Effect of interstitial fluid on the fraction of flow microstates that precede clogging in granular hoppers

3.1 MB·

by  Juha Koivisto

Effect of interstitial fluid on the fraction of flow microstates
that precede clogging in granular hoppers
Juha Koivisto and Douglas J. Durian
Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6396, USA
(Dated: February 27, 2017)
We report on the nature of flow events for the gravity-driven discharge of glass beads through a
hole that is small enough that the hopper is susceptible to clogging. In particular, we measure the
average and standard deviation of the distribution of discharged masses as a function of both hole
andgrainsizes. Wedosoinair,whichisusual,butalsowiththesystementirelysubmergedunder
water. Thisdampsthegraindynamicsandcouldbeexpectedtodramaticallyaffectthedistribution
oftheflowevents,whicharedescribedinpriorworkasavalanche-like. Thoughtheflowisslowerand
7
theeventslastlonger,wefindthattheaveragedischargemassisonlyslightlyreducedforsubmerged
1
grains. Furthermore, we find that the shape of the distribution remains exponential, implying that
0
clogging is still a Poisson process even for immersed grains. Per Thomas and Durian, Phys. Rev.
2
Lett. 114, 178001 (2015), this allows interpretation of the average discharge mass in terms of the

b fractionofflowmicrostatesthatprecede,i.e.thateffectivelycause,astableclogtoform. Sincethis
e fraction is barely altered by water, we conclude that the crucial microscopic variables are the grain
F
positions; grain momenta play only a secondary role in destabilizing weak incipient arches. These
4 insightsshouldaidon-goingeffortstounderstandthesusceptibilityofgranularhopperstoclogging.
2
PACSnumbers: 47.57.Gc,47.56.+r,47.55.Kf

]
t
f
o INTRODUCTION cohesive compact grains, in air or vacuum, there is gen-
s eral agreement that clogging statistics are Poissonian
.
at The flow of grains in an hourglass or hopper is an [6, 7, 13, 26, 27]. Namely, there is an exponential dis-
m iconicgranularphenomenon,strikinglydifferentfromthe tribution of flow times, and hence also an exponential
gravity-driven flow of fluid from a small hole in the bot- distribution for the amount of material discharged be-
-
d tom of a bucket [1]. At the top free surface, the grains tween successive clogs. Thus there is a well-defined av-
n are not level but form a conical depression. Below the erage “avalanche” size, as measured either from the av-
o
hole,thestreamofgrainsfansoutanddoesnotbreakup erage flow duration τ or from the average mass m
c (cid:104) (cid:105) (cid:104) (cid:105)
[ intodropletsbecausethereisnosurfacetension. Andthe discharged before a clog occurs. These are related by
growing mass of grains collected underneath is not level, m = ρAv τ where ρ is the mass density of the pack-
2 but forms a conical pile down which the added grains (cid:104)ing(cid:105),visthe(cid:104)ex(cid:105)itspeedofgrainsatthehole,A=π(D/2)2
v
5 avalanche. Another striking difference is that the dis- is the hole area, and D is the hole diameter.
8 charge rate of grains is constant, as long-described by Acrucialopenquestioniswhetherornotasharpclog-
9 the empirical Beverloo equation [1, 2], and sometimes gingtransitionexists: IsthereacriticalholediameterDc
2 can even increase with time [3, 4]; by contrast, the fluid above which the system will never clog, or instead does
0
discharge rate always decreases with time as the bucket clogging become so improbable for larger holes as to be
.
1 empties and the gravitational pressure head goes down. essentially unobservable on human time scales? This is
0
Butanevenbiggerdifferenceisthatgrainscanclog[5–7]. difficult to answer definitively by experiment or simula-
7
Thoughtheflowmayappearsmooth,itcansuddenlyand tion because m grows very rapidly with hole size, e.g.
1 (cid:104) (cid:105)
: unexpectedlyhaltduetotheformationofamechanically byfiveorderofmagnitudeastheholediameterincreases
v
stable arch or dome of grains spanning the hole. How to by a factor of three. In particular, data can be equally-
i
X predict the susceptibility of a given granular system to welldescribedbybothexponential[6,28,29]anddiverg-
r clogging [8], and how to anticipate that a clog is about ing power-law [7, 13, 28, 29] forms:
a
to form [9], are active research topics.
Cloggingisagenericfeatureofgranularhopperflow,as m = mgexp[c(D/d)3+b] (1)
(cid:104) (cid:105)
reviewed in Refs. [10, 11], which happens less frequently m = m /[α(D D)/d]γ (2)
g c
(cid:104) (cid:105) −
for larger holes and is unavoidable for holes smaller than
about 4-5 grains across. Details depend on grain shape, wherem istheaveragegrainmass,distheaveragegrain
g
e.g. [12–15], and similar phenomena arise in other con- diameter, and c,b,α,γ as well as the putative critical
{ }
texts ranging from transport in electronic [16] and par- hole size D are fitting parameters. This is also true for
c
ticulate [17, 18] systems with spatially-distributed pin- two-dimensional hoppers, where the average mass grows
ning sites to grains in channels and pipes [19, 20], grains as either a critical power-law or as an exponential in
drivenbyfluidflow[21,22],andevengrainswithbrains: (D/d)2 [6, 28]. While the competing fits may be equally
pedestrians [23], traffic [24], and livestock [25]. For non- good, we prefer exponential form for several reasons. As
2
Flow Flow Flow Preceding Clog
…
𝜏𝑜 2𝜏𝑜 3𝜏𝑜 𝑁𝜏𝑜 time
FIG. 1: When time advances by one sampling time, τ , there is a new configuration of grains near the hole and a new chance
o
to clog. In this cartoon, N distinct flow states are sampled prior to the formation of a stable clog. The fraction of microstates
that similarly cause clogging is F =1/(cid:104)N(cid:105), where (cid:104)N(cid:105) equals the average flow duration divided by τ .
o
negative points against the power-law, fitting results for known. As a different approach, we measure and com-
theexponentγ areoddlylargeandpublishedresultsvary pare clogging behavior versus hole size for experimental
widely. There is no established theory for the expected systemsthatareidenticalbutforonemajordifference: In
value of γ, or for the putative critical hole size D . In one the grains are in air and have collisional/inertial dy-
c
principle, stable arches can be constructed of arbitrar- namics,andintheotherthegrainsaretotallysubmerged
ily large size. Furthermore, in spite of explicit searches, in water and have overdamped viscous dynamics as well
no critical signature such as a kink in either the average asreducedfriction. Onceaclogforms,thestabilitycrite-
discharge rate [13] or in grain velocity fluctuations [30] riaforthegrainsinthearch/domearethesame;however,
has been found that could be used to locate D as the the dynamics of arch formation must be very different.
c
holesizeisdecreasedtowardtheputativetransitionfrom As shown below, we find that the clogging statistics are
above. Lastly, the exponential form and its dependence not strongly affected. Therefore, we conclude that grain
on dimensionality both follow naturally from a simple positions are key to predicting clogging probabilities.
model based on consideration of clogging as a Poisson
process in which microstates are randomly sampled [29].
The first step in the model of Ref. [29], and the in- MATERIALS AND METHODS
spiration for the present paper, is the realization that
the fraction F of all accessible microstates that precede Theexperimentalgranularsystemconsistofthreesizes
a clog can be found from measurement of m . Since of technical quality glass beads (Potters Industries A-
(cid:104) (cid:105)
clogging is a Poisson process, the act of flow can be in- series)withmaterialdensityρ =2.54 0.01g/cm3. The
g
terpreted as bringing new configurations into the region grain diameter distributions are measu±red using a Retch
ofthehole, sothatwithtimedifferentconfigurationsare Technology Camsizer. Results are displayed in Fig. 2
sampled at random until one arises that causes a clog to along with the mean, d, and standard deviation, σ . As
d
form. We call (cid:96) the sampling length, which is how far shown, the grains have a 5 10% polydispersity, and
the grains near the hole must flow in order to produce will be referred to by their n−ominal diameter values of
a new configuration. It is of order one grain diameter, d=0.5,1.0and2.0mm. Twentytothirtypercentofthe
and the corresponding sampling time is τo = v(cid:96). This d = 1.0 mm beads have multiple sharp edges, by visual
is illustrated in Fig. 1. The average discharge mass may inspection. The other beads can be described as round.
then be rewritten as m =ρAv τ =ρA(cid:96) τ /τo. In this This does not seem to affect the clogging results found
(cid:104) (cid:105) (cid:104) (cid:105) (cid:104) (cid:105)
expression,werecognize τ /τo asthenumberofdistinct below,andneitherdoesthelargerrelativepolydispersity
(cid:104) (cid:105)
configurations sampled in the average flow event, and for the d = 1.0 mm grains. Nevertheless, the clogging
hence F = τo/ τ as the fraction of flow configurations analysis is highly sensitive to the size of the particles
(cid:104) (cid:105)
thatprecedeaclog. Thusthefractionofflowmicrostates relativetothehole,inlightofEqs.(1-2). Inbothdryand
that cause a clog is submerged cases the grain volume fraction is measured
to be φ=0.58 0.04, which is close to expectation [31].
F =ρA(cid:96)/ m , (3) ±
(cid:104) (cid:105) Therefore, the mass density of the packing is ρ=φρ =
g
andcanbededucedfrommeasurementof m ,somewhat 1.47 0.10 g/cm3. The draining angle of repose is about
miraculously, without need to measure th(cid:104)e a(cid:105)ctual grain 24◦ w±hen the grains are dry, and about 21◦ when they
positions, momenta, or contact forces. are fully immersed in water [3].
In this paper we now ask about the nature of the mi- Inthedryexperiments,airconditionsarecontrolledby
crostates that causes clogging. For s spatial dimensions, standard laboratory air handling with humidity ranging
what is it about the positions, momenta, and/or contact between 20 50 rH and temperature between 20 25 C.
− −
forcesofthe (D/d)s grainsintheholeregionthatleads In submerged experiments, the fluid is filtered tap water
O
toaclog? Inprinciplethiscouldbeaddressedbysimula- with standard textbook properties: density ρ = 1.00
f
±
tion, where these microscopic quantities are all perfectly 0.01g/cm3andviscosityη =1.00 0.01mPa. Withthese
±
3
system parameters, the Reynolds number based on bead
size and single-grain terminal falling speed in water are
Re = ρ v d/η = 40,150,500 for the three grain sizes.
f t
{ }
The Stokes number is St = (1/9)ρ v d/η, which here is
g t
aboutRe/3. Itrefersspecificallytograininertia[31],the
importance of which is overestimated by using v since
t
the discharge speed is smaller in water than in air [3, 4]
and also since the relative speed of neighboring grains
in the coarse-grained flow field is smaller still. Even so,
Ref.[31]showsthatsedimentinggrainsneedtohaveSt>
30 in order for their inertia to jar a loose packing into a
dense packing. Therefore, we can expect very different
dynamics for submerged versus dry grains.
FIG. 3: Schematic illustration of the clogging apparatus,
1.4 showninverticalcrosssection. Thehopper(gray)iscylindri-
d =0.498mm
1.2 σd =0.048mm d =1.975mm caadljuasntdabhleandgisamfreotmer,aDdi,gfiittaslibnatloanacede(pbrleascski)o.nAinntohreifibceotwtoitmh
1.0 σd =0.046mm plate. Thewallsandbottomplatearemadeofpolycarbonate,
) 0.8 respectively 6 mm and 13 mm thick. The flowing grains are
d d =1.001mm
p( 0.6 σd =0.090mm indicated with brownish shading. The dry case is identical,
exceptthatthefluid(lightblue)isabsent. Notethatthetop
0.4 of the hopper is open, so that there is no back-flow of air or
0.2 water into the hopper as the grains exit.
0
0.3 0.5 0.7 1.0 2.0 3.0
d [mm]
stepper motor are connected to the same computer that
is interfaced to the balance, so that the system is fully
FIG.2: Normalizeddistributionofparticlediametersforglass
automatedasinRef.[26]. Athirdmethodistomanually
beads,measuredbyaRetchTechnologyCamsizerandlabeled
by average and standard deviation. tap to the side of the hopper. This is useful for the d =
0.5 mm grains, where the orifice size can be less than
2 mm in diameter and hence difficult to poke. In all
The apparatus for clogging measurements is shown
cases, the procedure is to initiate flow, to measure the
schematically in Figure 3, and is similar to that in
totalmassofgrainsthatisdischargedbeforeaclogforms,
Ref. [4]. The hopper consists of a flat bottomed cylin-
and to repeat as desired for a large number of discharge
der with inner diameter D = 195 0.5 mm and height
h
± events. No difference in behavior was noticed for the
h = 250 1 mm. The top is open, and is completely
± three unclogging methods or the two hole types.
underwater for the submerged experiments. The bottom
hasadepressionthatisfittedwithanadjustableiristhat
serves as the hole through which the grains exit. The
hole diameter is measured with a caliper, and is circular DISCHARGE MASS STATISTICS
to within ∆D = 0.1 mm. This is the largest source of
uncertainty,especiallyforsmallorificediameters. There- The first question is whether or not clogging remains
fore we also perform additional runs where the iris is re- a Poisson process when the grains are submerged. In
placed by an Aluminum disk with a precision-machined particular, the distribution of discharge masses is expo-
hole, to rule out systematic errors. The entire hopper nentialinairandvacuum,butcouldverywellbedifferent
hangs from a digital balance (Ohaus Valor 7000) that under water. To investigate this, as well as the fraction
records the change in mass with 10 Hz frequency and F of flow microstates that cause a clog, we measure the
1.0 g repeatability. Alternatively, for avalanches smaller dischargemassesforalargenumberofflowevents,forall
than about 10 g, the grains are collected in a cup that is three grain sizes, and for hole sizes ranging from slightly
weighedwithamoreaccuratebalance(OhausNavigator, larger than one grain to as large as was reasonably feasi-
0.1 g repeatability). ble. Theupperlimitonholesizeissuchthattheaverage
When a clog forms, the flow is re-started using one of discharge mass is (1000 g), which is ten times smaller
O
the three methods. In the submerged case, a stream of than the capacity of the hopper. This allows sampling
water is directed underneath the hole in order to break of long-duration events without need for re-filling, which
the clog. Alternatively, the clog is broken by poking it is infeasible for the submerged cases. In dry cases, the
with a stick either manually or via stepper motor. This upper limit is also set by the accuracy with which we
isusedinallthedrycases,andinsomeofthesubmerged could change the hole size. Only a very slight increase
cases. For these two methods, the water pump and the in hole diameter is needed to increase the average flow
4
duration from several hours to several days or weeks, i.e.
from barely feasible to not possible. Altogether for dry
1
andsubmergedcases,andforthethreegrainsizes,weex-
amined46differentholesizesandthousandsofdischarge
events. The goal of the experimental procedure was to ) 0.1
i y
measure at least 30 discharge events for each combina- /mh =exp(
tion of D and d. The average number of events is 104 m −x)
for each measurement point. Half of the measurements C(0.01
−
have more than 52 events and the largest measurement submerged, D =4.2mm
1
consists of 1172 events. There are only three measure- submerged, all
0.001
ment points with less than 10 discharge events, all for dry, all
very long duration (nearly infeasible) runs at the largest
0 1 2 3 4 5 6 7
D/d ratio.
m/ m
h i
To reveal the nature of all the discharge distributions,
we compute both the average m and the standard de- FIG. 5: One minus the cumulative distribution function ver-
(cid:104) (cid:105)
viation σm of the discharged masses, for a given set of susscaleddischargeeventmassford=1mmdiametergrains.
conditions, and plot one versus the other in Fig. 4. The Solidcirclesrepresentthesubmergeddatasetwiththelargest
datapointsfallonthelineσ = m ,where m variesby number of events (1172); crosses and diamonds represent all
m
(cid:104) (cid:105) (cid:104) (cid:105) datasetsforsubmergedanddrycases,respectively. Thesolid
almostsixordersofmagnitudeastheholeandgrainsizes
linelabeledy=exp(−x)istheexpectationforanexponential
are changed. This is consistent with an exponential dis-
distributionofdischargemasses;itsgoodagreementwiththe
tribution. Inaddition,wealsocollectedgreaterstatistics
data demonstrates that clogging is a Poisson process.
foracoupleofspecificgrain/holesizecombinationsand
directlyconfirmedthatthedistributionsarenearlyexpo-
nential. InFig.!5,weplotthecumulativedistributionfor
varieswithholediameter. Toinvestigate, resultsfor m
theonerepresentativesetwithalargenumberdischarges
(cid:104) (cid:105)
versusDareplottedtwodifferentwaysinFig.6. Thetop
andthecombinedsubmergedanddrycasesford=1mm
plot is a log-linear version of the raw data. For the dry
grains. The combined sets are first normalized with av-
grains, it shows a very rapid increase that can be well-fit
erage m before combining. The distributions are linear
by both the exponential and diverging power-law forms,
insemilogscaleoverawiderange,indicatingthatindeed
Eqs. (1-2), as expected. Fitting parameters are collected
we have exponential behavior. Thus, we conclude that
in Table I. Also as expected, doubling the grain size re-
clogging is also a Poisson process for submerged grains.
quiresdoublingtheholesizetoachievethesameaverage
This is the first such demonstration, to our knowledge.
discharge mass. For submerged grains, our new result is
Importantly, it permits us to analyze m in terms of F,
(cid:104) (cid:105) thattheaveragedischargemassisslightlyreduced,com-
below.
pared to the dry case at a given hole size. Furthermore,
thefunctionalformappearstobeunaltered,inthatgood
1000 fits are also obtained using Eqs. (1-2). To highlight the
exponentialform,theaveragemassdataarescaledbythe
100
grain mass m and are shown as a log-linear plot versus
g
[g] 10 = y D3 in Fig. 6b. This causes the data to fall onto straight
m 1 x dry sub. d [mm] lines, which is the expectation for the Eq. (1) form that
σ
0.5 grows exponentially in D3. Note, too, that these fits ex-
0.1
1.0 trapolate close to m /m =1 as the hole size decreases
g
0.01 2.0 toward zero: For(cid:104)the(cid:105) smallest holes, only a few grains
0.01 0.1 1 10 100 1000 escape before a clog forms.
m [g]
(cid:104) (cid:105) Asfurtherremarksonfitting,firstnotethatthediverg-
ing power-law fits in Fig. 6 assume the exponent to be
FIG. 4: Standard deviation versus mean for the distribution
γ =5. This value is taken from Ref. [13], and is roughly
ofdischargemasses,forallmeasuredcombinationsofholeand
grain sizes, under both dry and submerged conditions. The in the middle of the range of values reported by others.
datafallontheliney=x,whichimpliesthatthedistributions Similarly good fits can be obtained by adjusting γ at a
are exponential and that clogging is a Poisson process. This fixed critical hole size of D =5d for dry grains and 10%
c
figurehastwolessdatapointsthanseeninlaterfigures,where larger for submerged grains. Either way, the uncertain-
(cid:104)m(cid:105) but not σ were measured by collecting multiple events
m tiesinfittingparametersarequitelarge(andlargerthan
into a cup and weighing.
for c,b in the exponential fits). Even better-looking
{ }
power-law fits can be obtained by adjusting all three pa-
The next question concerns how the average mass rameters, D , γ, α ; however, the parameter uncertain-
c
{ }
5
d =0.5mm 1.0mm 2.0mm
TABLE I: Parameters for the fits displayed in Fig. 6 to
1000
(a) Eqs.(1-2),obtainedusingtheFortranODRPACKalgorithm
g]10100 dry sub.dry sub. dry sub. [e3le2m].eEnrtrsoorfetshtiemcaotveasriaarnecgeivmeantrbiyx.thFeorsqEuqa.re(1r)o,otthoefedxipaognoennatl
[
1 is fixed to γ =5.
(cid:105)
m
(cid:104)0.1
d (mm) dry submerged
0.01
m =m exp[c(D/d)3+b] 0.5 0.13±0.03 0.09±0.03
0.001 (cid:104) (cid:105) g
m =m /[α(D D)]5 c 1.0 0.13±0.01 0.11±0.03
10−4 (cid:104) (cid:105) g c− 2.0 0.16±0.01 0.14±0.01
0 2 4 6 8
D [mm] 0.5 10±2 11±3
b 1.0 8.8±0.6 9±2
106
(b) 2.0 7.2±0.6 7.7±0.6
105
0.5 2±1 3±2
104
/mg(cid:105)103 Dc 21..00 4.89±±10.8 59.0±±00.6.9
m100 dry sub. d [mm]
(cid:104) 0.5 0.03±0.09 0.02±0.10
0.5
10 α 1.0 0.05±0.04 0.04±0.05
1.0
1 2.0 2.0 0.07±0.05 0.06±0.04
0.1
0 100 200 300 400 500 600 700
D3 [mm3]
figuration and hence a new chance to clog. We take it
FIG.6: Theaveragedischargemassversusholediameter(a),
andscaledbygrainmassandplottedversusthecubeofhole to be (cid:96) = (0.75 0.20)d, as measured by two meth-
±
diameter (b). In both plots, the solid curves represent fits to ods in Ref. [29]. The resulting behavior for F versus
Eq. (1) while the dashed curves represent fits to a diverging (D/d)3 is shown by log-linear plot in Fig. 7. Note that
powerlawEq.(2)withexponenttakentobeγ =5. Thefor- this causes the data to collapse onto two straight lines,
mer is exponential in D3 and hence comes out as a straight
one for dry grains and one for submerged. These decay
line in the bottom plot. Fitting parameters are given in Ta-
rapidly,sincethesusceptibilitytocloggingdecreasesdra-
ble I.
matically with increasing hole size. Both cases may be
fit to F = exp C[(D/d)3 1] , which has the correct
{− − }
ties are unacceptably large. Similarly, the exponent may formandisalsocorrectlynormalizedtoF =1atD =d.
be adjusted in the form m exp[c(D/d)s]; however, By adjusting only the decay rate constant, C, we obtain
the fitted values are close(cid:104)to(cid:105)3∝, which is expected based very good fits as shown. The fitting uncertainty is about
on the model of Ref. [29] where s naturally equals the 3%, and the decay constant for the dry grains is about
number of spatial dimensions. Overall, the relative qual- 20% larger than for the submerged grains.
ityofthetwofittingformsiscomparable,butthesmaller
uncertainties and the clear physical meaning of the ex- 23 33 43 53
ponent point in favor of the exponential form. In con-
dry sub.d [mm]
1
sequence, we reinforce the belief that there is no sharp exp 0.5
cloggingtransition,i.e.thatallgranularhoppersaresus- m(cid:105)0.1 {−(0.107 1.0
cperpotbiabbleilittoy)c.logging (though perhaps with unobservable ρ(cid:96)/FA=(cid:104)1110000.0−−−1345 exp{−(0.130±0.00±4)[0(D.0/0d3)3[)(D/d)3−1}] 2.0
ANALYSIS OF FLOW MICROSTATES −1]
10−6 }
0 20 40 60 80 100 120
(D/d)3
We now use Eq. (3) to analyze the average discharge
mass data in terms of the fraction F = ρA(cid:96)/ m of
(cid:104) (cid:105) FIG. 7: The fraction of flow configurations that cause a clog
flow microstates that precede, i.e. that cause, the for-
versus the cube of hole diameter divided by grain diameter.
mation of a stable clog. In this expression, all quanti-
Experimentalresultsareforthreegrainsizes,andunderboth
ties on the right-hand side are known from the measure- dry and submerged conditions. The solid lines represent fits
ments discussed above except for the sampling length, toF =exp{−C[(D/d)3−1]}, andthedashedlinesrepresent
(cid:96). This is the average downward displacement of grains the range of fitting functions given by the quoted value and
in the hole region that is needed to create a new con- uncertainty in the fitting parameter C.
6
Since the decay of F is faster for the dry grains, they by computer simulation, or perhaps by experiments in a
are slightly less susceptible to clogging. This is counter quasi-two dimensional geometry.
to our initial expectation, which was that lubrication This work was supported by the Finnish Foundation’s
forces between approaching grains and reduced friction Post Doc Pool, Wihuri Foundation and Finnish Cultural
between contacting grains (vis-`a-vis the smaller repose Foundation (JK) and by the NSF through Grant No.
angle)wouldbothrendersubmergedgrainslesssuscepti- DMR-1305199 (DJD).
ble to clogging. This points to grain inertia, which has a
destabilizing effect on arch formation and is much larger
forthedrygrains. Intuitively,toformastableclog,anin-
cipientarchmuchbestrongenoughtowithstandcollision
[1] R. M. Nedderman, U. Tuzun, S. B. Savage, and G. T.
fromthegrainscollidingwithitfromabove. Ofallpossi-
Houlsby, Chem. Eng. Sci. 37, 1597 (1982).
blearches,fewercanbestablyformedinairbecausethey
[2] W. A. Beverloo, H. A. Leniger, and J. van de Velde,
must be stronger. Conversely, a greater variety of arches
Chem. Eng. Sci. 15, 260 (1961).
can be stably formed under water since weaker ones are [3] T. J. Wilson, C. R. Pfeifer, N. Mesyngier, and D. J.
additionally allowed, rendering submerged grains more Durian, Papers in Physics 6, 060009 (2014).
susceptible to clogging. This ties in with the conclusion [4] J. Koivisto and D. J. Durian, arXiv:1602.05627 (2016).
of Ref. [31] that the Stokes number controls the volume [5] S.S.MannaandH.J.Herrmann,Eur.Phys.J.E1,341
(2000).
fraction of random loose packings, such that looser more
[6] K.To,P.Y.Lai,andH.K.Pak,Phys.Rev.Lett.86,71
delicate packings may be formed when grain inertia is
(2001).
absent. This also ties in with the intuition of Ref. [8]
[7] I. Zuriguel, L. A. Pugnaloni, A. Garcimart´ın, and
that incipient arches must be strong enough to dissipate D. Maza, Phys. Rev. E 68, 030301 (2003).
thekineticenergyofthegrainsrainingdownfromabove. [8] R. Arevalo and I. Zuriguel, Soft Matter 12, 123 (2016).
[9] S. Tewari, M. Dichter, and B. Chakraborty, Soft Matter
9, 5016 (2013).
[10] I. Zuriguel, Papers in Physics 6, 060014 (2014).
CONCLUSIONS
[11] I. Zuriguel, D. R. Parisi, R. C. Hidalgo, C. Lozano,
A. Janda, P. A. Gago, J. P. Peralta, L. M. Ferrer, L. A.
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