This document summarizes research on jamming in a 2D hopper. Experiments used photoelastic disks to visualize jamming and arch formation. Results showed: 1) The probability of continuous flow decayed exponentially with time, confirming the hypothesis. 2) The characteristic time τ increased linearly with outlet diameter, matching predictions from Beverloo's equation for mass flow. 3) Visualization of arch formation provided insight into how jamming relates to stress chains within the material.

1.
Jamming and Flow
in 2-D Hopper
Sepehr Sadighpour (Duke University)
Paul Mort (Proctor & Gamble)
R. P. Behringer (Duke University)
Supported by IFPRI

2. Motivation
 How do grain size and
outlet diameter
influence jamming
probability?
 What does the
probability distribution
of jams look like?
Grains exiting silo often jam. Above:
Typical method of un-jamming a hopper

3. Anatomy of a Jam
 Grains self-support on
container walls by forming
“stress chains”
 Pressure maxes out with depth
 Hence, lower particles don’t
feel all the weight
 This is how a small arch near
outlet can hold up an entire
silo 2-D jam of photoelastic
disks in our apparatus

4. D
Prelude to the Beverloo Equation
The Variables of Flow:
particlesabovebyeduninfluenc
fall,freeinareoutletfrom~Grains
gravitytodueonaccelerati
outlettheofdiameter
diameterparticleeffective
densityapparent
openinghethrough t
dt
dm.
D
:AssumptionBasic
g
D
d
M
e
=
=
=
=
=
ρ

5. Beverloo Equation in 3-D: A Derivation
2
5
2
1
2
5
2
1
2
1
)(
.
0
..
)(
.
2
edkDgCM
DdMDgM
DDgAj
time
mass
M
v
timearea
mass
j
e
−=∴
→→∝
∝==
=
⋅
=
asbut
Aareaoutletthroughfluxmass
ρ
ρ
ρ
restfromdistanceafallingafter
outletthroughpassingparticlesofSpeed
D
Dgv 2
1
)(∝
(Boundary Layer Effect)

6. Beverloo Equation in 2D
2
3
2
1
2
)( edkDgC
.
M
DDA
−=
∝ :thatso,thanratherExcept
case.2DthetoovercarrieslogicSame

7. Founding our Hypothesis
 The screening effect explains
 Macroscopic effects
 Plateauing pressure
 Microscopic effects
 Jamming
 Free falling particles  The Beverloo equation
 So what does the insulation of the outlet region suggest
about the probability distribution of jamming?

8. Our Hypothesis
1)0(but)(
)(
1)(
)()(
)(
)(
)1()()(
===
−=
−=+
−=+
=
−
APeAtP
tP
dt
tPd
tP
dt
tPdt
dt
tPd
tP
dt
tPdttP
dt
dP
s
t
s
s
s
ss
s
s
ss
j
t
τ
dt
τ
τ
τ
τ
τ
:isjamawithouttimetosurvivingof flowyprobabilittheThen
time.sticcharacteriaiswhere
isjamaofyprobabilitthe,timeshortaOver
τ
t
etPs
−
=∴ )(

9. Sketch of our Apparatus
Flips like an hourglass to quickly run trials on both inclines.
Diameter of opening is variable.
Plug

10. Experimental Setup
 2 sheets of Plexiglass
 ~5000 bidisperse
photoelastic disks in between
(diameter = 3mm & 5mm)
 Polarizers taped to front and
back for high speed video, in
order to see arch formation
dynamics.

11. Actual Experiment
• We measured
duration of
continuous flow
for various outlet
sizes on the two
inclines
• 100s of runs per
opening size
gave probability
distribution for
survival time
2.3cm outlet, 45° incline. Slightly faster than real time.

12. Results: An Example
The survival probability exhibits expected exponential decay.
confirmedformτ/
)( t
etPs
−
=

13. Characteristic Time τ
What form could it have?
τ vs. Outlet Diameter
0
1
2
3
4
5
6
7
8
2.1 2.2 2.3 2.4 2.5 2.6
Outlet Diameter (cm)
τ(s)
45° τ (s)
30° τ (s)

14. A Reasonable Guess at τ
2
1
1
1
2
1
2
1
]
)(
[
)(
)(
g
dkD
Cτ
dkDD
g
D
v
D
Dgv
e
e
−
=
−→
∝=
∝
Making
betoscaletimenaturaltheexpectmightwe
wasscalevelocitynaturaltheasJust
τ

15. Verifying τ

16. Verifying τ
τ2
varies linearly with opening size, as posited form suggested.

17. Summary
 Photoelastic material connect jamming to
arch formation
 The probability of continuous flow decays
exponentially with time
 Reasonable results for τ(D)