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Unstable Drainage of Frictional Fluids and Magnetic Control of the Mechanical Behavior of Confined Granular Media

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Abstract

The transport of immersed grains in confined environments is found in many industrial and natural situations with notably the recovery and transport of oil in pipelines, the decontamination of soils, the transport of sediments or the circulation of physiological fluids. The study of the slow drainage of a model system composed of a mixture of water and glass beads sedimented in a Hele-Shaw cell or a capillary showed the emergence of instabilities, with the formation of labyrinthine patterns and granular plugs. In this thesis, we have identified the physical parameters and conditions necessary for the triggering of such instabilities, controlled by the meniscus pushing the grains at the liquid/air interface. We have also highlighted a new unstable drainage regime leading to the periodic formation of dunes along the capillary. As the friction of the grains with the walls of the tube governs the formation of the plugs, we have also studied an original method to control these frictional interactions by using ferromagnetic particles: submitted to a magnetic field, they acquire a magnetic moment, leading to tunable interactions of pairs of magnetic dipoles. Considering all these anisotropic interactions within a confined granular column, we were able to demonstrate the emergence of a radial force along the walls of the silo, whose amplitude and direction is completely determined by the applied magnetic field. This "magnetic Janssen effect" allows to control the apparent mass of the granular column, paving the way towards the design of granular meta-materials whose mechanical (un)blocking properties can be remotely controlled or even programmed.
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Unstable Drainage of Frictional Fluids and Magnetic
Control of the Mechanical Behavior of Conned
Granular Media
Louison Thorens
To cite this version:
Louison Thorens. Unstable Drainage of Frictional Fluids and Magnetic Control of the Mechanical
Behavior of Conned Granular Media. Soft Condensed Matter [cond-mat.soft]. Université de Lyon;
Universitetet i Oslo, 2021. English. �NNT : 2021LYSEN069�. �tel-03543092�
numéro national de thèse :2021lysen069
thèse de doctorat de luniversité de lyon
opérée par
lécole normale supérieure de lyon
en cotutelle avec
luniversité doslo
école doctorale n°52 physique et astrophysique de lyon (phast)
discipline :physique
soutenue publiquement le 17 novembre 2021,par :
louison thorens
UNSTABLE DRAINAGE OF FRICTIONAL FLUIDS AND
MAGNETIC CONTROL OF THE MECHANICAL BEHAVIOR
OF CONFINED GRANULAR MEDIA
DRAINAGE INSTABLE DE FLUIDES FRICTIONNELS ET CONTRÔLE
MAGNÉTIQUE DU COMPORTEMENT MÉCANIQUE DE MILIEUX
GRANULAIRES CONFINÉS
devant le jury composé de:
eric clément professeur espcipslsorbonne université
axelle amon maître de conférences université de rennes
pascale aussillous professeure aix-marseille université
nicolas taberlet maître de conférences université lyon 1 -ens de lyo n
dag kristian dysthe professeur université doslo
sous la direction de :
stéphane santucci chargé de recherche cnrs -ens de lyon directeur
knut jorgen måløy professeur université doslo co-tuteur
mickaël bourgoin directeur de recherche cnrs -ens d e ly on co-encadrant
Louison Thorens: Unstable Drainage of Frictional Fluids and Magnetic Control of
the Mechanical Behavior of Confined Granular Media, A dissertation submitted for the
degree Philosophiae Doctor at the Department of Physics of the University of Oslo, and
at the École Normale Supérieure de Lyon, © 2021
ABSTRACT
The transport of immersed grains in confined environments is found in many
industrial and natural situations with notably the recovery and transport of
oil in pipelines, the decontamination of soils, the transport of sediments or the
circulation of physiological fluids.
The study of the slow drainage of a model system composed of a mixture of
water and glass beads sedimented in a Hele-Shaw cell or a capillary showed
the emergence of instabilities, with the formation of labyrinthine patterns and
granular plugs.
In this thesis, we have identified the physical parameters and conditions neces-
sary for the triggering of such instabilities, controlled by the meniscus pushing
the grains at the liquid/air interface. We have also highlighted a new unstable
drainage regime leading to the periodic formation of dunes along the capillary.
As the friction of the grains with the walls of the tube governs the formation of
the plugs, we have also studied an original method to control these frictional
interactions by using ferromagnetic particles: submitted to a magnetic field, they
acquire a magnetic moment, leading to tunable interactions of pairs of magnetic
dipoles. Considering all these anisotropic interactions within a confined granu-
lar column, we were able to demonstrate the emergence of a radial force along
the walls of the silo, whose amplitude and direction is completely determined
by the applied magnetic field. This "magnetic Janssen effect" allows to control
the apparent mass of the granular column, paving the way towards the design
of granular meta-materials whose mechanical (un)blocking properties can be
remotely controlled or even programmed.
iii
RÉSUMÉ
Le transport de grains immergés en milieux confinés se retrouve dans de nom-
breuses situations industrielles et naturelles avec notamment la récupération et
le transport de pétrole en oléoducs, la décontamination de sols, le transport de
sédiments ou encore la circulation de fluides physiologiques.
L’étude du drainage lent d’un système modèle composé d’un mélange d’eau
et billes de verres ayant sédimentées dans une cellule de Hele-Shaw ou un
capillaire a démontré l’émergence d’instabilités, avec la formation de motifs
labyrinthiques et de bouchons granulaires.
Dans le cadre de cette thèse, nous avons identifié les paramètres physiques
et conditions nécessaires au déclenchement de telles instabilités, contrôlées
par le charriage des grains par le ménisque à l’interface liquide/air. Nous
avons d’autre part mis en évidence un nouveau régime de drainage instable
conduisant à la formation périodique de dunes le long du capillaire.
La friction des grains avec les parois du tube gouvernant la formation des
bouchons, nous avons également étudié une méthode originale permettant
de contrôler ces interactions frictionnelles en utilisant des particules ferroma-
gnétiques : soumises à un champ magnétique, elles acquièrent un moment
magnétique, conduisant à des interactions de paires de dipôles magnétiques,
ajustables. En considérant l’ensemble de ces interactions anisotropes au sein
d’une colonne granulaire confinée, nous avons pu démontrer l’émergence d’une
force radiale le long des parois du silo, dont l’amplitude et la direction est
complétement déterminée par le champ magnétique appliqué. Cet « effet Jans-
sen magnétique » permet alors de contrôler la masse apparente de la colonne
granulaire, ouvrant la voie à la conception de méta-matériaux granulaires dont
les propriétés mécaniques de (dé)blocage peuvent être contrôlées à distance,
voire programmées.
iv
ACKNOWLEDGMENTS
First of all, I would like to express my deepest gratitude to my main super-
visors. Stéphane Santucci, for being present all along these three years and
being always supportive, I hope this journey was only a beginning. Knut Jørgen
Måløy for trusting me to be part of Porelab, working with you is a real pleasure
and I am thankful to have crossed path with you. I would also like to thank
my co-supervisors Mickaël Bourgoin, for initiating this project and helping
me for several years now, and Eirik G. Flekkøy for the valuable discussions
and the clear interest in this project. You are the ones who made my doctoral
research possible by giving me the freedom to follow my thoughts and for
having unwavering belief in my abilities.
During my PhD I had the feeling of being a part of a wonderful community,
and I would like to thank all the amazing colleagues both in Porelab and in
Lyon. I would like to give a special thanks to the people who helped me work
in the laboratories, and more specifically to Mihailo Jankov. All of you made
this journey so enjoyable and full of sweet memories.
Finally, these three years, working in two countries, have not been without
hardships; and I would like to express my wholehearted thanks to my family
and my long-term friend Yannick Bertrand for their relentless support during
the crisis.
v
CONTENTS
1 confined granular media overview 1
1.1Mechanics of dry granular media......................................................... 1
1.2Grains interactions................................................................................... 10
1.3Multiphasic granular dynamics ............................................................ 16
2 magnetic janssen effect 21
2.1Experimental set-up ................................................................................ 21
2.2Results........................................................................................................ 23
2.3Ferromagnetic / non-ferromagnetic mixture...................................... 33
3 perspectives on magnetic granular medium dynamics 41
3.1Magnetic fatigue of the granular column............................................ 41
3.2Discharge of a magnetic 2D-silo............................................................ 46
4
unstable drainage of frictional fluid in a capillary tube
53
4.1Plugs formation........................................................................................ 53
4.2Onset of the bulldozing .......................................................................... 56
4.3Capillary washboarding ......................................................................... 60
4.4Ferromagnetic triggering........................................................................ 68
conclusion and perspectives 70
Appendix
a plug pressure 73
b capillary filling 75
bibliography 77
Papers
magnetic janssen effect 85
taming the janssen effect 91
magnetic janssen effect in hybrid granular media 95
discharge of a 2d magnetic silo 101
capillary washboarding during the slow drainage of a
frictional fluid 105
vii
1
CONFINED GRANULAR MEDIA OVERVIEW
Confined granular media in the presence of a liquid and gas phases may present
unstable dynamics during the slow drainage of the liquid. These instabilities
are the result of frictional interactions with the geometry walls as well as a
capillary forces at the liquid/gas interface.
In this chapter we give a review of confined granular media mechanics focused
on the interactions at play in these instabilities. First, we recall the dry grains
behaviour in a confined geometry. Second, in the perspective of a control of
the medium behaviour, we investigate the role of interactions between grains.
Finally, the multiphasic response of an immersed granular medium is described
when adding liquid and then air to the system.
1.1 mechanics of dry granular media
(a) Coffee grains (b) Gravel (c) Drug pills (d) Saturn
Figure 1.1: Examples of different granular material.
Coffee, gravel, pills or Saturn’s rings, presented in figure (1.1), all have in
common to be a collection of individual grains and form a part of what is
called granular matter. Thanks to their omnipresence in a variety of systems,
in both industrial or natural context, granular media have been the subject
of a recent upsurge within an active research community [13]. In Nature, or
more explicitly in geophysics, granular matter can take various forms, from
the formation of dunes in deserts, coastal regions where water comes into
play or even snow avalanches where phase transitions can occur. Granular
matter apprehension is therefore of high importance to prevent accidents or
predict the shape of future landscapes. From an industrial point of view, the
challenge of granular matter is mainly technological with the improvement of
granular processes and technical designs. Indeed, they represent the second
most manipulated materials after water [2], and as a striking example, the
building industry uses each year around 40 billion of tones of sand and gravel
[4]. In spite of granular media variety, their behaviour follow a set of common
fundamental laws some of which, of importance for the rest of this study, are
introduced and described in this chapter.
1
2 confined granular media overview
(a) Dune (b) Landslide (c) Sandstorm
Figure 1.2:
Depending on the system, granular matter can behave as a solid, a liquid
or a gas.
Before going any further, we recall that, usually, a distinction is made between
grains and colloids for which the thermal agitation plays a major role. A quick
energy comparison on a particle of radius
a
between the gravitational potential
energy and the thermal agitation gives a limit of
a1 µm
at room temperature:
ρg4
3πa4=1
2kBTa=4
s3kBT
8ρg1 µm , (1.1)
where
kB1×1023 J·K1
is the Boltzmann constant,
T300 K
is the room
temperature,
ρ2×103kg ·m3
the particle density and
g10 m ·s2
the
acceleration of gravity. In the rest of this study we will focus on grains of radius
far greater than 1 µm.
Despite their common presence and numerous studies, due to their com-
plexity, the mechanical behaviour of granular systems still remain challenging
for engineers and physicists. They are defined by an assembly of particles
for which individual mechanical interactions can be described. At the system
scale however, the granular assembly represents a colossal amount of particles
making a macroscopic description more intricate. For example, a single cup of
coffee made in a French press, well studied in [5], (
R=5 cm
,
h=2 cm
) contains
about
105
grounded coffee particles of size
500 µm
! Moreover, as figure (1.2)
represents, a granular system can behave as a solid, a liquid or a gas depending
on the system conditions. A sand pile at rest on a surface will behave as a solid,
characterized by an avalanche angle, while the granular assembly of a discharg-
ing silo flows as a liquid. At the extreme, grains in suspension in air will behave
as a dilute gas, for which sandstorm are a nice illustration. Moreover, several
or all these granular phases can coexist inside the same system. Think of an
hourglass where the sand is flowing out of a resting pile in the upper chamber.
All these considerations are the heart of the granular matter complexity.
Janssen Effect
One of the most renown behaviour of grains certainly is the Janssen effect
that we will describe here. Back in 1895, the German engineer H. A. Janssen
proposed a description of the stress redistribution inside grain silos. Janssen
noticed that the apparent mass at the bottom of a silo does not follow the hy-
1.1 mechanics of dry granular media 3
(a)
σz
σrσr
(b)
R
z
z+dz
h
0
σz|z+dz
σz|z
τ τ
[ June 13,2021 at 13:29 ]
Figure 1.3:
(a) Typical arch formation inside a confined granular packing, leading a
redistribution of the vertical stress along the radial direction. (b) Force
equilibrium on a horizontal slice of a confined granular medium in a silo.
drostatic pressure law of liquids for which the pressure is directly proportional
to the height of the column. In fact, a grain silo appears lighter than it really
is, suggesting that a part of the grains weight is supported by the walls of the
vessel. This model still remains a good description of silos’ mechanics and has
been used ever since to predict the stress distribution to take into account in
their construction.
Janssen’s description is based on three hypotheses:
(i)
the granular medium and the vertical stress
σz
in the radial direction
r
are
supposed uniform,
(ii)
the friction forces between grains and the walls are supposed to be fully
mobilized along the vertical direction
τ=µσr
, where
µ
is the friction
coefficient and σrthe radial stress acting on the wall,
(iii)
the vertical stress redirects partly to the radial direction following a propor-
tional law characterized by the Janssen constant
κ
,
σr=κσz
, as described
in figure (1.3.a).
Hypotheses (ii) and (iii) lead to the frictional stress at the tube wall
τ=µκσz
,
while (i) allows us to write the force equilibrium on a packing layer of height
dz, shown in figure (1.3.b):
πR2dzφρg+πR2σz|zσz|z+dz2πRdzµκ σz|z=0 , (1.2)
where the apparent density of the granular medium is
φρ
, with
φ
the packing
fraction typically around
0.6
for random close packing, and
ρ
the material
density of the grains, typically around
ρ2×103kg ·m3
for sand. Dividing
this equation by the infinitesimal height dzleads to the differential equation:
dσz
dz=2κµ
Rσz+φρg, (1.3)
4 confined granular media overview
Figure 1.4:
Janssen effect experimental apparatus. (a) Typical filling protocol where
the beads are poured inside a tube via a funnel. (b) Experimental set-up
inspired by [6] where the light grey area is free to move downward at a
constant velocity.
with a straight-forward solution:
σz(z) = φρgλ1eh
λ,λ=R
2κµ , (1.4)
where we defined the screening height
λ
depending on the tube radius
R
, the
Janssen constant
κ
usually ranging from 0to 1and depending on the grain
properties, and the friction coefficient
µ
between the grains and the silo wall.
The pressure at the bottom of the tube is usually expressed as the apparent
mass of the granular column m=σ(h)πR2/g:
m=m1em0
m,m=πR3φρ
2κµ . (1.5)
This exponential saturation law means that for a packing mass greater than
a few critical mass
m
(or a packing height greater than a few characteristic
length
λ
), the apparent mass at the bottom of the silo is constant and equals
the critical mass. In other terms, the upper part of the packing is screened by
the first layers of grains and the apparent mass for high enough silos does not
depend on the packing height. Typically, the characteristic length is about the
tube diameter λ2R.
1.1 mechanics of dry granular media 5
012345 6 7
0
0.2
0.4
0.6
0.8
1
m0/m(g)
m/m(g)
1exp(m0/m)
[ July 21,2021 at 11:05 ]
Figure 1.5:
Typical Janssen effect measurement, data from [7]. The apparent mass
of a granular column is measured for different packing parameters and
normalized by the critical mass m.
Experimentally, the Janssen effect is characterized by the measurement of
the apparent mass of grains poured inside a tube via a funnel as shown in
figure (1.4.a) . As shown in figure (1.4.b), the granular packing is resting on a
piston, slightly smaller than the tube to ensure no contact with the walls, in
order to measure directly the apparent mass of the packing using a force gauge.
Though seemingly simple, the Janssen system presents some experimental
difficulties. Indeed, when dealing with granular matter, one shall remember
that the friction forces are usually undetermined since the Coulomb criterion
gives an inequality between the normal and the resulting tangential friction
force [8]:
ftµfn, (1.6)
where
ft
and
fn
are respectively the tangential and normal forces. The friction
forces are usually in an undetermined state. The force network is undetermined
and imposed by the preparation of the granular packing. However, the Janssen
hypothesis (ii) requires that the friction forces at the wall are fully mobilized
in the vertical direction. In [6], L. Vanel and E. Clément proposed to reach
this condition by moving downward the lower part of the mass measurement
set-up, represented by the light grey area in figure (1.4.b). Figure (1.5) shows
the apparent mass measurement for different packing mass
m0
following this
experimental protocol. The experimental results are in very good accordance
with the Janssen prediction. Even though the Janssen description considers the
medium as continuous, it still remains nowadays a good prediction of the force
redistribution inside a confined granular packing.
Nevertheless, since 1895, some refinements to the Janssen model have been
brought to attention, one can think of the OSL (oriented stress linearity) model
taking into account the grain-scale forces [9,10]. Moreover, some limitations
6 confined granular media overview
010 20 30 40
0.8
0.9
1
1.1
h/σ
m/m0
D/σ=5.7
D/σ=8.2
D/σ=9.8
D/σ=17.5
0 5 10 15 20
0
5
10
(a) (b)
h (cm)
m(g)
Figure 1.6:
Janssen effect in a narrow tube. (a) Data from [11]. Two apparent mass
measurements for a tube diameter of about 5bead radii, the apparent mass
follows a two-parameters saturation law. The dashed line corresponds to the
classical Janssen prediction while the blue line corresponds to the modified
prediction. (b) Data from [12]. Apparent mass rescaled by the true mass
of grains for different narrow tube of diameter
D
, with a bead diameter
σ
.
One can notice that the apparent mass can be greater than the actual mass
of the packing.
to the Janssen model can be found for narrow granular column [11] where the
Janssen effect does not follow a one-parameter (
m
) exponential saturation
but depends on two parameters,
m=m(1exp (m/˜
m))
, where
˜
m
is a
screening mass. A typical example of this result is displayed in figure (1.6.a),
and can be understood as the formation of cell arrangement inside the packing.
Confined grains in a narrow tube can also display striking phenomena as
presented by [12] and shown in figure (1.6.b). In this case, the frustration of
mechanical stress at the grain level and the violation of Janssen hypothesis
(iii) leads to the apparition of additional compressive force resulting in a
packing apparent mass greater than the actual mass of the grains,
m/m0>1
,
called "reverse Janssen effect". On top of the intrinsic parameters of the system
(geometry, density, etc.), the Janssen effect depends strongly on the preparation
of the granular packing as described in [13].
To complete the description of confined granular media mechanics, we
present in the following sections some striking dynamics properties.
Discharge of a granular silo
We have described so far the static regime of grains silos. We propose here to
describe their dynamic discharge. An emptying silo is similar to an hourglass.
The most striking fact about them is that the output flow rate of beads does not
depend on the height of grains stored in the upper chamber. This surprising
behaviour is the reason why hourglass are filled with sand instead of water,
the passed time is proportional to the amount of sand that flowed through the
orifice. The origin of this independence over the height is still uncertain. For