# Lift at low Reynolds number

Lionel Bureau<EMAIL_ADDRESS>Université Grenoble Alpes,
CNRS, LIPhy, F-38000 Grenoble, France  Gwennou Coupier  gwennou.coupier@univ-
grenoble-alpes.fr Université Grenoble Alpes, CNRS, LIPhy, F-38000 Grenoble,
France  Thomas Salez<EMAIL_ADDRESS>Univ. Bordeaux, CNRS, LOMA, UMR
5798, 33400 Talence, France

(August 27, 2024)

###### Abstract

Lift forces are widespread in hydrodynamics. These are typically observed for
big and fast objects, and are often associated with a combination of fluid
inertia (i.e. large Reynolds numbers) and some specific symmetry-breaking
mechanism. In contrast, low-Reynolds-number flows are usually overdamped and
do not exhibit such peculiar and interesting features. However, the inclusion
of boundary effects qualitatively changes this picture. Indeed, in the context
of soft and biological matter, recent studies have revealed the emergence of
novel lift forces generated by boundary softness, flow gradients and/or
surface charges. The aim of the present review is to gather and analyse this
corpus of literature, in order to identify and unify the questioning within
the associated communities, and pave the way towards future research.

###### Contents

  1. I Introduction
    1. I.1 Context
    2. I.2 Prelude: from soft lubrication to bulk elastohydrodynamics
  2. II Soft-lubrication lift
    1. II.1 Context
    2. II.2 The key mechanism
    3. II.3 Theoretical developments
    4. II.4 Experimental pieces of evidence
  3. III Elastohydrodynamic lift in external flow
    1. III.1 Dynamics of deformable particles under simple shear flow
    2. III.2 Particle in a simple shear above a wall
    3. III.3 Lift in a channel flow
    4. III.4 Other configurations
  4. IV Elastohydrodynamic interactions between particles
    1. IV.1 Pair interaction
    2. IV.2 Diffusion in suspensions
    3. IV.3 Creation of cell free layers in blood flow
  5. V Flow-induced electrokinetic lift
    1. V.1 Context
    2. V.2 Experimental observations
    3. V.3 Origin
    4. V.4 Modelling
    5. V.5 Comparison with experiments
    6. V.6 Concluding remarks
  6. VI Conclusion and perspectives
  7. A Boundary integral and stresslet formulation

##  I Introduction

Figure 1: Reynolds’ slider. Despite the time-reversal symmetry of the steady
Stokes equations, the fore-aft asymmetry of a tilted immersed slider that
moves tangentially to a neighbouring wall generates a difference in the
hydrodynamic pressure magnitudes at the front and at the back of the slider.
As a consequence, the latter experiences a normal force. This seminal system
conceived by Reynolds highlights the leitmotiv of this review: at low Reynolds
numbers, a symmetry breaking in transverse motion may generate normal forces.
Figure taken from [203].

###  I.1 Context

We are all familiar with dynamically-induced lift forces in hydrodynamics.
These are typically observed for big and fast objects, e.g. in aeronautics or
ball sports, and result from fluid inertia (i.e. large Reynolds numbers) and a
symmetry-breaking mechanism, such as wing shape or ball rotation. We define
here a lift force as a force acting perpendicularly to the initial motion of
the object, which is generally due to its initial acceleration or to its
advection by external flows.

While more discrete in everyday life, lift effects do also exist in low-
Reynolds-number flows, and often result from a key role played by the flow
boundaries. Indeed, the confined hydrodynamic interaction between two objects
(e.g. a particle, a wall, etc.) or the bulk fluid-structure interaction may
break the flow symmetry. This was already understood by Reynolds [203] through
his famous tilted slider. In the latter example, a lift force exists due to a
fore-aft geometrical asymmetry between two immersed rigid objects in sliding
relative motion (see Fig. 1). However, for a rigid sphere moving along a rigid
wall, the time reversal-symmetry of the steady Stokes equations coupled to the
fore-aft symmetry of the contact warrants the absence of any emergent normal
force in the problem. To overcome this impossibility, in the absence of any
inertial effects, other symmetry-breaking mechanisms are thus required, as
schematized in Fig. 2.

A prominent example is that of the lift induced by elastic deformations, for
which we wish to bridge the gap between two aspects of the phenomenon that
were historically studied by separate research communities. These are: on the
one hand, the small-gap limit where the key mechanism is the deformation of
elastic surfaces mediated by the fluid separating the objects in relative
motion; and, on the other hand, the long-distance limit where the deformation
is directly set by an externally applied flow. In the next subsection, we
shall give a flavour of these regimes, and of the transition between them,
through a qualitative description of some selected works, before entering in
the details in sections II and III. Interactions between soft particles, that
may be brought in close contact by the flow, lead to situations where both
short-range and long-range effects come into play. This situation, which is
relevant to understand the structure and rheology of suspensions of soft
particles, is reviewed in section IV.

Among other mechanisms giving rise to the symmetry breaking at the heart of
lift forces at zero Reynolds number, electrokinetic effects are dominant in
the literature and are presented in section V. They involve ionic currents
taking place when fluid-immersed rigid objects carrying surface charges are in
relative motion. Such phenomena are typically of interest in microparticle-
sorting applications, and embody an illustration of cases where non-inertial
lift forces emerge between objects that are not necessarily deformable.

Those are important mechanisms since they demonstrate that inertial-like
effects can be triggered at microscopic and biological scales through a smart
role of boundaries. The three mechanisms discussed so far will be the topic of
the three main sections of the present review, respectively. Other mechanisms,
unexplored yet or marginally explored, will be also briefly mentioned in the
perspectives of the concluding section.

Figure 2: Required symmetry-breaking mechanisms for inertial and viscous lift
forces to arise, transversally to the main motion imposed by an external flow
or by a driving field acting on the particle (blue arrows), leading to tilted
trajectories (green arrows). (a) Inertial lift: a bottom-up asymmetry may
suffice to create a pressure difference across the particle, leading to the
apparition of the lift force [158]. Examples of such a symmetry breaking
include (a1) asymmetry in shape as for an airplane wing or (a2) rotation-
induced asymmetry, as for a rotating ball sport experiencing a Magnus effect
in free space [61], or a sphere in a channel according to the Segré-Silberberg
effect [216]. Note that wings often exhibit a fore-aft asymmetry as well, for
drag reduction requirements. (b) Viscous lift: even in the presence of bottom-
up asymmetry, a fore-aft symmetry prevents the apparition of a lift force. If
it was not the case, let us assume the particle has a velocity given by (i). A
left-right reversal of the boundary conditions (e.g. external flow, gravity
field,…) would lead, by fore-aft symmetry of the particle, to the symmetric
motion (ii). However, the time reversibility of the steady Stokes equations
imposes also that the motion would be according to (iii). Compatibility
between (ii) and (iii) leads to the impossibility of the existence of a lift
force. This marks the strong difference with situations where fluid inertia is
not negligible. The additional fore-aft asymmetry may be due to e.g. (b1) the
particle geometry/deformation or (b2) the boundary conditions at the surface
of the particle, that can be linked to charge distribution or slip properties,
among others. In most situations, the presence of a wall warrants the bottom-
up asymmetry, but we shall also explore cases where lift occurs in the absence
of nearby walls but with a vertical flow gradient. Figure 3: Transition
between the long-distance regime and the near-wall regime where the soft-
lubrication framework applies: the lift velocity $\hat{U}_{L}$ of a lipid
vesicle sheared above a wall is plotted against its distance $z$ from the
wall, located at $z=0$ (green dots). The scale is set by the typical size $R$
of the particle. Blue triangles show the near-wall correction
$\Delta\hat{U}_{L}$ that measures the difference between the $\sim 1/z^{2}$
long distance contribution and the velocity $\hat{U}_{L}$. This contribution
scales as $1/z^{4}$. Inset shows three shapes at three different positions.
Data from [286], Fig.2, for a vesicle with reduced volume of 0.95 and no
viscosity contrast – see section III for definitions.

###  I.2 Prelude: from soft lubrication to bulk elastohydrodynamics

Lipid vesicles, which are drops enclosed by a lipidic membrane, have been
studied both in the vicinity of a wall and far from it, when deformed by a
shear flow. We highlight here a selection of studies on these objects that
explored in particular the transition between the near-wall and the far-wall
regimes.

The lift force upon detachment acting on a heavy, quasi-spherical, lipid
vesicle lying on a substrate was determined experimentally by Lorz  _et al._
[152] using an interferometric technique, that allows one to accurately
determine the gap profile between a particle and a substrate, if it is on the
order of some hundreds of nanometers. A theoretical interpretation of their
results, displaying quantitative agreement with the experimental data, was
proposed by Seifert [217], while in a letter published simultaneously in the
same journal Cantat and Misbah [37] also provided an expression for the lift
force. Both groups considered a vesicle pinned to a rigid substrate by an
adhesive potential, and determined, using similar approaches, the lift force
acting on the vesicle as it is pinned to the wall at a given distance $h$ from
it (given by the location of the potential energy minimum), while being still
able to deform and open an asymmetric gap between its surface and the wall.
Cantat and Misbah considered a 2D vesicle whose asymmetry is essentially
described by its front and back curvatures (allowing for feedback between gap
shape and flow stress through the curvature energy of the membrane), and found
a $h^{-1/2}$ dependence for the lift force. Seifert considered a 3D vesicle
but assumed a linearly increasing gap, and used the adhesion energy as a
control parameter, rather than the curvature energy. He found a $h^{-1}$
dependence for the lift force. In both models, forces depend quadratically on
the shear rate, a signature of soft-lubrication-based mechanisms that will be
described in section II. Interestingly, Cantat and Misbah also ran numerical
simulations of the lift force as a function of shear rate and highlighted a
transition between this quadratic regime, when the vesicle is pinned, to a
regime where the force is linear with the shear rate, when the vesicle is
detached. This points to the fact that while the opening of an asymmetric gap
is, in the near-wall regime, the result of local balance between flow stress
and particle elasticity, the shape of the vesicle that is far from the wall is
governed by the sole interaction with the bulk flow. This regime of shear-
induced lift will be discussed in section III. More recently, a similar case
of coupling between adhesion forces and shear-induced lift has been studied
for drops on a deformable polymer brush, leading to complex phase diagrams
[142]. We stress here that these shear-induced lift phenomena require the
considered object to be deformed, which, in the vanishing-Reynolds-number
context that we focus on, implies rather soft objects such as biological
materials, drops and artificial capsules. We will exclude here the particular
case of filaments, which have generally a very complex shape dynamics even in
the absence of walls, thus rendering difficult our quest for universal
mechanisms.

The transition from the near-wall regime to the far-field regime can also be
illustrated through the work of Zhao et al. who simulated a lipid vesicle of
typical size $R$ whose center of mass is placed at different positions $z$
above a wall, and sheared by the flow [286]. As shown in Fig. 3, a vesicle
sheared far from the wall adopts a tilted shape characterized by a point-wise
symmetry, which is reminiscent of the symmetry of the imposed flow, if the
presence of the wall is omitted. The vesicle lifts away from the wall with a
velocity $\hat{U}_{L}(z)$ that scales like $1/z^{2}$, which we will show to be
a generic behaviour. Here, the presence of the wall induces a flow that repels
the vesicle, whose shape is essentially dictated by the shear flow. If the
vesicle is placed closer to the wall, a correction to the far-field lift
velocity must be considered, which we show here to scale like $1/z^{4}$. This
higher-order term (with respect to the inverse distance $z^{-1}$) comes
together with a change in shape of the vesicle, as a result of the interaction
with the wall, mediated by the fluid. Indeed, as seen in the inset of Fig. 3,
the bottom surface of the vesicle is deformed. In this regime, the wall not
only induces a lift on the vesicle by inducing a mean hydrodynamic stress on
it, but it also modifies its shape, which in turn modifies the way the flow
pushes away the vesicle. If, far from the wall, a description in terms of
center-of-mass-to-wall distance $z$ is relevant, considering the near-wall
regime ($z\simeq R$ and below) requires to carefully study the shape of the
gap between the particle and the wall. The relevant parameter becomes $h$, the
minimal value of the gap thickness. When $h/R\ll 1$, the relevant framework to
analyze the phenomenology is the lubrication theory, which will allow one to
solve the flow-structure interaction problem. This strong elastohydrodynamic
coupling becomes the dominant mechanism giving rise to lift in the absence of
imposed flow that deforms the particles.

##  II Soft-lubrication lift

###  II.1 Context

Soft and wet contacts are widespread in nature and technology. Their rich
history in science and engineering involves issues and scales as diverse as
the lubrication of roller bearings [99] after the industrial revolution, or
the catastrophic geological landslides [36]. The properties of these contacts
implicate the coupling between the local hydrodynamic pressure induced by
fluid flow and the deformation of the confining solids. Often as well, for
stiff surfaces associated with industrial devices, non-Newtonian lubricant
effects (e.g. piezoviscous and thermoviscous behaviours) are expected to play
an important role [85] and require multiscale numerical modeling [69]. These
interesting features for industrial lubricant flows may be considered as more
minor corrections in the context of soft materials and small velocities at
stake here, despite some potential interest for sorting strategies [271], and
are thus not addressed in details in the following. In the same spirit, if the
fluid itself is non-Newtonian (e.g. a polymer solution), the nonlinear
rheology may introduce the necessary symmetry breaking that gives rise to
particle lift, even for rigid boundaries [140]. In such cases, particles would
e.g. migrate to regions of low normal-stress differences (low-shear-rate
regions). These forces will not be addressed in detail in the present review.

Recently, such an elastohydrodynamic (EHD) coupling gained attention in the
context of confined, soft and biological matter, where very compliant solids
and tiny length scales are common [33]. In fact, this coupling could
conceivably play a crucial role in the motion of various physiological and
biological entities. Examples are numerous and include e.g. the incredible
frictional properties of mammalian joints [168] through the fine interplay
between soft cartilage and viscous synovial fluid, or the crucial influence of
vessel boundaries on the motion of deformable red blood cells [96]. The normal
motion towards a soft wall was investigated in particular [9, 144, 263, 119],
with a special attention given to the collision [55] and rebound [97, 239]
properties.

Furthermore, through surface-forces apparatus (SFA) [144, 145, 254, 260, 264,
261] and atomic-force microscopy (AFM) [40, 250, 122, 100, 265, 16], the near-
contact EHD (termed soft-lubrication in the following) coupling offers an
alternative strategy for micro and nanorheology of fragile soft materials,
with the key advantage of avoiding any solid-solid adhesive contact that could
alter their properties.

Figure 4: (left) In a surface-forces apparatus (SFA), a flow between an
oscillating sphere and an elastic film is created. (right) Real (blue) and
imaginary (red) parts of the force-distance impedance response $G_{\omega}(D)$
obtained for an elastomer (crosslinked PDMS) and compared to soft-lubrication
theory (dashed lines). Figure adapted from [145].

Indeed, as shown in Fig. 4, the motion of a spherical probe in a viscous
fluid, and near a soft material, generates a lubrication pressure field that
can in turn deform the soft material. As a consequence, the lubrication gap is
altered as compared to the rigid case, and the impedance response is directly
affected, with the appearance of elastic contributions.

###  II.2 The key mechanism

Despite the irrelevance of inertia, a soft-lubrication lift force emerges for
elastic bodies moving past each other within a viscous fluid. Essentially, any
fore-aft-symmetric object moving within such a fluid and along a nearby soft
wall is repelled from the latter by a dynamically-generated emergent normal
force.

Figure 5: Principle of the soft-lubrication lift force. For a non-deformed
wall (a), the classical lubrication pressure $p(x)$ induced in the viscous
fluid by the tangential motion of a sphere at velocity $V$ is antisymmetric in
the transverse direction $x$ (b), resulting in a null net force (integral of
the pressure along $x$) in the normal direction $z$. In contrast, a soft
surface is deformed by the pressure field (c). The latter then loses its
symmetry (d), which results in a finite emergent normal force: the soft-
lubrication lift force. Figure adapted from [226].

.

This force intimately arises from a symmetry breaking in the contact shape
(see Fig. 5), and thus the associated flow fields, due to the EHD coupling
described above. Qualitatively, the elastic deformation induced by the
hydrodynamic pressure generates a self-sustained asymmetric contact similar to
the one in Reynolds’ rigid slider (see Fig. 1), and thus a normal force. This
effect is well known at macroscopic scales, for relatively rigid materials
such as car tires undergoing aquaplaning, or industrial roller bearings
getting deformed in operating motors and machines.

Moving on to the context of mesoscale physics and soft matter, the earliest
theoretical descriptions of such a soft-lubrication lift effect are the ones
by [47], Dowson and Jin [60], Lequeux  _et al._ [143], as well as by Sekimoto
and Leibler [218], to the best of our knowledge. The underlying motivation
behind these similar approaches is the calculation of forces between soft
curved surfaces undergoing shear, which are important for the interpretation
of SFA measurements, the physics of cartilage, and the rheology of a variety
of complex fluids such as suspensions of colloidal particles protected by
grafted or adsorbed polymer chains, suspensions of gel microparticles, or
polymer emulsions and alloys on certain time scales.

The general idea can be illustrated from e.g. [218], through the calculation
of the soft-lubrication interaction between a cylindrical object of radius $R$
moving at transverse velocity $V$ past and nearby a flat wall (as in Fig. 5),
within a viscous fluid of dynamic viscosity $\mu$, both solids being covered
by polymer brushes. The latter are modeled as identical thin linear-elastic
compressible layers.

A thin linear-elastic compressible layer with Lamé coefficients of similar
magnitudes can be mapped onto a Winkler’s foundation, i.e. a mattress of
independent springs with a local and linear response to the external pressure
field $p(x)$ (see Fig. 6).

Figure 6: Schematic involving a Winkler’s foundation, i.e. a model and simple
type of elastic substrate characterized by an assembly of parallel,
independent and identical springs, leading to a local and linear response to
the external pressure field $p(x)$. For thin enough, compressible elastic
layers, such a toy model even provides a quantitative description of their
elastic Green’s function. Figure adapted from [57].

.

Interestingly, the Winkler’s foundation was proven to be of great modelling
power [57]. In such a description, the normal deformation field is given by
$\delta(x)=-Lp(x)/(2G)$, where we introduced an effective shear modulus $G$ as
well as an effective thickness $L$ of the mattress, and where we assumed for
simplicity a full compressibility (i.e. vanishing Poisson ratio). Near its
minimum $h_{0}$, the steady-state fluid-gap profile $h(x)$, along the
transverse direction $x$ of motion, is well approximated by a parabola (i.e.
second-order development of a spherical contact near the apex) corrected by
the elastic deformation of the elastic layer induced by the hydrodynamic
pressure. It thus follows that:

$h(x)\simeq h_{0}+\frac{x^{2}}{2R}+\frac{L}{2G}p(x)\ .$ (1)

#### Scaling analysis

The main idea is then based on a hierarchical scale separation, by considering
that the elastic deformation is small compared to the fluid-gap thickness,
which is itself small compared to the cylinder radius. By invoking the steady
Stokes equations in the lubrication approximation, it follows that the
leading-order pressure magnitude scales as $\sim\mu V\ell/h_{0}^{\,2}$, where
$\ell=\sqrt{2Rh_{0}}$ is the characteristic horizontal length scale, given by
the Hertz-like hydrodynamic radius emerging from the parabolic approximation
in Eq. (1). In such a framework, and in addition to the lubrication parameter
$h_{0}/R\ll 1$, one finds $\kappa\sim\mu\sqrt{R}VL/(Gh_{0}^{5/2})$ – the
dimensionless compliance – as a second natural small parameter of the problem.
Then, an expansion of the soft-lubrication flow problem is performed at order
$1$ in $\kappa$. The zeroth-order contribution corresponds to the purely rigid
case with gap profile $h^{(0)}(x)\simeq h_{0}+x^{2}/(2R)$ and a zeroth-order
pressure field $p^{(0)}(x)$ that can be computed analytically, and that is
found to be antisymmetric in $x$ (Fig. 5(b)). This is expected in view of the
time-reversal symmetry of the steady-Stokes equations and the fore-aft
symmetry of the contact shape. As the normal force per unit length $F_{z}$
exerted on the cylinder is dominated, in the lubrication approximation, by the
pressure contribution (i.e. the ratio between the viscous shear stress and the
pressure is of order $h_{0}/\ell\ll 1$), the latter antisymmetry of
$p^{(0)}(x)$ implies to evaluate the first correction $p^{(1)}(x)$ induced by
the elastic deformation. The magnitude of the latter scales as $\sim\kappa(\mu
V\ell/h_{0}^{\,2})$. Therefore, the resulting normal force per unit length
reads, at order 1 in $\kappa$:

$F_{z}\simeq\int_{-\infty}^{\infty}\textrm{d}x\,p^{(1)}(x)\sim\frac{\mu^{2}V^{2}R^{3/2}L}{Gh_{0}^{7/2}}\
.$ (2)

Interestingly, one sees that this soft-lubrication lift force increases with
the viscosity of the fluid, driving velocity, compliance, contact area, and
confinement. Then, Sekimoto and Leibler [218] confront the theoretical
predictions with the result of force measurements under shear between surfaces
covered with grafted polymer chains. While the observed normal forces in these
experiments are often attributed to brush swelling due to external flows, the
authors quantitatively argue here that the brush-deformation-induced soft-
lubrication lift force is instead the dominant mechanism behind the common
observations.

#### Soft-lubrication theory

Here, as an illustration of the typical method for the readers, we aim at
retrieving the scaling result above quantitatively. We place ourselves in the
rest frame of the cylinder. We introduce the fluid velocity field $u(x,z)$
along $x$, and the dimensionless variables: $z=Zh_{0}$, $h=Hh_{0}$, $x=X\ell$,
$u=UV$, $p=P\mu V\ell/h_{0}^{2}$, and $F_{z}=\mathcal{F}_{Z}\mu
V\ell^{2}/h_{0}^{2}$. Hence, the gap profile given by Eq. (1) is non-
dimensionalized as:

$H(X)=1+X^{2}+\kappa P(X)\ ,$ (3)

where:

$\kappa=\frac{L\mu VR^{1/2}}{\sqrt{2}Gh_{0}^{5/2}}\ .$ (4)

In the lubrication approximation where $h_{0}\ll R$, the incompressible steady
Stokes equations reduce to [203, 17, 185]:

$\partial_{ZZ}U=\partial_{X}P\ .$ (5)

with $\partial_{Z}P=0$, In addition, we impose no-slip boundary conditions,
through $U(X,Z=-\kappa P)=-1$ and $U(X,Z=H-\kappa P)=0$. Solving Eq. (5) with
these boundary conditions, and invoking the condition of volume conservation
yields the Reynolds equation:

$\partial_{X}\left(H^{3}\partial_{X}P+6H\right)=0\ .$ (6)

Solving the latter with vanishing pressure in the far field, one can then
calculate the dimensionless normal force (per unit length) exerted on the
cylinder, through:

$\mathcal{F}_{Z}=\int_{-\infty}^{\infty}\textrm{d}X\ P(X)\ .$ (7)

Since $\kappa\ll 1$, perturbation theory [226] using $P\simeq P^{(0)}+\kappa
P^{(1)}$, allows one to integrate Eq. (6) at first order in $\kappa$,
eventually leading to the dimensionless lift force:

$\mathcal{F}_{Z}\simeq\frac{3\pi\kappa}{8}\ ,$ (8)

and thus providing Eq. (2) as well as the missing prefactor therein.

###  II.3 Theoretical developments

Let us make a few comments about Eq. (2). This typical asymptotic expression
of the soft-lubrication lift force per unit length relies on several
assumptions: a 2D problem, a pure linear and local compressible elastic
rheology, a vanishing compliance, a near-contact/confinement situation, a
parabolic contact shape, etc. It is thus expected to find important
modifications of the lift force in more complex or realistic situations. First
of all, while the perturbative/asymptotic nature of the approach is expected
to hold at small elastic deformations, through the explicit factor $\sim(\mu
V)^{2}/G$ in the force expression, dimensionality and geometry are expected to
modify the dependencies on the various length scales of the problem.
Similarly, the exact elastic rheology (compressible vs incompressible, thin vs
thick) will modify the constitutive response between the pressure $p(x)$ and
the elastic deformation $\delta(x)$. Indeed, while a linear response is
expected to hold at small deformations, the simple Hookean proportionality
relation $\delta(x)/L\sim p(x)/G$ may be replaced by a nonlocal relation of
the type:

$\delta(x)\sim\frac{1}{G}\int_{-\infty}^{\infty}\textrm{d}x^{\prime}\,g(x-x^{\prime})p(x^{\prime})\
,$ (9)

Table 1: Different scalings of the soft-lubrication lift force for a sphere in
3D, for various geometries and rheologies of the elastic substrate [227].
Here, $\mu$ is the shear viscosity of the lubricant, $V$ the relative
tangential speed, $\omega$ the angular speed, $n$ the contact-degeneracy
parameter, $H_{\textrm{l}}$ the elastic substrate thickness, $R$ the sphere
radius, $G$ the shear modulus of the elastic substrate, $h_{0}$ the fluid-gap
thickness (noted $h$ in this manuscript) and $h_{\textrm{s}}$ the shell
thickness.

where $g$ is the dimensionless elastic Green’s function (in a 2D description
here), that simply reduces to a Dirac distribution in the Winkler’s case
discussed above. Qualitatively, $\delta$ is still the linear response to the
source $p(x)$ with a magnitude set by the compliance $1/G$ (and even a
proportionality in Fourier space). Quantitatively, we expect differences
depending on the exact Green’s function characterizing the response. To go one
step further along this line of thought, substrate viscoelasticy and
poroelasticity are expected to add one or several new time scale(s) in the
problem, rendering the response time-dependent, including memory effects.
Similarly, large deformability and/or elastic nonlinearities may induce a
saturation or even a non-monotonic behaviour of the force with gap distance,
beyond the small-deformation scaling in Eq. (2). Finally, adding non-Newtonian
effects, or conservative surface forces, such as van der Waals forces and
screened electrostatic interactions, is expected to lead to non-trivial
effects and coupling with the EHD picture above. One thus realizes that there
was room and need for further theoretical developments around Eq. (2).

Perhaps the most emblematic example of such developments, is the series of
work by Skotheim and Mahadevan [226, 227]. Therein, a systematic zoology of
various non-conforming and conforming contact geometries and elastic
responses, in 2D and 3D, was addressed analytically and numerically. This is
exemplified in Table 1, with a collection of asymptotic lift-force scalings
that are valid at large-enough distance (i.e. weak deformation) for the
particular case of a 3D sphere. This body of work applies the same soft-
lubrication framework as the one introduced above, and employs as well
numerical resolutions to go beyond the scaling expressions and the small-
compliance limit. Interestingly, thanks to the numerical resolution, a maximum
in the lift-force-vs-gap-distance behaviour was found in some cases. Note that
a capillary version of soft lubrication, analogous to the elastic one at stake
here, was not addressed therein, but was studied previously in the context of
rising bubbles [233].

Nearly at the same time, and importantly, Beaucourt  _et al._ [19] understood
the interest of such a lift force in a biophysical context. These authors
addressed in particular the case of vesicles, as model biological elastic
microparticles, during their motion in water near soft glycocalix layers.
Putting numbers on the lift expression, they found forces with magnitudes
lying in the physiological range. This work thus highlights the potential
importance of such soft-lubricated couplings for the dynamics of red blood
cells, and thus biological processes that are essential to life. Note that his
article is one of the few including both deformation of the substrate and
deformation of the particle by the shear flow (Fig. 7).

Figure 7: A deformable particle in a shear flow is elongated and tilted,
resulting in a lift away from the wall. In the meantime, deformation of the
neighboring elastic wall also induces lift, as in Fig. 5. Figure adapted from
[19].

To go beyond scaling symbols in the soft-lubrication lift expression for a
sphere in 3D is a more intricate task. An elegant solution based on Lorentz’s
reciprocal theorem was sketched by Stone et al. during an oral communication
at the 2004 APS-DFD meeting [231]. It was later on systematically explored by
Urzay  _et al._ [249] for the problem of a sphere translating and rotating
near a thin compressible elastic layer. Later on, Urzay [248] generalized the
scope to the added role of DLVO intermolecular interactions. There, the
competition of the hydrodynamic, intermolecular and deformation effects leads
to forces which do not scale linearly with the velocity, and produce a non-
additivity of the intermolecular effects. Mainly, the intensity of the
repulsive forces is reduced while the intensity of the attractive forces is
increased, collectively leading to an effective and reversible EHD adhesion
scenario. Besides, a more exotic irreversible EHD adhesion regime was also
found. Elastohydrodynamic corrections to the DLVO framework for the critical
coagulation concentration of electrolytes were obtained too.

Beyond global quantities, such as the net normal lift force, local details on
the contact shape matter as well, such as the fore-aft asymmetry in the space-
dependent elastic deformation profile. Furthermore, in 2D, the self-similar
properties of the shape of the soft-lubricated contact zone in a high-loading
case was investigated by Snoeijer  _et al._ [230]. Asymptotic results for the
liquid-gap thickness below a soft sphere in 3D pressed against a hard wall
were shown to agree with both experimental and numerical data. Later on,
Essink  _et al._ [68] managed to obtain analytical scaling laws in the high-
loading regime. In this work, the authors described various regimes of soft
lubrication for two-dimensional cylinders in lubricated contact with compliant
walls. They addressed and connected the limits of small and large entrainment
velocities, near thin elastic coatings, both compressible and incompressible.
The analysis relies intimately on the introduction of an elastohydrodynamic
boundary layer that appears at the edge of the contact region. Importantly, in
order to identify the proper regime in a given experimental setting, this
theoretical work reveals the importance of correctly estimating the ratio
between the elastic deformation and the fluid-gap thickness, as well as the
ratio between the thickness of the elastic material and the hydrodynamic
radius.

So far, the problems studied involved constant fluid-gap thickness and
transverse velocity. The case of a more general but prescribed motion in 2D
and 3D near a Winkler’s foundation was addressed analytically and numerically
by Weekley  _et al._ [267]. When the particle moves from rest towards the
wall, fluid trapping beneath the particle leads to an overshoot in the normal
force on the particle, with trapping at early times and fluid draining at late
times. When the particle is pulled from rest away from the wall, a transient
adhesive normal force emerges. When a cylinder moves from rest transversely
along the wall, an overshoot in the transverse drag appears. However, the case
of a free particle immersed in a viscous fluid and near a soft wall, with all
degrees of freedom allowed, is relevant to experiments and needed to be
addressed. The associated leading-order soft-lubrication interaction matrix
was derived by Salez and Mahadevan [212] in 2D, and later on by Bertin  _et
al._ [20] in 3D. Interestingly, a counterintuitive zoology of fluid-inertial-
like solutions emerges at low Reynolds number. These encompass: Magnus-like
effects, enhanced sedimentation, adhesive-like EHD forces, roll reversal,
oscillations, etc. In addition, the existence of a spontaneous soft-
lubrication torque, at next (i.e. second) order in $\kappa$, was revealed in
2D by Rallabandi  _et al._ [199], for compressible and incompressible
settings.

We have focused on purely elastic materials in the description above. In such
a framework, the softer the material, the larger the effect, until an optimum
or saturation eventually occurs. This suggests to employ rather soft materials
in practice. However soft gels and elastomers are inevitably prone to
poroelastic and viscoelastic effects. While the former have been sketched in
the lift context [82, 226, 227], the latter needed to be incorporated in
details. Pandey  _et al._ [187] thus analyzed soft-lubricated contacts with
viscoelastic walls. In particular, the authors focused on three canonical
viscoelastic descriptions, namely: Kelvin-Voigt, standard linear, and power-
law rheologies. They showed how viscoelasticity modifies the contact
properties when the time scales of both the substrate and the driving become
comparable. Mainly, they found modified asymptotic scaling laws for the lift
force, indicating a decrease of the magnitude of the EHD effect due to inner
viscous contributions within the viscoelastic material. Later on, Kargar-
Estahbanati and Rallabandi [120] employed Lorentz’s reciprocal theorem to
derive a general integral relation between the soft-lubrication lift force and
the linear response function of the soft substrate. They first analyzed the
lift force as a function of Poisson’s ratio and thickness of the elastic
material. Moreover, they found a superposition of steady and oscillating modes
for a lubricated object moving near a viscoelastic material. The amplitudes
and phases of these modes contain information about the elastic and viscous
components of the material response, thus opening the way to the fine
characterization of the mechanical properties of materials via lift force
measurements. Compared to normal mode excitations [40, 250, 144, 145, 254,
122, 260, 264, 100, 265, 16, 261], the interest of the transverse mode for
such a purpose is rooted in Eq. (2), where a $\sim V^{2}$ dependence appears.
Therefore, with a sinusoidal excitation, a frequency doubling is expected,
thus enabling the use of a region of the spectrum that is distinct from the
one at the driving frequency.

As introduced above, Winkler’s foundation is the simplest linear and local
elastic model (see Fig. 6) [57]. Since it avoids the complication of nonlocal
responses associated with elastic materials, it is often used as a simplified
model for thought. A natural question emerging from that is how valid such a
model is to describe actual physical systems, with a particular focus on the
lift problem. In particular, in the limit of strictly incompressible and thin
elastic layers, one expects an infinite resistance to deformation, and hence
the Winkler’s approach breaks down. Chandler and Vella [42] provided an answer
to such a question by formally deriving a lift force that interpolates between
the Winkler and incompressible limits for thin elastic layers. They found that
the applicability of the Winkler model is not determined by the value of the
Poisson ratio alone, but by some compressibility parameter that combines the
Poisson ratio with a measure of the layer slenderness, which depends on the
problem under consideration. Essentially, for Poisson ratios strictly smaller
than 0.5, the crossover to Winkler’s model as the thickness is reduced is
rooted in the elastic Green’s function itself [144, 120].

Finally, the effective compliance of a material, and the lift force as a
consequence, can be increased tremendously by using slender geometries, such
as membranes and plates. The EHD coupling in such systems was addressed by
Daddi-Moussa-Ider and collaborators [50, 51]. In the first article, the
authors computed the leading-order frequency-dependent translational and
rotational mobilities of an axisymmetric particle immersed in a viscous fluid
and moving near an elastic cell boundary allowed to stretch and bend. The
authors found that the translation-rotation coupling mobility is primarily
determined by bending, whereas shearing (i.e. wall-bounded shear flow) mostly
affects the rotational mobility. In the second article, the authors derived
the lift force exerted on a rigid spherical particle translating parallel to a
finite-sized membrane. Specifically, the Lorentz reciprocal theorem was
employed, as well as a perturbative expansion for small deformations of the
membrane. The authors reported interesting attractive and repulsive regimes
depending on the dominant elastic mode (i.e. shearing vs bending) at play.

As a concluding remark, we expect no qualitative difference between the two
dual situations of: i) a rigid particle near a soft wall; and ii) a soft
particle near a rigid wall. This is reminiscent of the situation in dry
elastic contacts [114, 162].

###  II.4 Experimental pieces of evidence

Despite the above abundant theoretical literature, experimental evidence for
such a soft-lubrication lift force in soft matter is recent and scarce.

A possible preliminary qualitative observation may have been reported in the
context of smart lubricants and adsorbed polyelectrolytes by Bouchet  _et al._
[28]. The authors investigated the lubricant properties of a strong
polyelectrolyte, in aqueous solutions of different salt concentrations. They
first studied how the morphology of the adsorbed layer could be modified by
increasing the salt concentration. Then, a complex velocity dependence of the
friction was observed, with a maximum value at intermediate velocities and
even some hysteresis. A progressive increase in separation between the rubbing
surfaces with velocity was also observed (see Fig. 8).

Figure 8: Shear stress as a function of driving velocity measured with a SFA
covered by strongly adhesive polyelectrolyte layers and in presence of a
lubricant. Figure taken from [28].

These observations were qualitatively discussed in terms of an hypothetical
indication of the presence of a soft-lubrication lift force.

A first quantitative study, by Saintyves  _et al._ [210], showed an effective
reduction of friction induced by the soft-lubrication lift force. The authors
employed a fluid-immersed negatively buoyant macroscopic cylinder moving along
a soft inclined wall. They observed a steady-state sliding regime with an
effective friction that was significantly reduced relative to the rigid case
(see Fig. 9).

Figure 9: (left) A rigid cylinder immersed in a viscous bath slides along an
inclined plane covered with a thin elastic layer. Fluorescent particles
embedded in the latter make it possible to observe its deformation using a
laser and a camera. (right) Sliding speed $V$ as a function of normalized time
$T=t/t_{\textrm{tot}}=tV_{\infty}/L$, for several shear moduli of the elastic
coating. Here, $V_{\infty}$ is the time-averaged steady-state sliding speed,
$L$ is the total length of the substrate, and $t$ is the time. The dotted line
corresponds to the case of a bare glass substrate. Figure adapted from [210].

The observations were rationalized by invoking the soft-lubrication lift. This
study was followed up by a work dedicated to the rotational motion of the
cylinder [211]. The authors experimentally quantified the steady spinning of
the cylinder and theoretically showed that it is due to an aspect-ratio
dependent combination of a soft-lubrication torque generated by the flow and
the viscous friction on the edges of the finite-length cylinder. We note that
the contribution of edge effects within the lubricated motion of a cylinder
moving near flat rigid walls was then revisited in more details [240]. The
experimental results of [211] were consistent with a transition from an edge-
effect dominated regime for short cylinders to a gap-dominated soft-
lubrication regime when the cylinder is very long. A puzzling feature about
these two studies is the fact that the Winkler’s foundation describes best the
observations, despite the rather incompressible character of the elastomers
used. The answer to that puzzle might be given Chandler and Vella [42] in the
lift context. Indeed, for Poisson’s ratios strictly smaller than $1/2$, an
incompressible layer will eventually behave as a compressible one for small-
enough film thicknesses.

Subsequently, an experimental study by Davies  _et al._ [54] revealed the
significance of the soft-lubrication lift force in biological and microscopic
settings. The authors addressed the motion of glass microbeads in a linear
shear flow close to a wall bearing a thin soft biomimetic polymer brush.
Combining microfluidics and optical tracking, they demonstrated that the
steady-state bead-to-surface distance increased with the imposed shear rate
(see Fig. 10).

Figure 10: (left) A glass microbead is advected in an aqueous environment
within a microfluidic chamber whose walls are decorated by a biomimetic
polymer brush. (right) The distance to the wall is measured versus the imposed
shear rate, for three brush elastic moduli (increasing from orange to pink).
Theoretical lines including the soft-lubrication lift contribution are fitting
the data. Figure adapted from [54].

The article is concluded by physiological estimates, indicating the potential
relevance of the effect for the transport of red blood cells – and thus for
life processes.

The same year, a macroscopic study by Rallabandi  _et al._ [198] demonstrated
the large amplification of the soft-lubrication lift for very compliant
boundaries associated with slender geometries (see Fig. 11).

Figure 11: Gravitational sedimentation of a macroscopic sphere immersed in a
viscous fluid, along a vertical membrane under tension, exhibits an important
normal drift induced by the soft-lubrication lift. Figure taken from [198].

The authors combined theory and experiments in order to show that a small
particle moving along an elastic membrane through a viscous fluid is repelled
from the membrane due to soft-lubrication forces. An analytic expression for
the particle trajectory is derived, including a normal migration velocity of
the particle that is quadratic in speed and depends on a combination of the
tension and bending resistances of the membrane. The quantitative agreement
with the theoretical predictions with no fitting parameter indicates once
again the presence and relevance of the soft-lubrication lift force.
Furthermore, due to the slenderness of the membrane, the effective compliance
is large and the effect is strong enough for separation and sorting of
particles on the basis of both their size and density. Once again, the
relevance for biology – where membranes are widespread – is discussed.

The above recent experimental literature provides confidence in the existence
of the soft-lubrication lift force, as well as in its importance at small
scales and for biology. However, in these works, the quantitative evidence for
the soft-lubrication lift is only indirect since only trajectories and
effective friction coefficients are typically measured. A direct measurement
was thus needed. The first SFA and AFM direct force measurements of the soft-
lubrication lift force at the nanoscale were performed by Vialar  _et al._
[253] and by Zhang  _et al._ [284], respectively. On the one hand, in the
former SFA study, the authors investigated the behavior of mica surfaces
coated with microgels under shear and compression. The emergence of velocity-
dependent, shear-induced normal forces was observed and quantified (see Fig.
12).

Figure 12: (left) Shear-induced force as a function of driving velocity
measured with a SFA covered by microgel layers and in presence of a lubricant,
for various gap thicknesses increasing from brown (130 nm) to grey (300 nm)
symbols. (right) Measured lift force as a function of lubricant thickness.
Figure adapted from [253].

Moreover, the data are in agreement with the soft-lubrication lift force but
revealed a counterintuitive value of the microgel elastic modulus. On the
other hand, Zhang et al. employed an AFM colloidal probe near an horizontally-
oscillated elastomeric layer and measured the average lift force as a function
of the gap size (see Fig. 13), for various driving velocities, viscosities,
and stiffnesses.

Figure 13: (left) A soft substrate is fixed atop a rigid piezo stage that is
transversally oscillated along time $t$, at angular frequency $\omega$ and
with amplitude $A$. A rigid sphere is glued to an AFM cantilever and immersed
in a viscous liquid lubricant near the substrate. The normal force
$F_{\textrm{N}}$ exerted on the sphere at a given distance from the surface is
directly measured from the deflection of the cantilever along $z$. (right)
Temporal average $F$ of $F_{\textrm{N}}$ as a function of the gap distance $d$
to the substrate, for both rigid (silicon wafer) and soft polydimethylsiloxane
(PDMS) substrates. The inset shows a log-log representation of the data for
the soft substrate where the solid line indicates a $-5/2$ power law
characteristic of an EHD lift force in the case of a semi-infinite
incompressible elastic substrate [227]. Figure adapted from [284].

The results are in agreement with a quantitative model developed from the
soft-lubrication theory for small compliances [20]. For larger compliances, or
equivalently for smaller confinement length scales, an empirical scaling law
for the observed saturation of the lift force is proposed and discussed. This
high-loading conjecture should be compared in future to recent theoretical
developments [68].

##  III Elastohydrodynamic lift in external flow

Even when not directly deformed by the presence of a wall, deformable
particles will exhibit a specific dynamics when interacting with flow velocity
gradients. This will in turn create an additional flow which is often
compatible with the top-bottom asymmetric boundary conditions only if the
particle has a normal motion relatively to the flow direction.

A reference situation is that of a particle in a simple shear flow above a
wall (Sec. III.2), but other configurations give also rise to lift, among
which the flow of a particle in a channel (Sec. III.3). The interaction
between two flowing particles is also an example of lift, whose study is often
of interest because it is the fundamental mechanism of non-Brownian diffusion
in a suspension (Sec. IV). Other, more marginal, situations for the lift of a
single particle are examined in Sec. III.4.

Before going further, a quick overview on the dynamics of soft particle under
simple shear flow is needed here, with the aim to focus on the main
information needed to understand the dependence of the amplitude of the lift
force with the mechanical properties of the considered particle.

###  III.1 Dynamics of deformable particles under simple shear flow

A non-spherical rigid particle cannot maintain a steady orientation under
shear flow, due the rotational component. By contrast, particles with liquid
cores can maintain a fixed shape under shear flow while accommodating the
rotational stresses by a rotation of their inner fluid. The way they do so
depends on the detail of their mechanical properties, and a combined motion
with whole rotation at fixed shape, coupled with shape oscillations, can be
met.

In this review, the term vesicle will be used to designate a drop of liquid
encapsulated by an incompressible elastic layer. The sole elastic deformation
energy of this layer is thus associated to bending. In particular, initially
spherical vesicles are non-deformable, by virtue of incompressibility of the
membrane and of the encapsulated fluid. Vesicles have long been considered as
model systems to mimic the main behaviour of more complex cells such a red
blood cells. By neglecting the elastic shear contribution of the underlying
cytoskeleton, researchers have gained simplicity, that was needed for
theoretical modeling purposes and to reduce cost and complexity of numerical
simulations. In parallel, models for extensible shells — which are often named
capsules — have been developed. Finally, The last ten years have seen the
development of more accurate models to describe membranes of cells, that
includes the possibility for shear at constant surface in the membrane plane.

The literature on the dynamics under shear flow of liquid drops [200, 232],
vesicles (see the [255] review and some subsequent articles [73, 21, 280, 72,
74]), capsules (see the [13] review and also [228, 258, 87, 65, 64, 14, 283]),
or (models of) red blood cells (see a thorough introduction in [167] and some
subsequent articles [102, 166, 89]) is extended and is still being enriched by
studies of increasing refinement in the modeling and in the experimental
approach. Adding more complexity to the rheological properties of the membrane
leads in general to more complexity in the diagrams of dynamical states. The
parameters of the system are usually combined in a set of dimensionless
numbers such as reduced volume $\nu$ (characterizing initial deflation of the
object), viscosity ratio $\lambda$ between the inner and the outer fluid, and
capillary number(s) $Ca$, that compare hydrodynamic shear with either surface
tension or bending rigidity or shear elasticity of the membrane.

An oversimplified picture is that for low $\lambda$ and high $Ca$, particles
adopt a drop-like behaviour, called tank-treading (TT), where the particle
keeps a constant angle $\theta$ (see Fig. 14 for notation) relatively to flow
direction. For high $\lambda$ or low $Ca$, particles behave more like solids
and tumble under flow, with a periodic evolution of $\theta$. In between, a
rich zoology of motions has been described, from small oscillations around a
given angle to off-plane orbital motion. The oscillations of the main axis of
the particle is often accompanied by shape oscillations of more or less
important amplitude. For ellipsoidal rigid particles, orbits have been exactly
described by Jeffery [113]; this description is also a good proxy for tumbling
motion of not too deformable particles like red blood cells [167] or capsules
[65] under moderate shear flow.

###  III.2 Particle in a simple shear above a wall

Once the dynamics of particle in unbounded flow is solved, the flow produced
in presence of the wall can be calculated, and the resulting lift velocity
determined. A question that arises naturally is what is the domain of validity
of this approximation in the particle-to-wall distance domain ? To understand
how the correction terms emerge, it is convenient to introduce the boundary
integral formalism to solve the Stokes flow due to the force distribution on
the particle surface. From this formalism will also naturally emerge the
notion of stresslet, the key ingredient to quantify far-field lift. In order
to focus on the underlying physics we leave the technical details to the
appendix.

To facilitate the notations and summations, we introduce the position vector
$\mathbf{x}=(x_{1},x_{2},x_{3})$, where $x_{1}$ corresponds to the flow
direction, and $x_{3}$ to the direction perpendicular to the wall (located at
$x_{3}=0$), i.e. to $z$ in Fig. 2.

We consider particles made of a 2D interface delimiting their interior, filled
with a fluid of viscosity $\mu^{\prime}\equiv\lambda\mu$. In that case, the
flow field at any point $\mathbf{x}_{0}$ outside the particle reads [194]

$\displaystyle u_{j}(\mathbf{x}_{0})$ $\displaystyle=$ $\displaystyle
u_{j}^{\infty}(\mathbf{x}_{0})-\frac{1}{8\pi\mu}\int_{S}\Delta
f_{i}(\mathbf{x})G_{ij}(\mathbf{x},\mathbf{x}_{0})dS$ (10) $\displaystyle+$
$\displaystyle\quad\frac{1-\lambda}{8\pi}\int_{S}u_{i}(\mathbf{x})T_{ijk}(\mathbf{x},\mathbf{x}_{0})n_{k}(\mathbf{x})dS.$

The imposed flow is $\mathbf{u}^{\infty}$ and
$\Delta\mathbf{f}=\mathbf{f}^{ext}-\mathbf{f}^{int}=(\mathbf{\sigma}^{ext}-\mathbf{\sigma}^{int})\cdot\mathbf{n}$
is the discontinuity in the interfacial surface force. The tensors
$\mathbf{\sigma}$ are the fluid stress tensors; in order to account for the
presence of body forces, they can be replaced by the modified stress tensors
such that
$\sigma^{MOD}_{ij}=\sigma_{ij}+\rho\mathbf{g}\cdot\mathbf{x}\delta_{ij}$
[194]. Here, $\mathbf{g}$ is the acceleration field, like gravity, and $\rho$
is the associated quantity, like fluid density. Keeping this in mind, we will
drop the $MOD$ superscript from now on. Finally $\Delta\mathbf{f}$ can be
written as
$\Delta\mathbf{f}=(\rho^{ext}-\rho^{in})\mathbf{g}\cdot\mathbf{x}\,\mathbf{n}+\Delta\xi$,
where $\Delta\xi$ is the discontinuity in the surface force that depends only
on the interface mechanical properties. For a given model of particle (e.g. a
drop, a vesicle, a capsule), and in the absence of significant inertia of the
membrane, it can be calculated according to the chosen constitutive law for
the surface, as it must equal the opposite of the membrane load.

$\mathbf{G}$ is the Green’s function that is adapted to the boundary condition
of the problem and $\mathbf{T}$ is the associated stress tensor.

Regarding numerical simulations, Eq. (10) can be implemented to compute the
particle dynamics according to a two-step process where first the displacement
of the particle membrane is calculated according to Eq. (10), after what the
force $\mathbf{\xi}$ can be calculated in this new configuration, and so on.
This so-called boundary integral method has given rise to several developments
regarding numerical schemes be used, following the seminal work of Pozrikidis
[195]. In particular, it has successfully been used to describe the motion of
drops [247], vesicles [234, 45, 165, 286, 75] or capsules [258, 178, 177] in
the vicinity of walls. To do this, Green’s function that are adapted to the
considered boundary conditions must be used, which we describe below. Note
that Eq. (10) does not provide a direct expression for the velocity as it
appears on both sides of the equation, when $\lambda\neq 1$. This requires to
implement adapted numerical schemes to ensure convergence.

For an unbounded domain, the Green’s function
$\mathbf{G}^{\infty}(\mathbf{x},\mathbf{x}_{0})$ is called the Stokeslet and
describes the flow field created at position $\mathbf{x}_{0}$ by a point force
located at position $\mathbf{x}$. This flow field is anisotropic and decreases
as the inverse of the distance to the point force.

The Green’s function we need here is that satisfying the no slip condition on
the wall. Blake [26] proposed a calculation of this semi-infinite Green’s
function, using Fourier transform. It can be thought as the Green’s function
associated with other point singularities located at the reflection point
$\mathbf{x}^{IM}=(x_{1},x_{2},-x_{3})$ of the initial force. Interestingly, in
Blake [26], this interpretation in terms of singularities is obtained a
posteriori, after the direct calculation is lead. An alternative, more
intuitive construction of the fundamental solutions of Stokes flow in a half-
space have been proposed by Gimbutas  _et al._ [92].

The semi-infinite Green’s function reads
$\mathbf{G}=\mathbf{G}^{\infty}+\mathbf{G}^{w}$, where the wall Green’s
function $\mathbf{G}^{w}(\mathbf{x},\mathbf{x}_{0})$ is a function of
$\mathbf{x}^{IM}-\mathbf{x}_{0}$. It introduces a velocity field in
$\mathbf{x}_{0}$ that has several contributions that depend on the inverse of
the distance $|\mathbf{x}^{IM}-\mathbf{x}_{0}|$ to the power 1,2 and 3 (see
Appendix). Similar decomposition exist for the stress tensor
$T_{ijk}=T^{\infty}_{ijk}+T^{w}_{ijk}$.

The lift velocity $U_{L}$ can be then thought as the averaged velocity in the
$x_{3}=z$ direction over the particle volume, which can be transformed into a
surface integral for incompressible particles, noting that
$\divergence{(x_{3}\mathbf{u})}=u_{3}$:

$U_{L}=\frac{1}{V}\int_{V}u_{3}dV=\frac{1}{V}\int_{S}x_{3}\mathbf{u}\cdot\mathbf{n}\,dS,$
(11)

where $\mathbf{n}$ is the unit vector normal to the surface.

#### Far-field contribution and corrections

In the absence of walls, the dynamics of the particle is governed by Eq. (10)
where $\mathbf{G}=\mathbf{G}^{\infty}$ and $\mathbf{T}=\mathbf{T}^{\infty}$.
This leads to the dynamics described in section III.1.

The semi-infinite Green’s function and the associated stress tensor introduce
a dependence of the velocity field with the distance to the wall. This
velocity field may induce a displacement of the particle far or towards the
wall, but also modify its shape dynamics. Therefore, the velocity field
created by the presence of the particle will be modified, hence possibly the
lift velocity.

We discuss below how these potential contributions scale with the distance $z$
of the particle to the wall, in view of identifying the domains of validity of
the far-field formalism and of the soft-lubrication regime.

Following Nix  _et al._ [178] who proposed it in the case $\lambda=1$, we
decompose the flow velocity $u_{3}$ into two contributions: (a) a self term
$U^{s}$ that arises (through Eq. (11)) from the flow created $\mathbf{u^{s}}$
by the particle, obtained from Eq. (10) by considering only
$\mathbf{G}^{\infty}$ and $\mathbf{T}^{\infty}$, and a wall-term term $U^{w}$,
obtained when considering $\mathbf{G}^{w}$ and $\mathbf{T}^{w}$.

In an infinite simple shear flow, the whole configuration has a point symmetry
with respect to the centre of the particle. Considering two opposite points on
the membrane, one can see that the $\mathbf{u^{s}}\cdot\mathbf{n}$ terms are
equal while the $x_{3}$ term has opposite sign, whence $U^{s}=0$, as expected.
Note that this argument holds only in the absence of some symmetry-breaking
mechanism, like buckling, which we are not aware of in unbounded simple shear
flow, unless the particle itself presents some heterogeneities [149]. The self
term $U^{s}$ can be non zero if point symmetry is lost, in particular if the
particle is deformed by the wall.

We now consider the leading order of the wall term, which is often the only
one considered in models. Far from the wall, this velocity $U^{w}$ may be
approximated by that of its center, that we set to be located at position
$\mathbf{x}_{0}=(0,0,z)$. This far-field velocity $U^{w,ff}$ thus reads

$\displaystyle U^{w,ff}$ $\displaystyle=$
$\displaystyle-\frac{1}{8\pi\mu}\int_{S}\Delta
f_{i}(\mathbf{x})G^{w}_{i3}(\mathbf{x},\mathbf{x}_{0})\,dS$ (12)
$\displaystyle+\frac{1-\lambda}{8\pi}\int_{S}u^{0}_{i}(\mathbf{x})T^{w}_{i3k}(\mathbf{x},\mathbf{x}_{0})n_{k}(\mathbf{x})dS,$

where $\mathbf{u}^{0}$ is the leading order term in the velocity on the
particle surface. For $|\mathbf{x}-\mathbf{x_{0}}|\ll z$, one can expand
$G^{w}(\mathbf{x},\mathbf{x}_{0})$ and $T^{w}(\mathbf{x},\mathbf{x}_{0})$
around $\mathbf{x}_{0}$. In the absence of external forces and torque and
considering symmetry properties of the tensors as well as fluid
incompressibility, one eventually finds (see Appendix for details):

$U^{w,ff}=-\frac{9}{64\pi\mu}\frac{\Sigma_{33}}{z^{2}}=\frac{AR^{3}|\dot{\gamma}|}{z^{2}},$
(13)

where

$\displaystyle\Sigma_{ik}=$ $\displaystyle\int_{S}\Big{[}\frac{1}{2}(\Delta
f_{i}(x-x_{0})_{k}+\Delta f_{k}(x-x_{0})_{i})-\frac{1}{3}\Delta
f_{j}(x-x_{0})_{j}\delta_{ik}\Big{]}\,dS$ (14)
$\displaystyle+(\lambda-1)\mu\int_{S}(u^{0}_{i}(\mathbf{x})n_{k}(\mathbf{x})+u^{0}_{k}(\mathbf{x})n_{i}(\mathbf{x}))\,dS$

is called the stresslet, while the second expression in Eq. (13) provides a
convenient dimensionless form to discuss results obtained for different
particles. It should be noted that the second term vanishes not only for
particles with no viscosity contrast but also for rigid particles [18].
Equation (13) was proposed following a more straightforward approach by Smart
and Leighton [229]. By coherence with the leading order approximation we made
here, one must keep in mind that the stresslet $\Sigma_{33}$ is that created
by the interaction with the external flow, in the absence of wall. Checking
the validity of this approximation (and the associated domain in the $z$-axis)
can be made through full numerical simulations or experiments. Doing so, one
must keep in mind that while the theoretical approach of Eq. (13) through the
determination of the stresslet will provide the lift velocity at a given
position for a particle in its stationary dynamics, simulations or experiments
provide full trajectories along which the shape at given position might not be
the stationary one. Comparing both approach requires to ensure that the
typical time needed for shape change is much smaller than the migration time.
A priori , this will be achieved for sufficiently high capillary numbers. In
practice, this conditions holds in the situations we will describe below.
Also, in simple shear flow, the shape depends only weakly on the position. The
situation will be more complex in quadratic flows, which will be shown to
trigger more complex couplings between shape and lift direction. An agreement
between direct numerical simulations and Eq. (13) using an independently
calculated stresslet was found for vesicles in [286, 75] and for capsules in
[178].

An explicit discussion on the domain of validity of the far-field expression,
as well on the scaling for the next order terms, was carried out by Nix  _et
al._ [178], through numerical simulations of a capsule in a quite narrow range
of capillary numbers (of order 0.1-1), and $\lambda=1$.

They first showed that the self term $U_{s}$ is negative (that is, the
particle is attracted towards the wall), and decays as $(z/R)^{-4}$, on a
large range $z/R\gtrsim 1.2$. This shows that the modification of the lift
velocity due to shape distortion under the influence of the flow created by
the stresslet in interaction with the wall, which exhibits contributions
scaling like $(z/R)^{-2}$ as in Eq. (13), is not of the same order as the
direct stresslet contribution. We make the remark here that, to our knowledge,
this result has never been derived formally and generalised to a larger class
of particles.

Regarding the wall term, Nix  _et al._ [178] highlighted that the far-field
expression can be corrected by two terms; first, replacing the stresslet by
its value at the considered distance from the wall enhances the lift velocity.
Second, the overall contribution of the wall is shown to be smaller, due to
the asymmetric deformation (with respect to the point-wise symmetric shape)
that has a negative contribution to the lift velocity, as for the self term.
It has been shown that these two corrections of the wall term also varies like
$(z/R)^{-4}$ and that their overall contribution added to the self term is
negative.

We re-analyzed some results of the literature, which points to a potential
generality of this result. As shown already in Fig. 3, the correction to the
far-field expression is also negative for moderately deflated vesicles, with a
$(z/R)^{-4}$ scaling. It becomes significant for $z/R\lesssim 3$. These
features seem to hold also for drops, as we found by revisiting the data from
[247], Fig. 5.

These results help identifying the transition zone between the far field
regime and the soft lubrication regime, where the lift results from modified
flow due to the deformation of the interface because of the flow between the
particle and the wall. We discuss in next sections how this transition depends
on the mechanical parameters involved. Noteworthy, the soft lubrication
correction seems to be always negative for sheared particles.

We propose a geometrical interpretation of this negative contribution, based
on the observed shape sequences upon unbinding, which are illustrated in Figs.
3 or 14 but also seen in [234, 178]. Far from the wall, the particle adopts an
orientation that roughly follows that of the elongational component of the
flow, i.e. 45∘ relatively to the flow direction. So does its bottom membrane,
on average. Closer to the wall, the shape of the bottom end of the particle,
which is close to the wall, is controlled by the local interaction with the
wall, which creates, as in the soft-lubrication framework depicted in Fig. 5,
a quasi-horizontal gap with a small opening angle. Compared to the high
opening angle induced by the bulk flow, this reduces the fore-aft asymmetry of
the particle, hence a negative correction to the lift velocity. In future
works, it would be interesting to examine this hypothesis by comparing the
amplitude and scaling of this negative contribution to the lift force of a
similar particle moving along the wall without external shear flow (i.e., in
the typical soft-lubrication configuration).

In the soft lubrication framework, results are often presented in terms of
lift force; on the other hand, for particles deformed under external flow,
lift velocities is the usual output. This probably relates to the main
motivation of these studies, which is to calculate flux of particles. These
two quantities can be related through the drag acting on the particle. It is
insightful to examine how this can be achieved in practice. To that aim, we
first review the case of vesicles, which is interesting for comparing results
from different studies as there are only few parameters involved. Furthermore,
results focusing on either lift force or lift velocity are available.

#### Lift velocity and lift force, the vesicle case

For large enough capillary numbers such that the hydrodynamic stress overcomes
bending forces, the dynamics of vesicles depends only on their initial
deflation and on the viscosity contrast [73]. In the range of parameters where
tank-treading motion occurs, the shape and angle of inclination of the vesicle
are therefore independent of the shear rate, so is the prefactor $A$ in Eq.
(13). We are not aware of studies on lift of vesicles at small capillary
numbers, when wrinkles appear due to hydrodynamic forces or Brownian
fluctuations and modify the nature and the transition between dynamical
regimes in unbounded shear flow [180, 56, 3].

Two different sets of experiments are available in the literature that
describe the lift of vesicles of radius of order 10 microns: Callens  _et al._
[35] studied the lift velocity of vesicles in the absence of gravity, while
Abkarian  _et al._ [1], Abkarian and Viallat [2] examined the close wall lift
force by balancing it by the vesicle weight.

Experiments for lift velocity were performed in parabolic flights, allowing
for successions of normal gravity phases and low gravity phases. In the first
phase, sedimentation of vesicles on the bottom plate of a shear chamber
allowed for the creation of a well-defined initial condition, while the lift
velocity of the vesicle could be measured in the low gravity phase, without
being screened by sedimentation. Thanks to this, distances to the wall of up
to 7 times the vesicle radius could be explored. Vesicles with inner fluid
having the same viscosity inside and outside were studied in [35] while more
viscous inner fluids were considered in [34]. In the range $3\lesssim
z/R\lesssim 7$, the distance to the wall is found to scale with time to the
power $1/3$, indicating agreement with the far-field scaling. They also found
a prefactor $A$ that is independent from the shear rate, that was varied by a
factor 10. Eventually, the prefactor $A(\nu,\lambda)$ has been determined for
reduced volumes $\nu\gtrsim 0.95$ and $\lambda=1$ [35], 4 and 6.5 [34]. The
reduced volume $\nu\leq 1$ characterizes the deflation of the vesicle,
therefore its ability to get deformed, and reads
$\nu=3\mathcal{V}/4\pi(\mathcal{A}/(4\pi))^{3/2}$, where $\mathcal{V}$ and
$\mathcal{A}$ are the vesicle volume and surface area, respectively. They are
both constant due to the inner fluid and membrane incompressibility.

In the experiments by Abkarian  _et al._ [1], Abkarian and Viallat [2],
vesicles with no viscosity contrast but with density contrast, and $\nu\gtrsim
0.92$, are sheared close to the wall, with shear rates varying by a factor 5
such that different equilibrium positions can be scanned. It is found that at
equilibrium the gap $h$ between the vesicle and the wall scales linearly with
the shear rate, suggesting the following expression the lift force $F_{L}$:

$F_{L}=B(\nu)\mu R^{3}\dot{\gamma}/h.$ (15)

Simulations of vesicles with no viscosity contrast were proposed in several
studies by using boundary integral method [234, 286, 75]. All simulations, as
well as far-field theoretical calculations led in [256] or [75], consider
mostly vesicles with reduced volumes higher than 0.95. This corresponds, e.g.,
to a prolate ellipsoid of long axis 1 and short axis 0.63.

Using both the results of Callens  _et al._ [35] and Abkarian  _et al._ [1] to
make comparisons with these existing simulations and theories requires first
to establish a link between lift velocity and lift force, which we did not
find in the literature.

In presence of a body force (which is often, in practical cases, gravity), the
lift motion is modified and can even vanish if this body force acts opposite
to the lift force. Regarding boundary integral framework, the presence of the
body force induces an additional term $U^{g}$ in the migration velocity
$U^{g}=\frac{1}{V}\int_{S}x_{3}\mathbf{u}^{g}\cdot\mathbf{n}\,dS$, where

$u^{g}_{j}(\mathbf{x_{0}})=-\frac{1}{8\pi\mu}\int_{S}(\rho^{ext}-\rho^{in})\mathbf{g}\cdot\mathbf{x}\,n_{i}G_{ij}(\mathbf{x},\mathbf{x}_{0})\,dS$
(16)

is the flow field created by the particle due to the presence of a density
difference across its membrane. For a sphere of radius $R$ in an unbounded
flow, with a gravity field in the direction $-x_{3}$, solving Eq. (16) would
lead to the well-known Stokes law $6\pi\mu RU^{g}=-P$, where $P=\frac{4}{3}\pi
R^{3}(\rho^{in}-\rho^{ext})$ is the weight of the particle minus the
Archimedes force. In presence of a wall at $x_{3}=0$, the expression of
$G_{ij}^{w}$ shows that the first order correction to the velocity would scale
like $1/z$. For a sphere settling towards a wall, the full correction for the
modified drag force as a function of distance to the wall has been solved by
Brenner [32].

$\displaystyle 6\pi\mu RU^{g}\Lambda(\cosh^{-1}[z/R])=-P,\qquad\mbox{ with}$
$\displaystyle\Lambda(\xi)$ $\displaystyle=$
$\displaystyle\frac{4}{3}\sinh(\xi)\sum_{n=1}^{\infty}\frac{n(n+1)}{(2n-1)(2n+3)}$
$\displaystyle\Big{[}\frac{2\sinh((2n+1)\xi)+(2n+1)\sinh(2\xi)}{4\sinh^{2}((n+1/2)\xi)-(2n+1)^{2}\sinh^{2}(\xi)}-1\Big{]}.$

The first order of this correction is given by

$\Lambda(\cosh^{-1}[z/R])\simeq 1+\frac{9}{8}\frac{R}{z},$ (18)

indicating, in agreement with the intuition, that the presence of the wall
increases the drag on the particle. We hypothesize that for deformable
particles, this expression of the drag force could be considered as a first
approximation.

In the presence of a body force that acts against lift, an equilibrium
position in the $z$ direction is found by the particle, corresponding to
$U^{g}+U_{L}=0$. Defining the lift force $F_{L}$ as the force that balances
the weight $P$, one eventually finds, that

$F_{L}\simeq 6\pi\mu RU_{L}\Lambda(\cosh^{-1}[z/R]),$ (19)

where the lift velocity is given by the appropriate expression, and the
correction on drag is assumed to be that for a sphere. This expression will
serve us as a basis for discussion.

Figure 14: Sequence of shapes taken by a vesicle as it unbinds and lift away
from the wall under simple shear flow. Figure taken from [1].

We can now address the question of the comparison of simulations, theory and
experiments, while also considering the question of the link between lift
force and lift velocity. To that aim, we focus on vesicles with no viscosity
contrast and a reduced volume of 0.97, which is documented in experiments [35,
1], simulations by Zhao  _et al._ [286], an historical far-field model by Olla
[183] that assumes an ellipsoidal shape and the far-field model by Farutin and
Misbah [75] that makes no assumption on the shape and calculate directly the
stresslet 111Under the same hypothesis as Farutin and Misbah [75], Vlahovska
and Serral Gracia [256] proposed an expression for the stresslet whose
numerical prefactor differs by some 20%. Since the model by Farutin and Misbah
[75] is more recent and is in perfect agreement with their own numerical
simulation, we hypothesize that their result is more likely to be correct..
For all above mentioned studies, but that of Abkarian  _et al._ [1], the lift
velocity as a function of distance $z$ between the vesicle centroid and the
wall is given, in the absence of gravity. For the sake of comparison with the
results of Abkarian  _et al._ [1], we make the assumption that $z=h+R$ (see
Fig. 2) and calculate the lift force through Eq. (III.2). In order to support
discussion, the lift force arising from the simulations of Zhao et al. is also
calculated using the 1st order approximation of the drag coefficient (Eq.
(18)) as well as 0th order, i.e. the drag of a sphere in an unbounded flow.
The results are presented in Fig. 15.

Figure 15: Dimensionless lift force $\hat{F}_{L}=F_{L}/(\mu\dot{\gamma}R^{2})$
on a vesicle characterized by $\nu=0.97$ and $\lambda=1$ sheared above a wall,
as a function of reduced gap size $\hat{h}=h/R$ between the vesicle and the
wall. $R$ is the radius of a sphere having the same volume as the vesicle.
Lines and dots extend on a range where the law has been established in the
different studies. $\blacksquare$: $\hat{F}_{L}=1.1/\hat{h}$ according to the
close-wall experiments of Abkarian  _et al._ [1], Fig. 8. Other data are
obtained through $\hat{F}_{L}=D\hat{U}_{L}$ where $\hat{U}_{L}$ is the
dimensionless lift velocity $U_{L}/(\dot{\gamma}R)$ and $D$ the dimensionless
drag coefficient, which is taken either as $D_{0}=6\pi$ (unbounded domain),
$D_{1}=6\pi(1+9/[8(\hat{h}+1)])$ (first order approximation of Brenner’s
drag), or $D_{\infty}=6\pi\Lambda(\cosh^{-1}[\hat{h}+1])$ (Brenner’s drag).
$\bullet$: $D=D_{\infty}$ and $\hat{U}_{L}=A(0.97)/(\hat{h}+1)^{2}$ with
$A(0.97)=0.11$ according to the far-field experiments of Callens  _et al._
[35], Fig. 7; Red line: the same with $A(0.97)$ given by Olla’s model [183]
(ellipsoid of aspect ratio 0.705); Orange line: the same with $A(0.97)$ given
by the theoretical model of Farutin et al. (Eq. 5 in [75], for $\Gamma=0.021$,
$\Gamma$ being defined in the referred article). Note that in [35], the volume
and reduced volumes of the vesicle are known only indirectly through the
measurement of its short axis in the vorticity direction and of the projection
of its long axis parallel to the $z$ direction. These data are converted into
volume and reduced volume through the direct calculation of Farutin and Misbah
[75], which have shown excellent agreement with these experiments [34]. Blue
lines: $\hat{F}_{L}=D\hat{U}_{L}$, where the lift velocity is obtained from
direct simulation in [286], Fig. 2 and $D=D_{0}$ (dotted line), $D_{1}$
(dashed line), or $D_{\infty}$ ( full line).

Regarding far-field behaviour, simulations by Zhao  _et al._ [286], modeling
by Farutin and Misbah [75] and experiments by Callens  _et al._ [35] show very
good agreement. By contrast, the lift velocity predicted by Olla [183] is
about 30% larger. Comparison between Fig. 7 in [35] and Fig. 1 in [75] shows
that a vesicle whose shape is not prescribed a priori has a (long axis) /
(short axis) ratio that is smaller than that given by the prolate ellipsoid
assumption, made by Olla. This may qualitatively explain why, for a given
reduced volume, the lift predicted by Olla is larger than that measured in
other studies.

Remarkably, the far field measurements of the lift force by Abkarian  _et al._
[1] are perfectly described by the gravity-free simulations of Zhao et al.,
who scanned the same range of particle-to-wall distance, providing the drag
contribution is that of Brenner for a sphere of equal volume. This agreement
is far from being reached if one considers only the 0th (as used in [234]) or
1st order of the drag force.

The conclusion of this aggregation of data is that the $1/\hat{h}$ scaling for
the drag force, proposed by Abkarian et al., is therefore recovered, in the
close-wall range, by multiplying the multipolar drag coefficient of Brenner
with a lift velocity that is composed of a repulsive stresslet contribution
that scales as $(\hat{h}+1)^{-2}$ and of an attractive contribution that is
the consequence of wall-induced deformation. This contribution scales like
$(\hat{h}+1)^{-4}$.

Zhao  _et al._ [286] also ran simulations including gravity and found good
agreement with the experimental data of Abkarian  _et al._ [1]. In the same
article, they compared their simulations of the lift force with that
calculated from the lift velocity times a first order expression for the drag
(as in Eq. (18)). Unfortunately, we are not able to comment on this approach
by lack of definition of some parameters and of detail of calculation by the
authors. Other direct simulations of vesicles under gravity have been
performed in by Meßlinger  _et al._ [165], but they consider a 2D system. On a
quite narrow range of distances, the authors found that the force scales like
$1/z^{2}$ even for $z/R$ close to 1.

As a conclusion, our review highlights the necessity to better understand
interactions in the crossover zone $z\sim R$ between soft-lubrication effects
and shear-induced lift. While the main contribution to lift, the stresslet
created by the unperturbed particle, has been widely commented, the correction
terms did not benefit from such an attention. We have also highlighted the
difficulty in comparing results focused on the velocity with that focused on
the force. While using Brenner’s drag for a rigid sphere seems satisfactory,
studies focused on the drag exerted on a deformed particle whose dynamics is
imposed by the external flow would certainly be helpful.

#### Dependence with mechanical properties

The relative contribution of each of the $1/z^{2}$ and $1/z^{4}$ terms depends
on the detail of shape and dynamics of the particle.

Increasing the viscosity contrast seems to always make the lift velocity
decrease, for vesicles [165, 286, 75, 34] or for capsules, for which Singh
_et al._ [224] showed that this is the direct result of the increase of the
absolute value of the second term in the stresslet expression (Eq. (14)),
although the absolute value of the first term decreases. It should be noted
that for capsules, we have not found any experimental study regarding their
lift under simple shear flow.

At least for vesicles, the range of validity of the far-field expression
increases with the viscosity contrast, a consequence of the increasing cost
for interface deformation under the influence of the wall [286].

The lift velocity also depends on the ability of the particle to deform, which
is essentially related to reduced volume for vesicles, and to capillary number
for drops and capsules. It is natural to start our discussion with the
simplest of these objects, i.e. drops.

[229] measured experimentally the lift on drops of viscosity contrast 0.083.
Since then, we have found no record of another attempt to measure directly
this lift in a simple shear flow; other works include drop-drop interactions,
that will be discussed further. They found that the far-wall velocity follows
Eq. (13), with the prefactor $A$ being proportional to the capillary number
$Ca$, defined as $Ca=\mu\dot{\gamma}R/\sigma$, where $\sigma$ is the surface
tension. This indicates that upon an increase of the flow stress (compared to
the elastic restoring force), the drop elongates more, thus inducing a larger
lift force. This result was theoretically derived by a direct calculation of
the stresslet, both by Chan and Leal [41] and Smart and Leighton [229], under
the hypothesis of small deformation (hence the linear dependence with $Ca$).
They both proposed an expression for the prefactor, which depends on the
remaining characteristic parameters $\lambda$, the viscosity contrast between
the inner and the outer fluids. Both expressions are different but vary by at
most 2 % on the whole range of viscosity contrast. $\alpha(\lambda)$ was found
to be an increasing function of $\lambda$, yet with small variations, from
0.58 when $\lambda\to 0$ to 0.69 when $\lambda\to\infty$. These theoretical
results do not match the experimental results, where the lift amplitude is
found to be almost 2 times larger.

Numerical simulations generally find a lift velocity that is slightly smaller
than in theoretical works, therefore they agree even less with experiments
[247, 246, 124, 224].

We mention here that with drops, another configuration of interest has been
studied by Smart and Leighton [229], that of a free surface instead of a rigid
wall. In this case, the overall expression for the far-surface lift velocity
(Eq. (13)) is unchanged, but a multiplicative factor $2/3$ is introduced. As
the stresslet is that given by the bulk flow, we expect the expression given
by the authors to be valid not only for the drops they study. Experimentally,
as for the lift due to a rigid wall, their experimental results are above the
theoretical prediction.

An increase of the absolute value of the stresslet, therefore of the lift
velocity, is also mentioned for capsules in [178] (see Fig. 16). Regarding the
relative contribution of the far-field term and of the first soft lubrication
correction term, Nix  _et al._ [178] showed that, while the far-field
approximation is valid even for particles in contact with the wall at low
capillary number, this approximation is valid only when $h/R\gtrsim 3$ for a
capillary number one order of magnitude larger (Fig. 16). This impact of the
capillary number on the locus of the transition zone is similar to that
previously discussed for vesicles of different viscosity contrast: less
ability to deform lead to a smaller contribution of the soft-lubrication term.

Singh  _et al._ [224] highlighted the same year the same range of transition
distance between the far-field regime and a more complex set of contributions,
where capsules with different viscosity contrasts are also considered. At a
distance to the wall where both contributions are comparable, a
phenomenological single scaling is proposed by the authors.

For a vesicle with no viscosity contrast, the more deflated the vesicle, the
faster it migrates [1, 35, 286, 75]. However, for high enough viscosity
contrast, increasing asphericity can lead to transition towards tumbling
motion, which is preceded by a decrease of the inclination angle, therefore of
the lift velocity [183, 34]. To our knowledge, this decrease has not been
observed experimentally, but similar phenomenon has been exhibited in
Poiseuille flow [45], as will be discussed later on.

From a geometrical view point, the decrease of the inclination angle
(potentially until zero), leads the configuration towards a fore-aft symmetric
state, hence the vanishing lift velocity. A similar decrease of inclination
angle is observed for capsules by Nix  _et al._ [178] upon an increase of
capillary number; in that case though, they also elongate [257, 134], which is
probably the reason why the lift velocity does not decrease.

In their 2D simulations, Meßlinger  _et al._ [165] have considered the case of
tumbling vesicles and shown a non-zero lift force on average, by contrast with
the case of a purely rigid object. The reason stems from the elongational
component of the flow that makes the vesicle be more elongated when its long
axis is along this component than when it is orthogonal to it. Over one
rotation period the mean shape is therefore not fore-aft symmetric. Still the
mean lift velocity remains much smaller than in the tank-treading regime. A
more formal discussion on this aspect can be found in [184]. Similar results
were found for capsules by Hariprasad and Secomb [105], who maintained a 2D
capsule that would tumble in an unbounded flow close to a wall by a gravity
force of varying intensity. The proximity with the wall induces a change in
the dynamics that switches from tumbling to tank treading as the force is
increased. As a result, the lift force increases, such that, in the rather
narrow range of parameters explored by the authors, a quasi constant
equilibrium height is observed while the force is multiplied by a factor 4.

Figure 16: Lift velocity in simple shear flow for a capsule with a membrane
following Neo-Hookean law. Dots and full lines correspond to simulations while
the doted line is the far-field approximation (Eq. (13)). Figure adapted from
[178].

#### Red blood cells

While more complex than the above mentioned particles, red blood cells exhibit
similar lift properties.

In physiological conditions, red blood cells flow in vessels where the maximal
shear rate ranges from 20 to 1500 s-1. In these conditions, cells will
essentially exhibit a tumbling-like motion that couples with strong
deformations [167, 161].

As seen in [165] for tumbling vesicles, isolated red blood cells may still
migrate. This has been confirmed experimentally in simple shear - like flow
(in a large pipe in reality) in the 70’ by Goldsmith, who highlighted a 4
$\mu$m transverse drift for cells having travelled 1 cm in the flow direction
[96]. More than 40 years later, a study in microgravity conditions, similar to
that carried out for vesicles in [35], has quantified the lift of red blood
cells in a narrow range of shear rates $10\leq\dot{\gamma}\leq 50$ s-1 [98].
The far-field scaling has been confirmed, with a prefactor $AR^{3}=0.36$
$\mu$m3 (following the notations of Eq. (13)) in physiological conditions
(i.e. an external fluid of viscosity 1.4 mPa.s close to that of plasma). This
factor is increased by a factor 15 if the viscosity is multiplied by 9,
indicating the impact of flow stress on the deformation of the cells, that are
more elongated and can even make a transition towards a tank-treading like
motion [86, 167]. Considering that red blood cells have a mean volume of 90
$\mu$m3 [15], whence $R=2.8\,\mu$m, this leads to a prefactor $A$ between
0.016 (tumbling-like regime) and 0.15: this latter value is comparable to that
found for a vesicle in tank-treading regime.

The value for the lift velocity found in physiological conditions has been
calculated to be compatible with the pioneering result of Goldsmith [96]. It
should be noted that, at a distance e.g. 4 $\mu$m from the wall the lift
velocity is about 2 $\mu$m/s at a shear rate of 100 s-1, which is comparable
to sedimentation velocity [159]. This illustrates the difficulty in measuring
experimentally far-field lift velocities.

While numerical simulations of red blood cells under flow are now numerous, we
have found no records for this geometrically simple configuration of simple
shear flow near a wall, as far as realistic 3D simulations of cells are
concerned. We shall come back to this point later on while discussing
collective effects.

###  III.3 Lift in a channel flow

Lift in channel flow is the other configuration explored in the literature, as
it is relevant for particle handling in microfluidic devices, and to
understand biological flows such as blood flow especially.

These flows are characterized by the increased presence of walls but also by
linear variations of the shear rate. For large channels (compared to particle
size) and in order to gain fundamental understanding of the lift mechanism, it
is insightful to consider first the case of an unbounded Poiseuille flow that
is, a parabolic velocity profile with no walls imposing a condition of zero
velocity.

#### Lift in unbounded Poiseuille flow

In the soft lubrication framework, the necessary breaking of the bottom-up
symmetry in terms of pressure (see Fig. 2 for convention on directions) is
obtained by the presence of a wall. The pressure gradient that is created
between the particle and the wall is different from that on the other side of
the particle, leading to different deformations patterns at the bottom and at
the top of particles, eventually leading to the fore-aft asymmetry that
induces an overpressure below the particle and makes the appearance of the
lift force possible.

This sequence is indeed also possible in the configuration of an unbounded
Poiseuille flow where a particle is surrounded by different shear rates on
both sides, except when it is located on the central line of the flow. This
situation can create an asymmetry in the deformation patterns along the flow
direction, making thus possible the appearance of a net lift force. Notably,
while in the soft-lubrication framework the creation of a gap resembling that
of the Reynolds slider makes it intuitive the sign of the lift force, the
deformation arising on both sides of the particle leads to a less clear
situation.

Indeed, several scenarii have been predicted. For a drop in a 2D parabolic
flow, Chan and Leal [41], Leal [141] have predicted that a drop would migrate
outward for a viscosity ratio $0.71\lesssim\lambda\lesssim 11.35$ but towards
the central line for other values of $\lambda$. For an axisymmetric Poiseuille
flow, the interval for outward migration becomes $0.56\lesssim\lambda\lesssim
10.2$.

Theoretical and numerical studies on vesicles have shown that vesicles
apparently behave differently: Kaoui  _et al._ [117] ran 2D numerical
simulations of vesicles with $\lambda=1$ and evidenced inward migration at
almost constant velocity along the trajectory but at the very end, when the
particle meets the central line. Danker  _et al._ [52] confirmed these results
through a theoretical approach valid in the small deformation approximation.
For all $\lambda$ such that a tank-treading motion takes place, an inward
migration is predicted. A tentative physical argumentation in favor of this
migration has been given in this article, which is reported in Fig. 17.
Similarly, Helmy and Barthès-Biesel [106] theoretically showed that capsules
migrate towards the centerline, in the limit of small deformation (small
capillary number).

Farutin and Misbah [75, 76] performed more systematic theoretical studies for
a 3D, axisymmetric flow. They revealed a much more complex situation, for a
vesicle of given reduced volume $\nu=0.9$. The behaviour strongly depends on
the capillary number $Ca$ and on the viscosity ratio $\lambda$. For
$\lambda=1$, vesicles migrate towards the center at high capillary number, at
a constant velocity but in the vicinity of the center. This situation
corresponds to the case most easily studied by theory as a stationary shape
can be considered (due to large $Ca$), as in [52, 75]. At lower $Ca$, i.e.
when the particle has no time to adjust its shape to the surrounding local
flow before being advected further, an equilibrium position away from the
center is found. For larger viscosity contrasts, outward motion is observed at
large $Ca$ whereas metastability was observed at smaller $Ca$, the vesicle
migrating outward or toward a position close to (but not on) the center,
depending on its initial position.

Kaoui  _et al._ [115] reached similar conclusions for 2D vesicles with no
viscosity contrast: upon a decrease of capillary number a deflated enough
vesicle does not converge towards the center but stays at a finite distance
from it, adopting an asymmetric shape described as a slipper shape. This
situation is also favored by a deflation of the vesicle. A tentative
explanation for this phenomenon is proposed by the authors, based on their
numerical observation: the transition towards an off-center, asymmetric shape
is accompanied by a reduction of the lag, which is anticipated to be a
favorable configuration by the authors, despite the increase in internal
dissipation inside the particle not being symmetric anymore.

However, arguments based on a minimization of energy or of dissipation are not
supported by any fundamental principle in this viscoelastic problem with
moving boundaries. Indeed, Farutin and Misbah [76] discard both possibilities
by showing they are not compatible with their numerical observations — though
in the meantime arguments of that kind are still being used in [236]. Improper
use of arguments based on dissipation consideration is also discussed in [53].

Figure 17: Creation of an inward lift force in Poiseuille flow for a tank-
treading vesicle located at a position $y>0$ from the central line, according
to Danker  _et al._ [52]. The local velocity field can be decomposed into a
local shear that dictates the shape of the vesicle and local quadratic
correction (blue arrows). The normal contribution of this flow field points
downward on segments AB and CD and upward on the two other segments, which are
smaller. Hence a negative lift force acting on the particle. This argument is
debatable as it does not take into account either the part of the flow that
modifies the particle shape but not its position, or the relative intensity of
the normal contribution on each segment. In addition, it should also apply per
se for liquid drops, and would contradict the finding of Chan and Leal [41].
This illustrates the difficulty in getting an intuitive picture for migration
in quadratic flows. Figure taken from [52].

A question naturally arises: does this complex behaviour survives in more
realistic situations where walls are present ? Walls induce additional lift
forces but also additional space dependence of the shape. A secondary question
is, how do lift forces due to flow curvature and lift forces due to the
presence of a wall compare to each other in intensity ? An attempt to answer
partly this question can be found in the 2D numerical simulations of Kaoui
_et al._ [116] where a vesicle with no viscosity contrast is placed in a semi-
bounded parabolic flow, i.e. where only one wall is present (say, at position
$z=-z_{0}$ while the centerline is at $z=0$). In this case, a vesicle placed
at a distance $z>0$ from the center larger than its typical radius migrates
outwards, while it would migrate inwards in the absence of the opposite wall.
This indicates, at least in this specific situation, that the lift due to the
presence of the wall overcomes that due to the flow curvature, at a distance
from the center large than a particle radius.

Nix  _et al._ [177] carried out a more detailed study regarding this question,
with capsules. They quantified the ratio of the contributions arising from the
shear gradient and from the presence of a wall, that grows as the particle
approaches the center of a channel. An interesting output of their study is
that the drift velocity due to shear gradient hardly depends on the viscosity
ratio (in the explored range $1<\lambda<5$) while the effect of the wall
diminishes with increasing $\lambda$ – as already discussed here. As a result,
the effect of shear gradient is predominant on a larger area within the
channel for more viscous particles.

#### Migration in a channel

In a channel, using boundary integral method to solve the flow field requires
to incorporate more complex Green’s function. Even in a 2D case where only two
opposite walls are needed, this requires to incorporate multiple image systems
[241, 172]. An alternative method consists in considering walls as soft
boundaries of known rheological property, such that additional integrals must
be considered, with the advantage to handle only the Green’s functions for
unbounded flow [118]. In all cases, the strong impact of continuous shape
evolution due to non-homogeneous shear rates makes it difficult to exhibit a
simple scaling for the lift velocity: one cannot simply plunged a particle of
given shape into the desired geometry. Yet, several experiments and
simulations tend to prove that a scaling
$U_{L}\propto\dot{\gamma}(z)/z^{\alpha}$, with the exponent $\alpha$ close to
1, holds for several types of particles. In the following, we will denote by
$r$ the radius of the channel, and $\hat{r}=r/R$ its dimensionless form, that
accounts for the degree of confinement of the particle.

Figure 18: Lift of a red blood cell in a microchannel: snapshots at different
positions along the channel. Figure taken from Ref. [153].

Coupier  _et al._ [45] first proposed such a scaling for lipid vesicles. By
varying experimentally the confinement ($2\lesssim\hat{r}\lesssim 9$), they
showed that, in the range of viscosity contrast $1\leq\lambda\leq 10$ and of
reduced volume $\nu\geq 0.92$, trajectories from the wall to the center are
well described by the law

$\dot{z}=\xi\frac{R^{\delta+1}\dot{\gamma}(z)}{(z-z_{w})^{\delta}},$ (20)

where $\delta$ is close to 1 and $\xi$ a dimensionless parameter that depends
on the vesicle properties, similar to $A$ for the drift under simple shear
rate (Eq. (13)). $z_{w}$ is the position of the center of mass when the
particle is as close as possible to the wall. For the quasi-spherical vesicles
considered in Coupier  _et al._ [45], $z_{w}\sim R$ but in general, it may
depend on particle deformability. 2D numerical simulations have provided a
similar scaling [45]. Having $\delta$ close to 1 can somehow be viewed as an
intermediate case between the unbounded parabolic flow (where $\delta=0$) and
the shear flow in presence of a wall (where $\delta=2$).

The alternative law

$\dot{z}=\xi\frac{R_{0}^{\delta+1}\dot{\gamma}(z)}{z^{\delta}},$ (21)

is formally simpler and has been proposed in subsequent articles ([197] then
[153]) to allow for comparison between different situations with no need to
take into account the detail of the near-wall interactions.

Simulations of red blood cells yet having a non-physiological viscosity
contrast of 1 (and therefore in tank-treading regime) have highlighted an
exponent that is essentially in the range 1.2-1.3 for $\hat{r}=6$ and 8.8
[197].

Similarly to what was found for simple shear flows, the more complex dynamics
followed by red blood cells in physiological conditions does not prevent them
from following similar law. Losserand  _et al._ [153] found experimentally
through in vitro experiments that on a large range of confinements
($1.5\lesssim\hat{r}\lesssim 10$), Eq. (21) was followed with an exponent
$\delta\simeq 1.3$ (Fig. 18). They also mentioned that a fit of experimental
data by a trajectory obtained through Eq. (21) poses practical issue as
parameters $\xi$ and $\delta$ are strongly correlated: several pairs of values
for these parameters indeed lead to reasonable fits. Discussions on the exact
value of exponent $\delta$ should probably be considered with care.

Regarding the dependence with the particle mechanical properties, the overall
picture is that an increase in deformability (through an increase of capillary
number or a decrease of the viscosity ratio) leads to an increase in migration
velocity towards the center, be it for capsules [58, 197], vesicles [45] or
red blood cells [153]. For vesicles, varying the reduced volume of viscous
enough particle have an interesting effect: starting from the sphere,
deflating the vesicle allows for the fore-aft symmetry breaking that leads to
migration, but below a given reduced volume, the more elongated particle
aligns in the flow (approaching then the tumbling transition) and recover a
fort-aft symmetry; the lift velocity then drops to 0 [45].

The above mentioned studies focus on the migration from the wall towards the
center. As pointed out in [177, 116], this migration is dominated by the wall
effect in its vicinity. When the particle approaches the center, shear
gradient contribution will become dominant. As discussed previously, the
direction of the transverse migration might be reversed. In addition, since
the shear rate decreases as the particle approaches the center, the capillary
number decreases therefore the particle shape is not any more in a quasi-
steady configuration, leading to a more complex coupling between shape and
migration. This aspect is discussed in particular in the last pages of [146].

#### Shape-lift coupling and instability in channels

While particles approach the centerline, the presence of walls seem not to
destroy the complexity seen in unbounded Poiseuille flow. It rather
complexifies the picture, at least for deformable enough particles. Kaoui  _et
al._ [118] considered 2D vesicles with no viscosity contrast, and scrutinized
their behaviour while the confinement and the capillary number are varied. As
shown in Fig. 19, increasing the confinement leads to the appearance of
another kind of behaviour, which is an oscillation in the lateral position,
which can be centered or not, which is called snaking. The possibility to de-
stabilize this state towards a stationary shape through a time-varying flow
has been explored by Boujja  _et al._ [29].

Remarkably, while a transition from symmetric, centered shape to off-center
slipper shape is observed upon a decrease of the capillary number, a
transition from symmetric shape towards slipper is also observed upon an
increase of the capillary number, as long as a more viscous inner fluid is
considered: in Ref. [236], a 2D vesicle with a viscosity contrast of 5
exhibits such a behaviour, which encourages the authors to establish a
similitude with experimental observations on red blood cells. The latter
indeed exhibit the appearance of slipper shapes upon an increase of flow
velocity, in very confined situations [244, 101].

This behaviour for high viscosity contrast particles was later on confirmed by
3D numerical simulations of red blood cells [101, 238, 53], but also of
vesicles [4] — thus disregarding shear elasticity as an important parameter in
this problem. Guckenberger  _et al._ [101] explored the full range of
parameters relevant for microcirculation, and furthermore showed that most
configurations in the parameter space lead to bistability between the
centered, parachute-like shape and the slipper shape, whose existence depend
on the initial condition. Noteworthy, the snaking behaviour observed in 2D
simulations, but also in another 3D study [80] does not seem to take place in
their study. Similarly, a study of a vesicle placed in a confined simple-shear
flow has exhibited similar centered/off-centered transition without snaking
dynamics. Recktenwald  _et al._ [201] however highlighted slight oscillation
in lateral position of slipper-shaped red blood cells through experiments and
numerical simulations. They could be interpreted as a signature of off-
centered snaking. These oscillations are obtained after the flow velocity has
been increased progressively, and they tend to disappear after a while, on a
time scale that depends on the viscosity contrast. These results suggest that
the discussions on the existence of stable states should include the question
of the relaxation time needed to exit or enter a given state, which calls for
longer simulation times and also makes the comparison with experiments more
delicate.

More recently, several experimental developments have taken place for better
characterization of shape zoology, including careful design of chips to
control initial conditions [202], 3D tomography of flowing cells [223], and
machine-learning based methods for high throughput classification [127, 157,
223]

While the behaviour of these confined cells may be used as a tool to
characterize their individual mechanical properties, it should also be noted
that it has direct consequences on the collective behaviour, since the cell
shape will directly influence the flow pattern around it, therefore the
aggregation-disaggregation dynamics of a train of cells [163, 91, 243, 44,
237, 7, 276].

Figure 19: Diagram for the behaviour of 2D vesicles in a channel, as a
function of confinement ratio $\hat{r}$ and capillary number $Ca$. Figure
taken from [118].

###  III.4 Other configurations

#### Curved streamlines

Curved channels are frequent in microsystems. While more marginally studied,
this configuration has attracted some attention, in particular because this
geometry leads to interesting features when inertia comes into play (Dean
vortices).

Before considering this complex geometry, Ghigliotti  _et al._ [90] first
considered a model configuration with an unbounded flow consisting in circular
streamlines. When placing a 2D vesicles is this flow, they observed that tank-
treading vesicle migrate towards the center while tumbling one hardly
migrated. They demonstrated that the inward migration velocity is proportional
to $NR^{2}\dot{\gamma}/(r-R)$, where $r$ is the radial position of the
vesicle, and $N$ is the normal stress difference, that is related to the cell
mechanical properties. Chan and Leal [41] also predicted such a result for a
drop. In real systems, be it a curved channel or a Couette device, a wall
would be present at some point, therefore inducing outward migration.

Ebrahimi, Balogh and Bagchi have recently demonstrated that indeed a capsule
would converge towards an intermediate position between the wall and the
centre line, for a channel of circular [67] or rectangular [66] cross section.
Being the result of two competing effects lying on the same mechanism, the
resulting position is independent from the capillary number. Higher curvature
leads to a final position closer to the inner wall.

These studies are, to our knowledge, the sole ones accounting for migration in
curved channels at zero Reynolds number. It remains to be determined whether
this would greatly affect the flow of particles in channels, where the curved
part has necessarily a finite length. In [66], Fig. 3b, a capsule starting on
the central line of the channel is shown to have moved by 5% of the distance
to the inner wall after the channel has turned by 180∘, for a very sharp turn
of curvature radius of 5 times the cell radius. While this will probably lead
to negligible effect in most channels of interest, one may still use this
effect to induce particle separation by considering channels in spirals. Such
a geometry may also be used to validate the above mentioned numerical studies.

#### Oscillating flows

The case of near centerline migration has shown us that the interplay between
migration and shape leads to complex behaviour when a stationary shape cannot
be reached. Another way to produce a time lag between shape relaxation and
particle migration is to force changes in the applied flow. These changes can
be triggered by time varying boundary conditions or by the geometry, a typical
situation being structured microchannels.

Following a series of studies on particles dynamics under oscillating
unbounded flows [173, 125, 62, 179, 287, 160], this more recent field of
research has now been explored through several kind of particles and
geometries and will probably meet growing interest in the next years, for the
rich behaviours that emerge and the potential applications that could be
developed.

We first consider a particle placed in a time-periodic harmonic shear flow
bounded by a wall. In this problem, a new dimensionless parameter must be
added, which is the ratio $\hat{\omega}$ between the oscillation pulsation and
the maximal shear rate. At high enough capillary number such that the particle
shape relaxation time is set by the shear rate, the intuitive picture is that,
if $\hat{\omega}$ is increased from 0 (corresponding to stationary flow), the
particle will face situations where it does not have enough time to re-orient
itself after flow reversal, such that it will migrate towards the wall. On
average though, the net migration should be positive. In the large
$\hat{\omega}$ limit, the picture is that of a fixed shape in a time varying
flow, which recovers a fore-aft symmetry when time-averaging is made.

Zhu  _et al._ [289] considered 3D simulations of a capsule with no viscosity
contrast. In addition to confirming the decrease of the mean migration
velocity with $\hat{\omega}$, they also highlighted a non monotonous evolution
of this mean velocity, when rescaled by the typical flow velocity, with the
capillary number, at given $\hat{\omega}$. This is due to the plateauing of
the mean deformation upon an increase in capillary number (in practice, upon
an increase in maximal shear rate), because the capsule fails to reach its
potential maximal deformation, due to flow reversal. It is found that the
optimal capillary number scales linearly with $\hat{\omega}^{-1}$, in line
with the idea that at high capillary number the deformation is not limited by
its own deformability but rather by the time $\propto\omega^{-1}$ during which
the shear is applied in a given direction.

It is interesting to observe that the notion of time lag between shape
deformation and surrounding flow is sufficient to create an effective
asymmetry leading to migration even in an unbounded shear flow, providing the
particle presents an initial asymmetry, as discussed by [137] where a wide
class of particle is considered.

We are not aware of studies with flow reversal in a Poiseuille flow. Periodic
spatial modulation of the channel have instead attracted some attention in the
last 10 years, but most focus was on shape changes of centered vesicles or red
blood cells [181, 31]. Yet, such a sawtooth channel is argued to be an
efficient way to center cells in a microfluidic devices in [6]. However, the
picture might turn out to be more complex, according to recent numerical
simulations of vesicles in a wavy channel, that have exhibited off-centered
equilibrium positions in a configuration where the same vesicles would be
centered if the channel was straight [138].

##  IV Elastohydrodynamic interactions between particles

Figure 20: Fundamental configuration for pair interaction studies

As two particles cross each other in a flow, they may experience a lift force
of similar nature as that induced by the presence of a neighbouring wall. The
induced normal displacement has been documented by several experimental and
numerical studies. Such fluid-mediated scattering events in a suspension
induce a diffusion in all directions. This diffusion has two consequences:
mixing in the suspension, and flux along concentration gradients. Contrary to
Brownian diffusion, these two phenomena are characterized by coefficients that
are a priori independent [49], the down-gradient diffusion coefficient being
expected to be several times larger than the self-diffusion coefficient [49,
110, 98]. They can, in principle, be deduced from the knowledge of the
displacement map of the two particles as a function of their initial relative
position [49, 151]. However, this approach poses convergence issues due to the
slow decrease of the interaction force with lateral distance between particles
[151, 262].

Figure 21: Pair interaction between two droplets in unbounded shear flow. (A)
Evolution of the shift $z/R$ in the shear direction versus time, for a fixed
capillary number and various viscosity contrasts $\lambda$ indicated on the
figure. Particles are located in the same shear plane ($y_{0}=0$). The fine
solid curve corresponds to rigid sphere, and illustrates the absence of net
lift, for symmetry reasons similar to that discussed in Fig. 2. The fine dash-
dotted curve refers to a configuration not relevant here. The representation
in (B) is commonly used in these problems to depict the final position in the
$z$ (shear gradient) and $y$ (vorticity) directions as a function of initial
position. The vertices of the light gray grid indicate the initial positions
that were tested, while the vertices of the thick, deformed, one indicate the
final positions far from the reference particle. Here, $\lambda=1$ and in (b)
the capillary number is larger than in (a). In (b), for an initial position
$y/R=2,z/R=1$, the grid is slightly deformed towards smaller $y$, indicating
an attraction in the vorticity direction. Adapted from [151].

###  IV.1 Pair interaction

By comparison with the lift of a particle close to a wall, the finite extent
in all directions of the interacting particles makes this interaction problem
even richer. As they approach each other also in the flow direction, soft
lubrication effects will also take place when they collide. We will first
consider two identical particles in an unbounded shear flow of flow direction
$x$, shear direction $z$ and vorticity direction $y$, one of them placed at an
initial position $(x_{0},y_{0},z_{0})$ from the other, whose center of mass is
taken as the origin, with $x_{0}<0$ of large absolute value (see Fig 20). If
$z_{0}\neq 0$ (for further discussion, we will consider $z_{0}>0$), the two
particles will eventually cross each other, which may result in a net
displacement $(\Delta y,\Delta z)$ in the two directions perpendicular to
flow. This displacement depends a priori on both initial coordinates of the
first particle. Along the flow direction, an additional displacement $\Delta
x$ will also be found. Compared to the differential displacement between the
particles due to advection, it is however quite small and is seldom commented.

Before turning to deformable particles, it is worth mentioning that the finite
duration of the interaction between particles makes it possible to obtain a
net separation between solid particles. Da Cunha and Hinch [49] proposed a
model for the interaction between rough spherical particles, assuming that the
approach phase builds up a repulsive force while the separation phase does
not. However simple this assumption might seem, the existence of this
separation effect was later on proved experimentally [27].

Loewenberg and Hinch [151] studied numerically pair interaction of identical
drops in simple shear flow, for different values of viscosity contrast and
capillary number. A representative set of their data is shown in Fig. 21. The
relative trajectories shown in Fig. 21A show that, for particles not separated
in the vorticity direction, a significant shift of order one radius is
observed, if the initial position in the shear direction is also of the order
one radius. For drops, this shift decreases upon an increase of viscosity
contrast, which marks a strong difference with the case of a drop above a flat
wall, for which the dependence of the lift velocity with $\lambda$ is weak.
The synthesis of the final positions reached depending on the initial
positions shows several interesting features that illustrate the complexity of
this problem. First, a larger capillary number do not necessarily induce a
larger displacement. For small initial $z$, the contrary even occurs (see Fig.
5 in [151]). Again, this is in marked contrast with the results for a particle
near a wall. This points to a more complex situation from the geometrical
point of view, since the incoming particle do not only flow above the other
one but also hits it initially: Loewenberg and Hinch [151] argued that the
increased deformability reduces somehow the cross section for near contact
interaction at collision.

Though not commented by the authors, a weak attraction in the vorticity
direction can be seen for a drop initially located at position $(2R,R)$, in
Fig. 21B(b). This phenomenon also appeared later on in other studies.

Fig. 21B also shows that net displacements are much larger in the shear
gradient than in the vorticity direction. This implies that diffusion due to
collisions is strongly anisotropic, as will be discussed later.

Figure 22: Succesive snapshots of interacting drops with $\lambda=1.4$ and
$R=20\,\mu$m. The scale bar is $25\,\mu$m. Adapted from [103].

As for lift above a wall, experimental studies on droplet interaction are
scarce. Guido and Simeone [103] and Wang  _et al._ [259] showed some
trajectories, and confirmed the typical trajectories shown in Fig. 21A, and
the weak net displacement as soon as the initial distance is larger than a few
radii, as in Fig. 21B. As can be seen in Fig. 22, the collision between the
particles result in the creation in an extended and long-lasting contact
between the particles, with the creation of lubrication film. [151] exhibited
different scalings for the duration of the approach sequence and have showed
that the separation stage is much shorter.

Capsules are again widely ignored by experimentalists, while several numerical
simulations shed light on the detail of their interactions under shear flow.
Lac  _et al._ [135] asked themselves whether the presence of a membrane
modifies the drift observed for droplets. Their numerical simulations showed
that for capsules with no viscosity contrast placed in the same $xz$ plane,
the net displacement is smaller. They also found a weak effect of the
capillary number and of the initial deflation. The same team published a
complementary study where they studied the consequence of the capsules not
being in the same shear plane (i.e. $y_{0}\neq 0$) [133]. By contrast with
what is observed on drops, a clear attraction in the vorticity direction is
observed for initial separation in the vorticity direction of order the
capsule radius and small separation in the shear direction (i.e., small
velocity difference). In this situation, the displacement depends strongly on
the capillary number, and increases with it. The authors provided no
explanation for this phenomenon.

Lac et al. also highlighted the fact that even if the capsules are placed on
the same streamline of the unperturbed flow, they may still interact and cross
each other. This is due to the flow perturbation associated with the tank-
treading motion of the membrane, which have the effect to shift the particles
apart: the particle located at $x_{0}<0$ and $z_{0}=0$ will see its $z$
position increasing because of the clockwise tank-treading motion of the
membrane of the particle located at the origin.

Singh and Sarkar [225] confirmed the difference between drops and capsules,
the latter showing less cross-stream separation, for viscosity contrast equal
to 1. The difference turns out to decrease upon an increase of this viscosity
contrast.

Le and Chiam [139] considered a refined model for capsules that includes
bending elasticity (while previous one only included shear elasticity). They
also considered different initial shapes, one of them being the biconcave
shape of red blood cells. Finally they considered a viscosity ratio of 4,
closer to physiological values for red blood cells. The capsules were kept in
the same shear plane. Instead of particle crossing, they observed for small
initial $z_{0}$ a motion called spiraling, that consists in oscillations of
the particle positions at finite distance from each other. In this article,
the origin of this effect is particularly unclear, as the positions around
which oscillations take place correspond to an equilibrium configuration
induced by the periodic boundary condition. Moreover, the viscosity ratio
apart, the configuration is quite similar to that studied in [135], where no
such motion was described.

Fore small capillary numbers, more deflated capsules exhibit more complexe
interaction patterns, like a swapping of positions as they collide, or a
pairing followed by a rotation of the couple as a whole. This happens in
conditions where the capsule would tumble, if isolated in the flow. As these
behaviours take place in the middle of the simulation box, they are probably
more trustful. This possibility of more complex interactions has been
confirmed by other numerical simulations of capsules with no viscosity
contrast [109]: for an initial position defined by small enough $|x_{0}|$,
$z_{0}$ close to 0 and $y_{0}\neq 0$, the initial shift in the $z$ direction
due to the flow induced by the reference capsule is not large, essentially
because the studied capsule can flow straight. However, the attraction after
interaction, as observed in Fig. 21A, is maintained, such that the sign of $z$
is changed, implying a backward motion of the capsule and a new crossing. In
the meantime, it is, on average, attracted in the $y$ direction (as already
seen in [133]). Depending on deformability, the interaction might end up there
(thus resembling the swapping motion described by Le and Chiam [139]), or go
on for one or more additional interaction, leading to what the authors called
minuet motion. In agreement with Le and Chiam [139], swapping or multiple
swapping (i.e. minuet) is favored by low capillary number.

Pair interaction of vesicles has been studied through experiments, numerical
simulations and theory. Using the far field perturbation due to one vesicle
(which is proportional to its stresslet and decays as the inverse of the
distance squared), Farutin and Misbah [75] proved that, for weakly deformed
vesicles with no viscosity contrast, placed in the same shear plane, the net
displacement scales as the inverse of the initial position $z_{0}$ squared.
They also provided an expression for the prefactor, as a function of reduced
volume. Their result agrees quantitatively with their own simulations. They
did not study the case of initial offset in the $y$ direction. Following the
same theoretical framework, Gires  _et al._ [93] showed that vesicles with
high viscosity contrast (but not tumbling) exhibit attraction in both
directions as soon as $|y_{0}|>|z_{0}|$. The range of validity of this theory
makes it however weakly amenable to experimental check.

Gires  _et al._ [94] studied the case of vesicles with no viscosity contrast
not initially placed in the same shear plane was studied through numerical
simulations. They found that for vesicles with initial position $y_{0}$ larger
than a threshold which is of order the vesicle radius, attraction in the
vorticity direction takes place, while almost no net displacement is observed
in the shear direction. The author notes that, while in [93] the attraction
can be interpreted in terms of contribution of the far field perturbation due
to the other vesicle, here the small distance between the vesicles makes it
necessary to consider additional forces due to the fluid flow in the thin film
created between the vesicle, that are deformed by their interaction. This
soft-lubrication approach may follow the guidelines of Loewenberg and Hinch
[151]. Numerically, the additional pressure that builds up was particularly
discussed in the study of capsules dancing menuet [109].

Figure 23: Net displacement $\Delta z$ as a function of initial offset $z_{0}$
for vesicles places in the same shear plane. Dots correspond to experimental
data for vesicles with $0.98>\nu\gtrsim 0.75$ and $\lambda=0.28$ (black
squares), $\lambda=1$ (red disks), and $\lambda=3.8$ (open squares). $Ca$ lies
between 10 and 100. Dashed line ($Ca=10$), full line ($Ca=50$) and dotted line
($Ca=100$): simulations for $\lambda=1$ and $\nu=0.95$. Dash-dotted line:
simplified model (Eq. (22)). Adapted from [94].

Gires  _et al._ [94] also presented experiments between vesicles with
$0.28<\lambda<3.8$, and reduced volume $\nu\gtrsim 0.75$ placed in the same
shear plane. Their main results are shown in Fig. 23. As for most previous
studies, a maximal shift of the order one radius is found. Remarkably, and in
the limit of the experimental uncertainty the net displacement does not seem
to depend much either on the reduced volume or on the viscosity contrast,
though both are varied in a large range. The experimental results match well
with simulations of vesicles with no viscosity contrast.

In an attempt to rationalize this weak dependence on the mechanical properties
of the vesicles, the authors proposed to model the interaction between the
vesicles as the lift of one vesicle above a wall of finite length $2R$.
Assuming the vesicle starting at $z_{0}$ moves with velocity $\dot{\gamma}z$
relatively to the vesicle of reference, Eq. (13) leads to
$dz/dx=AR^{3}/z^{3}$. Integration along the trajectory leads to the net
displacement

$\Delta z=(z_{0}^{4}/R^{4}+8A)^{1/4}-z_{0}/R.$ (22)

A fit of experimental data with this rough model, shown in Fig. 23, showed
good agreement with single fitting parameter $A$ close to the typical values
found by Olla. The overall amplitude of the interaction curve is set by the
maximal displacement $(8A)^{1/4}$. This $1/4$ exponent explains why the
variations of $A$ with cell mechanical property are smoothed out when net
displacement is considered. Note that the long distance limit $z_{0}/R\gg 1$
of Eq. (22) is $\Delta z\sim 2AR^{3}/z^{3}$, which is not in agreement with
more accurate theoretical derivations [75]. Yet, this law can serve as a good
proxy for estimating the drift due to interaction.

In channels, the interaction between particles becomes more complex, due to
the varying shear rate. Also, the presence of walls strongly modifies the
perturbing field creating by one particle, and attraction/repulsion in the
flow direction becomes a dominant feature, as it exists also for rigid spheres
[290]. This leads to complex structuring mechanisms in the flow direction
[163, 91, 243, 44, 237, 7, 276], which we will not study here.

Finally, the collision between particles of different properties is of great
interest to understand segregation mechanisms within a suspension.

Kumar and Graham [130], Kumar  _et al._ [132], Singh and Sarkar [225] studied
pairs of capsules with different rigidities; the key finding is that the
stiffer particle is more displaced, although the relative displacement remains
quasi constant. In line with the result for similar capsules, the net
displacement depends only weakly on the capillary number. Závodszky  _et al._