Flow and clogging in bottlenecks: simulations and experiments
- Iker Zuriguel (University of Navarra, Spain)
- Ignacio Pagonabarraga (CECAM EPFL, Switzerland)
Controlling the development of clogging in bottlenecks is crucial in fields such as engineering, medicine and architecture. Clogging of granular materials in conduits or industrial silos may lead to stop a line of production causing significant economic loss. In the same way, clogging of suspended particles is a major issue in oil and gas transport through pipelines. Indeed, floating gas hydrate particles can plug oil lines with the consequent safety and environmental hazards. In a much smaller spatial scale, clogging leads to intermittent flow when a dense suspension of microparticles passes through a constriction in a microchannel [1, 2]. Therefore, a more deep and fundamental understanding and use of clogging will be most beneficial. As an example, during the last years, clogging with microparticles has been used in medicine to provoke embolization of blood vessels to shrink a tumor or block an aneurysm. Minimizing clogging in suspensions is also of critical importance in ecological engineering. Nowadays, the use of subsurface flow treatment in wetlands is common worldwide for removing pollutants from wastewaters. This solution has important advantages due to its low cost of maintenance and its mechanical simplicity; the major drawback of this technology is its unpredictable life time, mostly limited by clogs that obstruct the pores . Probably, the most dramatic example of clogging is the one occurring when crowds in panic are evacuated through emergency exits that cannot absorb the amount of people approaching the doors . Sadly, in the last century clogging in narrow passages has provoked hundreds of deaths like in Hillsborough stadium (Sheffield, England) in 1989, or the most recent one happening during Halloween 2012 in Madrid Arena (Spain).
All these examples of clogging occur in systems where both, the interactions among particles and the interaction between particles and the surrounding media, are rather different. For the case of inert grains, gravity and contact forces are the only at play. For particle suspensions, however, the hydrodynamics of the flowing fluid as well as the capillary effects should be also 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 behaviour. In the last decade, more than one hundred works with clogging as the key ingredient have been published, which constitutes a gauge of the interest and relevance of this phenomenon. Nevertheless, a reasonable understanding of the physical mechanisms behind clogging is lacking.
Several arguments can be given to justify the lack of fundamental understanding in clogging; probably the most important stands on its local character when compared, for example, with the global nature of jamming . This fact seems to complicate the definition of global extensive variables within the system which could be used to characterize the phenomenology of clogging. As a consequence, most of the researchers working in multiple-particle systems have approached the problem of clogging and flow through bottlenecks as a collateral investigation. In addition it is remarkable that most scientists treat clogging as a particular problem occurring in a given media (inert particles, suspensions, active matter…) focusing primarily in the properties of the media and overlooking the general aspects of the clogging itself. Therefore, there is a need to develop common methodologies to understand, explain and predict the structural and dynamical properties of clogging. Indeed, in the last years, several striking analogies between clogging in different systems have been identified. For example, having a sufficiently high density of particles per unit area near the bottleneck has been shown to be necessary to observe clogging in suspensions [1,6], silos  and crowd dynamics . Pressure is known to play a relevant role in clogging; for example, clogs of humans are exclusively reported in panic situations, when people impulsively push each other in their wish to reach the exit. A practical generic solution which seems to release such pressure is the placement of an obstacle just in front of the outlet. Indeed, this strategy has been proved to be efficient in clogging prevention for both silos  and room evacuation [10,11,12]. These are only mere examples indicating that a general framework for clogging should be achievable.
Computer simulations are ideally positioned to complement experimental efforts in situations where real conditions are not easily achievable. Indeed, one of the most important difficulties arising in the crowd evacuation situation is the difficulty of performing experiments without putting people in danger. We anticipate the key role that simulations will have in the forthcoming years, but they are still far from being completely reliable in this area.
Three works have been performed in different fields at the beginning of the last decade that can be considered as the pioneers in dealing with this subject [1,4,13]. In , Haw observed important modifications in the concentration of suspended particles in a fluid, after being sucked with a syringe. The author attributed this phenomenon to the clogging of particles in the neighborhood of the syringe end. In addition, it was reported the relevance of parameters such as the size of suspended particles, their concentration and the applied pressure (measured as velocity of suck). Some years before, Helbing et al  published their famous work in Nature where, among other things, it was shown the faster-is-slower effect. This behavior was attributed to the increase of exit blockages (and injured people) with the desired pedestrian velocity. The same blockages were proposed as the reason for which a wider area in a corridor has a prejudicial effect on the evacuation process. Almost at the same time of the work of Helbing et al, K. To and coworkers faced a simplified version of the clogging problem using inert grains flowing through a hopper . With the spirit of providing a simplified explanation to this process, they presented a restricted random walk model that captured some of the experimental findings in terms of the shape of the blocking arches.
After these works, the number of studies concerning flow through bottlenecks and clogging has steadily increased. In the field of crowd dynamics, Saloma et al.  studied by means of numerical simulations and experiments with mice, the distribution of ‘clogging times’ measured as the elapsed time between the passages of two consecutive mice. These distributions revealed power law tails for long times. The same scale free behavior was obtained for the distribution of burst sizes with the exception of small doors where an exponential decay was observed. Parisi et al. have shown, both numerically  and experimentally using ants  the relationship of clogging with the faster is slower behavior. Ants have become a widely used model to simulate egress  as important similarities with humans have been identified. For instance, Altshuler et al. observed that, under panic conditions, ants perform a nonsymmetrical use of two identical exit doors . Nevertheless the use of this animal to represent pedestrian behavior is still under discussion as it seems that their collective behavior displays important differences with humans. A key aspect seems to be the fact that jamming, as understood in vehicular traffic, is not observed ; i.e. there is not a velocity dependence on density as observed in both, pedestrians and cars.
Probably the scenario where more attention has been paid to the development of clogging is the case of inert grains flowing out of a silo by gravity. After the work of To et al. , Zuriguel et al reported in two consecutive works [19,20] that the probability of clogging was constant during the whole avalanche, leading to an exponential distribution of the avalanche sizes. The first moment of this distribution (average avalanche size) was then easily calculated and used to show a strong dependence with the size ratio between the outlet and the particles. Indeed for a 3D silo it was reported a divergence of the avalanche size for an outlet diameter around 5 times the beads diameter. Although this clogging transition has been identified even for inclined silos and orifices and different outlet shapes , a physical explanation is lacking to support it. Recent experiments of the discharge of a silo in the presence of an obstacle above the outlet shed light on the apparently crucial role that a variable such as pressure (or any kind of confinement) has on the clogging process . In addition the presence of the obstacle was shown to slightly increasing the flow rate, a behavior which was analyzed in detail in . Concerning the flow rate, it is remarkable that since 1961, it exists a widely known expression  where the flow rate scales with the square root of gravity as recently observed experimentally in . Nevertheless, the physical meaning of other parameters included in such expression is not fully understood .
Despite its undeniable interest, clogging in suspensions of particles has been less profusely studied than using inert particles or in crowds. There has been a long standing interest in the understanding of the behavior of dense suspensions and the relationship between glasses and gels. In particular, the jamming transition has attracted attention as a generic way to understand the behavior of dense suspensions and its connections with other types of phase transitions. When driven out of equilibrium, dense suspensions can also jam. Geometric constraints and defects are known to play an important role in promoting jamming. However, the connection of pinning in driven suspensions, the underlying jamming transition and clogging has not been explored systematically. The work of Liu  proposed a phase diagram to unify the behavior of jamming in colloids and granular materials, but the analogy in the behavior of these systems when forced out of equilibrium and in confined geometries has not been yet properly analyzed.
A system narrowly related with suspensions is the one composed of self-propelled particles like microswimmers or bacteria. At this respect, the peculiar behavior of bacteria when passing through a bottleneck has been recently described . Even though the density of bacteria was far away of the one at which clogging may be expected, the already counter-intuitive behavior observed at low densities supports the idea that improving our knowledge on this phenomena is necessary.
Unfortunately, only a few of the above mentioned works have attempted to draw a relationship among the behavior observed in different systems. In  the exponential
decay of the avalanche sizes observed in silos was generalized to other systems of groups of particles escaping from a bottleneck. The origin of this trend was proposed to be result of a random alternation between particle and gap propagations. In [9,22] the placement of an obstacle in silos and its effect on both, clogging and flow rate, was investigated in an attempt to emulate egress from a room. Nevertheless, in a static silo like the one used in these works, the flow rate properties are completely different to the ones observed in pedestrians: if the flow is arrested the clogging structure remains forever stable. On the contrary, pedestrians or other animals have their own internal energy capable of destroying the blockages and resume the flow. Hence, the use of a vibrated silo  –where arches can be also broken- seems a more suitable strategy to look for analogies with the pedestrians case.
Summarizing, in the last decade, several works have demonstrated the striking properties of the clogging phenomenon. Even though these works have focused in particular scenarios, it seems rather obvious that a general framework could be achievable. In particular, a good understanding of the variables that prevent or promote clogging in the different systems may lead to place the foundations that would allow building up a phase diagram like the one that Liu et al.  proposed for jamming after the work of Cates et al. .
 M. D. Haw. Jamming, Two-fluid behavior, and self-filtration in concentrated particulate suspensions. Phys. Rev. Lett. 92, 185506 (2004).
 D. Genovese & J. Sprakel. Crystallization and intermittent dynamics in constricted microfluidic flows of dense suspensions. Soft Matter 7, 3889-3896 (2011).
 P. Knowles, G. Dotro, J. Nivala, & J. García. Clogging in subsurface-flow treatment wetlands: Occurrence and contributing factors. Ecological Engineering 37 99-112 (2011).
 D. Helbing, I. Farkas, & T. Vicsek. Simulating dynamic features of escape panic. Nature 407, 487-490 (2000).
 A.J. Liu, & S.R. Nagel. Jamming is not just cool anymore. Nature 396, 21-22 (1998).
 P. G. Lafond, M. W. Gilmer, C.A. Koh, E.D. Sloan, D.T. Wu, & A.K. Sum. Orifice jamming of fluid-driven granular flow. Phys. Rev. E, 87, 042204 (2013).
 N. Roussel, T.L.H. Nguyen, & P. Coussot. General probabilistic approach to the filtration process. Phys. Rev. Lett. 98, 114502 (2007).
 D. Helbing, I.J. Farkas, P. Molnar, & T. Vicsek. Simulation of pedestrian crowds in normal and evacuation situations. Pedestrian and evacuation dynamics 21, 21-58 (2002).
 I. Zuriguel, A. Janda, A. Garcimartín, C. Lozano ,R. Arévalo, & D. Maza. Silo clogging reduction by the presence of an obstacle. Phys. Rev. Lett. 107, 278001 (2011).
 D. Helbing, L. Buzna, A. Johansson, & T. Werneret. Self-organized pedestrian crowd dynamics: Experiments, simulations, and design solutions. Transportation Science 39, 1 (2005).
 N. Shiwakoti, M. Sarvi, G. Rose, & M. Burd. Biologically Inspired Modelling Approach for Collective Pedestrian Dynamics Under Emergency Conditions. Transportation Research Record 2196, 176-184 (2010). D.R. Parisi, & C.O. Dorso. Microscopic dynamics of pedestrian evacuation. Physica A 354, 606 (2005).
 K. To, P.Y. Lai, & H.K. Pak. Jamming of granular flow in a two-dimensional hopper. Phys. Rev. Lett. 86, 71-74 (2001).
 C. Saloma, G.J. Perez, G. Tapang, M. Lim, & C. P. Saloma. Self-organized queuing and scale-free behavior in real escape panic. Proceedings of the National Academy of Sciences 100, 11947-11952 (2003).
 S.A. Soria, R. Josens, & D.R. Parisi, Experimental evidence of the Faster is Slower effect in the evacuation of ants. Safety Science 50, 15841588 (2012).
 N. Shiwakoti, & M. Sarvi. Understanding pedestrian crowd panic: a review on model organisms approach. Journal of Transport Geography 26, 1217 (2013).
 E. Altshuler, O. Ramos, Y. Nuñez, J. Fernández, A. J. Batista-Leyva, & C. Noda. Symmetry Breaking in Escaping Ants. The American Naturalist 166, 643-649 (2005).
 A. John, A. Schadschneider, D. Chowdhury, & K. Nishinari. Trafficlike collective movement of ants on trails: absence of jammed phase. Phys. Rev. Lett. 102, 108001 (2009).
 I. Zuriguel, L.A. Pugnaloni, A. Garcimartin, & D. Maza. Jamming during the discharge of grains from a silo described as a percolating transition. Phys. Rev. E 68, 030301 (2003).
 I. Zuriguel, A. Garcimartín, D. Maza, L. A. Pugnaloni, & J. M. Pastor. Jamming during the discharge of granular matter from a silo. Phys. Rev. E 71, 051303 (2005).
 C.C. Thomas, & D.J. Durian. Geometry dependence of the clogging transition in tilted hoppers. Phys. Rev. E 87, 052201 (2013).
 F. Alonso-Marroquin, S.I. Azeezullah, S.A. Galindo-Torres, & L.M. Olsen-Kettle.
Bottlenecks in granular ﬂow: When does an obstacle increase the ﬂow rate in an hourglass?. Phys. Rev. E 85, 020301(R) (2012).
 W. A. Beverloo, H. A. Leniger, & J. van de Velde. The flow of granular solids through orifices. Chem. Eng. Sci. 15, 260 (1961).
 S. Dorbolo, L. Maquet, M. Brandenbourger, F. Ludewig, G. Lumay, H. Caps, N. Vandewalle, S. Rondia, M. Mélard, J. van Loon, A. Dowson, & S. Vincent-Bonnieu. Influence of the gravity on the discharge of a silo. Granular Matter 15, 263-273 (2013).
 A. Janda, I. Zuriguel & D. Maza. Flow Rate of Particles through Apertures Obtained from Self-Similar Density and Velocity Profiles. Phys. Rev. Lett. 108, 248001 (2012).
 E. Altshuler, G. Miño, C. Pérez-Penichet, L. del Río, A. Lindner, A. Rousselet & E. Clément. Flow-controlled densification and anomalous dispersion of E. coli through a constriction. Soft Matter 9, 1864 (2013).
 D. Helbing, A. Johansson, J. Mathiesen, M.H. Jensen, & A. Hansen. Analytical approach to continuous and intermittent bottleneck flows. Phys. Rev. Lett. 97, 168001 (2006).
 A. Janda, D. Maza, A. Garcimartín, E. Kolb, J. Lanuza, & E. Clément. Unjamming a granular hopper by vibration. EPL 87, 24002 (2009).
 M.E. Cates, J.P. Wittmer, J.P. Bouchaud, & P. Claudin, Jamming, force chains, and fragile matter. Phys. Rev. Lett. 81, 1841-1844 (1998).