b}}}=\left(\bm{\tilde{f}_{\text{elastic}}}+\bm{\tilde{f}_{\text{bending}}}\right)\tilde{\delta}(\bm{\tilde{x}}-\bm{\tilde{x}_{\Gamma}}),$
(3)

where $\tilde{\delta}(\bm{\tilde{x}}-\bm{\tilde{x}_{\Gamma}})$ is a Dirac
distribution that is non-zero on the surface of the membrane.

The shear and area-dilatation membrane stresses are described using the thin-
shell theory, and are briefly summarized here. The interested reader is
referred to Green & Adkins [15] as well as to the analytical study of Barthès-
Biesel & Rallison [16] for more details. We adopt a neo-Hookean law [15],
which surface strain-energy function is expressed as:

$\tilde{W_{s}}^{NH}=\frac{\tilde{E_{s}}}{2}\left(\lambda_{1}^{2}\lambda_{2}^{2}+\frac{1}{\lambda_{1}^{2}\lambda_{2}^{2}}\right),$
(4)

where $\lambda_{1,2}$ are the principal stretches in the two tangential
directions, and $\tilde{E_{s}}$ is a shear modulus. The principal stresses
$\tilde{\sigma}_{1,2}$ are given by:

$\tilde{\sigma}_{i}=\frac{1}{\lambda_{j}}\frac{\partial\tilde{W_{s}}^{NH}}{\partial\lambda_{i}},\qquad
i,j\in\\{1,2\\},\quad i\neq j.$ (5)

The bending stresses for biological membrane are governed by the Helfrich’s
bending energy $\mathcal{E}_{b}$ [35, 36]:

$\tilde{\mathcal{E}}_{b}=\frac{\tilde{E_{b}}}{2}\int_{\Gamma}\left(2\tilde{\kappa}-\tilde{\kappa}_{0}\right)^{2}dS,$
(6)

where $\tilde{E_{b}}$ is the bending modulus, $\tilde{\kappa}$ is the mean
curvature and $\tilde{\kappa}_{0}$ is a reference curvature. Taking the
variational formulation of Eq. (6) leads to the bending force per unit area
$\tilde{A}$:

$\bm{\tilde{f}_{\text{bending}}}/\tilde{A}=-2\tilde{E}_{b}(\Delta_{s}(\tilde{\kappa})+2(\tilde{\kappa}-\tilde{\kappa}_{0})(\tilde{\kappa}^{2}-\tilde{\kappa}_{g}+\tilde{\kappa}_{0}\tilde{\kappa}))\bm{n},$
(7)

where $\tilde{\kappa}_{g}$ is the Gaussian curvature and $\bm{n}$ is the outer
normal vector.

(a)

(b)

Figure 1: (a) Schematic of the geometry of the fluid domain. The channel has a
square cross-section of side length $3\tilde{a}$. (b) Visualization of the
full channel and the computational grid over the symmetry plane of the
channel.

At $t=0$, an initially spherical capsule of radius $\tilde{a}$ is placed in a
square channel of width $\tilde{W}=3\tilde{a}$ at a distance
$\tilde{h}_{0}=30\tilde{a}$ from a sharp corner, as represented in Fig. 1. An
average cross-section velocity $\tilde{U_{0}}$ is imposed at the inlet
boundary, while the outflow boundary condition
$\partial\bm{\tilde{u}}_{n}/\partial\bm{n}=0$ is imposed at the outlet
boundary. When several capsules are considered, we use the same initial
conditions as Lu et al. [10]: a trailing capsule is inserted in the simulation
only after the centroid of its preceding capsule has advanced by a distance
$\tilde{d}$. Our problem is governed by the trailing dimensionless numbers:

  1. 1.

The channel Reynolds number
$Re=\tilde{\rho}\tilde{U_{0}}\tilde{W}/\tilde{\mu}$,

  2. 2.

The Capillary number $Ca=\tilde{\mu}\tilde{U_{0}}\tilde{a}/\tilde{E_{s}}$,
representing the ratio of viscous stresses over elastic stresses,

  3. 3.

The reduced bending stiffness coefficient
$E_{b}=\tilde{E_{b}}/(\tilde{E_{s}}a^{2})$,

  4. 4.

The confinement ratio $\beta=2\tilde{a}/\tilde{W}$,

  5. 5.

The reduced initial gap between capsules $d_{0}=\tilde{d}/2\tilde{a}-1$.

In this study, the Reynolds number $Re$ ranges from 0.01 to 50, the Capillary
number $Ca$ varies from $0.075$ to $0.35$, and the reduced initial gap $d_{0}$
is chosen from 0.125 to 1. The reduced bending stiffness $E_{b}$ and the
confinement ratio $\beta$ are both kept constant, with $\beta=2/3$ and
$E_{b}=5\cdot 10^{-3}$ as proposed by Pozrikidis [37]. The reference curvature
$\tilde{\kappa_{0}}$ is equal to $-2.09/\tilde{a}$ in this study, as is common
for some biological membranes such as RBC membranes [38, 39]. In the rest of
this study, we use the capsule radius $\tilde{a}$ as the characteristic length
scale, and we define the characteristic time scale as the radio of the capsule
radius over the average cross-section velocity, i.e.
$t=\tilde{a}/\tilde{U_{0}}$.

##  III Numerical method and validations

We use our adaptive Front-Tracking Method (FTM) to solve the above equations:
we provide below a brief overview of the numerical method, while an in-depth
description is available in [40]. Eq. (1) and Eq. (2) are solved using the
Finite Volume method on an adaptive octree grid using the open-source software
Basilisk [41]. The membrane is discretized using an unstructured triangulation
and Eq. (5) is solved using a linear Finite Element Method, while Eq. (7) is
solved using a paraboloid-fitting method. The membrane triangulation and the
octree grid communicate by means of the immersed boundary method [42, 43],
where the Dirac distribution in Eq. (3) is regularized using a cosine-based
formulation:

$\tilde{\delta}(\bm{x_{0}}-\bm{x})=\begin{cases}\begin{aligned}
&\frac{1}{64\tilde{\Delta}^{3}}\prod_{i=1}^{3}\left(1+\cos\left(\frac{\pi}{2\tilde{\Delta}}(x_{0,i}-x_{i})\right)\right)\quad\text{if}\quad|x_{0,i}-x_{i}|<2\tilde{\Delta}\\\
&0\quad\text{otherwise}\end{aligned}\end{cases},$ (8)

where $\bm{x_{0}}=[x_{0,1}\;x_{0,2}\;x_{0,3}]$ is the location of a Lagrangian
node on the surface discretization of the membrane, and $\tilde{\Delta}$ is
the local mesh size of the Eulerian octree grid. Extensive validation of the
present numerical method was the focus of our previous study [40] and is
therefore not presented here. Nonetheless, the convergence with respect to the
Eulerian grid as well as the release distance of the capsule from the corner
are investigated below.

In the immersed boundary method, it is well known that the support of the
regularized Dirac distribution may extend outside of the fluid domain if the
immersed object of interest becomes very close to the domain walls [44, 10,
9]. In order to avoid unphysical loss of momentum for the specific membrane
nodes close to the wall, it is important to ensure that none of the supports
of the regularized Dirac distribution extend outside of the fluid domain, i.e.
that there always exist more than two grid cells between membrane nodes and
the domain boundaries. As such, we simulate the dynamics of a capsule for two
different grid resolutions in the configuration where it is most deformed and
is the closest to the channel wall, as shown in Fig. 2b. Figure 2a shows the
velocity of the capsule $\tilde{V}$ inside and downstream of the corner for
Eulerian resolutions equivalent to 32 and 64 grid cells per initial capsule
diameter, as well as the deviation of the velocities in these two
configurations. Excellent agreement is found between the velocities computed
using the two grid resolutions, with the maximum discrepancy lower than 1% and
the average discrepancy over the considered time range of about 0.5%.
Moreover, in both configurations it was found that more than 3 grid cells are
present in the lubrication layer between the capsule tail and the upper corner
wall. These results indicate that an equivalent grid resolution of 32 grid
cells per capsule initial diameter is sufficient to obtain converged
solutions, and that the present simulations do not suffer from immersed
boundary stencils extending outside of the fluid domain.

(a)

(b)

Figure 2: (a) Centroid velocity of a capsule at $Ca=0.35$ and $Re=50$ for two
grid resolutions: 32 grid cells per initial diameter (red dotted line) and 64
grid cells per initial diameter (red solid line). The blue curve denotes the
deviation in the centroid velocities for these two grid resolutions. (b)
Corresponding shape and grid resolutions of the capsule and the flow field:
blue means zero velocity and red means large velocity.

Next we investigate the influence of the normalized release distance $D_{c}$
between the initial position of the capsule centroid and the corner. Indeed,
after its release the capsule relaxes from a spherical to an equilibrium
steady shape and it is important that this steady state is reached before the
capsule enters the corner. As such, we consider three initial distances
$D_{c}=15$, 30 and 60 in the most challenging configuration at $Re=50$ and
$Ca=0.35$, i.e. the capsule is highly deformable and placed in a highly
inertial flow. The inlet boundary is located at a distance of 90$a$ away from
the corner and is therefore sufficiently far away from the capsule to not
alter its response. The norm of the capsule centroid velocity $\tilde{V}$ and
the reduced capsule surface area $\mathcal{A}=\tilde{\mathcal{A}}/4\pi a^{2}$
are shown in Fig. 3, where the origin of the reduced time $t$ is chosen at the
time the capsule reaches a minimum velocity $\tilde{V}_{min}$. In Fig. 3(a) we
remark that the capsule velocity $\tilde{V}$ at $D_{c}=15$ decreases
significantly prior to entering the corner: this is because the initially
spherical capsule is located farther away from the channel walls and is
therefore advected faster than when it has reached a steady shape. We observe
that neither the capsule velocity shown in Fig. 3(a) nor the normalized
surface area shown in Fig. 3(b) present a steady state before the capsule
enters the corner in the case $D_{c}=15$. Therefore a larger initial distance
$D_{c}$ should be used. When considering $D_{c}=30$, both the velocity and the
normalized surface area present steady values before the corner.
Interestingly, inside and after the corner the capsule velocity and surface
area almost overlap when the capsule is released 15 and 30 initial radii away
from the corner, suggesting that the corner resets the dynamics of the capsule
regardless of its previous state. The fact that steady values for the velocity
and the surface area of the capsule are reached before the corner for
$D_{c}=30$ suggests that this initial release distance is suitable for the
rest of this study. Interestingly, releasing the capsule at $D_{c}=60$ leads
to an unexpected result: the capsule seems to no longer be in a steady motion
as its velocity (respectively its normalized surface area) is slightly
decreasing (respectively slightly increasing) prior to entering the corner.
This suggests that in this challenging configuration, the relaxation of the
capsule from a fixed spherical shape to a steady “parachute” shape occurs over
very long time scales. However, the magnitude of the deviations between the
capsule velocity and surface area in the cases $D_{c}=30$ and 60 is at most
3%. As the capsule has already reached a pseudo steady state by the time it
reaches the corner in the case of $D_{c}=30$, and as the aforementioned
discrepancies are small, we choose $D_{c}=30$ in the rest of this study.
Again, this short study of the impact of the initial release distance on the
capsule dynamics was performed in our most challenging configuration as we
considered our highest Reynolds number and highest Capillary number. The
discrepancy between the cases $D_{c}=30$ and 60 is less pronounced $-$
sometimes nonexistent $-$ for less deformable membranes and less inertial
flows.

(a)

(b)

Figure 3: Centroid velocity (a) and normalized surface area (b) of a capsule
flowing through a corner from three distinct normalized release distances
$D_{c}=15$, 30 and 60, at $Re=50$ and $Ca=0.35$.

##  IV Motion and deformation of a single capsule

We consider the motion of a single capsule through a square duct at
$Ca=0.075,0.15,0.25,0.35$ and $Re=0.01,1,25,50$, extending the investigation
carried out in a non-inertial framework by Zhu & Brandt [33]. In order to
establish the influence of the increasing effect of inertia on the motion and
the deformation of a single capsule, we first recall the overall dynamics of a
capsule moving through a duct corner in the Stokes regime, as detailed in
[33]. The capsule once released from its initial position moves along the
center of the channel due to the symmetry of the flow far from the corner.
While approaching the corner, the capsule velocity decreases until reaching a
minimum in the corner region. The capsule experiences moderate to high
deformation (depending on the Capillary number considered) due to the flow
acceleration, and its velocity strongly increases; this phenomenon being
referred to as the overshoot of velocity. Further away from the corner, the
capsule moves in the downstream branch of the duct, relaxing to a steady state
(shape and velocity), and moving along the center of the duct.

We investigate the influence of the Reynolds number $Re$ and the Capillary
number $Ca$ on the dynamics and the deformation of the capsule, reporting the
time evolution of its surface area $\mathcal{A}$ scaled by the initial surface
area of the capsule $\mathcal{A}_{\text{sphere}}=4\pi\tilde{a}^{2}$, as well
as the velocity $V$ of the capsule centroid scaled by its equilibrium velocity
$V_{eq}$ before the capsule enters the corner region. In the remainder of this
study and unless otherwise stated, the time origin is chosen such that $t=0$
when capsule velocity reaches a global minimum, i.e. $V_{min}=V(t=0)$. We
borrow this convention from Zhu & Brandt [33], as it corresponds to setting
the time origin when the capsule is located at the heart of the corner.

###  IV.1 Influence of the Reynolds and Capillary numbers

To characterize the dynamics of the capsule as it flows through the corner, we
analyze the time evolution of the centroid velocity $V$ and the surface area
$\mathcal{A}$. Figure 4 shows the velocity of the capsule centroid for $Ca$
ranging from 0.075 to 0.35. $Re$ is constant for each subfigure of Fig. 4.
Conversely, Fig. 5 shows the same data as Fig. 4, but with each subfigure
corresponding to a constant $Ca$. From both figures, we observe a general
trend for all cases: the capsule approaches the corner with a steady velocity
$V_{eq}$, then reaches a global minimum $V_{min}$ and a global maximum
$V_{max}$ as it flows through the corner, and relaxes back to $V_{eq}$
downstream of the corner. Moreover, we observe in Fig. 4 that the velocity
extrema increase with increasing $Ca$. In the more inertial regimes
especially, the maximum velocity deviation of the capsule at $Ca=0.35$ is
close to three times that of the capsule at $Ca=0.075$.

We note from Fig. 5 that the curves corresponding to $Re=0.01$ and $Re=1$
practically overlap, indicating that the capsule motion in low inertial
regimes is very similar to that in the non-inertial regime. As the Reynolds
number is increased to 25 and 50, major deviations from the non-inertial
regime appear. First, as the capsule enters the corner zone, a local maximum
appears in the capsule velocity, which is independent of the Capillary number,
and is about 1% greater than $V_{eq}$ at $Re=25$ and 2% greater than $V_{eq}$
at $Re=50$. This local maximum is due to the migration of the capsule across
the centerline of the secondary channel: in this process the capsule is
located far away from the channel walls and is therefore less subject to their
confinement effect. Then, the minimum velocity $V_{min}$ is reached in the
heart of the corner. Interestingly, at small $Ca$, $V_{min}$ is observed to be
independent of the $Re$, as can be seen in Fig. 5(a) at $Ca=0.075$. In
contrast, in the case of larger $Ca$ the minimum velocity of the capsule
increases slightly with $Re$. A difference of about 4% is observed for
$V_{min}$ as $Re$ increases from 0.01 to 50 for both $Ca=0.25$ and $Ca=0.35$.

As the capsule exits the corner zone and migrates to the channel centerline,
its velocity reaches its maximum value $V_{max}$ which increases with
increasing $Re$ and $Ca$: at $Ca=0.075$, $V_{max}$ increases by 3% between
$Re=0.01$ and $Re=50$ while at $Ca=0.35$, $V_{max}$ increases by about 8%
between $Re=0.01$ and $Re=50$. Then, the capsule velocity relaxes back to its
equilibrium value and its relaxation time increases with increasing $Re$.
Interestingly, velocity undershoots are observed during the relaxation stage
in the inertial regime, which magnitude increases with $Re$. The relaxation
time does not depend on $Ca$.

(a) $Re=0.01$

(b) $Re=1$

(c) $Re=25$

(d) $Re=50$

Figure 4: Temporal evolution of the capsule centroid velocity $V$ at fixed
Reynolds numbers.

(a) $Ca=0.075$

(b) $Ca=0.15$

(c) $Ca=0.25$

(d) $Ca=0.35$

Figure 5: Temporal evolution of the capsule centroid velocity $V$ at fixed
Capillary numbers.

The time evolution of the normalized capsule surface area $\mathcal{A}$ is
shown in Fig. 6 for fixed $Re$ and in Fig. 7 for fixed $Ca$. We observe that
the surface area presents a maximum $\mathcal{A}_{max}$ at around $t=1$ before
relaxing to its equilibrium value $\mathcal{A}_{eq}$. Unsurprisingly, Fig. 6
confirms that a large $Ca$, i.e. a highly deformable capsule, results in a
greater surface area than for lower $Ca$. Figure 6 also shows that the
magnitude of the maximum surface area increases with $Ca$. Moreover, when
large $Ca$ are considered the time evolution of the capsule surface area
presents some undershoots that are more pronounced as $Re$ is increased.
Additionally, Fig. 7 reveals that $Re$ has a very strong influence on the
deformation of the capsule, especially at large $Ca$: at $Ca=0.075$,
$\mathcal{A}_{max}/\mathcal{A}_{eq}$ increases from 2% to 8% between $Re=0.01$
and $Re=50$, and at $Ca=0.35$ it increases from 8% to a staggering 22% between
$Re=0.01$ to $Re=0.35$. In particular, at $Ca=0.35$ the maximum capsule
surface area increases from 9% to 40% of the surface area of a sphere between
the non-inertial and the highly inertial regimes. These surface area
deviations are very large and are discussed further in the next section.

(a) $Re=0.01$

(b) $Re=1$

(c) $Re=25$

(d) $Re=50$

Figure 6: Temporal evolution of the capsule surface area $\mathcal{A}$ at
fixed Reynolds numbers.

(a) $Ca=0.075$

(b) $Ca=0.15$

(c) $Ca=0.25$

(d) $Ca=0.35$

Figure 7: Temporal evolution of the capsule surface area $\mathcal{A}$ at
fixed Capillary numbers.

###  IV.2 Maximum deformation of the capsule

(a) $\mathcal{A}_{max}$ as a function of $Re$

(b) $\mathcal{A}_{max}$ as a function of $Ca$

Figure 8: Maximum surface area $\mathcal{A}_{max}$ as a function of $Re$ and
$Ca$ for a single capsule passing through the corner.

The maximum surface area $\mathcal{A}_{max}$ of the capsule is presented in
Fig. 8, as a function of both the Reynolds number and the Capillary number. To
better analyze the trends in this figure, we also report the maximum area at
intermediate Reynolds numbers, namely at $Re=12.5$ and $37.5$. The data
reported in Fig. 8 clearly exhibits a double linear scaling of
$\mathcal{A}_{max}$ with both $Ca$ and $Re$ as long as $Ca$ is below 0.35 $-$
at $Ca=0.35$, the shape of the curve $\mathcal{A}_{max}(Re)$ is slightly
concave. The slope of the scaling is about $0.003$ for $\mathcal{A}_{max}(Ca)$
and $1.12$ for $\mathcal{A}_{max}(Re)$. This means that the capsule maximum
deformation responds proportionally to the Capillary number, but also to the
Reynolds number. To our knowledge, this is the first time such a trend has
been reported and established for low ($Re=1$) to moderate
($Re=12.5,25,37.5,50$) inertial regimes. We believe that this result can be
used as a predictive tool for many studies involving single capsules
travelling through duct corners, as the maximum deformation observed for a
capsule is a measure of its mechanical integrity, which is of major interest
in many microfluidic applications.

(a) $V_{max}$

(b) $V_{min}$

Figure 9: Maximum (minimum) velocity $V_{max}$ ($V_{min}$) as a function of
$Re$ and $Ca$ for a single capsule passing through the corner.

Additionally, we present in Fig. 9 the maximum and minimum velocity of the
single capsule flowing through the corner. In the non-inertial regime, the
maximum velocity of the capsule increases with $Ca$, as shown in Fig. 9(a). In
inertial conditions we observe that $V_{max}$ increases for $Re$ ranging from
1 to 50. The increase in $V_{max}$ between $Re=1$ and $Re=50$ is significant
in Fig. 9(a), especially for large $Ca$. For instance, at $Ca=0.35$, $V_{max}$
increases by about 8% between the non-inertial and the highly inertial
regimes. We then consider the evolution of the minimum velocity $V_{min}$ for
a single capsule at various $Ca$ and $Re$ in Fig. 9(b). In general, we observe
that the minimum velocity decreases with $Ca$ in both the non-inertial and the
inertial regimes for $Re\leq 25$. In Fig. 9(b), we also observe a non-
monotonous behavior of $V_{min}$ at low inertia and at sufficiently high $Ca$:
for $Ca\geq 0.15$, $V_{min}$ first decreases with increasing $Re$, reaching a
minimum for $Re=12.5$, before increasing sharply at $Re>12.5$. Overall, we
observe from Fig. 9 that the presence of inertia tends to increase both
velocity extrema of the capsule, especially at large $Ca$.

A quantity of practical interest to experimentalists is the maximum stress
experienced by the capsule, as it can be used to predict a priori if a given
geometry can induce plastic deformation or even breakup of the capsule
membrane [5]. More specifically, it is the largest eigenvalue
$\tilde{\sigma}_{2}$ of the stress tensor $\bm{\tilde{\sigma}}$ that can bring
insight into the mechanical integrity of the membrane. In Fig. 10, we show the
maximum and average values of $\tilde{\sigma}_{2}$ over the membrane surface
as the capsule approaches and flows through the corner at $Ca=0.35$ and
$Re=1$, 25 and 50. We observe that $\tilde{\sigma}_{2,\,avg}$ follows a trend
very similar to that of the capsule surface area observed in Fig. 7(d):
$\tilde{\sigma}_{2,\,avg}$ varies smoothly with time, presents a maximum near
$t=1$ and a local minimum near $t=2.5$, and the value of the maximum deviation
from steady state nearly doubles between the low and moderate inertial cases
$Re=1$ and $Re=50$. We also note that the steady state value of
$\tilde{\sigma}_{2,\,avg}$ prior to entering the corner is independent of
$Re$, as was observed in the case of the capsule surface area in Fig. 7(d). In
particular, we find by comparing Fig. 7(d) and 10 that at $Ca=0.35$, a non-
dimensional surface area $\mathcal{A}$ of about 1.14 leads to an average non-
dimensional membrane stress of about 0.4. The steady state of the maximum
stress $\tilde{\sigma}_{2,\,max}$, however, increases by about 40% between the
low inertial case ($Re=1$) and the moderate inertial cases ($Re=25$, 50).
Inside the corner, $\tilde{\sigma}_{2,\,max}$ increases by nearly 75% between
$Re=1$ and $Re=50$, confirming that a capsule in a moderate inertial regime
has a higher risk of breakup than in a low inertial regime.

It is worth noting that for all $Re$, the value of the maximum stress
$\tilde{\sigma}_{2,\,max}$ is about double that of the average stress
$\tilde{\sigma}_{2,\,avg}$: since we showed previously that
$\tilde{\sigma}_{2,\,avg}$ is closely related to the capsule surface area $-$
a quantity that is relatively easy to measure experimentally $-$, this
observation can be used by experimentalists as a rule of thumb to estimate the
maximum stress in the capsule membrane and assess the mechanical integrity of
the membrane.

$Re=50$

$Re=25$

$Re=1$

Figure 10: Left: Maximum and average tensions in the capsule at $Ca=0.35$ and
$Re=1$, 25 and 50. Right: Capsule shape colored by $\tilde{\sigma}_{2}$ when
$\tilde{\sigma}_{2,\,max}$ reaches its maximum.

###  IV.3 Evolution of the capsule shape

(a)

(b)

(c)

(d)

(e)

(f)

Figure 11: Sequence of Capsule outlines for different $Ca$ and $Re$. The time
between each frame is $t=1.5$.

We now illustrate the temporal evolution of the capsule travelling through the
corner. Figure 11 shows the outline of the capsule in the symmetry plane $z=0$
for successive discrete times. The capsule outlines are given for $Ca=0.075$
and $Ca=0.35$ and $Re=0.01$, 25 and 50. Prior to entering the corner, the
capsule adopts a steady shape that is determined by the confinement of the
walls. In the case of $Ca=0.35$, we observe the well-known “parachute” shape.
Upstream of the corner, the trajectory of the capsule coincides with the
centerline of the primary (vertical) channel. As the capsule flows through the
corner, the capsule deviates from the channel centerline: in the non-inertial
regime, Zhu & Brandt [33] showed that the capsule trajectory closely matches
the flow streamlines. We obtain the same conclusion in the inertial regime.
When inertia is considered, the capsule trajectory crosses the horizontal
centerline of the secondary channel and comes increasingly close to the upper
wall as $Re$ increases, before relaxing to the channel centerline.

Figures 11(a) and 11(b) show clear differences in the effects of $Ca$ in the
Stokes regime. Increasing $Ca$ from $0.075$ to $0.35$ causes the equilibrium
shape of the capsule to change from an slightly deformed spheroid to a concave
“parachute” shape. For a small $Ca=0.075$, the equilibrium shapes of the
capsule remain similar as $Re$ increases from $Re=0.01$ to $Re=50$ (see Fig.
11(a), 11(c), and 11(e)). However, the deformation of the capsule becomes more
evident inside the corner at higher $Re$, particularly in Fig. 11(e). After
passing the corner, the capsule shape returns to its steady spheroid shape
observed in the Stokes regime for all values of $Re$. In the case of a high
$Ca=0.35$, we observe that the equilibrium shape of the capsule is more and
more concave as $Re$ increases. Inside the corner, the capsule is highly
elongated and presents an increasingly long tail for increasing $Re$ $-$ e.g.
Fig. 11(f) in the case of $Re=50$. In the highly inertial regime, strong
lubrication interactions occur between the capsule and the top wall, resulting
in a flat top surface.

(a)

(b)

Figure 12: Outlines of a single capsule passing a corner with (a) maximal
surface area $\mathcal{A}$ and (b) maximal velocity $v_{max}$ at various $Re$
and $Ca$.

In figures 12(a) and 12(b), we present the single capsule outline with the
maximum surface area $\mathcal{A}_{max}$ and the maximum velocity $V_{max}$
inside the corner for all the cases investigated in this section. Inside the
corner, the maximum surface area of the single capsule is reached when it
approaches the upper wall and it is quickly followed by the maximum velocity.
From figures 12(a) and 12(b), we observe in particular that a high $Re$ leads
to an elongation of the capsule in the streamwise direction, while a high $Ca$
increases the concavity of the capsule. Moreover, we note that the centroid of
the capsule moves closer to the rim of the outline at high values of $Ca$:
note that the centroid drawn in figures Fig. 11-12(b) corresponds to the
centroid of the three-dimensional capsule, not to that of the two-dimensional
outline. The results shown in figures 11-12(b) indicate that $Ca$ has a
significant effect on capsule deformation, while $Re$ has a more pronounced
effect on the trajectory of the capsule as well as its deformation resulting
from the lubrication layer against the top wall of the corner. In particular,
at high $Re$, the capsule undergoes significant stretching, which may cause
damage or even rupture in microfluidic devices. Understanding the effects of
$Re$ on capsule deformation and the resulting damage is crucial in designing
efficient and reliable microfluidic devices.

###  IV.4 Discussion on the Stokes regime

We observe in figures 7 a surprising, non-monotonous behavior of the capsule
surface area around $Re=1$: at large $Ca$, the surface area of the capsule is
smaller at $Re=1$ than at $Re=0.01$ and $Re=25$. Additionally, in Fig. 7(a)
the steady surface area of the capsule at $Re=0.01$ and $Ca=0.075$ downstream
of the corner is about 1% lower than the initial spherical surface area of the
capsule, indicating a small loss of the internal capsule volume. The cause of
these observations may be related to the limitations of the FTM coupled with a
sub-optimal choice of numerical parameters in the case of $Re=0.01$ only.
Indeed, the immersed boundary method is known to conserve volume
asymptotically rather than to machine precision. In earlier IBM studies
involving capsules, the volume loss is always small, typically below 1% [24,
10, 8, 9]. Moreover, Stokes conditions are known to be challenging for PDE-
based incompressible Navier-Stokes solvers, as the matrix inverted in the
velocity viscous Poisson problem is less well conditioned at low $Re$. While
it is worth noting that the capsule surface area in the Stokes regime should
be interpreted with caution, these limitations only affect the capsule surface
area and not the centroid velocity. Moreover, our solver was extensively
validated in Stokes conditions in [40] and showed excellent agreement with the
BIM as well as other FTM solvers. As such, while further investigation should
be conducted in the Stokes regime, it cannot be excluded that at high $Ca$ the
capsule surface area at $Re=0.01$ is physically slightly greater than that at
$Re=1$. Finally, the main focus of the present work is to investigate the
inertial motion and deformation of capsules through a sharp corner, i.e. in
conditions where our FTM solver does not suffer from the limitations outlined
above.

##  V System of two capsules

In this section, we consider two identical capsules flowing through the corner
as we vary the normalized interspacing distance $d=\tilde{d}/2\tilde{a}-1$
between the capsules as well as the Reynolds and Capillary numbers. Lu et al.
[10] previously considered the binary interaction of capsules flowing through
a T-junction: they showed that when $d_{0}\geq 1.3$ the trailing capsule has
minimal impact on the motion of the leading capsule. In contrast, in their
T-junction geometry Lu et al. observed that the motion of the trailing capsule
is significantly affected by the presence of the leading capsule. To gain
insight into the physical features relevant to capsule interactions through a
corner in the inertial and non-inertial regimes, we select small values for
the normalized interspacing distance $d_{0}=1,1/2$, and $1/4$ and we examine
phenomena such as migration, dynamics and deformation of the leading and the
trailing capsules.

###  V.1 Qualitative analysis: trajectory and capsule shape

(a)

(b)

Figure 13: (a) Trajectory of the two capsules at different $Ca$ and $Re$. (b)
Outlines of the leading and trailing capsules at $Ca=0.35,Re=50,d_{0}=0.25$,
with comparison to a single capsule.

We first analyze the trajectory and the qualitative shapes of the pair of
capsules as they flow through the corner. Figure 13(a) shows the trajectory of
the capsules at $Re=0.01,$ 25 and 50 and $Ca=0.15$ and 0.35. We note that all
curves corresponding to the same $Ca$ overlap: $Ca$ has no impact on the path
of either the leading or the trailing capsule. Likewise, we observe no
significant difference in the trajectories of the leading and the trailing
capsules, unlike the strikingly different paths reported in the case of a
T-junction [10]. In fact, the key parameter that controls the capsule
trajectory is the Reynolds number. As $Re$ increases, the inertia drives the
capsule closer to the upper channel wall, as observed in Section IV in the
case of a single capsule. We then illustrate the capsule shape on the symmetry
plane $z=0$ in Fig. 13(b) for the most deformed capsule configuration
corresponding to $Ca=0.35$ and $Re=50$ with an initial interspacing distance
$d_{0}=0.25$. We compare the outlines of the leading and the trailing capsules
to that of a single capsule in the same conditions. Qualitatively, the
deformation of interacting capsules is not significantly different than that
observed in the case of a single capsule. Perhaps more surprisingly, the
qualitative outlines of the leading and the trailing capsules are also very
similar, almost overlapped, even in the strongly interacting configuration
corresponding to $d_{0}=0.25$. Note that this qualitative shape analysis
relies on the outline of the capsule in the plane of symmetry $z=0$, while the
actual three-dimensional shape of the leading and trailing capsules may differ
more strongly.

###  V.2 Quantitative analysis: velocity and membrane surface area

(a)

(b)

Figure 14: Temporal evolution of the velocity $V$ and the surface area
$\mathcal{A}$ of capsules at $Ca=0.35,Re=50$ with $d_{0}=0.25$: a comparison
of the leading, trailing and a single capsule.

We now compare the temporal evolution of the velocity of the centroids of the
capsules as well as the time evolution of their surface areas, as plotted in
Fig. 14. To simplify the identification of interaction features, we first
focus on the most deformed configuration corresponding to $Ca=0.35$, $Re=50$
and $d_{0}=0.25$. For reference, we also plot the evolution of a single
capsule under the same conditions in red. Throughout the remainder of this
study, and unless otherwise stated, the velocity of interacting capsules is
normalized by the equilibrium velocity $V_{eq}$ of a single capsule for the
same Capillary and Reynolds numbers. This normalization choice allows for an
unbiased comparison between the velocities of the leading and the trailing
capsules. In this section we also denote the reduced velocity of the single
capsule by $V_{s}$, that of the leading capsule by $V_{l}$ and that of the
trailing capsule by $V_{t}$. Similarly, we denote by $\mathcal{A}_{s}$,
$\mathcal{A}_{l}$, $\mathcal{A}_{t}$ the normalized surface areas of
respectively the single, leading and trailing capsules.

In Fig. 14(a), we observe that the velocity of the leading capsule is affected
by the presence of the trailing capsule before it reaches the corner, as it is
about 1% higher than that of a single capsule. However, the extrema of $V_{l}$
as it flows through the corner closely match those of $V_{s}$. After the
corner, $V_{l}$ is about 2% larger than $V_{s}$ but slowly relaxes back to
$V_{s}$ further downstream. With regards to the trailing capsule, we note that
its velocity is more markedly affected by the presence of the leading capsule.
Prior to reaching the corner, $V_{t}$ is about 1% lower than $V_{s}$, but
inside the corner its minimum value is 4% lower than $V_{s}$. However, the
maximum of $V_{t}$ is identical to that of both $V_{l}$ and $V_{s}$.
Downstream of the corner, $V_{t}$ quickly relaxes back to $V_{s}$ and
maintains a similar value thereafter, eventually converging to $V_{eq}$. The
time evolution of the surface areas of the pair of capsules is shown in Fig.
14(b). The normalized surface area of the leading capsule $\mathcal{A}_{l}$ is
clearly influenced by the presence of the trailing capsule, as was observed
above in the case of its velocity. The steady and maximum surface areas of the
leading capsule are about 2% lower than that of the single capsule. In
contrast, the steady surface area of the trailing capsule closely matches that
of the single capsule upstream and downstream of the corner, while its maximum
value is about 1% higher than that of the single capsule. We postulate that
the small interspacing distance between the two capsules disturbs the wake
behind the leading capsule, which tends to mitigate its deformation and
therefore decreases its surface area. Conversely, as the wake of the trailing
capsule is unaffected, the discrepancies between its surface area and that of
the single capsule are less pronounced.

(a) $Re=0.01$

(b) $d_{0}=1$

(c) $d_{0}=0.25$

Figure 15: Temporal evolution of $V$ of the leading and trailing capsules at
different $Ca$, $Re$ and $d_{0}$.

We then present the time evolution of the velocity and surface area of the
leading and the trailing capsules at various $Ca$, $Re$ and $d_{0}$. We first
focus on the velocity of the capsules, displayed in Fig. 15 for $Ca=0.15$ and
0.35 and for $d_{0}=0.5$ and 1. The velocity of both capsules displays a
minimum at $t=0$ and a maximum at $t\approx 2$ at $Ca=0.15$ and $Ca=0.35$. The
extrema of the velocity are more pronounced as $Ca$ increases. The effects of
the initial interspacing distance $d_{0}$ on these extrema are less evident
but still present: the velocity maxima of both the leading and the trailing
capsules are increased by about 1% as $d_{0}$ is halved from 1 to 0.5.
Interestingly, the relaxation time of $V_{t}$ to $V_{eq}$ is significantly
reduced when compared to that of $V_{l}$: about 3 time units in the case of
$V_{t}$ with respect to more than 10 time units in the case of $V_{l}$.
Capsule velocities in the inertial regimes at $Re=25$ and 50 and at $Ca=0.15$
and $Ca=0.35$ are plotted in Fig. 15(b) and Fig. 15(c) for $d_{0}=1$ and 0.25,
respectively. The results are similar to that of the non-inertial regime: $Ca$
enhances the velocity deviations and the extrema are more pronounced in the
case of the trailing capsule. Surprisingly, we note that Fig. 15(b) and Fig.
15(c) display very similar behaviors: therefore, the interspacing distance
does not seem to impact the capsule velocities inside the corner: its effects
are bounded to the capsule velocities upstream and downstream from the corner.
We will come back to this observation in Section V.3.

When analyzing the capsule surface areas for varying $Re$, $Ca$ and $d_{0}$, a
similar behavior is found: the surface area of the trailing capsule is
consistently greater than that of the leading capsule, and increasing
Capillary and Reynolds numbers and decreasing the initial interspacing
distance enhance this phenomenon. In particular we report in Table 1 the
maximum surface areas of the leading capsule and in Table 2 that of the
trailing capsule. As can be seen from Table 1 and Table 2, the maximum surface
area of the leading capsule exceeds that of the trailing capsule by up to 5%.
The full time-dependant data is provided in Appendix A.

$d_{0}$ |  | $Re=0.01$ | $Re=25$ | $Re=50$
---|---|---|---|---
$1$ | $Ca=0.15$ | 1.065 | 1.138 | 1.193
$Ca=0.35$ | 1.263 | 1.334 | 1.399
$0.5$ | $Ca=0.15$ | 1.065 | 1.135 | 1.186
$Ca=0.35$ | 1.247 | 1.323 | 1.383
$0.25$ | $Ca=0.15$ | 1.068 | 1.129 | 1.180
$Ca=0.35$ | 1.236 | 1.308 | 1.379
Table 1: Maximum surface area $\mathcal{A}_{max}$ of the leading capsule at different $Ca$, $Re$ and $d_{0}$. $d_{0}$ |  | $Re=0.01$ | $Re=25$ | $Re=50$
---|---|---|---|---
$1$ | $Ca=0.15$ | 1.068 | 1.143 | 1.201
$Ca=0.35$ | 1.271 | 1.342 | 1.417
$0.5$ | $Ca=0.15$ | 1.070 | 1.144 | 1.204
$Ca=0.35$ | 1.277 | 1.345 | 1.414
$0.25$ | $Ca=0.15$ | 1.069 | 1.148 | 1.204
$Ca=0.35$ | 1.277 | 1.344 | 1.41
Table 2: Maximum surface area $\mathcal{A}_{max}$ of the trailing capsule at
different $Ca$, $Re$ and $d_{0}$.

###  V.3 Time evolution of the interspacing distance

We now analyze the time evolution of the interspacing distance between the two
confined capsules considered in this section. Figure 16 shows the time-
dependent interspacing distance for $Ca=0.15$ and 0.35, $Re=25$ and 50 and
$d_{0}=1$, 0.5 and 0.25. In this figure, we note that in all cases, the
interspacing distance decrease immediately after the trailing capsule is
released. This is due to the fact that upon release, the trailing capsule is
spherical and therefore located farther away from the channel walls than is
the leading capsule, resulting in its initial acceleration before a steady
shape is found $-$ typically within less than five time units. In the case
where $d_{0}=1$, the interspacing distance $d$ is steady until the leading
capsule approaches the corner, reaches a minimum then a maximum value inside
the corner and becomes steady again as the trailing capsule leaves the corner
region. Interestingly, the steady interspacing distance after the corner is up
to 10% greater than its steady value prior to the corner, suggesting that the
corner separates the two capsules. Moreover, the initial interspacing distance
is greater in the case $Re=25$ than in the case $Re=50$: this is only an
artifact of our release mechanism. Indeed, the steady “parachute” shape of the
capsule is deployed faster at $Re=50$ than at $Re=25$, leading to a shorter
initial acceleration phase of the trailing capsule towards the leading capsule
at $Re=50$ than at $Re=25$. When $d_{0}=0.5$ and $d_{0}=0.25$, we observe that
the interspacing distance steadily increases until the capsules reach the
corner region where it displays the same behavior as in the case of $d_{0}=1$,
and continues to increase downstream of the corner. While a steady value of
$d$ is not clearly reached within the considered time range, we can
extrapolate the trend and conclude that the interspacing distance seems to
saturate to values ranging from 0.6 to 0.8 depending on $Re$, $Ca$ and
$d_{0}$. Therefore, the pair of confined capsules we consider exhibit a
minimum stable interspacing distance $d_{min}$. Moreover, we note that the
slope of $d$ is greater in the case of lower initial interspacing distances,
suggesting that the relative velocity of the capsules is a function of their
interspacing distance. To investigate further this behavior, we show in Fig.
17 the velocity of the two capsules at $Ca=0.35$, $Re=50$ and $d_{0}$ ranging
from 0.25 to 1. We observe that the velocity of the trailing capsule is lower
than that of the leading capsule prior to entering and downstream of the
corner, and that the velocity difference increases with decreasing
interspacing distance. This velocity difference confirms the above
observations in terms of interspacing distance, in particular that a lower
interspacing distance results in a greater relative velocity between the two
capsules, i.e. an enhanced repulsive behavior. Moreover, we note in Fig. 17
that the difference in velocity minima between the leading and the trailing
capsules is always greater than the difference between their velocity maxima.
As a result, the residence time of the trailing capsule inside the corner
region is always greater than that of the leading capsule, and the corner
tends to separate the pair of capsules. The present analysis of the binary
interaction of capsules through a corner reveals that the two considered
capsules do interact in this geometry, affecting their motion and deformation.
In particular, the trailing capsule tends to be more deformed than the leading
capsule, and the corner tends to separate the pair of capsules. A natural
question that arises is that of the accumulation of such effects if more than
two capsules are considered.

(a) $Ca=0.15$

(b) $Ca=0.35$

Figure 16: Temporal evolution of $d$ for different initial interspacing
distance $d_{0}$ and Reynolds number $Re$. Figure 17: Effects of the initial
interspacing distance $d_{0}$ on the evolution of the capsules velocities $V$
at $Ca=0.35,Re=50$.

##  VI Train of ten capsules

(a) $Ca=0.15$

(b) $Ca=0.15$

(c) $Ca=0.25$

(d) $Ca=0.25$

(e) $Ca=0.35$

(f) $Ca=0.35$

Figure 18: Time evolution of the reduced surface areas and velocities of ten
capsules at $Re=50$ and $d_{0}=1/8$ for $Ca=0.15$, 0.25 and 0.35.

(a) $Ca=0.15$

(b) $Ca=0.25$

(c) $Ca=0.35$

Figure 19: Temporal evolution of $d$ for a train of 10 capsules at $Re=50$ and
$d_{0}=0.125$ for (a) $Ca=0.15$, (b) $Ca=0.25$ and (c) $Ca=0.35$. Figure 20:
$\mathcal{A}_{max}$ as a function of the capsule number.

(a)

(b)

Figure 21: (a) $V_{max}$ and (b) $V_{min}$ as a function of the capsule
number.

In this last section, we investigate the behavior of a train of ten capsules
flowing through the corner. We insert each capsule using the same procedure
employed in the previous section: a new initially spherical capsule appears at
a distance $D_{c}=30$ radii from the corner as soon as the preceding capsule
has advanced by a reduced distance $\tilde{d}=2\tilde{a}(1+d_{0})$. The
capsules are removed from the computational domain when they are less than one
initial diameter away from the outflow boundary. Our goal is to determine if
the findings of the previous binary capsule analysis accumulate when more than
two capsules are considered, especially with regards to the increased surface
area of the capsules and the separating effect reported in Section V. As such,
we plot in Fig. 18 the normalized surface area and velocity of each capsule of
the train at $Re=50$, $d_{0}=0.125$ and $Ca$ ranging from 0.15 to 0.35. The
same figure obtained in the case of $d_{0}=1$ is provided in Appendix B. In
Fig. 18, the darkness of the color corresponds to the position of the capsule
in the train: darker means increasing capsule number i.e. further downstream
along the capsule train. As mentioned in Section V, the initial peaks in the
surface area and velocity of the capsule are insertion artifacts and do not
contribute to the physics that is the focus of this section. We observe in
Fig. 18 that the behavior of the last capsule is significantly different than
that of the rest of the train. In Section V we hypothesized that the
difference in surface areas of the leading and the trailing capsules is due to
the fact that the wake of the leading capsule is significantly affected by the
presence of the trailing capsule. The present observation in Fig. 18
corroborates this statement: all of the capsules in the train see their wake
affected by a trailing capsule, except in the case of the last capsule. As a
result, its deformation is greater and extends closer to the channel walls,
thus decreasing its velocity. We also remark in Fig. 18 that this effect is
enhanced with increasing $Ca$. While noteworthy in the case of a pair of
capsules, this effect is less pertinent to the study of a train of capsules,
as only the core of the capsule train is relevant to typical microfluidic
applications. As such, in the remainder of this section our analysis is
focused on the first ninth capsules of the train.

As expected, a steady state is reached in the straight channel prior to the
corner for each capsule and for all $Ca$. While the steady surface area
remains constant with increasing capsule number, i.e. as we move further
downstream in the train of capsules, we observe that the velocity of the
capsules decreases. In particular the difference between the steady velocity
of the first and ninth capsules increases with increasing $Ca$. As the
capsules enter the corner region, they display the familiar pattern previously
described in Section IV and Section V, before relaxing to steady values. The
shape of the deviation pattern is strikingly similar across different capsules
of the train, regarding both the velocity and the surface area of the
capsules, except that they are shifted in time and magnitude. More precisely,
the surface area curves are shifted upwards with increasing capsule number
while the velocity curves are shifted downwards with increasing capsule
number. As a result, the maximum surface area of the capsule increases and the
velocity extrema decrease with increasing capsule number. This behavior is
more pronounced as $Ca$ increases. Additionally, we compare in Fig. 19 the
normalized interspacing distance $d$ between each pair of capsules in the
train. In Fig. 19, each curve is shifted in time such that $t=0$ corresponds
to $d_{min}$ inside the corner. For all $Ca$, we observe that the interspacing
distance $d(1,2)$ between the first and the second capsules increases to a
steady value close to 0.5, and that the corner has marginal effects on the
downstream evolution of $d(1,2)$: this behavior is identical to the case of
two capsules studied in the previous section. However, as we move downsteam in
the train of capsules, $d$ increases slower and slower prior to the corner
until it remains constant for capsule numbers greater than 7, at a steady
value $d\approx 0.7$ that decreases only marginally with increasing $Ca$.
After the transient regime due to the corner, $d(i,i+1)$ for capsule numbers
$i$ greater than 7 reaches a steady state that is slighly higher than prior to
entering the corner. In other words, the corner tends to increase the
interspacing distance, and therefore exhibits a separating effect. This
seperating effect is observed regardless of the initial interspacing distance
$d_{0}$, as was the case in the previous section when only two capsules were
considered.

Finally, in order to investigate further the influence of the capsule number
on the capsule dynamics, we plot in figures 20-21 the maximum surface area as
well as the maximum and minimum velocities of each capsule of the train for
varying Capillary numbers and interspacing distances. The difference in
minimum velocity (respectively, maximum velocity) between the first and the
ninth capsule is about 15% (respectively, about 7%) at $Ca=0.35$ while it is
about 11% (respectively, 2%) at $Ca=0.15$. Similarly, the difference in
maximum surface area between the first and the ninth capsule is about 4% at
$Ca=0.35$ and less than 1% at $Ca=0.15$. These results correspond to
$d_{0}=0.125$, while in the case of $d_{0}=1$ only deviations lower than 1%
are observed in the extrema of the capsule surface area and velocity (except
in the case of $Ca=0.35$ for which velocity deviations of 2% are observed).
The very small deviations observed in the case $d_{0}=1$ indicates that for
this interspacing distance the capsules interact very weakly. As such, there
exist a critical interspacing distance $d_{c}$ below which capsule
interactions are observed, with $0.125<d_{c}<1$.

The fact that $d_{c}$ is less than 1 can be surprising, as a normalized
interspacing distance of $d_{0}=1$ would typically be classified as a strongly
interacting regime in other geometries, e.g. in the T-junction investigated by
Lu et al. [10]. The main reason for the low interaction we observed is likely
due to the short residence time of the capsules in the corner region. Indeed,
Lu et al. showed that the residence time is determinant in the path selected
by the capsules in a T-junction geometry. Another reason for such a low
critical interspacing distance is related to the very confined configuration
we study: the capsule shape and behavior is primarily due to the presence of
the walls, while the small disturbances of the flow field due to the other
capsules only marginally contribute to each capsule dynamics. Future studies
could explore the dynamics of a train of capsules in a wider channel, i.e. in
a less confined configuration, where each capsule could be more influenced by
the wake disturbances of their preceding neighbor.

##  VII Conclusion

In the present work, the inertial and non-inertial dynamics of three-
dimensional elastic capsules flowing through a sharp corner are investigated.
The capsule trajectory, surface area, velocity and membrane stress are
analyzed in the cases of one, two and a train of ten capsules released
upstream of the corner. The channel Reynolds number ranges from 0.01 to 50,
the Capillary number representing the ratio of viscous stresses over elastic
stresses ranges from 0.075 to 0.35 and the initial normalized interspacing
distance between two capsules is varied from 1 to 0.125. The goal of this
study is to help provide practical guidelines in order to anticipate capsule
breakup and estimate throughput in inertial microchannels.

The case of a single capsule with no inertia was previously studied by Zhu &
Brandt [33], who reported that the capsule follows the flow streamlines
closely regardless of the Capillary number. In inertial flows, we found that
this statement is still valid for all considered Reynolds and Capillary
numbers. As the streamlines of the inertial flow cross the centerline of the
secondary channel $-$ the horizontal channel downstream of the corner $-$, the
capsule position is increasingly close to the top wall for increasing Reynolds
number, especially in the case of large Capillary numbers. However no
collision between the capsule and the wall of the secondary channel was
observed thanks to strong lubrication forces. In their study, Zhu & Brandt
also analyzed the velocity of the capsule centroid and the surface area of the
capsule membrane: they found that the capsule velocity decreases in the corner
and increases immediately after the corner, with an overshoot increasing with
membrane deformability. The surface area of the capsule was also found to
reach a maximum slightly shifted in time with respect to the minimum of
velocity. In the inertial regime, we observed that this behavior is enhanced
as the Reynolds number increases. However our results at $Re=1$ do not differ
significantly from results obtained in the non-inertial regime, which
corroborates the same observation made by Wang et al. [8, 9]. Moreover, at
sufficiently high inertia, capsule surface areas lower to equilibrium surface
areas are observed as the capsule relaxes to its steady state. In other words,
immediately after the corner the capsule oscillates around its steady shape.
This phenomenon is enhanced as the Capillary number increases. Additionally,
we reported that the relationship between the maximum surface area
$\mathcal{A}_{max}$ of the capsule and the Reynolds number is linear as long
as the Capillary number is kept below 0.35. At $Ca=0.35$, the relationship
between $\mathcal{A}_{max}$ and $Re$ is not perfectly linear and the curve
$\mathcal{A}_{max}(Re)$ is slightly concave. Moreover, from $Re=1$ to $Re=50$,
the maximum surface area increases nearly linearly over the full range of
$Ca$. At $Ca=0.35$, we compared the membrane stress to the capsule surface
area and found that (i) the time evolution of the average stress presents a
strong correlation to that of the membrane surface area, and (ii) in our
configuration, the value of the maximum stress is double that of the average
stress. As a result, observing the capsule surface area experimentally can
provide reliable insight into the average stress as well as an estimate of the
capsule maximum stress. This finding is of primary importance in the design of
microfluidic devices where capsule breakup is to be avoided, as well as in the
development of targeted drug delivery methods for which a controlled capsule
breakup is sought.

We then investigated the interaction of several capsules in the corner
geometry. First, two capsules are considered with varying initial interspacing
distances. Similar to the case of a single capsule, neither the trajectory of
the leading nor of the trailing capsule is observed to significantly deviate
from the flow streamlines. In the range of initial interspacing distance
considered, the velocity of the trailing capsule is found to be generally
lower than that of the leading capsule as well as that of a single capsule at
the same Reynolds and Capillary numbers. Similarly, the velocity of the
leading capsule is greater than that of a single capsule in the same
conditions. This velocity difference is also visible in the time evolution of
the interspacing distance $d$ between the pair of capsules. In particular, we
found that capsules initially located at $d_{0}\leq 0.5$ tend to separate.
This suggests that there exists a minimum stable gap $d_{min}>0.5$ between two
confined capsules. A systematic analysis of this effect is left for future
studies. In contrast, inside the corner the surface area of the trailing
capsule is found to be larger than that of the leading capsule and of the
single capsule in the same conditions. However, in the configuration we
consider where confinement is strong, the magnitude of these effects is small
even for capsules located very close to each other: the velocity of the
leading and trailing capsules only deviates by a few percents from that of a
single capsule. Next, we examined the case of a train of capsules and sought
to determine whether the effects observed with a pair of capsule accumulate.
While no interaction occurs for a large initial interspacing distance
$d_{0}=1$, we found that in the case $d_{0}=1/8$, the steady and extremum
surface areas of the trailing capsules increase by up to 5% and eventually
saturate at the tail of the train, around the ninth capsule. In all cases, the
corner is found to separate the pair of capsules as well as the capsule train,
which can be further evidenced from the analysis of the time evolution of the
capsule velocity inside the corner region.

We believe that the present work is a step forward towards providing practical
guidelines to avoid capsule breakup in inertial and non-inertial microfluidic
experiments. Future works could study capsule membranes exhibiting a strain-
hardening elastic behavior, e.g. as described by the Skalak law [45], as well
as vary the confinement ratio $\beta=2\tilde{a}/\tilde{W}$ in order to
consider high-throughput microfluidic devices. In the case of lower
confinement ratios in particular, we expect to see stronger capsule
interactions along with cross-stream capsule migration inside and downstream
of the corner. Finally, the present work could also be useful to develop
membrane characterization techniques, where viscoelastic membrane properties
could be inferred from the time-dependant evolution of a capsule of interest
through a corner.

##  Appendix A Time evolution of capsule surface areas in the case of two
interacting capsules

Figure 22 shows the evolution of the surface area of the leading and the
trailing capsules in the non-inertial regime (Fig. 22(a)), as well as at
$Re=25$ and $Re=50$ where the initial interspacing $d_{0}$ is 1 (Fig. 22(b))
and 0.25 (Fig. 22(c)).

(a) $Re=0.01$

(b) $d_{0}=1$

(c) $d_{0}=0.25$

Figure 22: Temporal evolution of the reduced surface area $\mathcal{A}$ of the
leading and the trailing capsules at various interspacings $d_{0}$ and various
Capillary and Reynolds numbers $Ca$ and $Re$.

##  Appendix B Train of capsules at large initial interspacings

We provide in Fig. 23 the time evolution of the surface area and velocity of
each capsule in a train of 10 capsules flowing through a corner at $Re=50$,
$Ca=0.35$ and a reduced initial interspacing distance between each capsule
$d_{0}=0.125$. As can be noted in this figure, the capsules in this regime do
not interact as the surface area and velocity evolution of each capsule is
almost identical.

(a) $Ca=0.15$

(b) $Ca=0.15$

(c) $Ca=0.25$

(d) $Ca=0.25$

(e) $Ca=0.35$

(f) $Ca=0.35$

Figure 23: Time evolution of the reduced surface areas and velocities of ten
capsules at $Re=50$ and $d_{0}=1$ for $Ca=0.15$, 0.25 and 0.35.

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