A42-1–A42-19 jfm.2022.730

# Phoresis in cellular flows:
from enhanced dispersion to blockage

Romain Volk1<EMAIL_ADDRESS>Michaël Bourgoin1  Charles-Édouard
Bréhier2
Florence Raynal3<EMAIL_ADDRESS>1Laboratoire de Physique, ENS de
Lyon, CNRS, 69364 Lyon CEDEX 07, France 2Institut Camille Jordan, Univ Lyon,
Université Claude Bernard Lyon 1, CNRS, F-69622 Villeurbanne CEDEX, France
3Laboratoire de Mécanique des Fluides et d’Acoustique, Univ Lyon, École
Centrale Lyon, INSA Lyon, Université Claude Bernard Lyon 1, CNRS, F-69134
Écully, France.

(Received 25 March 2022; revised 12 July 2022; accepted 20 August 2022; 2022)

###### Abstract

In this article, we study numerically the dispersion of colloids in a two-
dimensional cellular flow in the presence of an imposed mean salt gradient.
Owing to the additional scalar, the colloids do not follow exactly the
Eulerian flow field, but have a (small) extra-velocity proportional to the
salt gradient, $\mathbf{v}_{\mathrm{dp}}=\alpha\nabla S$, where $\alpha$ is
the phoretic constant and $S$ the salt concentration. We study the demixing of
an homogenous distribution of colloids and how their long-term mean velocity
$\mathbf{V_{m}}$ and effective diffusivity $D_{\mathrm{eff}}$ are influenced
by the phoretic drift. We observe two regimes of colloids dynamics depending
on a blockage criterion $R=\alpha GL/\sqrt{4D_{c}D_{s}}$, where $G$ is the
mean salt gradient amplitude, $L$ the length scale of the flow and $D_{c}$ and
$D_{s}$ the molecular diffusivities of colloids and salt. When $R<1$, the mean
velocity is strongly enhanced with $V_{m}\propto\alpha G\sqrt{\Pen_{s}}$,
$\Pen_{s}$ being the salt Péclet number. When $R>1$, the compressibility
effect due to the phoretic drift is so strong that a depletion of colloids
occurs along the separatrices inhibiting cell-to-cell transport.

††volume: 948, A42

##  1 Introduction

The transport of a passive contaminant or a magnetic field in flows with
closed streamlines has been the focus of many studies in the past few decades,
see, for instance, Solomon & Gollub (1988); Shraiman (1987); Soward (1987);
Young et al. (1989). In such configurations, it has been shown that the
transport of Brownian particles is greatly enhanced by advection: their
dynamics at long time becomes diffusive, with the ratio of their effective
diffusivity to their molecular diffusivity growing as the square root of the
Péclet number, $\Pen=UL/D$, where $U$ is the velocity scale, $L$ its the
length scale, and $D$ the molecular diffusivity of the particles. Since these
pioneering studies, modified versions of this problem have been addressed in
the context of non-tracer particles i.e., particles which do not strictly
follow the fluid motions because of their response time to flow modifications
(Maxey, 1987a). The question of long time dynamics of very heavy, weakly
inertial, particles settling under the action of gravity was addressed by
Pavliotis & Stuart (2005) and Afonso (2008) who derived expressions for the
mean particle settling velocity, which was found to be larger than the
settling velocity in a quiescent fluid. More recently, such study was extended
to the case of arbitrary density ratio by Renaud & Vanneste (2020); in
particular, the authors pointed out the role of compressibility effects due to
particle inertia, which can lead to reduced transport in the case of light
particles that tend to be trapped in vortical flow regions (Maxey, 1987b; Wang
& Maxey, 1993). Reduction of effective diffusivity was also observed
numerically by Li et al. (2021) who studied the case of passive Brownian
particles submitted to the action of a mean transverse force in an array of
vortex similar that of Solomon & Gollub (1988), and in the case of active
particles moving in a laminar vortical flow as transport barriers develop
(Berman et al., 2021).

Phoretic particles such as colloids are another class of particles which do
not follow exactly the fluid motion; indeed, they are submitted to a drift
velocity that originates from heterogeneities in the background of chemical
(diffusiophoresis) or thermal (thermophoresis) origin. Anderson (1989) showed
that the drift velocity for diffusiophoresis is of the type
$\mathbf{v}_{\mathrm{dp}}=D_{\mathbf{dp}}\nabla\log S$, where
$D_{\mathrm{dp}}$ is the diffusiophoretic mobility and $S$ is the
concentration in background chemical (e.g. salt), whereas Gupta et al. (2020)
determined experimentally that the dependence can be rather more complex (in
particular, $D_{dp}$ may depend on the salt concentration). These observations
are in accordance with a recent more complex expression for the phoretic
velocity proposed in Menolascina et al. (2017); Salek et al. (2019). Thus,
depending on the species considered and its concentration, a drift velocity of
the type $\mathbf{v}_{\mathrm{dp}}=\alpha\nabla S$ can also be observed in
practice (corresponding to $D_{\mathrm{dp}}\propto S$), as discussed in Chu et
al. (2022); this is the expression we consider in this article. Note finally
that in a view of simplicity, in the following we mainly refer to
diffusiophoresis. However, other mechanisms like chemotaxis (for instance,
movement of a motile organism in a direction corresponding to a gradient
concentration of a nutriment) lead to a similar expression for the drift
velocity, but with a much larger drift coefficient $\alpha$, as also discussed
in Chu et al. (2022).

Mixing of phoretic particles in the presence of salt heterogeneities was first
studied experimentally in a $\Psi$-shaped channel (Abécassis et al., 2009),
and later in chaotic advection at the micro- or macro-scale (Deseigne et al.,
2014; Mauger et al., 2016). In particular, the authors showed that mixing is
delayed when the colloids are introduced together with the salt, and enhanced
when introduced in salted water. In this context, it has been shown that
although the drift velocity is usually very weak as compared with the flow
velocity, mixing is strongly modified because, due to diffusiophoresis, the
particle velocity field is compressible (Volk et al., 2014; Raynal et al.,
2018; Raynal & Volk, 2019; Chu et al., 2020, 2021). One may thus wonder how
the long time transport of such phoretic particles by a cellular flow is
modified when a mean scalar gradient such as salt or temperature is imposed to
the system. It can be anticipated that in the large time regime, a stationary
state is reached and we are interested here in the behaviour of the effective
velocity and diffusivity in this asymptotic regime: for instance, we wonder
whether the average colloids velocity will be enhanced or reduced as compared
with the case without advection. Answering this question is not trivial
because the presence of strongly localised gradients may lead to enhanced
cell-to-cell transport as well as arrested transport as observed previously in
a simpler geometry (Chu et al., 2020).

In this article, we study the dispersion of colloids in a 2-dimensional
cellular flow in the presence of an additional scalar (which we call salt)
forced with an imposed mean salt gradient. The flow satisfies

$\mathbf{u}(x,y)=u_{0}(\sin kx\cos ky,-\cos kx\sin ky)\,,$ (1)

with $k=\pi/L$. It is $2L$ periodic in space and has closed streamlines in
square cells of side $L$. It is a good model for Rayleigh-Bénard flows close
to the instability threshold (Solomon & Gollub, 1988); the same flow was also
realised using an electrolyte, a network of magnets and a DC current to drive
the flow using the Lorentz force in order to address particle settling
(Bergougnoux et al., 2014).

Owing to the presence of salt gradients, the colloids do not follow exactly
the Eulerian flow field $\mathbf{u}$, but have a (small) extra velocity of the
type

$\mathbf{v}_{\mathrm{dp}}=\alpha\nabla S\,,$ (2)

where $S$ is the salt concentration and $\alpha$ is the (small) phoretic
coefficient taken constant and independent of $S$. Salt and colloids
concentration fields ($S$ and $C$) satisfy the advection diffusion equations

$\displaystyle\partial_{t}S+\nabla\cdot S\mathbf{u}=D_{s}\nabla^{2}S\,,$ (3)
$\displaystyle\partial_{t}C+\nabla\cdot
C(\mathbf{u}+\mathbf{v}_{\mathrm{dp}})=D_{c}\,\nabla^{2}C\,,$ (4)

where $D_{s}$ and $D_{c}$ are the molecular diffusion coefficients of salt and
colloids, respectively. Note the one-way coupling: from equation (3) the salt
evolves freely, whilst colloids are coupled to salt via the phoretic velocity
$\mathbf{v}_{\mathrm{dp}}$ (equations (4) and (2)). Note also that equation
(4) is linear in $C$; we suppose that the mean colloids concentration $C_{0}$
is such that the mean distance between colloids is sufficiently large so that
colloid–colloid interactions can be neglected.

The salt concentration is forced by an imposed mean salt gradient in the
$x$-direction

$\mathbf{G}=\langle\nabla S\rangle=G\,\mathbf{e}_{x}\,,$ (5)

where $\langle\cdot\rangle$ represents a spatial average over a square of side
$2L=2\pi/k$. Using the $\pi$-theorem (Vaschy, 1892; Buckingham, 1914) shows
that the problem is governed by three independent non-dimensional parameters:
we choose the salt and colloids Péclet numbers $\Pen_{s}=u_{0}L/D_{s}$ and
$\Pen_{c}=u_{0}L/D_{c}$ respectively, and a new additional number, a criterion
of blockage, which will be shown to be the ratio $R=\alpha
GL/\sqrt{4D_{c}D_{s}}$, see section 4.2. Therefore, without loss of generality
we can fix $u_{0}$, $L$ and $G$ and only vary $\alpha$, $D_{s}$ and $D_{c}$.
In the following, we set $u_{0}=1$, $L=\pi$ and $G=1$ for the numerical
simulations.

By means of high-resolution numerical simulations performed both in the
Eulerian and Lagrangian frameworks, we study the demixing of an homogenous
distribution of colloids and how their mean velocity $\mathbf{V}_{m}$ and
effective diffusivity $D_{\mathrm{eff}}$ (obtained in the long time regime,
see section 3) are influenced by the phoretic drift for a wide range of
parameters.

This paper is organised as follows. In Section 2 we address the well-
documented case of mixing of diffusing salt in a cellular flow in the presence
of an imposed mean salt gradient $\mathbf{G}$, which converges towards a
stationary state. In Section 3 we describe the demixing of diffusing colloids
in this stationary salt concentration field; here again the colloids
concentration field reaches a stationary state. We consider Eulerian
simulations, and explain the typical concentration fields observed. Then we
consider Lagrangian simulations for higher colloids Péclet number $\Pen_{c}$,
and study the net mean velocity $V_{m}$ reached by the colloids, together with
their effective diffusion; in particular, we show that, depending on the
parameters considered, $V_{m}$ can be much higher than what could be expected,
but also much less, up to vanishing, a situation referred to as “blockage”.
Finally in Section 4 we go into more detail concerning the mean colloids
velocity analysis. We deduce the order of magnitude of $V_{m}$ when no
blockage occurs, obtain the condition of blockage and compare with numerical
simulations in a wide range of parameters.

##  2 Case of salt

The transport of a scalar or a vector in a cellular flow has been the focus of
many studies in the past decades (Shraiman, 1987; Soward, 1987; Young et al.,
1989; Afonso, 2008; Renaud & Vanneste, 2020). We propose here to briefly
outline the subject in the context of an imposed mean gradient which
corresponds to the original configuration of Shraiman (1987).

The salt concentration $S(x,y,t)$ evolves independently of the colloids and is
forced by the imposed gradient $\mathbf{G}=(G_{x},G_{y})$ following equations
(3) and (5). In practice, we solve numerically the equation for
$S^{\prime}=S-\mathbf{G}\cdot\mathbf{x}$, which satisfies the equation

$\partial_{t}S^{\prime}+\nabla\cdot
S^{\prime}\mathbf{u}=D_{s}\nabla^{2}S^{\prime}-\mathbf{G}\cdot\mathbf{u}\,.$
(6)

Because the velocity field is periodic in space with zero average

$\langle\mathbf{u}\rangle=\frac{1}{4L^{2}}\iint_{[0,2L]^{2}}\mathbf{u}(x,y)\,dxdy=0\,,$
(7)

equation (6) for $S^{\prime}$ can be solved with periodic boundary conditions
with $\langle S^{\prime}\rangle=0$ (Holzer & Siggia, 1994). This ensures that
the total salt concentration field $S=S^{\prime}+\mathbf{G}\cdot\mathbf{x}$
satisfies boundary conditions compatible with the imposed mean gradient:

$\displaystyle S(x+2L,y)$ $\displaystyle=$ $\displaystyle S(x,y)+2G_{x}L$ (8)
$\displaystyle S(x,y+2L)$ $\displaystyle=$ $\displaystyle S(x,y)+2G_{y}L\,.$
(9)

In the following, without any loss of generality, we set $L=\pi$ for the
numerical simulations. The equation for $S^{\prime}$ is solved in spectral
space in a square domain $(x,y)\in[0,2\pi]^{2}$ using the method described in
Volk et al. (2014) with a mean gradient in the $x$ direction
($\mathbf{G}=(G,0)$). Because equation (6) is linear, $S^{\prime}$ is
proportional to $G$ so that we set $G=1$ without loss of generality. We set
$u_{0}=1$ and adjust the spatial resolution to $256^{2}$ for Péclet number
$\Pen_{s}=u_{0}L/D_{s}\in[30,300]$ and $512^{2}$ for
$\Pen_{s}=u_{0}L/D_{s}\in[300,3000]$. Such very high resolutions are chosen to
ensure high precision when interpolating the salt gradient in real space which
is needed for the Lagrangian dynamics of colloids.

Integrating in time from an initial condition $S^{\prime}=0$, the salt
concentration develops strongly localised gradients and reaches a stationary
state ($\partial_{t}S=0$) thanks to the stationarity of the forcing term
$-G\,u_{x}$ (equation 6). Such localisation can be understood in the limit of
infinite Péclet numbers. Indeed, in the case of a non-diffusing scalar
($D_{s}=0)$, equation 3 leads to $\mathbf{u}\cdot\nabla S=0$, which admits a
solution for which $\nabla S$ is zero almost everywhere. Given the forcing, it
implies that $S$ is piecewise constant and increases by jumps by a value $GL$
in the direction $x$ of the forcing; the jump occurs at the vertical boundary
of a cell, where the velocity is vertical and perpendicular to the salt
gradient so that $\mathbf{u}\cdot\nabla S=0$.

Figure 1: Salt case, with a positive mean imposed gradient
$\mathbf{G}=\pi/L\,\mathbf{e}_{x}$ ($L=\pi$). $(a)$: Two-dimensional
concentration field for $\Pen_{s}=314$; the concentration is almost constant
in a given cell. $(b)$: One-dimensional cut along the dotted line in figure 1
$a$ $(y=\pi/2)$ for three different salt Péclet numbers: - - -:
$\Pen_{s}=31.4$; -$\cdot$-$\cdot$-, $\Pen_{s}=314$; —, $\Pen_{s}=3\,141$; for
$\Pen_{s}\geq 314$, the profile is flat except for a very small layer around
the vertical separatrix where the concentration jumps abruptly.

This is indeed the picture that we obtain at large Péclet number, as shown in
figure 1, which displays the Eulerian salt concentration field obtained for
$\Pen_{s}=314$ (Fig 1 $a$) together with a one-dimensional horizontal profile
at mid-height of a cell for different Péclet numbers (Fig 1 $b$): the
concentration is almost constant in a given cell, and jumps abruptly (but on a
finite slope) by a value $\Delta S=GL$ from one cell to the next one in the
$x$-direction.

The typical width $\ell_{s}$ on which the concentration increases by a value
$\Delta S$ (the salt Batchelor scale (Batchelor, 1971)) results from a
competition between contraction and diffusion through the equation of salt
gradient $\vec{\gamma}^{s}=\nabla S$:

$\frac{D\gamma_{i}^{s}}{Dt}=-\partial_{i}u_{j}\,\gamma^{s}_{j}+D_{s}\nabla^{2}\gamma^{s}_{i}\,,$
(10)

where $D/Dt=\partial_{t}+u_{k}\partial_{k}$ is the material derivative. This
horizontal gradient is built by bringing the scalar to the hyperbolic point
$(\pi,0)$ by contraction by the velocity field along the horizontal separatrix
($i=j=x)$; because the concentration field is stationary its width is set when
the two terms on the right-hand side of equation (10) are of the same order of
magnitude (Raynal & Gence, 1997), that is, $u_{0}/L\sim D_{s}/\ell_{s}^{2}$,
therefore

$\ell_{s}\sim\frac{L}{\sqrt{\Pen_{s}}}\,.$ (11)

Indeed, as the horizontal velocity reverses near the point $(\pi,0)$, the
advection term $\mathbf{u}\cdot\nabla\gamma_{x}$ vanishes when averaged over
the width $\ell_{s}$ while the contraction term remains. This concentration
gradient is then advected upwards by the velocity field along the vertical
separatrix $x=L=\pi$. It reaches the top of the cell after a time
$\tau^{s}_{\mathit{adv}}\sim L/u_{0}$, time after which it is dissipated by
diffusion because the characteristic dissipation time is
$\tau^{s}_{\mathit{diff}}\sim\ell_{s}^{2}/D_{s}$, and
$\tau^{s}_{\mathit{adv}}=\tau^{s}_{\mathit{diff}}$ because of equation (11).
Thus, this concentration discontinuity is only visible around the vertical
separatrix, with an associated gradient

$\vec{\gamma}^{s}\sim\frac{\Delta S}{\ell_{s}}\,\mathbf{e}_{x}\sim
G\,\sqrt{\Pen_{s}}\,\mathbf{e}_{x}\,.$ (12)

This scenario supposes that the salt can be transported on a large scale
before diffusing, i.e. $\Pen_{s}\gg 1$, in agreement with figure 1 $b$.

##  3 Case of colloids

In the following we suppose that the salt has reached its stationary state,
and focus on the dynamics of the colloids which are advected by the velocity
field $\mathbf{v}=\mathbf{u}+\mathbf{v}_{\mathrm{dp}}$ (with
$\mathbf{v}_{\mathrm{dp}}=\alpha\nabla S$) and diffuse with a diffusion
coefficient $D_{c}$. Starting from a uniform distribution, colloids will demix
because $\nabla\cdot\mathbf{v}_{\mathrm{dp}}\neq 0$ (Volk et al., 2014) and
reach a non-uniform stationary state that we will characterise thereafter.

This problem can be addressed either in the Eulerian or in the Lagrangian
frameworks. On the one hand, one may use the Lagrangian approach by solving
the stochastic differential equation governing the position $\mathbf{X}_{c}$
of the colloids:

$d\mathbf{X}_{c}=\bigl{(}\mathbf{u}(\mathbf{X}_{c})+\alpha\nabla
S(\mathbf{X}_{c})\bigr{)}dt+\sqrt{2D_{c}}\,d\mathbf{W}\,,$ (13)

where $\mathbf{W}$ is a two-dimensional Wiener process (Van Kampen, 2007).
Such framework is used in the following in cases when the colloid Péclet
number, $\Pen_{c}=u_{0}L/D_{c}$, is very large, in order to compute the mean
velocity of the colloids and their effective diffusivity at long times
$t\in[\tau_{c}/10,\tau_{c}]$, where $\tau_{c}=L^{2}/D_{c}$ is the diffusion
time of the colloids. In practice, we compute a three cubic periodic
interpolant of the salt gradient, resolved with very high precision, for each
value of $Pe_{s}$, which is then used for all cases when varying
$\alpha\in[0,~{}10^{-2}]$ and $D_{c}\in[10^{-6},~{}10^{-1}]$; starting from at
least $\mathcal{O}(10^{4})$ uniformly distributed colloidal particles, we
solve equation (13) numerically for each trajectory by splitting the
deterministic and stochastic parts at each time step. We use a fourth-order
Runge-Kutta method for the deterministic part with $dt=10^{-2}$ and a standard
Euler scheme Euler scheme for the stochastic part (Higham, 2001).

On the other hand, the problem may also be addressed in an Eulerian framework
by investigating the colloids concentration field $C$, as
$C(\mathbf{x},t)/\langle C\rangle$ is the probability of finding a colloid at
location $\mathbf{x}$ at time $t$. The concentration $C$ is coupled with that
of the salt through the diffusiophoretic velocity
$\mathbf{v}_{\mathrm{dp}}=\alpha\nabla S$. We use the Eulerian approach and
solve the partial differential equation (4) in the case of moderate Péclet
numbers ($Pe_{c}\leq 3\cdot 10^{4}$, $D_{c}\in[10^{-4},~{}10^{-1}]$) using the
same code as for the salt; without loss of generality we set the initial
uniform condition $C(x,y,t)=C_{0}=1$ as the equation is linear in $C$. We run
the simulation until $C$ has reached the stationary regime
($\partial_{t}C=0$).

In practice in this study, we restricted Eulerian simulations to cases with
moderate colloids Péclet numbers: indeed, when the Péclet number increases,
the colloid concentration fronts become steeper, even in the absence of
diffusiophoresis. This is also the case when blockage occurs. Because steep
fronts can only be caught at a very high numerical price with a spectral code,
we limited the Eulerian cases to $\Pen_{c}\leq 3100$. We tested the
equivalence between the two methods on a range of intermediate Péclet numbers
by comparing the mean Lagrangian colloid velocity to its Eulerian counterpart
$\langle C(\mathbf{u}+\alpha\nabla S)\rangle$: we successfully obtained the
same value with both methods for all cases tested.

When dealing with diffusiophoresis, it is usual to consider separately the
salt-attracting case where the colloids move toward the salt gradient
($\alpha>0$), and reversely the salt-repelling case ($\alpha<0$) (Deseigne et
al., 2014; Volk et al., 2014). Indeed, changing the sign of the
diffusiophoretic parameter may fully change the physics and the scaling
involved (Raynal & Volk, 2019; Chu et al., 2020). This is not the case here:
the stationary colloids concentration field is actually invariant under
transformation $\alpha\rightarrow-\alpha$, $x\rightarrow-x$ and
$y\rightarrow-y$, see Appendix A. This implies that the physics of the problem
is unchanged when considering salt-attracted or salt-repelled colloids. Thus
without loss of generality we set

$\alpha\geq 0\,.$ (14)

Finally, as the drift velocity ($\mathbf{v}_{\mathrm{dp}}=\alpha\nabla S$) is
linear in $\alpha\,G$, equations are unchanged if the imposed salt gradient
$\mathbf{G}$ is reversed and $\alpha$ changed into $-\alpha$.

Because colloids are expected to have a small diffusion coefficient as
compared with the salt, we consider only cases for which

$D_{c}\ll D_{s}$ (15)

(equivalently, $\Pen_{c}\gg\Pen_{s}$). In the case $\alpha>0$, the colloids
start to demix due to diffusiophoresis, and gradients of colloid concentration
appear. In the stationary state, the width $\ell_{c}$ of these gradients is
governed by the competition between the dominant mechanism of gradient
creation (contraction either by the velocity field or diffusiophoresis) and
destruction by diffusion. In the case when contraction by the velocity field
is dominant, we obtain as for the salt a heterogeneity of width
$\ell_{c}=L/\sqrt{\Pen_{c}}$ along the vertical separatrix, with
$\ell_{c}\ll\ell_{s}$. When diffusiophoresis is dominant, it acts as an
additional mechanism of creation of gradient, which leads to an even smaller
Batchelor scale $\ell_{c}$. Finally, we have whatever $\alpha$

$\ell_{c}\ll\ell_{s}\,.$ (16)

###  3.1 Eulerian results: colloidal concentration field

The salt field is uniform, except in the vicinity of the vertical separatrices
where there is a very strong salt gradient. It is therefore understood that
far from the separatrices, diffusiophoresis does not operate and the colloids
move only along the current lines; as a result, the colloids concentration
remains constant in the central vortex. Because non-trivial behaviour takes
place around the vertical separatrices, we consider the equation of evolution
of the gradient in colloids $\vec{\gamma}^{c}=\nabla C$ (Raynal & Volk, 2019)
111Compared with our previous work (Raynal & Volk, 2019), the diffusiophoretic
velocity is here proportional to the salt gradient.:

$\frac{D\gamma^{c}_{i}}{Dt}=\underbrace{-\partial_{i}u_{j}\,\gamma^{c}_{j}}_{(a)}+\underbrace{D_{c}\nabla^{2}\gamma^{c}_{i}}_{(b)}-\underbrace{\alpha\partial_{j}\gamma^{c}_{i}\partial_{j}S}_{(c)}-\underbrace{\alpha\gamma^{c}_{j}\partial_{i}\partial_{j}S}_{(d)}-\underbrace{\alpha\gamma_{i}^{c}\partial_{j}^{2}S}_{(e)}-\underbrace{\alpha
C\partial_{i}\partial_{j}^{2}S}_{(f)}\,.$ (17)

Let us evaluate each of those terms on the separatrix:

$\displaystyle(a)$ $\displaystyle\sim$ $\displaystyle u_{0}\gamma^{c}/L$ (18)
$\displaystyle(b)$ $\displaystyle\sim$ $\displaystyle
D_{c}\gamma^{c}/\ell_{c}^{2}$ (19) $\displaystyle(c)$ $\displaystyle\sim$
$\displaystyle\alpha\gamma^{c}\Delta S/(\ell_{c}\ell_{s})$ (20)
$\displaystyle(d)$ $\displaystyle\sim$ $\displaystyle\alpha\gamma^{c}\Delta
S/\ell_{s}^{2}\ll(c)$ (21) $\displaystyle(e)$ $\displaystyle\sim$
$\displaystyle\alpha\gamma^{c}\Delta S/\ell_{s}^{2}\ll(c)$ (22)
$\displaystyle(f)$ $\displaystyle\sim$ $\displaystyle\alpha\gamma^{c}\Delta
S\ell_{c}/\ell_{s}^{3}\ll(c)\,,$ (23)

where the three last terms are negligible compared with term $(c)$ because of
relation (16). As the velocity field $\mathbf{u}$ is perpendicular to the salt
or colloids gradients along the vertical separatrix, as in Section 2 the left
hand-side term of equation (17) is zero in this stationary regime. We consider
the case when the mechanism of creation of gradients by diffusiophoresis is
stronger than by advection, which gives $(c)\gtrsim(a)$. The width of the
heterogeneity in colloids is then given by the competition between the
creation of gradients by diffusiophoresis, and its destruction by diffusion
($(c)\sim(b)$). Therefore, we obtain

$\ell_{c}\sim\frac{D_{c}\ell_{s}}{\alpha\Delta S}\sim\frac{D_{c}}{\alpha
G\sqrt{\Pen_{s}}}\,,$ (24)

given equation (12). Combining condition $(c)\gtrsim(a)$ with scaling (24)
gives the following condition for the parameter $\alpha$ to ensure
diffusiophoresis has a stronger effect than advection:

$\alpha\gtrsim\frac{\sqrt{D_{c}D_{s}}}{GL}\,.$ (25)

In the case of salt, the gradient is advected along the vertical separatrix
and then dissipated (see Section 2); what about colloids? The gradient of
colloids is created in the vicinity of the vertical separatrix and then
advected in the cell over a distance $d$ for a time $\tau^{c}_{adv}\sim
d/u_{0}$, until it is dissipated by diffusion, which happens roughly at time
$\tau^{c}_{\mathit{diff}}\sim\ell_{c}^{2}/D_{c}$. We obtain the travelled
distance $d$ before dissipation by equaling the two times, i.e.

$d\sim\frac{D_{c}D_{s}}{\alpha^{2}G^{2}L}\,,$ (26)

where we have used equation (24) for $\ell_{c}$. Owing to condition (25), this
distance verifies

$d\lesssim L\,.$ (27)

In practice, because the gradient is present along the entire vertical, it is
advected along the upper edge of the cell, then rapidly diffuses. If the
condition (25) is not met, and if the diffusiophoretic effect is weak compared
with contraction by the flow field, then the phenomenology for colloids goes
back to that of salt where the equilibrium state corresponds to a balance
between advection and diffusion with $(a)\sim(b)$, and the inhomogeneity is
located only on the vertical separatrix. Therefore, the colloidal
heterogeneity would essentially be dissipated somewhere on the upper edge or
rapidly after, depending on how the left- and right-hand side of condition
(25) compare. This is indeed what we observe in the Eulerian simulations when
varying the different parameters $\alpha$, $\Pen_{s}$ and $\Pen_{c}$.

Figure 2: Eulerian colloid field $c(y,y)$ in four adjacent cells for
$\alpha=10^{-3}$ and $\Pen_{c}=3\,141$: $(a)$ $\Pen_{s}=31.4$, advection and
diffusiophoresis have comparable strength in the creation of gradients, the
two terms in relation (25) are of the same order of magnitude; $(b)$
$\Pen_{s}=314$, diffusiophoresis is more efficient and relation (25) is met.

As an illustration, Figure 2 shows two typical colloidal concentration fields
for $\alpha=10^{-3}$ and $\Pen_{c}=3\,141$ and two different salt Péclet
numbers $\Pen_{s}=31.4$ $(a)$ and $\Pen_{s}=314$ $(b)$: in figure 2 $a$, the
two terms in equation (25) are of the same order of magnitude, whereas
condition (25) is satisfied in figure 2 $b$. Note the different scales in the
colourbars: except in the cores of cells which seem more homogeneous, the
heterogeneities are more pronounced when the diffusiophoretic effects are
stronger, with more asymmetric positive and negative deviations from the mean
$C_{0}=1$.

###  3.2 Lagrangian results: velocity and effective diffusion

The Eulerian simulations showed how the colloids are demixed due to
diffusiophoresis in the stationary regime. As a result, it is expected that
their averaged velocity is modified. In order to explore this phenomenon when
$\Pen_{c}$ is large, we now perform Lagrangian simulations and investigate the
statistics of colloids displacement at long times, denoted
$\Delta\mathbf{X}_{c}=\mathbf{X}_{c}(t)-\mathbf{X}_{c}(0)$. We denote an
ensemble average over trajectories by $\langle\cdot\rangle_{\mathcal{L}}$;
this allows to define the mean Lagrangian velocity
$\langle\mathbf{V}\rangle_{\mathcal{L}}$ through the relation
$\langle\Delta\mathbf{X}_{c}\rangle_{\mathcal{L}}\sim\langle\mathbf{V}\rangle_{\mathcal{L}}\times
t$ at large times. We also measure an effective diffusion $D_{\mathrm{eff}}$:
we have $\langle(\Delta X_{c})^{2}\rangle_{\mathcal{L}}-\langle\Delta
X_{c}\rangle^{2}_{\mathcal{L}}\sim 2D_{\mathrm{eff}}^{\,x}\times t$ and
$\langle(\Delta Y_{c})^{2}\rangle_{\mathcal{L}}-\langle\Delta
Y_{c}\rangle^{2}_{\mathcal{L}}\sim 2D_{\mathrm{eff}}^{\,y}\times t$ at large
times, where we have treated separately the $x$ and $y$ directions in order to
take into account a possible anisotropy.

Because the mean Eulerian velocity field $\langle\mathbf{u}\rangle$ is zero,
one could reasonably think that the mean Lagrangian velocity
$\langle\mathbf{V}\rangle_{\mathcal{L}}$ is of the same order of magnitude as
the mean salt gradient $\alpha\mathbf{G}$. However, this is not at all the
case when looking at figure 3 $a$, that shows the ratio of mean Lagrangian $x$
and $y$ velocity components of $\langle\mathbf{V}\rangle_{\mathcal{L}}$ to
$\alpha G$ as a function of the Péclet number $\Pen_{c}$, at fixed
$\Pen_{s}=314$ and $\alpha=10^{-3}$: whereas the $y$-component remains zero as
expected, the $x$-component is much larger than $1$ for most of the points.
Moreover, the ratio has a non-trivial behaviour: at low to moderate Péclet
numbers $\Pen_{c}$, the velocity increases, but eventually decreases to zero
at larger Péclet numbers.

Figure 3: $(a)$ Non-dimensional mean Lagrangian velocity
$\langle\mathbf{V}\rangle$ and $(b)$ non-dimensional effective diffusion
$D_{\mathrm{eff}}$, for $\Pen_{s}=314$ and $\alpha=10^{-3}$ as a function of
the Péclet number $\Pen_{c}$: $\bullet$ $x$-direction; $\diamond$
$y$-direction. The velocity is made non-dimensional using the diffusiophoretic
velocity $\alpha G$ based on the mean gradient; the effective diffusion is
compared with the diffusion coefficient of the colloids $D_{c}$. The solid
black line is the analytical solution proposed by Shraiman (1987):
$D_{\mathrm{eff}}/D_{c}=Pe_{c}^{1/2}/\sqrt{2\pi}$.

Figure 3 $b$ shows the effective diffusion coefficient in the $x$ and $y$
directions. The line is the prediction by Shraiman (1987) for a diffusing
scalar without diffusiophoresis in a cellular flow: he showed that the
effective diffusion grows as the square root of the Péclet number, and becomes
orders of magnitude higher than the diffusivity of the scalar. Very
interestingly, we find that the effective diffusion at small to moderate
Péclet number ($\Pen_{c}\leq 3000$) is isotropic, and follows the prediction
by Shraiman, although the effects of diffusiophoresis are macroscopically
visible on the velocity at those Péclet numbers. For $\Pen_{c}>3000$, while
the mean Lagrangian velocity begins to decay (figure 3 $a$), the effective
diffusivity becomes highly anisotropic, with an increase in the $x$-direction
compared with Shraiman’s prediction, and a decrease in the $y$-direction, At
even higher Péclet numbers, the effective diffusivity in both directions
finally decays to zero, although for larger Péclet numbers than for the
velocity.

Figure 4: $(a)$ The $x-$component of the Lagrangian velocity $\langle
V_{x}\rangle$ and $(b)$ non-dimensional effective diffusion
$D_{\mathrm{eff}}/D_{c}$, for $\Pen_{s}=314$, $\Pen_{c}=15\,708$ as a function
of $\alpha$. $\bullet$: $x$-direction; $\diamond$: $y$-direction.

Figure 4 shows the mean Lagrangian velocity and effective diffusion as a
function of the diffusiophoretic coefficient $\alpha$ for the same salt Péclet
number and $\Pen_{c}=15\,708$, a Péclet number for which the effective
diffusion is anisotropic (figure 3 $b$), more realistic of the large Péclet
numbers encountered for colloids. The mean velocity in the $y$-direction is
again zero and is not shown here. For small $\alpha$, $\langle V_{x}\rangle$
increases linearly, before vanishing also at higher values. In figure 4 $b$,
we recover at $\alpha=0$ the value predicted by Shraiman (1987) for the
effective diffusion in both directions. Then $D_{\mathrm{eff}}^{\,x}$
increases rapidly for small $\alpha$, before decreasing to zero at larger
values; in the $y$-direction, the effective diffusivity decreases to zero as
$\alpha$ increases.

In the next section we deepen the analysis of the colloids velocity and study
the physical mechanisms at stake which explain the observations we just
reported based on the Lagrangian simulations.

##  4 From enhanced dispersion to blockage

###  4.1 Enhanced dispersion

Figures 3 $a$ and 4 $a$ showed that transport is strongly enhanced at small
diffusiophoretic forcing, and blocked at higher forcing. In order to explain
these observations, we shall return to the behaviour of colloids in the salt
gradient along the vertical separatrices. Colloids that are in this region are
shifted to the right under the effect of diffusiophoresis: they move with a
velocity $\mathbf{v}=\mathbf{u}+\mathbf{v}_{\mathrm{dp}}$ with
$\mathbf{v}_{\mathrm{dp}}=\alpha\nabla S$. The velocity $\mathbf{u}$ is
roughly vertical, whereas the diffusiophoretic velocity is roughly horizontal,
see figure 5 $a$ around the separatrix $x=L$ (upward velocity).

Figure 5: $(a)$ Schematic displacement of a particle in the salt gradient
(light blue region) around the vertical ascending separatrix $x=L$. The
Eulerian velocity field $\mathbf{u}$ is vertical ascending, and
$\mathbf{v}_{\mathrm{dp}}$ is horizontal; the total displacement is therefore
$\mathbf{v}_{t}=\mathbf{u}+\mathbf{v}_{\mathrm{dp}}$, with $\mathbf{u}\approx
u_{0}\,\mathbf{e}_{y}$. The particles that change or will change cell are
located on the grey strip of width $\ell_{d}$. $(b)$ Schematic location of
particles that change cell, in grey, for cells located between $x=0$ and
$x=2L$; an example of a trajectory is shown in red. Outside this grey stripe,
the particles remain trapped in a cell.

We denote by $\ell_{d}$ the horizontal displacement travelled by a colloid
under the effect of the salt gradient as it rises or falls along the vertical
separatrix ; in a first step, we suppose $\ell_{d}\leq\ell_{s}$. Along the
vertical separatrix we have $u\sim u_{0}$, so that

$\frac{\ell_{d}}{v_{\mathrm{dp}}}\sim\frac{L}{u_{0}}\,,$ (28)

see figure 5 $a$. Because of equation (12) the diffusiophoretic velocity in
the salt gradient can be written

$v_{\mathrm{dp}}\sim\alpha G\sqrt{\Pen_{s}}$ (29)

and, finally,

$\ell_{d}\sim\frac{\alpha GL}{u_{0}}\sqrt{\Pen_{s}}\,.$ (30)

Hence, the particles which change vortex during their ascent (or descent) in
the salt gradient are located in a strip of width $\ell_{d}$, indicated in
grey in figure 5 $a$. In figure 5 $b$ we follow those particles on several
vortices: in the horizontal strips, the salt gradient is negligible, so that
the colloids follow the Eulerian flow field, mostly horizontal here and such
that $\mathbf{u}\sim u_{0}\,\mathbf{e}_{x}$. Therefore, the total displacement
of particles is horizontal, as found in our Eulerian and Lagrangian
simulations, see for instance figure 3 $a$. Note also that the strips coloured
in grey in figure 5 $b$ are in good qualitative agreement with the regions of
high concentration in the Eulerian colloidal fields in figure 2. The regions
not coloured in figure 5 $b$ are the locations of the particles that are
trapped inside a cell: by construction, the Lagrangian velocity of these
trapped particles is zero on average. We can thus evaluate the spatial average
horizontal velocity $V_{m}$ on a cell of size $L^{2}$:

$\displaystyle V_{m}$ $\displaystyle\sim$
$\displaystyle\frac{1}{L^{2}}\left[(\ell_{d}\times
L)\,v_{\mathrm{dp}}+(\ell_{d}\times L)\,u_{0}\right]$ (31) $\displaystyle\sim$
$\displaystyle\frac{\ell_{d}}{L}\,(v_{\mathrm{dp}}+u_{0})\,.$ (32)

With equation (28), we obtain $v_{\mathrm{dp}}/u_{0}\sim\ell_{d}/L$, so that
$v_{\mathrm{dp}}$ can be neglected, and, using equation (30),

$V_{m}\sim\alpha G\sqrt{\Pen_{s}}\,.$ (33)

This simple result leads to some remarks.

  * $-$

The mean colloids velocity is predicted to be much larger than that
corresponding to the mean salt gradient ($\alpha G$), in accordance with
figure 3 $a$.

  * $-$

We recover the proportionality in $\alpha$ observed in figure 4 $a$ for small
$\alpha$.

  * $-$

Relation (33) is expected to slightly overestimate the actual velocity.
Indeed, we assumed $u\sim u_{0}$ on the separatrices; however, the mean of
$u_{0}\sin kx$ on the side of a cell is rather $u_{0}/\pi$; changing $u_{0}$
into $u_{0}/\pi$ leads to

$V_{m}\sim\alpha G\sqrt{\frac{\Pen_{s}}{\pi}}\,.$ (34)

Anyway, the velocity of a colloid is actually still lower when not laying
exactly on the separatrix, so that even equation (34) slightly overestimates
the velocity.

Let us test this prediction by performing Lagrangian numerical experiments for
many different values of $D_{s}$, $D_{c}$ and $\alpha$. For all those values
we measure the $x$-components of the velocity and display $\langle
V_{x}\rangle_{\mathcal{L}}/(\alpha G\sqrt{\Pen_{s}})$ as a function of
$Pe_{c}$ in figure 6.

Figure 6: Normalised horizontal velocity $\langle
V_{x}\rangle_{\mathcal{L}}/(\alpha G\sqrt{\Pen_{s}})$ as a function of
$\Pen_{c}$ for the different numerical cases performed. Symbols: $(\circ)$
$\Pen_{s}=31.4$; $(\square)$ $\Pen_{s}=314$; $(\diamond)$ $\Pen_{s}=3144$. The
colour of symbol codes for the value of $\alpha\in[5\times
10^{-6},~{}1.6\times 10^{-3}]$, the darker the larger the value.

It can be observed that the formula predicts the correct order of magnitude
for the maximal value of the velocity ($1/\sqrt{\pi}\simeq 0.56$) at a given
colloid Péclet number. However, the expression (34) does not depend on
$\Pen_{c}$ disagreeing with figure 3 $a$, and fails to predict the rapid decay
of the mean velocity observed in figures 3 $a$ and 4 $a$ when increasing
$\Pen_{c}$ and $\alpha$. There is thus a need to refine the analysis and
figure out why the dispersion of the particles stops, both for the velocity
and the effective diffusion.

###  4.2 Blockage

As described previously, the grey stripe in figure 5 $b$ is the location of
particles that can change cell, whilst particles outside of this stripe remain
trapped in a cell. Therefore, blockage occurs when the grey stripe becomes
empty of colloids. In order to understand how this can happen, let us first
come back to the salt profile already described in section 2. The horizontal
salt gradient is located in the vicinity of the vertical separatrices; because
the mean gradient $\mathbf{G}$ points toward positive $x$, one goes from low
salt concentrations to high salt concentrations when crossing vertical
separatrices from left to right, see figure 1. As explained previously, the
horizontal salt gradient at $x=0$ is built by bringing (from right to left)
the scalar to the hyperbolic point $(0,L)$, on the high salt concentration
side, by contraction by the velocity field along the horizontal separatrix:
this implies that this separatrix should correspond to a local maximum of salt
concentration. Symmetrically, the horizontal separatrix that relates point
$(0,0)$ to $(L,0)$ builds the salt gradient at $x=L$, on the low concentration
side, by bringing a deficit of salt, and should correspond to a local minimum.

Figure 7: $(a)$ One-dimensional cut of the salt concentration field along the
vertical lines $x=L/2$ and $x=3L/2$ with $L=\pi$ (dotted lines in figure 7
$b$) in the whole computational domain $[0,2L]^{2}$ for three different salt
Péclet numbers: - - -, $\Pen_{s}=31.4$; -$\cdot$-$\cdot$-, $\Pen_{s}=314$; —,
$\Pen_{s}=3\,141$. $(b)$ The horizontal parts of the stripe correspond to
regions where the salt concentration gradient points vertically inwards
(figure 7 $a$). Although the salt gradients involved are less important than
around the vertical separatrices, colloids in these places are deviated
outside the grey stripe. The red arrows indicate schematically their total
velocities $\mathbf{v}_{t}=\mathbf{u}+\mathbf{v}_{\mathrm{dp}}$.

This is exactly what is observed when plotting a one-dimensional cut of the
salt concentration field at $x=L/2$, as shown in figure 7 $a$ for different
Péclet numbers (see also the whole salt concentration field for $\Pen_{s}=314$
in figure 1 $a$). The situation is logically reversed when considering the
one-dimensional cut at $x=3L/2$. Now let us come back to the displacement of
colloids, also depicted in figure 7 $b$: in the horizontal grey stripes, the
salt gradient is vertical and always points inwards, and so does the
diffusiophoretic velocity $\mathbf{v}_{\mathrm{dp}}=\alpha\nabla S$; the total
velocity of the colloids, $\mathbf{v}_{t}=\mathbf{u}+\mathbf{v}_{\mathrm{dp}}$
with $\mathbf{u}\approx u_{0}\,\mathbf{e}_{x}$, deviates the colloids inwards
the vortices. Therefore, when diffusiophoretic effects are strong enough, the
colloids rapidly leave the grey stripe and get trapped into the vortices.
Note, however, that the vertical salt gradients involved here are smaller than
the horizontal salt gradients considered previously around the vertical
separatrices; therefore, when the diffusiophoretic coefficient $\alpha$ is
small, the total velocity is nearly horizontal.

The process described previously explains how colloid particles can get
trapped and lead to blockage. With this single mechanism at stake, however,
all situations considered should lead to blockage; indeed, even though the
velocity is nearly horizontal in the horizontal stripes when diffusiophoretic
effects are weak, the small vertical deviation should lead to blockage on very
long times. This is obviously not what we observe in the numerical
simulations, where we also observe enhanced transport at large times. The
reason is that we have up to now neglected molecular diffusion in our
schematic description: to effectively observe a depleted zone, and eventually
for it to become a real barrier to transport from one cell to another, this
band without colloids must be advected all around the cell, without having the
time to diffuse. Blockage therefore only occurs if the diffusion time of the
depleted stripe, $\ell_{d}^{2}/D_{c}$, is large compared with the advection
time over the entire perimeter of cell $4L/u_{0}$. This competition is
governed by the ratio $R^{2}$ of the two time scales, such that

$R=\sqrt{\frac{\ell_{d}^{2}/D_{c}}{4L/u_{0}}}\,.$ (35)

Using relation (30) for $\ell_{d}$, one obtains the condition of blockage:

$R=\frac{\alpha\,GL}{2\sqrt{D_{c}D_{s}}}\gg 1\,,$ (36)

where we recognise $\Delta S=GL$, the jump in concentration from one cell to
the other (section 2). Note that we recover a condition similar to what we had
found before (equation (25)). Interestingly, this condition is independent of
the Eulerian velocity field $\mathbf{u}$, although the blockage would not
occur without the velocity field.

Throughout this section, we have assumed so far that $\ell_{d}\leq\ell_{s}$.
However, using equations (11) and (30), we obtain

$\frac{\ell_{d}}{\ell_{s}}\sim\frac{\alpha G}{u_{0}}\,\Pen_{s}=\frac{\alpha
GL}{D_{s}}\,;$ (37)

therefore $\ell_{d}$, as calculated using (30), may be larger than $\ell_{s}$.
Because $\ell_{d}$ measures the deviation of colloids by diffusiophoresis, and
because the diffusiophoretic velocity is zero outside the salt gradient, this
implies that at most

$\ell_{d}\sim\ell_{s}$ (38)

(see also figure 5 $a$). We can therefore wonder whether the expressions
obtained previously for the velocity and the blockage condition remain valid
in this case. Indeed, if equation (30) overestimates $\ell_{d}$, then
condition (36) could falsely lead to a condition of blockage. We show
thereafter that in that case condition (36) remains a valid criterion for
blockage.

As explained previously a salt heterogeneity of width $\ell_{s}$ diffuses
after having travelled a distance $L$ along the vertical separatrix; hence, a
colloidal heterogeneity of width $\ell_{s}$ should travel much longer in space
and time before diffusing, because $D_{c}\ll D_{s}$. This can be checked using
the same criterion as before, the ratio $R^{\prime 2}$ between the diffusing
time and the travelling time on the perimeter of a cell, i.e., taking
$\ell_{d}=\ell_{s}$ in equation (35); we obtain

$R^{\prime}=\sqrt{\frac{\ell_{s}^{2}/D_{c}}{4L/u_{0}}}=\frac{1}{2}\sqrt{\frac{D_{s}}{D_{c}}}\gg
1\,,$ (39)

where $\ell_{s}$ was evaluated using equation (11). Therefore, this case when
$\ell_{d}\sim\ell_{s}$ also corresponds to a situation of blockage, with no
transport from one cell to the other. Let us show that in this situation of
blockage we also have $R\gg 1$: we denote by $\ell_{d}^{\mathit{est}}$ the
overestimated width of the depleted band $\ell_{d}$ given by relation (30).
Then $\ell_{d}^{\mathit{est}}\geq\ell_{d}\sim\ell_{s}$, and according to
equations (35) and (39), $R\geq R^{\prime}$. Thus, we have both $R\geq
R^{\prime}$ and $R^{\prime}\gg 1$, and, finally, $R\gg 1$. The criterion (36)
on $R$:

  * $(i)$

$R\ll 1$, enhanced dispersion;

  * $(ii)$

$R\gg 1$, blockage;

is valid for all situations.

Figure 8: Successive positions of $N=100$ long trajectories of colloids,
plotted modulo $2L$, in 4 adjacent cells. Particles were initially distributed
uniformly and randomly inside the cells with positions dumped long after
release, each colour coding for one trajectory. Parameters are $\Pen_{s}=314$
for each figure and different values of $\Pen_{c}$ and $\alpha$. The criterion
$R$ is increasing from $(a)$ to $(f)$: $(a)$ $\alpha=10^{-3}$,
$\Pen_{c}=3\,141$, $R=0.497$; $(b)$ $\alpha=10^{-3}$, $\Pen_{c}=15\,700$,
$R=1.11$; $(c)$ $\alpha=10^{-3}$, $\Pen_{c}=157\,000$, $R=3.51$; $(d)$
$\alpha=4.\,10^{-3}$, $\Pen_{c}=15\,700$, $R=4.44$; $(e)$ $\alpha=10^{-3}$,
$\Pen_{c}=314\,000$, $R=4.97$; $(f)$ $\alpha=8\times 10^{-3}$,
$\Pen_{c}=15\,708$, $R=8.88$.

We first test the blockage criterion for different values of the parameters
$D_{s}$, $D_{c}$ and $\alpha$ performing simulations in the Lagrangian
framework: as a proxy for the stationary distribution of colloids, we display
in figure 8 successive positions of $N=100$ long trajectories of colloids,
initially randomly and uniformly distributed inside four adjacent cells, for
growing values of the criterion $R$. These trajectories, plotted modulo $2L$
and dumped long after release so that their position distribution is
stationary, allow to see if particles sample uniformly the cells or regroup in
some localised regions. Although the cells are hardly distinguishable in
figure 8 $a$, they become more distinct as $R$ increases, and finally become
completely disjoint. In this last regime, unfolded trajectories are trapped in
cells and their mean velocity drop to zero.

Figure 9: Normalised horizontal velocity $\langle
V_{x}\rangle_{\mathcal{L}}/(\alpha G\sqrt{\Pen_{s}})$ versus the blockage
criterion $R=\alpha GL/2\sqrt{D_{c}D_{s}}$ for the same numerical cases as in
figure 6. Symbols: $\circ$, $\Pen_{s}=31$; $\square$, $Pe_{s}=314$;
$\diamond$, $\Pen_{s}=3144$. The colour of symbol codes for the value of
$\Pen_{c}\in[31\,;3.14\times 10^{5}]$, the darker the larger the value.

This proves that $R$ is indeed a good indicator of trapping so that we choose
to plot the rescaled mean velocity $\langle V_{x}\rangle_{\mathcal{L}}/(\alpha
G\sqrt{\Pen_{s}})$ of figure 6 as a function of the blockage criterion
$R=\alpha GL/2\sqrt{D_{c}D_{s}}$ instead of $\Pen_{c}$. The result is
displayed in figure 9: points are no longer scattered but collapse reasonably
well, and fall to zero provided $R$ is large enough, when blockage dominates
over other mechanisms of transport. Finally, we note that the separation at
small $R$ coincides with the (weak) logarithmic dependence on the colloidal
Péclet number, as pointed out from figure 3 $a$, and which was not taken into
account in the present model. Despite this observation, the order of magnitude
obtained for $\langle V_{x}\rangle_{\mathcal{L}}$ is rather good.

##  5 Summary and conclusion

We have studied the joint mixing/demixing of salt and colloids in a cellular
flow with closed streamlines, and investigated how the long-term dispersion of
the colloids is modified by a linear phoretic drift
$\mathbf{v}_{\mathrm{dp}}=\alpha\nabla S$. In the chosen configuration of an
imposed salt mean gradient along the $x$-direction $\langle\nabla
S\rangle=G\,\mathbf{e}_{x}$, it is known that the salt concentration reaches a
stationary state characterised by the presence of strong gradients localised
along the vertical separatrices, so that the ratio of its effective
diffusivity to the molecular diffusivity grows as the square root of the salt
Péclet number (Shraiman, 1987).

By means of high-resolution numerical simulations performed both in the
Eulerian and Lagrangian frameworks, we have shown that, starting from a
uniform colloid concentration field (with no imposed colloid gradient), the
colloids will demix. We have considered the long time dynamics of this
demixing process, characterised by their mean velocity $\mathbf{V}_{m}$ and
their effective diffusivity $D_{\mathrm{eff}}$; in particular, we have studied
how $\mathbf{V}_{m}$ and $D_{\mathrm{eff}}$ are influenced by the phoretic
drift for a wide range of parameters $(\alpha
G,\Pen_{s}=UL/D_{s},\Pen_{c}=UL/D_{c})$ with $\alpha$ and $G>0$.

The main finding is that we observe substantially two regimes of colloids
dynamics depending on the blockage criterion $R=\alpha GL/\sqrt{4D_{c}D_{s}}$.
When $R<1$, the colloids are free to move and a strong demixing occurs at high
$\Pen_{c}$; the concentration becomes homogeneous in the core of the flow
cells whereas it is higher along the vertical separatrices, and in regions
along the horizontal separatrices where the velocity $\mathbf{u}$ of the flow
is in the same direction as $\mathbf{G}$ ($\mathbf{G}\cdot\mathbf{u}>0$). This
is well explained as a combination of the non divergence free drift velocity,
advection from the mean flow $\mathbf{u}$ and molecular diffusion. In this
regime, the effective diffusivity is very close to $D_{c}$ but the mean
colloids velocity is strongly enhanced in the direction of the salt gradient.
Using a model which takes into account the thickness of the high concentration
region with an enhanced cell-to-cell transport due to the phoretic drift, we
have shown that $\langle V_{x}\rangle_{\mathcal{L}}\propto\alpha
G\sqrt{\Pen_{s}}$. When $R=\alpha GL/\sqrt{4D_{c}D_{s}}>1$ the phoretic drift
is so strong that a depletion of colloids occurs along the separatrices so
that colloids can no longer migrate from one cell to another and transport is
suppressed. In this regime of blockage, all transport observables (mean
velocity, effective diffusivity) go to zero when $R$ is large enough. We also
made an interesting observation in the transition regime $R\sim 1$ (although
we could not explain it), for which the effective diffusivity along the mean
gradient, $D_{\mathrm{eff,x}}$, first strongly increases when increasing $R$
while $D_{\mathrm{eff,y}}$ and $\langle V_{x}\rangle_{\mathcal{L}}$ are
decreasing functions of $R$. Note finally that the regime of blockage,
starting from a uniform distribution, is typical of compressible effects.

One may wonder if such behaviours as enhanced dispersion or blockage could be
observed in practice. In the case of chemotaxis, because of the very large
values of the drift coefficient $\alpha$ involved (Chu et al., 2022), a
situation of blockage would be likely to happen in such flow configurations,
where the separatrices would become real transport barriers (Berman et al.,
2021). However, when the nutriments would lack, and the gradient $G$ would
decrease sufficiently, the motile organisms would become free to leave their
cell and mperhaps find some food somewhere else. The case of diffusiophoresis
could be studied in a similar configuration to that of Solomon & Gollub
(1988), for which a laminar flow with closed streamlines is created in an
elongated volume ($H=0.75$ cm height, $L_{\mathrm{t}}=15\,\mathrm{cm}$ long,
$e=1.5\,\mathrm{cm}$ deep) by creating a laminar and stationary Rayleigh-
Bénard flow with a small temperature difference $\Delta T\simeq 1^{\circ}$ C.
Whether enhanced transport or blockage would be observed in such a
configuration by introducing salt and colloids at the same side (salt-in
configuration) or at the opposite sides (salt-out configuration) remains an
open question. If we choose LiCl for the salt as in Mauger et al. (2016), with
$D_{s}=1360\,\mu\mathrm{m}^{2}$/s and a typical velocity $U\sim
0.5\,\mathrm{mm/s}$, we obtain a salt Péclet number $\Pen_{s}\sim 2760$ of the
same order of magnitude as the highest value used in this study. Considering a
drift velocity of the type
$\mathbf{v}_{\mathrm{drift}}=D_{\mathrm{dp}}\nabla\log S$ with
$D_{\mathrm{dp}}=290\,\mu\mathrm{m}^{2}$/s for LiCl, one may obtain an
estimate of the blockage criterion $R_{\mathrm{exp}}$ for the experiments by
replacing $\alpha$ with $D_{\mathrm{dp}}/S$ in $R$ to obtain
$R_{\mathrm{exp}}=D_{dp}GL/S\sqrt{4D_{c}D_{c}}$. In the case of the
aforementioned configuration one would have long after salt injection
$G=\Delta S/L_{\mathrm{t}}$ and $L=H$ so that an estimate of the criterion
becomes

$R_{\mathrm{exp}}\simeq\frac{1}{40}\frac{D_{dp}}{\sqrt{D_{c}D_{c}}}\frac{\Delta
S}{S}.$ (40)

Taking roughly $\Delta S/S\sim 1$, one would then have
$R_{\mathrm{exp}}=\mathcal{O}(0.1)$ which corresponds to enhanced transport
velocity and a weak modification of the effective diffusivity. One may also
argue that before developing a salt gradient over the whole width, it is
developed over a much limited length at shorter times which increases the
value of the criterion. One may then observe blockage at short time followed
by enhanced transport at long time depending on the time delay between
injection of salt and colloids. We let such experimental study for future
work.

## Acknowledgments

This work was supported by the French research programs ANR-16-CE30-0028, and
IDEXLYON of the University of Lyon in the framework of the French program
“Programme Investissements d’Avenir” (ANR-16-IDEX-0005).

Declaration of Interests. The authors report no conflict of interest.

##  Appendix A Symmetry $\alpha\rightarrow-\alpha$

Figure 10: Colloids concentration fields on the domain $[0:2L]^{2}$ for
$\Pen_{s}=314$, $\Pen_{c}=3141$ and opposite values of $\alpha$: $(a)$
$\alpha=10^{-3}$; $(b)$ $\alpha=-10^{-3}$. Note the identical colourbar for
the two cases. A colloid located at a point $(x,y)$ in the case $\alpha>0$ has
the opposite total velocity of the colloid located at $(mL-x,nL-y)$ where $m$
and $n$ are integers ($m=n=2$ on the figure) with $\alpha<0$.

Let us consider two different situations for which only the sign of $\alpha$
is changed. The velocity field is the same for both cases, as well as the salt
concentration field. The equation for the gradient of salt can be obtained
from equation 3; in the stationary state considered here, it may be written:

$\nabla S\cdot\mathbf{u}=D_{s}\nabla\cdot\nabla S\,.$ (41)

The velocity $\mathbf{u}$ is changed into $-\mathbf{u}$ by point symmetry
around a point of coordinates $(mL,nL)$, where $m$ and $n$ are integers.
Therefore, the gradient of salt $\nabla S$ is unchanged by this symmetry.
Physically, a colloid which moves with a total velocity
$\mathbf{v}_{t}=\mathbf{u}+\mathbf{v}_{\mathrm{dp}}$ in the case $\alpha>0$
has the opposite velocity of the symmetric colloid (with the symmetry point
mentioned previously) when $\alpha<0$, as depicted in figure 10. This is also
visible from the equation of concentration in the stationary case

$\nabla\cdot C(\mathbf{u}+\alpha\nabla S)=D_{c}\,\nabla^{2}C$ (42)

which is invariant under the point symmetries mentioned above when $\alpha$ is
changed into $-\alpha$. As an illustration, figure 10 shows two concentration
fields with opposite values of $\alpha$.

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