PHYSICS OF FLUIDS
VOLUME 10, NUMBER 12
DECEMBER 1998
Silo hiccups: Dynamic effects of dilatancy in granular flow
Thierry Le Pennec
Department of Physics, University of Oslo, P.O. Box 1048, Blindern, 0316 Oslo 3, Norway
and Groupe Matiére Condensée et Matériaux, Université de Rennes 1, F-35042 Rennes Cedex, France
Knut Jo” rgen Målo” y and Eirik G. Flekko” y
Department of Physics, University of Oslo, P.O. Box 1048, Blindern, 0316 Oslo 3, Norway
Jean Claude Messager and Madani Ammi
Groupe Matière Condensée et Matériaux, Université de Rennes 1, F-35042 Rennes Cedex, France
~Received 4 December 1997; accepted 19 August 1998!
The granular flow through an open silo is investigated experimentally. A mechanism based both on
the dilation of the granular medium and an interaction with the interstitial gas causes the flow to stop
at regular intervals. The experiments are carried out at different surrounding pressures P 0 , and it is
found that the intermittent flow becomes continuous at sufficiently low P 0 , showing that the
intermittency is linked to the interaction the gas. The scaling of the average flow rate with particle
size further supports our view of the gas-grain interaction. © 1998 American Institute of Physics.
@S1070-6631~98!00912-X#
allow the grains to pass by each other.12 This expansion, or
dilation, will create more void space between the grains, thus
leaving more room for the interstitial fluid. Depending on the
time scale of the process, this in turn will lead to a local
lowering of the pressure, and when there is a granular flow,
this pressure drop will couple back on the motion of the
grains. This effect is expected to be general.
The main mechanism, as it is realized in the present
experiment, is illustrated in Fig. 1. The sand, which starts out
with approximately constant density, moves down as a block
through the upper pipe until it reaches the constriction or
shear zone. Here the sand must dilate in order to accommodate the shear which is caused by the deformation. The dilation ~expansion! will create a local pressure drop in the shear
zone which causes both a drag force on the moving sand and
an air flow into hopper. In the present experiment we propose that the pressure drop ~together with wall friction,
which is always present! entirely stops the flow at a regular
time interval T, as long as the ambient pressure is above a
certain threshold value. The flow intermittency is caused by
a periodic sucking in of air, hence the term ‘‘silo hiccups.’’
I. INTRODUCTION
Granular materials exhibit several flow properties which
are qualitatively different from what is observed in simple
fluids. These differences are linked to the fact that moving
granular materials may continuously change their behavior
from that of solids to that of liquids or highly compressible
gases, thus making general descriptions in terms of continuum equations notoriously difficult. Recent research has
focused on dynamic studies of granular materials in confined
geometries, a field of broad industrial importance.1 Examples
of the complex and intriguing phenomena arising from the
granular nature of these systems2 include shocklike density
waves in hoppers3 and tubes,4,5 1/f noise,6 and pattern formation in the density profiles of sand flowing out of a
hopper.3,7–9 A fundamental understanding of the detailed
mechanisms governing these granular flows is still mostly
lacking.
In recent experiments periodic intermittency in the mass
flow rate have been observed in closed ‘‘ticking’’ hour
glasses10,11 where gas-grain interactions are a dominant
mechanism. Gas-grain interactions are generally important
when grains are small or the interstitial fluid sufficiently viscous. In the present paper we investigate the intermittent
flow of grains from an open model silo. In contrast to the
‘‘ticking’’ hour glass, which is governed by the ~nonlocal!
increase of a pressure difference between two closed chambers, the key mechanism behind the intermittency in the
present experiments is local and linked to the dynamic dilation of the granular medium. As such, it is another uniquely
granular phenomenon which has no analogue in simple fluids.
The term dilatancy was first introduced by Reynolds12 in
1885. It is the effect seen when one steps on a wet beach and
the sand appears to dry around the foot. In general when a
granular medium is subject to a local shear it must expand to
1070-6631/98/10(12)/3072/8/$15.00
II. EXPERIMENTAL SETUP
The experiments are illustrated in Figs. 2 and 3. The silo
consists of a cylindrical upper part, open to the surrounding
pressure at the top, with a conical edge with an orifice at the
bottom.
In contrast to experiments in straight tubes,4 the present
phenomenon is easily observed as the constriction in the silo
geometry ~the hopper! serves to fix its location and scale.
This allows for precisely controlled visual observations and
pressure measurements.
The silo was filled with spherical glass beads with diameters ranging from 35 mm to 1 mm. To obtain the same
porosity f 0 50.3860.01 of the different powders, the silo
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© 1998 American Institute of Physics
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Le Pennec et al.
Phys. Fluids, Vol. 10, No. 12, December 1998
3073
FIG. 1. A sketch illustrating the proposed effect. The sand moves down as
a block through the upper pipe until it reaches the shear zone. Here a local
pressure drop caused by the dilation of the sand causes air to be sucked into
the shear zone from the surroundings with an upward velocity relative to the
moving grains.
was tapped carefully on the sidewalls. The height of the silo
is 600 mm, with an internal diameter of D t 516 mm, and 30
mm. Experiments were performed with silos of different
conical angle 2 a 5180°, 19°, 14°, and 10°. The silos were
made of brass, Plexiglas, and glass with a diameter of orifice
D ranging from 2 to 15 mm. The silos made of brass have
the advantage of preventing electrostatic charges. The importance of electrostatic interactions was further investigated by
performing experiments with silver coated particles. No
qualitative difference was observed in these experiments.
The experiments were all done under dry conditions with a
relative humidity within the range 20%–40%.
In order to visualize the detailed movement of the sand
in the vicinity of the orifice, a transparent glass silo with
FIG. 3. Sequence of images which show the granular flow at the orifice
within one oscillation. Images 1, 2, and 3 show an interface ~separation
between the dense and dilute zone! which moves up. Images 4, 5, 6, and 7
show the the collapse of this interface followed by a strong increase in the
flow rate. Images 8 and 9 show the flow just before it stops.
conical angle 2a of 10° was used together with a video camera.
III. INTERMITTENCY CYCLES
FIG. 2. Drawing of the hopper with the laser and the photodetectors ~PD!. D
is the orifice diameter, D t the diameter of the cylindrical part, h the height,
and 2a is the opening angle of the conical part.
We now describe one of the flow intermittency cycles
which is shown in Fig. 3. The particles were observed to fall
from the closely packed phase at the lower interface seen in
Fig. 3. In image 1, when the interface is localized at the
orifice, the absence of powder just below suggests that the
flow is completely stopped for a short time. This is also
easily seen in a direct video visualization, where a somewhat
smeared video image of the grains becomes sharply con-
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3074
Le Pennec et al.
Phys. Fluids, Vol. 10, No. 12, December 1998
FIG. 4. The intensity at the top ~a! and bottom ~b! photodetectors as a
function of time. The integers correspond to the images in Fig. 3.
trasted for about 0.05 s. Just after this stop the front between
the falling particles and the closed packed region propagates
upward in the conical part ~images 2 and 3! until it collapses
~image 4!. In general, the speed with which the front moves
upward will depend on the force networks in the packing, the
weights of the grains, and the hydrodynamic drag force acting on the particles. After the interface reaches the critical
height were it collapse and suddenly falls down ~images 4
and 5!, a significant increase in the mass flow rate ~images 5,
6, and 7! is observed. The strong increase in the mass flow
rate is followed by a decrease in the width of the powder
beam which finally snaps off ~images 8 and 9!.
The correlations between the movement of the sand in
the vicinity of the orifice and at the top surface was studied
using two 5-mW He–Ne lasers with expanded beams as
shown in Fig. 2. The upper laser beam was partly screened
by the upper granular surface, and the transmitted intensity,
measured by photo-diode PD1 @see Fig. 4~a!#, is thus linearly
related to the height of the interface of the sand. At the
conical opening we measured the transmitted intensity of an
expanded laser beam passing through the conical part of the
silo @Fig. 4~b!#. Due to a more efficient screening of the light
by the dense packing than from the free falling particles, the
recorded intensity decreases when the interface moves down.
The increase in the slope of curve ~a! starts at the same
time as a fast decrease in curve ~b! in Fig. 4. This corresponds to image 4 in Fig. 3 and shows that the upper and
lower interfaces start to move down simultaneously. The upper interface will continue to move quickly, with a corresponding outflow, for a short time after the lower interface
has reached the height of the orifice ~images 6–8!. In this
period the lower signal does not directly reflect the flow velocity, but rather the filling of the constriction. Just before
the lower interface, which is observed in images 1–4 in Fig.
3, starts to move upward there is a short stop in the bulk
movement as described above. This is too short to be observed in these measurements, but can be seen in the video
visualization experiments. In the visualization experiments
no bubble formation10 was observed in or above the conical
part of the silo. After the stop the lower interface starts to
move upward in the conical part until the interface again
reaches a height where it collapses. In this part the upper
interface is moving only slowly, as seen in the low slope of
FIG. 5. The average flow rate ^W& ~s! and the period T ~d! of the intermittency as a function of the ambient pressure P 0 in units of the atmospheric
pressure P a .
curve ~a! in Fig. 4. The fluctuations in the high levels of
curve ~b! in Fig. 4 reflect the fluctuations in the particle flow
from the interface.
IV. QUANTIFICATION AND DISCUSSION
As will be seen below the intermittency is governed by
internal pressure gradients which can only arise from variations in the density of the granular packing. In order to argue
that the intermittency is really due to dynamic dilation, we
need to rule out other mechanisms of interior expansion in
the granular packing. The simultaneous motion of the upper
and lower interfaces as deduced from Fig. 4, and the lack of
observed bubbles, support the picture that the grain packing
in the tube moves without expansion. Furthermore, the tube
walls will exert a stronger vertical force per unit area on the
granular packing in the constriction than above. Hence, there
is no mechanical reason for the packing to open up or form
bubbles as it moves down into the constriction. The final
dilation due to shearing in the constriction cannot be
avoided.
A. The pressure evolution and the critical pressure
To investigate the importance of the ambient pressure on
the flow, we performed experiments in a chamber with a
reduced ambient pressure P 0 . In these experiments we used
d550 m m particles, and 2 a 5180°. As seen in Fig. 5, both
the flow rate ^W&, averaged over the entire duration of the
flow, and the period T between stops in the granular flow
were roughly constant for pressures higher than 0.2 bar.
When the pressure becomes lower than a critical pressure
P c 50.1 bar, a transition from the intermittent to a continuous flow regime was observed. Correspondingly the period T
diverges and the flow rate ^W& increases by nearly a factor of
2. In Fig. 5 both the remarkable constancy of the period and
the flow rate over different pressures, and the pronounced
transition from intermittent to continuous flow are striking,
and in need of an explanation.
To investigate the gas-grain interactions in more detail,
the average pressure difference d P between the local pres-
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Le Pennec et al.
Phys. Fluids, Vol. 10, No. 12, December 1998
sure in the constriction and the ambient pressure was measured. The measurements were carried out using a pressure
sensor connected to a small hole at a height D55 mm above
the orifice of a hopper with opening angle 2 a 510°. The
surrounding pressure was 1 bar. For small particles, as used
here, the pressure drop building up between stops in the flow
was measured as d P'0.001 bar' r gD, where r is the mass
density of the granular packing and g is the acceleration of
gravity. Hence, in the intermittent regime the pressure drop
d P was observed to balance the weight of the particles occupying a region of linear dimension D above the orifice. In
fact, for the observed halt of granular motion to take place,
the pressure drop must be able to support the weight of the
grains in the orifice for a short time. This can be used to
estimate the critical pressure P c . The dilatancy, measured as
the specific expansion d V/V, will cause a pressure drop that
depends only on the compressibility of the gas. Lowering
P 0 , the gas will eventually be so compressible that the pressure force resulting from the expansion of the gas inside the
orifice is unable to balance the weight of the falling grains.
Using the ideal gas law for isothermal gas expansion to get
the pressure drop we can write 2 d P/ P 0 5 d V/V @an adiabatic expansion, would only lead to the replacement d V/V
→(5/3) d V/V#. To obtain the critical value P 0 5 P c we set
d P5 r gD in the above equation. This gives directly P c
5 r gD/( d V/V). To get an estimate of d V/V, as it results
from the reconfigurations of the grains, we carried out an
independent measurement. A funnel with the narrow pipe
pointing up, closed on the wide bottom end with an elastic
rubber membrane, was filled with grains. The funnel was
then filled with water. Pushing the membrane from the bottom a dilatancy d V/V51% – 2% was obtained from the observation of the sinking water in the top pipe. Using this
result in the above equation we get directly that P c
'(100– 200) r gD'(0.1– 0.2) bar. This estimate is perhaps
in better agreement with the observed value of P c ~Fig. 5!
than could be expected. Note that the choice of the size of
the volume, defined by D, over which the pressure falls is
rather arbitrary. The above estimate of P c only considers the
pressure force needed to balance the weight of the grains at
the orifice, and not the forces needed to overcome the inertia
of the grains.
To understand the process that stops the granular outflow, one must consider the granular inertia and the time
over which the pressure forces act. Moreover, as the granular
medium is continuously deformed as it passes through the
orifice, the dilatancy must also be considered as a dynamic
process, as in the study of Lee et al.8 For this purpose we
need a pressure evolution equation.
Assuming that the gas is isothermal one may derive the
following equation
f
S
D S
D
]P
k
1u•¹ P 5¹• f P ¹ P 2 P¹•u.
]t
m
~1!
Here f is the local porosity, u the grain velocity, k the ~density dependent! permeability, and m the viscosity of the air.
The above equation is derived from the mass conservation
equations of the gas and the grains and a local Darcy law in
3075
the Appendix. The two terms on the left hand side taken
together are simply the material derivative of the pressure
~multiplied by f!. They describe the time rate of change of
the pressure in a local reference frame that moves with the
sand. The first term on the right hand side is a diffusive term
describing the Darcy flow. The second term on the right hand
side is a source term that describes the pressure increase due
to compression of the void space in the granular packing.
The permeability k may be related to f by the so-called
Carman Cozeny expression,13 which has the following form:
k ~ r̂ s ! 5
f3
a2
.
45 ~ 12 f ! 2
~2!
This is the well-studied relation, which is known to hold
relatively well for random packings of spheres, as long as the
packing density is not too small. The effective diffusion coefficient
D̂[
P k~ f ! P 0k~ f 0 !
'
,
m
m
~3!
where f 0 '0.4 is the closed pack porosity, may thus be
evaluated. Using the particle radius a525 m m and taking P
to be the atmospheric pressure and m the viscosity of air, we
find D̂'100 cm2/s. The different terms in Eq. ~1! may thus
be evaluated when ¹ is replaced by an inverse characteristic
distance, which we take to be D. This gives
S
¹• f 0
D
k0
D̂
¹ P ' 2 P'1000 s21 P,
m
D
u
P¹•u'u•¹ P' P'10 s21 P,
D
~4!
where we used D50.3 cm and we have evaluated u from the
measured values of the average flow rate W. The diffusive
term is then seen to dominate the terms containing u in Eq.
~1!. By dropping these terms we are left with a simple diffusion equation describing the evolution of P. The characteristic diffusion time over a distance D, t D 5D 2 /D̂'0.5 ms is
almost two orders of magnitude smaller than the time a grain
spends in passing a distance D at the orifice. If one knew the
instantaneous pressure drop D P inside the dilating regions, it
would then be possible to check if the pressure forces alone
were sufficient to stop the flow. This could be achieved by
comparing the necessary momentum change r D 3 u with the
impulse D PD 2 t D at the orifice. Taking D P as the measured
value d P described above ~which is smaller than the real
value since the measured value results from some diffusive
smearing!, the impulse is too small to balance the momentum change by roughly a factor of 10. However, it is likely
that a series of dilatancy events is responsible for the stopping of the flow. This is consistent with the picture of a
series of discrete shear bands reported by Lee et al.8 In a
radiographic study of hopper outflow ~with weak gas interaction! they observed that during the flow the dilating regions form in series of narrow bands. These bands, which
form in the constriction, have an angle of 45 degrees to the
flow direction. The fact that granular packings dilate and
form shear bands when flowing out of a hopper has been
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3076
Le Pennec et al.
Phys. Fluids, Vol. 10, No. 12, December 1998
FIG. 6. A qualitative picture showing how the various forces acting on the
sand in the constriction are correlated with the velocity u there. Here f is the
friction force from the walls, r g is the gravity force per unit volume and D P
the pressure difference between the inside and the outside of the constriction. The integers correspond to the images in Fig. 3.
observed by several authors.3,7–9 While these observations
are all, in a way, concerned with dynamic effects of dilatancy, they did not include the coupling of hydrodynamics
and granular dynamics studied here.
While Eq. ~1! does not give a definite magnitude of the
pressure impulse, it does indicate an explanation for the pressure independence of T and W in the intermittent regime.
Assume that the stopping process is governed by a series of
dilation events. Each such event will have a certain pressure
drop D P depending on the local expansion of the packing,
and D P will relax diffusively over a time t D . The impulse is
proportional to D Pt D . The relaxation time t D }1/D̂}1/P 0 .
The pressure drop D P, on the other hand, will be proportional to the relative granular density change in the dilatancy
process, i.e., D P} P 0 D r / r if the process happens on a faster
time scale than t D . In this case the impulse and the momentum change associated with each dilatancy event will be P 0
independent as the P 0 dependence in t D cancels the P 0 dependence in D P.
When t D becomes too large, so that the impulse D Pt D is
dominated by the impulse of gravity during the period T, the
flow becomes continuous. This will eventually happen as P 0
is decreased. In principle this could be used to compute P c .
However, t D goes as the square of the distance between the
opening and the dilatancy event. This distance is not well
known, and such a computation would therefore be highly
unreliable.
Figure 6 summarizes our picture of the forces acting on
the granular material in the constriction and their correlation
with the local average of the velocity u in the constriction.
The forces are taken as forces per unit area, and they are the
gravity ( r gD), the friction f from the walls, and finally the
pressure drop D P. The latter is the most important force for
the stopping of the flow. Due to the finite diffusive relaxation
time t D , the pressure force will remain finite even as the
velocity goes to zero. In the figure the dashed line shows
how further relaxation of the pressure may have proceeded if
the granular motion had not started again. The friction force
f will, in general, fluctuate strongly and also depend on the
density. It will, in general, have a larger static than dynamic
value, and when the flow is at rest, arching effects between
the grains will transfer friction forces over many grain diameters. These effects are illustrated by the steps in the friction
force associated with vanishing u. However, this sudden increase cannot explain the stopping process of the flow since
it does not set in before after the flow has stopped. Moreover,
we have no reason to expect that r gD or f depends on the
background pressure P 0 . The observation that the flow becomes continuous and never stops when P 0 is below its critical value thus indicates that the pressure force is crucial in
stopping the flow.
To test the dependence of the period T on the initial
compactification, experiments were performed for two different porosities f 50.38 and f 50.44 ~using d551 m m
particles!. A shorter period was observed for the more compactified than the less compactified powder. For the more
compact powder T50.17 s while T50.24 s for the the less
compact powder. This is in qualitative agreement with what
we expect, since the compactified powder will have to expand more by dilatation than the less compact powder, thus
creating a larger pressure drop. This means that fewer dilatency events should be required to stop the flow. However,
the change in T is likely to also reflect changes in intergranular friction, grain-wall friction and changes in the location
and magnitude of the pressure-dilatency coupling. Hence,
while there is a basis for predicting the sign of the effect, a
good quantitative theory appears out of reach.
B. Possible effects of molecular discreteness
An isothermal decrease in pressure could affect both the
compressibility and the mobility k/m of the gas. The intergrain spacing is only about 20 times larger than the mean
free path at the lowest pressures, and the hydrodynamic description will receive significant corrections due to the molecular nature of the gas.14,15 Klinkenberg modeled the correction due to the wall-slip gas flow in an idealized porous
medium consisting of capillary tubes with random orientation, and found that the velocity wall-slip results in a correction term in the permeability14,15
k 8 5 k ~ 118cl/b ! .
~5!
Here k is the permeability measured by liquid flow, where
finite mean free path effects are supposed to be negligible, l
is the mean free path of the gas, b is the pore size, and c is a
constant, which is experimentally determined to be close to
unity. As an estimate we will use b5d/3, where d is the
particle size. The mean free path of the gas molecules is
estimated to 0.08 mm at P 0 51 bar. This gives the correction
term k 8 / k 51.04 at P 0 51 bar, and k 8 / k 51.4 at P 0
50.1 bar. It is possible that the crossover behavior of T near
P c is governed by the effect that the impulse D Pt D decreases
with increasing k 8 .
The critical pressure P c is in the pressure range where
the Klinkenberg effect Eq. ~5! is expected to be important.
However, while t D should depend on the Klinkenberg effect,
the pressure force resulting from a given dilatancy will not.
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Phys. Fluids, Vol. 10, No. 12, December 1998
FIG. 7. The flow rate W5W a as a function of the particle diameter d on a
log–log scale for the hopper angles 2 a 510°(s), 14°~n!, 19°~1!,
180°~h!. The insert shows the mass flow rate W as a function of the column
height h for particle diameter d589 m m. Curve ~b! corresponds to a slight
compactification of the packing relative to curve ~a!.
This means that while molecular discreteness may affect the
value of P c , it has no consequence for our interpretation that
dilatancy created pressures stop the flow.
C. Dependence on particle size and hopper opening
diameter
To investigate the particle size dependence of the flow,
we used powders consisting of spherical glass beads with
diameters ranging from 35 mm to 1 mm. To obtain the same
porosity f 0 50.3860.01 of the different powders, the silo
was tapped carefully on the side-walls. Figure 7 shows the
asymptotic mass flow rate W a normalized with r as a function of particle size d. The insert in Fig. 7 shows how the
average flow rate depends on the height of the granular column. The asymptotic flow rate W a is the time averaged flow
rate corresponding to the h→` limit.
The two curves in the insert correspond to slightly different initial states of the packing. For the upper curve the
system was more loosely packed with a porosity f which is
8% larger than f 0 . As seen in the insert, the effect of this
initial difference in compactification lasted almost throughout the experiments. This again indicates that the grains
move coherently as a solid block relative to the smooth silo
walls. To confirm this point, experiments with porosity f 0
were also performed with smaller initial filling heights. No
changes in the subsequent mass flow rate W(h) were observed relative to the larger filling heights. This shows that
the initial porosity is conserved in the packing as it moves
down the main part of the tube.
For flows of large particles that are not governed by
interaction with the gas, the flow rate W is independent of the
column height h until the upper interface reaches the conical
part of the silo.16 In the present case, however, the small
particle flow rate is observed to vary with h at much larger h
values. This effect is assumed to be hydrodynamic. It is
caused by the air coming in from the top, thus decreasing the
pressure difference at the opening when h becomes sufficiently small.
When the particle flow is controlled by inertia of the
particles rather than drag forces, the flow rate is
Le Pennec et al.
3077
d-independent.1,17 Figure 7, on the other hand, shows that in
the intermittent (d,100 m m) regime the flow is consistent
with W}d 2 , where d is the particle diameter.
The data collapse observed for W a in the intermittent
regime over different hopper angles is rather striking as it
indicates that the gas-grain interactions are unaffected by the
hopper angle a.
It is well known from hopper experiments that stagnant
zones will form near the outlet.3 In these zones which form
along the inclined walls, there is no particle motion. It has
been found that the distance d c from the opening to the stagnant zone scales as d c ; a 22.2. 3 The length of the inclined
section measured from the hopper opening to the bottom of
the vertical cylinder walls, l, goes as l;1/a for small a. At
a sufficient small angle d c will become larger than l, and the
stagnant zone will disappear. In the continuous flow regime,
it is seen from Fig. 7 that the volume flow rate becomes
constant for angles smaller than a 515°. This indicates pure
‘‘mass flow’’ with no stagnant zones. For larger angles the
volume flow rate was found to decrease with increasing a as
seen in Fig. 7. This decrease is probably due to the fact that
the stagnant zones form boundaries that are effectively
rough, whereas the walls are smooth. Hence a larger friction
is felt from the boundaries of the stagnant regions than from
the smooth walls. Our observations are consistent with the
prediction by Rose and Tanaka.18 In the oscillatory, gas
dominated regime the volume flow rate was found to be
quite insensitive to the hopper angle. This is somewhat surprising, since a should affect both friction forces and the
local volume over which the pressure may relax. However,
for nonsteady flows little is known about the stagnant zones,
and one must be careful in interpreting the nonsteady results
in terms of the time-independent picture.
For particles larger than 100 mm, a slight decrease in
particle flux with particle size is observed. This decrease can
be explained by an effective reduction in the orifice diameter
D→(D2bd), where b is a dimensionless constant between
1 and 3.1
The W}d 2 behavior in the intermittent regime can be
justified by the following argument. When the particles are
small the drag forces will control the granular outflow. If the
column is sufficiently high such that the main air flow takes
place through the orifice, the slip velocity v a 2 v p between
the speed of the air v a and the local speed of the particles v p
will be proportional to v p . This follows under the assumption that the dilatancy expansion rate is proportional to the
volume flux of grains. The particle velocity then follows
from the local Darcy law v p } v a 2 v p 5 k ( f ) r g/ m , where
k ( f ) is the permeability of the packing and m is the dynamic
viscosity of the air. Since in general we have k}d 2 , we get
the mass flow rate W} v p D 2 }d 2 D 2 , in agreement with the
measurements of Fig. 7.
The dependence of the mass flow rate on D 2 has been
checked in separate small particle experiments using different diameters of the orifice D, as seen in Fig. 8. As seen in
Fig. 8 the mass flow rate is consistent with D 2 dependence.
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3078
Le Pennec et al.
Phys. Fluids, Vol. 10, No. 12, December 1998
ments and valuable input during the writing of the manuscript. This work has been in part supported by the CNRS
and the NFR through a Programme International de Coopération Scientifique grant.
APPENDIX: DERIVATION OF THE PRESSURE
EQUATION
Here we derive Eq. ~1! from the conservation of granular
mass and gas mass, using the assumption of an isothermal
gas to relate the gas density and gas pressure.
The conservation of the granular mass density r s may be
written
]r̂ s
1¹• ~ r̂ s u! 50,
]t
FIG. 8. The dependence of W a / r on the diameter of the orifice.
V. CONCLUSION
In conclusion we summarize the essential argument of
the paper. By enclosing the hopper, sketched in Fig. 2, in an
airtight chamber where the pressure could be varied, we were
able to observe that the granular flow exhibits a crossover
from intermittent to continuous flow at a critical ambient
pressure P c . This made it clear that the intermittency is governed by internal pressure gradients. In the absence of an
externally imposed pressure differences ~the silo is open in
both ends!, these pressure gradients can only arise from
variations in the density of the granular packing. The question we addressed was if these density variations could be
caused by any other effects than the dilatancy we know must
take place in the constriction of the hopper. Spontaneous
formations of air bubbles would represent such an alternative
mechanism for density variations. However, our observations strongly support that prior to the stopping of the flow,
the granular packing moves as a single rigid object with the
exception of the deformation in the hopper region. Hence, no
alternative mechanisms appear active, and we are left with
the interpretation that it is the local pressure drop caused by
the dilatancy that stops the flow. This argument has been
substantiated by experimental measurements.
By comparing terms in the pressure evolution equation
~1! it was found that relaxation of the pressure gradients was
mainly diffusive. This was used to explain the observed pressure independence of the period T.
From a practical point of view, the understanding of the
flow of granular materials with interstitial gas is important
for designing durable and efficient silos. Using a silo geometry we have examined the rather complex granular dynamics in the outflow of a model silo. Of possible practical utility
is the data collapse of Fig. 7 which shows that the small
particle flow rate is rather independent of hopper angle.
Moreover, Fig. 5 demonstrates how the outflow of a hopper
may be increased by roughly a factor of 2 by lowering the
pressure.
ACKNOWLEDGMENTS
It is a pleasure to thank Bob Behringer, Daniel Bideau,
Alex Hansen, Steve Pride, and Erik Skjetne for helpful com-
~A1!
where u is the granular velocity and the density compacity
r̂ s 5 r s / r g has been normalized by the density r g of the material in the grains. Hence r̂ s 512 f where f is the porosity.
The conservation of the gas mass density r a may be written
S F
]r a
k ~ r̂ s !
1¹• r a u2
¹P
]t
m
GD
50.
~A2!
Here the air current has both an advective term caused by the
motion of the grains, and a diffusive term describing the
Darcy flow in the local rest frame of reference for the sand.
The permeability k is given as a function of f s in Eq. ~2!. By
using the relation r̂ s 512 f in Eq. ~A1! we get
2
]f
1¹• ~~ 12 f ! u! 50.
]t
~A3!
Using the isothermal equation of state for an ideal gas, r a
} f P, we can write Eq. ~A2! in the form
S F
]~ f P !
k
1¹• f P u2 ¹ P
]t
m
GD
50.
~A4!
Eliminating ] f / ] t between Eq. ~A4! and Eq. ~A3! gives
P¹•u2 P¹• ~ f u! 1 f
S
D
]P
k
1¹• ~ f Pu! 2¹• f P ¹ P 50.
]t
m
~A5!
The sum of the second and the fourth terms of Eq. ~A5!
equals f u•¹ P, which gives
f
S
D S
D
]P
k
1u•¹ P 5¹• f P ¹ P 2 P¹•u,
]t
m
~A6!
which is nothing but Eq. ~1!. This ends our derivation.
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