060014 2014 L. A. Pugnaloni K. To, Institute of Physics, Academia Sinica,
Taipei, Taiwan. 060014

During the past decades, notable improvements have been achieved in the
understanding of static and dynamic properties of granular materials, giving
rise to appealing new concepts like jamming, force chains, non-local rheology
or the inertial number. The ‘saltcellar’ can be seen as a canonical example of
the characteristic features displayed by granular materials: an apparently
smooth flow is interrupted by the formation of a mesoscopic structure (arch)
above the outlet that causes a quick dissipation of all the kinetic energy
within the system. In this manuscript, I will give an overview of this field
paying special attention to the features of statistical distributions
appearing in the clogging and unclogging processes. These distributions are
essential to understand the problem and allow subsequent study of topics such
as the influence of particle shape, the structure of the clogging arches and
the possible existence of a critical outlet size above which the outpouring
will never stop. I shall finally offer some hints about general ideas that can
be explored in the next few years.

# Invited review: Clogging of granular materials in bottlenecks

Iker Zuriguel[inst1] E-mail<EMAIL_ADDRESS>
(21 November 2014; 5 December 2014)

††volume: 6

99 inst1 Departamento de Física, Facultad de Ciencias, Universidad de Navarra,
31080 Pamplona, Spain.

##  1 Clogging in bottlenecks, a multiscale and multidisciplinary problem.

When a system of discrete bodies passes through a constriction, the
interactions among the particles might lead to the development of clogging
structures that, eventually, completely arrest the flow. This phenomenon is
observed in a wide range of systems with relevant consequences. Clogging of
granular materials in silos may force a production line to be stopped. Much in
the same way, clogging of suspended hydrated particles is a major issue
concerning oil and gas transport through pipelines [1]. At a smaller spatial
scale, clogging leads to intermittent flow when a dense suspension of
colloidal particles passes through a constriction in a microchannel [2, 3]. A
straightforward application of the understanding that could be gained
concerning clogging in suspensions is found in ecological engineering.
Nowadays, an alternative that is becoming widely used for removing pollutants
from wastewater is the use of subsurface flow treatment. The most important
drawback of this technique is its unpredictable lifetime, mostly limited by
clogs that obstruct the pores [4]. At an even smaller scale, intermittent
flows are observed when electrons on the liquid helium surface pass through
nanoconstrictions [5]. Finally, clogging can also develop when crowds in panic
are evacuated through emergency exits that cannot absorb the amount of people
approaching the doors [6, 7, 8].

All these examples of clogging take place in a broad range of systems where
widely different forces are at play: those concerning the interactions among
particles as well as those related to the interaction between particles and
the surrounding media. For the case of non-cohesive inert grains, gravity and
contact forces are the only relevant ones. For particle suspensions, however,
the hydrodynamics of the flowing fluid as well as the capillary effects must
be taken into account. Dynamics of crowds through bottlenecks are even more
difficult to approach theoretically, yet a social force model has been proved
to adequately reproduce the observed behavior in some circumstances.

In the last decade, the number of works published about clogging has
experienced a sudden increase, which constitutes a gauge of the interest and
relevance of this phenomenon. Despite this, the physical mechanisms behind
clogging are still not well understood. Several issues contribute to this, but
probably the most important one concerns the local character of clogging when
compared, for example, with the global nature of jamming [9], a scenario that
has attracted much more attention over the last years. This ‘local character
of clogging’ seems to complicate the definition of global extensive variables
within the system which could be used to characterize the phenomenology.

In this manuscript I shall give a brief summary of the major advances achieved
in the understanding of clogging of non-cohesive inert grains at a bottleneck.
I will start by presenting results on the static silo, then I will move to the
case of a vibrated silo, and finally I will present some conclusions and
mention several questions that —from my point of view— should be addressed in
the forthcoming years.

##  2 Clogging in hoppers or silos

The development of clogs that obstruct the flow in the discharge of bins or
silos by gravity is a problem that has always worried the engineering
community [10, 11, 12, 13]. The goal in all these works was finding a ratio of
the outlet to particle size that could guarantee the absence of clogging. For
non-cohesive materials, it was known that this value is around $5$ although,
depending on the particle properties, it could increase up to $10$.
Nevertheless, little was understood about the mechanism that triggers a clog
and the physical variables that control its development. Indeed, it was not
until the beginning of this century when the scientific community started to
carefully investigate this problem [14].

###  2.1 Avalanche size distribution

Figure 1: Histogram $n_{R}(s)$ for the number of grains $s$ that flow between
two successive clogs. Data correspond to a circular orifice of $6\;mm$
diameter and glass beads with a diameter of $2\;mm$. More than 4000 events
were recorded. In the inset, a semilogarithmic plot with the solid line
indicating an exponential fit.

One of the first questions that was tackled concerns the statistics of the
avalanche sizes (the usual measured magnitude is the number of grains that
flow out of the silo from the breakage of a clogging arch, until the
development of a new one). The key feature of avalanches, which is now well
accepted, is that the distribution of their sizes follows an exponential decay
(see Fig. 1), a result reported for the first time in [15]. This trend was
explained in [16] by assuming that the probability of clogging is constant
during the whole avalanche, a behavior observed for all the outlet sizes
explored. Afterward, a many-particle-inspired theory was proposed based on a
continuity equation in polar coordinates [17]. Although friction, force
networks or inelastic collapse of the particles were not taken into account,
the authors reproduced the exponential decay of the avalanche (or burst)
sizes. The intermittent flows reported were explained in terms of a random
alternation between particle propagation and gap propagation. Very recently, a
probabilistic model —in which the arches were modeled by a one-dimensional
stochastic cellular automaton— also sheds light on the origin of the
exponential character of the avalanche size distribution [18]. The stochastic
nature of the clogging process was also suggested in [19] where it was shown
that the particles that end up forming the clog were totally uncorrelated at
the beginning of the avalanche.

The exponential decay of the avalanche size distribution has been reported in
several arrangements: 2D and 3D silos [16, 20, 21], 2D hoppers [22, 23], 2D
and 3D tilted hoppers and silos [24, 25], silos with the presence of obstacles
[26, 27], 2D silos where the particles were driven by different gravity forces
[28], and fluid driven particles in 2D and 3D [29, 30]. However, there are
some examples where this exponential tail breaks down. Those situations are
typically related to a breaking of the symmetry of the problem as in the
following cases: 1) usage of particles with shapes that are not spherical
[31]; 2) emplacement of multiple orifices [32]; 3) implementation of slots in
3D silos instead of the normal circular orifices [33]. In the latter case, a
power law distribution was observed as it will be explained in section 2.4
Incidentally, power law distributions were also numerically obtained when
considering internal avalanches, defined as the number of grains that move
inside the silo between consecutive clogs. This result was compatible with the
idea of Self Organized Criticality, in analogy with avalanches developed at
the surface of a pile [34]. This work was, indeed, one of the first approaches
to the avalanche statistics in the silo problem.

###  2.2 Does a critical outlet size exist?

Considering the exponential character of the avalanche size distribution, its
first moment (the average avalanche size $\langle s\rangle$) can be easily
calculated and used to study the dependence of clogging on the size ratio
between the outlet and the particles. For spherical beads in a 3D silo, a
divergence of the avalanche size was reported for an outlet diameter about $5$
times the bead diameter (see Fig. 2) [31]. This divergence was shown to be
robust, as it holds for particles with widely different properties. Among
these, the shape of the particle was reported to be the most influential on
the critical outlet size value. Nevertheless, in a subsequent work, the
existence of such a critical outlet size was challenged by K. To [22]. In a
two dimensional silo, it was shown that several empirical fits agree
reasonably well with the experimental data: some were compatible with the
existence of a critical outlet, but others were not (see Fig. 3).

Following this idea, Janda et al. [20] demonstrated that one of the non-
divergent expressions proposed in [22] could be analytically deduced using
both, the probability that a given number of particles meet above the outlet
—as suggested by Roussel et al. [35]— and the probability of finding arches of
a given size within a granular deposit —as found in [36, 37]. Unfortunately,
the reasoning used in two dimensions was not applicable to three dimensional
silos, where the transition seems to actually exist. Interestingly, this
clogging transition has been also identified for inclined silos and orifices
[24, 25], as well as in the discharge of granular piles through an orifice
below its apex [38]. Very recently, the mean avalanche size has been put on
relation with the fraction of clogging configurations that are sampled by the
orifice, suggesting that $\langle s\rangle$ should increase exponentially with
the hole width raised to the system dimensionality [39]. According to these
results, clogging is akin to the jamming and glass transitions in the sense
that there is not any sharp discontinuity in the behavior, but a dramatic
increase of the relaxation times as the orifice size is enlarged.

Figure 2: Mean avalanche size $\langle s\rangle$ vs. $R$, the ratio between
the outlet and particle diameters. The solid line is a fit with the equation:
$\langle s\rangle=A(R_{C}-R)^{-\gamma}$; with $R_{C}=4.94\pm 0.03$,
$\gamma=6.9\pm 0.2$, and $A=9900\pm 100$. Inset: mean avalanche size $\langle
s\rangle$ vs. $1/(R_{C}-R)$. Note the logarithmic scale. Figure reprinted with
permission from Ref. [31]. Copyright (2005) by the American Physical Society
[65]. Figure 3: Variation of the decay rate $\alpha$ with hopper or silo exit
$d$. $\alpha$ is obtained from the fittings of the avalanche sizes with
$F(s)=e^{-\alpha(s-s_{0})}$. The solid line is the fitted curve
$\alpha=Ae^{-Bd^{2}}$ with $A=0.846$ and $B=0.275$. The inset shows the same
data and the fitted curve with $d^{2}$ plotted in the x axis. Figure reprinted
with permission from Ref. [22]. Copyright (2005) by the American Physical
Society [65].

###  2.3 Clogging arches

Complementary to the analysis of the avalanche sizes, some authors have paid
special attention to the arches that clog the orifices. Clogging arches are
structures of several mutually stabilizing particles that have to span, at
least, the size of the constriction. In his seminal work, To et al. introduced
a simple model to explain the clogging probability based in the geometry of
the clogging arches (see Fig. 4) [14]. They proposed that the position of
particles in a clogging arch is the result of a random walk model with some
restrictions: 1) the horizontal span of the arch should be larger than the
orifice; 2) the arch has to be convex everywhere; 3) particles conforming the
arch should be in contact with each other. This model nicely reproduced the
clogging probability for hopper angles below $75^{\circ}$. In a subsequent
work [40], the same authors introduced an approximation of the arch shape to a
circular arc centered at the apex of the hopper cone. From this, they
calculated detailed properties of the clogging arches, such as the number of
disks conforming them, finding good agreement with the experimental results.
Some of the ideas proposed by To et al. were corroborated in Ref. [41] where
it was reported that the aspect ratio of the arches (the height divided by
half the span) tends to one, a result that is compatible with a semicircular
shape. In addition, it was shown that the convexity condition assumed by To is
not necessarily fulfilled in all the particles. Indeed, $17\%$ of the
particles had an associated angle with their two neighbors above
$180^{\circ}$. Hence, arches were locally concave at that particle, a
situation which was named ‘defect’. Despite this seemingly mismatch with the
restricted random walk model, a strong inverse correlation of the angle
associated to a particle and the one of their neighbors was also shown.
Apparently, this inverse correlation compensates the apparition of defects and
preserves the validity of the restricted random walk model.

Figure 4: (a) Image of a typical clogging arch. (b) Configuration of the arch
where $r_{i}$ illustrate the $i$ steps of the restricted random walk model
proposed in [14]. Figure reprinted with permission from Ref. [14]. Copyright
(2001) by the American Physical Society [65].

Apart from the works mentioned above, where only the geometry of the arches
was evaluated, there have been preliminary attempts to consider the forces
involved within the particles conforming the arches. In [42], force analysis
was used to calculate the jamming probability of mixed sizes disks that move
downwards under gravity in a two-dimensional hopper. The authors focused in
the simplest case of arches formed by three discs which, for the outlet size
employed, were the most common. Finally, Hidalgo et al. [43] performed
numerical simulations and found that —in clogging arches— the tangential
forces in the ‘defects’ were very high, while normal forces were abnormally
low. The outcome concerning tangential forces is somehow expected as friction
is necessary to stabilize defects. On the contrary, the result concerning the
normal forces is rather counterintuitive, but it was in accord with a previous
forecasted prediction based on experimental works on the stability of arches
in a vibrated silo [44].

###  2.4 Orifice geometry.

As stated above, clogging is a local phenomenon in the sense that it always
takes place at the constriction. According to this, it seems rather obvious
that the properties of the confining geometry would importantly affect the
clogging process. An evidence of this can be found in [14] where it is shown
that the clogging probability in a hopper is notably reduced as the hopper
angle increases from $60^{\circ}$ to $75^{\circ}$. On the contrary, hopper
angles below $60^{\circ}$ give rise to similar clogging probabilities. The
reason for this seems to be founded on the fact that, for sufficiently flat
hoppers, the grains develop a spontaneous internal angle of repose which acts
as an internal hopper.

The effect of this internal angle of repose is also relevant in the works of
Durian’s group who implemented inclined orifices and silos [24, 25]. This
practice is relatively common in industrial hoppers and hence, knowing the way
in which clogging is affected, becomes significant. The authors have proved
that increasing the tilting angle of the orifice or silo augments the
propensity to clog according to a reduction in the projection of the aperture
area against the average flow direction. In addition, a clogging phase diagram
is proposed combining tilting angle and outlet size. For circular apertures,
the same diagram is found for four grain types (including prolate and oblate
ones). For slots, however, the shape of the phase diagram for the case of
lentils and rice seems to be different than for more isotropic grains, an
effect attributed to an alignment between grains and slit axes.

The use of slots instead of circular orifices was already proved to be
beneficial to prevent clogging [45]. In this work, some conservative
guidelines are given to select the minimum outlet size that assures no flow
interruption. While for horizontal and vertical slots the ratio of slot width
to particles size are $3.3$ and $4.6$ respectively, for horizontal circular
outlets the ratio of orifice diameter to particle size is $6.4$. This number
may seem considerably larger than the ones reported in [31], but it should be
taken into account that particles of different properties (including
anisotropic ones) were employed. Even more importantly, in a subsequent work,
it was reported that, as the length of the slot increases, the avalanche size
distribution departs from the exponential behavior displaying a power law
decay [33]. This behavior is explained in terms of a model where a slot is
represented by a series of statistically independent cells whose length is
related with a hypothetical distance along which particles’ movement is
correlated. Interestingly, the model matches experimental outcomes for a
correlation distance of around $10$ particle diameters. Nevertheless, this
result needs to be confirmed as in other experiments using slots, the
avalanche size distribution has been found to be exponential for different
types of grains [25].

A configuration which is closely related to the slot geometry is the placement
of several aligned orifices. Very recently it has been reported that clogging
can be significantly reduced by having more than one exit orifice. In this
situation, when one of the orifices jams, the flow through the adjacent
unjammed orifice might cause perturbations in the clogging arch, destroying it
and leading to a sequence of jamming and unjamming events [46]. The necessary
condition to observe this behavior is, of course, that orifices are close
enough to each other. In the same line, Mondal and Sharma [32] have shown that
adjacent outlets start affecting each other when the distance is approximately
three times the diameter of the particles. These authors point toward the
importance of stable particles (adjacent to the arches) resting on the base of
the silo. Remarkably, the role of these particles was in fact overseen in
previous works that analyzed the properties of clogging arches [14, 40, 41,
42, 43, 44].

A smart alternative to alter the clogging process consists of placing an
obstacle just above the outlet. In [26], it was reported that the clogging
probability may be reduced up to $100$ times if the obstacle position is
properly selected. This dramatic effect was attributed to a reduction in the
pressure (or particle confinement) in the orifice neighborhood, which
apparently favors arch destabilization. The explanation given is that
particles colliding above the orifice —which eventually could form a clogging
arch— are not easily stabilized if there is not a certain confinement that
facilitates energy dissipation. This idea was supported by the observation of
a sudden increase on the number of particles ejected upwards in the outlet
proximities when the obstacle was placed. In the same work, simulations of a
silo filled with a few layers of grains revealed the same kind of clogging
reduction as the layer of grains above the orifice was reduced, then
confirming the important role of pressure in the process. In a subsequent work
[27], it was shown that the effect of the obstacle is enhanced as the outlet
size enlarges. It is noteworthy that, in all the cases, the clogging reduction
is achieved with just a tiny alteration of the flow rate (up to $10\%$ in the
worst situation). In these works, an issue that remains unclear is the role of
the packing fraction above the orifice. Clearly, the placement of the obstacle
affects this variable which should be, indeed, related to pressure.
Nevertheless, robust measurements of volume fraction are extremely difficult
near the outlet due to the existence of strong gradients.

As far as I know, there has been only one attempt to unveil the role of volume
fraction on clogging in dry granular media [47]. In this work, a pseudo-
dynamic model was implemented to prepare samples with different initial
configurations by means of a tapping procedure. Although packing fraction
affects clogging, their main conclusion is that this is not a good macroscopic
parameter to predict the size of the avalanches that would flow through a
given aperture, suggesting that further information about the packing
properties is necessary. A nice alternative to study the effect of packing
fraction on the ability of a system to develop clogs is the use of solid
particles suspended in a fluid. This is precisely what it was done in [48]
where it was proved that the probability of bridge formation increased with
the volume fraction. Note that this system has the advantage of allowing a
better control of the volume fraction than just varying the initial
configuration as done in [47].

###  2.5 Effect of polydispersity and particle shape

A recursive topic that arises in the granular community is the roles that size
polydispersity and particle shape play on the behavior of such materials. For
the case of clogging in silos, in [31] it was reported that polydisperse
samples displayed the same exponential decay of the avalanche size than
monodisperse ones. Furthermore, it was revealed that polydispersity had a
negligible effect in the critical outlet size above which clogging would not
occur. In [49], clogging of bidisperse samples was also shown to be similar to
the monodisperse case as long as segregation is prevented. In addition, the
authors propose that the parameter that should be considered to characterize
the mixture is the particles volume-average diameter.

Contrary to polydispersity, particle shape seems to play a major role in
clogging development as evidenced using prolate (rice) and oblate (lentils)
particles [31]. The critical outlet size increases (i.e., clogging is more
likely) when anisotropic particles are employed, a result coherent with that
obtained in fluid driven suspensions of mica flakes when compared with glass
beads [48]. An issue that is still open concerning anisotropic particles in
the discharge of a silo is the characteristic particle length that should be
chosen to compare with that of the orifice. In addition, there is a lack of
experiments or simulations about the effect of using faceted particles in the
clogging probability. Some words have been written suggesting that faceted
particles dramatically increase the clogging ability due to their tendency to
align [50, 51], yet there are not systematic results on this interesting
topic.

###  2.6 Dynamic signatures of clogging

Provided that clogging in bottlenecks is a consequence of the sudden formation
of a stable arch at the very narrowing, it is a big challenge to find
dynamical descriptors in the flowing state that can be used to predict an
eventual arrest of the flow. The first important result about this challenge
was reported by Longhi et al. [52] who studied the impulses recorded by a
force transducer at the hopper boundary near the orifice. Although the
distribution of impulses does not reveal any static signature of jamming, the
distribution of the time intervals between collisions ($\tau$) produces
interesting distinctive features as the outlet size is reduced and approaches
the clogging region (see Fig. 5). In fact, this distribution tends to a power-
law $P(\tau)\sim\tau^{-3/2}$ implying that the mean time interval tends to
diverge as the outlet is reduced. This is so even when the average time
computed from a finite (albeit large) data set shows a relatively negligible
dependence on the outlet size.

Figure 5: (A) Probability distributions, $P(\tau)$, of the time intervals
$\tau$ between collisions, on a log-log scale for an outlet length $3.3$ times
the diameter of the particles. The solid line corresponds to a power law
$P(\tau)\sim\tau^{-3/2}$. The average time interval between impulses is marked
in the figure. (B) $P(\tau)$ on a log-log scale for different opening sizes
ranging from $3$ to $16$ times the diameter of the particles (the curves on
the top correspond to the smaller outlet sizes). The curves are displaced
vertically for clarity. The solid line is the power law:
$P(\tau)\sim\tau^{-3/2}$. Figure reprinted with permission from Ref. [52].
Copyright (2002) by the American Physical Society [65].

In [53], the flow rate properties were carefully examined in a two-dimensional
silo for outlet sizes both, above and below the supposed critical outlet size.
Even though the average flow rate behaved smoothly and did not display any
characteristic property near the critical size, it was observed that the flow
rate fluctuations are non symmetric for small apertures. For large orifices,
the measurements of the instantaneous flow rate ($q$) display Gaussian-like
distribution of the fluctuations around the average. Nevertheless, as the
outlet size is reduced, temporal interruptions of the flow are evidenced by
the development of a peak at $q=0$ in addition to the one that corresponds to
the flowing regime.

In this direction, a step further was performed by Tewari et al. [54] who
implemented event-driven simulations to analyze the velocity fluctuations of
grains flowing through a hopper. The analysis in this work was not restricted
to the region of the orifice, as all the grains of the silo were studied.
Interestingly, although the kinetic temperatures are always higher at the
boundaries of the silo, the correlation times display a tendency that reverses
as the outlet size is reduced: whereas for high flow rates (far above the
critical outlet size) the flow at the center has longer autocorrelation times
than at the boundary, the opposite is valid for low flow rates as fluctuations
relax more slowly at the boundaries. In this work, it is also suggested that
clogging is preceded by the appearance of vortices that nucleate at the
corners of the hopper and extend inwards.

##  3 Vibrated silos: clogging and unclogging

Up to now, I have described investigations related to the clogging process
presuming that, once a clogging bridge is formed, all the kinetic energy is
dissipated and the structure is forever stable. Nevertheless, an alternative
approach can be implemented, which consists on applying an external input of
energy and study its effect on clogging. This strategy gives rise to a
dramatic change in the observed dynamics when the orifice is small, i.e., in
the region where clogging is frequent. Unlike the case of a static silo, the
flow in the vibrated silo is characterized by the alternation of jamming and
unjamming events. Indeed, apart from the flow rate fluctuations found in a
static silo [53], in the vibrated case long flow interruptions were present.
These were attributed to arches that form and were initially stable, but
destabilize as a consequence of vibrations (see Fig. 6). This behavior
suggests that the intermittent flow in vibrated silos can be split in two
different, independent processes: clogging and unclogging. Following this line
of reasoning, Mankoc et al. [55] reported that the probability that a system
clogs does not depend on the vibration, which only introduces a non-zero
probability of unclogging once an arch has blocked the orifice. This
probability of unclogging was measured in three different ways which led to
consistent results whose most conspicuous feature was an increase of the
unclogging probability with the outlet size.

Figure 6: Signal from a photosensor at the exit of a silo: a value of $1$
indicates that a particle is blocking the beam, zero means that the beam is
unobstructed. (a) Static silo. (b) Vibrated silo. (c) A zoom of the signal
shown in (b) during the first three seconds, the same time stretch as in (a).
All the data were obtained using an orifice of diameter $3.05$ times the beads
diameter. Figure reprinted with permission from Ref. [55]. Copyright (2009) by
the American Physical Society [65].

Janda et al. [56] devised a similar experiment in which the hopper wall of an
eccentrically discharged silo was a piezoelectric, allowing a local
perturbation of the clogging arch. In this sense, this work is conceptually
different than that of Mankoc et al. where the whole silo was vibrated. The
most interesting result revealed by Janda et al. was that the distribution of
times that the system takes to get unclogged exhibits a power law decay. At
low vibration accelerations, anomalous statistics for the jamming times were
evidenced as the exponent $\alpha$ of the power law was below $2$ and the
first moment could not be calculated (see Fig. 7). This property is, indeed,
strongly reminiscent of the anomalous dynamics usually observed for creeping
flows of glassy materials. In a recent work, this behavior has been shown to
be universal in other systems of macroscopic particles flowing through a
bottleneck like sheep, a model of pedestrians, and colloids [57]. Furthermore,
for the case of inert grains, several variables have been shown to affect the
value of the exponent going from $\alpha>2$ to $\alpha\leq 2$, i.e., from an
unclogged situation (where averages can be defined) to a clogged scenario
(where the average flow rate would tend to zero as the measuring time
increases). These variables are: the intensity of vibration, the outlet size,
the height of the layer of grains above the outlet, and the inclination of the
2D silo with respect to the vertical which modifies the component of the
gravity affecting the grains. Increasing the intensity of vibration and
enlarging the outlet size favors the development of unclogged situations,
while increasing the layer of grains or the silo verticality facilitates the
transition to clogging. A similar idea was anticipated by Valdés and
Santamarina [58] who suggested that the acceleration that would be required
for unclogging increases with increasing skeletal forces in the particles
forming the bridge. Furthermore, they related this prediction to the higher
stability exhibited by the arches formed in a suspension when subjected to
high fluid velocities [59].

Figure 7: Histogram, in logarithmic scale, for the time lapses that the
orifice remains blocked in a vibrated silo for different vibration
accelerations as indicated in the legend. Data correspond to an outlet size
$1.78$ times the particle diameter. The dashed line has a slope of two
evidencing that, for the smallest acceleration displayed, the slope is smaller
than two. Figure reprinted with permission from [56]. Copyright (2009) by IOP.

Finally, in a 2D vibrated silo which allowed observation of the clogging
structures, it was established a relationship among the bridge geometry and
its resistance to vibration [44]. In particular, it was revealed that the
intensity of vibration at which the arches collapse is inversely correlated
with the maximum angle among the particles conforming it. For the particular
case of angles above $180^{0}$ (the so called defects), this dependence was
explained in terms of a very simple force analysis. In summary, from this work
it was concluded that arches break at defects and, the larger the maximum
angle, the weaker is the arch.

##  4 Perspectives

After more than a decade of research, significant advance has been achieved in
the understanding of clogging. Despite all that, the relevance of the
remaining open issues and the importance of the consequences that clogging has
from an applied point of view, hint about an augment of activity on this topic
in the forthcoming years. Probably, a sensible approach that should be
investigated is isolating the dynamic and geometric contributions in the
development of clogs. Effectively, a clogging arch should have a structure
compatible with the confined geometry. But in addition, this structure has to
be able to persist until all the kinetic energy of the system is dissipated.
Unfortunately, increasing the outlet size leads to a modification of the
geometry of the problem (as the span of clogging arches has to increase), but
also affects to the velocity of the particles (which increases with the square
root of the orifice diameter). In a recent work, Arévalo et al. made initial
progress in this direction by exploring clogging when reducing the driving
force up to $10^{-3}g$ where $g$ is the gravity; but undoubtedly, new
strategies should be devised to understand the effect of dynamics in the
clogging process.

A situation which seems simpler as dynamic effects are removed is the study of
unclogging as explained in section 3 The power law decays observed in the time
that the system needs to become unclogged, suggest a creeping process where
the bridges would age with time, increasing their endurance. A straightforward
way of testing the validity of these ideas would be an analysis of this
process using photoelastic particles to evaluate temporal evolution of the
forces within the arch [60]. The usage of this kind of particles could also be
implemented in order to unveil an old question concerning the relationship
among clogging arches and force chains.

Another issue that remains unsolved is, whether or not, clogging can be seen
as a phase transition and, if so, what kind of transition clogging is. As
explained above, from the measurements of unclogging times in a vibrated silo,
a divergence has been found that can be used to rigourously characterize the
clogged state through the definition of a ‘flowing parameter’ [57]. A thorough
inspection of the dependence of this parameter on different variables becomes
necessary to corroborate its usefulness. Nonetheless, as this approach is
based on the unclogging times, it cannot be used for the ‘singular’ case of a
static silo where, if formed, clogs last forever. In such scenario, instead of
the traditional way of studying the divergence of the avalanche size as the
outlet is enlarged, I believe that it is pertinent to approach the transition
from the flowing region. There, it should exist some parameter that reveals
distinctive behavior when the outlet size is reduced as explained in section
22.6 A reminiscent problem of this alternative is the difficulty of choosing a
region where to perform the analysis as the silo is precisely characterized by
the existence of strong spatial and temporal gradients. In this sense, a
geometry that becomes promising is a narrow pipe without any constriction
where clogs may develop at any place [61, 62, 63]. Apart from a zone at the
top of the pipe where the pressure increases with depth, a rather homogeneous
behavior should be observed within the rest of the system, allowing clean
measurements of variables like velocity, density and so on. The study of this
geometry can also be seen as an intermediate stage between clogging and
jamming as it is also the case of ‘jamming by pinning’ [64] where the increase
of the number of obstacles in the system (and so the characteristic distance
between them) was shown to reduce the density at which the system jams.

###### Acknowledgements.

I would like to thank the referee Kiwing To whose comments have, undoubtedly,
helped to improve the quality of this manuscript. I am very grateful to Angel
Garcimartín, Diego Maza, Carlos Pérez-García and Luis Pugnaloni, without whom
this work would never have been possible.

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