Simulating granular flow with molecular dynamics

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Simulating granular flow with molecular dynamics

Gerald Ristow

To cite this version:

Gerald Ristow. Simulating granular flow with molecular dynamics. Journal de Physique I, EDP

Sciences, 1992, 2 (5), pp.649-662. �10.1051/jp1:1992159�. �jpa-00246574�

tllassification Physics Abstracts

05.60-46.10

Shnulating granular flow with molecular dynalrdcs

Gerald H. llistow

HLRZ, KFA J61ich, Postfach 1913, W 5170 Jfilich, Germany

(Received

IS January 1992, accepted 22 January

1992)

Abstract. We investigate by _means of Molecular Dynamics simulations _an assembly of

spheres to model

a granular medium flowing from _{an upper} rectangular chamber through _a hole into a lower chamber. Two different two dimensional models _are discussed _one of them including

rotations of the individual spheres. The outflow properties _are investigated and

iompared

_to

experimental data. The qualitative agreement suggests that _our models contain the _necessary

ingredients to describe the outflow properties of granular media.

I. Introduction.

A

granular medium, consisting

of cohesionless

particles

like lead beads _or

sand,

_can behave in many different _ways. The extreme cases are solidlike behaviour when _a

granular

medium resist shear

by undergoing plastic

deformation and fluidlike behaviour _{as seen} in

avalanches,

sand dunes and under vibration

(see

_e.g. ^{Clement and}

Rajchenbach [I]).

The transition between these two

regimes

is _seen in

geophysical phenomena

like rock

sliding

and earth

quakes

and leads to _a

variety

of industrial and

pharmaceutical applications ([2, 3]).

Recently,

there has been

quite

_some interest in this field

especially

from _a

computational point

of view. One

approach by Hong

and McLennan [4] among others _uses hard

spheres

and

a collision

table,

^another cellular automata

(see

_e-g- Baxter and

Behringer _[5]).

Our model considers soft

spheres

where the softness _can be varied via _a parameter ^{which is}

given by

the material

properties

and dates back to _our

knowledge

to Cundall and Strack [6].

In the

following

_we consider two models to describe the flow of

granular

materials. In order to compare the results to

experimentally

measured

quantities

from industrial bins and bunkers

([2, 3])

we restrict ourselves in this _paper to the outflow of _a

granular

material from _an upper chamber

through

_a hole of diameter D into _a lower chamber of

equal

size in two dimensions.

This setup is sketched in

figure

I at two different times

during

_a

typical

simulation. The

dynamic

of the system is obtained

by chosing

_an initial

configuration

and

solving

Newton's

equations

of motion for each individual

particle

for all later times. Due to the interaction of the

particles

the time axis is discrete and the

procedure

is in

principle

the

following.

For _one time step the forces _on all

particles

_are calculated and their

positions, velocities,

accelerations

650 JOURNAL DE PHYSIQUE I N°5

a) b)

Fig.I.

Outflow of 900 spherical particles from _an upper chamber through _a hole of size D

= lla

for ₇

" 30

(snapshots

after

(a)

10000 and

(b)

20000

iterations).

etc. _are

predicted

for the _next time step

according

to their

Taylor expansions.

The forces _are recalculated for the _new

positions

and

compared

to the old values. The

positions,

velocities etc. are corrected

by taking

a mean value of the forces for this time step. This is

repeated

over and _over

again

and is known _as the

predictor-corrector

method in Molecular

Dynamics

simulations where the details

might

differ due to numerical considerations. We have chosen this method since _we want to be able to describe

grains

^that _can penetrate ^{each other}

slightly

and

thereby transfering

translational into

rotitional

_energy and vice _versa. Due to

gravity

which is the

driving

force in _our simulation the system cannot be treated _as in usual molecular

dynamics (MD)

where _one

keeps

a constant internal temperature.

During

^{each collision there}

is _a loss in kinetic _energy which is converted into internal

degrees

of freedom of each

grain

and leads to _a reduction of the total kinetic and rotational

energies.

The main

ingredient

ⁱⁿ our

models in addition to

gravity

will therefore be _a friction force which

only

acts

during

^collisions

and which is

proportional

to the velocities of the

particles.

In section 2 _we

investigate

_a

simple

model

describing

cohesionless

particles

of

equal

diameter.

Only

normal forces between

interacting particles

_are taken into _account

thereby neglecting

rotational

degrees

of freedom.

In section 3 _we extend this model in the

spirit

of Cundall and Strack [6] to include rotations of the

spheres

^and _we ^considered a Hertzian contact force. Our model includes the

possibility

to

study

the effect of

particles

taken from _an

assembly

of

differing sphere

diameters.

Last not least _we summarize _our results in section 4.

2. Model with cohesionless

spheres.

In _our first model _we consider N

spherical particles

of

equal

diameter _a in two dimensions

having

the

potential

V(r)

=

~~

~~~

~~ _~ ~ ~~

(l)

0 otherwise

motivated

by

the

repulsive

part of the Lennard-Jones

potential.

The parameter _r~ ^{is the cut-ofl} distance of the

potential

^which we fixed at _r~

=

1.3«,

_a

= I _mm and _we have chosen d

= I Nm.

Since all

particles

have

equal

diameter _{we measure} the _mass in units of

particles.

The time evolution of the system was done

by

MD

using

_a fifth-order

predictor-corrector algorithm (see

e.g. Allen and

Tildesley [7]).

^{Whenever the distance between} two

particles

is less than _r~

their interaction

gives

rise _to

a

repulsive

force to which _we add _a friction force

proportional

to the

particle's velocity thereby taking

into _account the _energy loss

during

collisions between

particles

and

particles

with walls. This leads _{to an} additive _term of -7vj in the force

acting

on

particle

I. To compare our results to

existing experimental

data _we consider the

following

setup. ^{A box of dimensions}

(50a)

_x

(100a)

is

separated by

_a ^{wall into} two chambers of

equal

size

(see Fig. I).

In the _upper

chamber,

_we put 900

particles

at random

positions

with _zero

initial velocities. Due to

gravity,

all

particles

will fAll downwards. We let the system ^evolve for

approximately

100000 time steps ^chosen as At

= 0.00006 _s and monitor the total _energy to

ensure that the system ^relaxes to a stable

starting configuration.

^The outer walls of the box _are

chosen _as flat

giving

rise to a

potential

that _was

approximated by integrating equation (I) along infinitely long

walls. The middle wall consists of smaller

particles

of diameter

a/3

to avoid

numerical difficulties. The

packing

of the

particles

is

nearly hexagonal

with _some distortions introduced

by

^{the walls} of the box.

We _{now open a} shutter of diameter D

by deleting

_some

particles

in the centre of the middle wall. When D is greater ^{than 2.6a}

particles

will flow from the _upper box into the lower _one,

resembling

_a sand

glass,

until _a

given angle

of _repose of the

remaining particles

in the _upper chamber is reached. This behaviour is shown in

figure

I for _a friction coefficient of ₇

= 30

and _a hole size of D ₌ lla. The

spherical particles

_are denoted

by

dots of smaller size than their actual diameter. The line attached _to each

point

shows the linear

interpolated path

of each

particle during

the _next time step

(the

line

length

_was

magnified by

_a factor of 30 to make the

particle's

motion

visible).

We varied both parameters ^{and found that for}

increasing

7 a dome will build _up in the lower chamber which becomes

sharper

for smaller hole sizes.

This is

easily

understood since the friction coefficient controls the distance _a

particle

_can travel

freely

after each interaction. The greater ₇ ^the more kinetic _energy looses _a

particle during

a collision. This leads _to less

higher

bounces and therefore _to

a smaller traveled distance. In

figure

2 _we show the summed kinetic _energy

(E,.(t)

=

£)~~ £$~ _Ei(j),

_where

_Ei(j)

_{is the}

total translational _energy of the ith

particle

for the

jth

time

step) during

the first and second 10000 iterations. Two

adjacent equipotential

lines

(lines

of constant E~~,) ^indicate a

drop

of

lie

in the kinetic _energy. The _arrows indicate the

strength

and the direction of the energy

gradient.

Our model

predicts

_very well the

experimentally

found behaviour where _{one sees no} convective flow in the _corners of the _upper box _as found in [2].

Figures

16 and 2b show the system ^at a later

stage

^{when there} _{are no more}

particles

above the middle of the

opening.

The

remaining particles

in the _upper chamber form two side _cones with the walls and the

spheres

slide down the incline towards the

edges

of the

opening.

652 JOURNAL DE PHYSIQUE I N°5

, 1 ,

~ _~

a)

'

~ ₎ /

'~

/

' j,

' b)

Fig.2.

Same simulation

as in

figure

I. Equipotential fines for the summed kinetic energies of the particles in the _upper chamber

(averaged

_over the first

(a)

and second

(b)

10000

iterations).

In

figure

3 we look at the

averaged outflowing

velocities of the

particles just

below the hole

as a function of time where the

averaging

_was done _over 500 time steps. In this

section,

all velocities _are measured in ^~~' One _{sees two}

regimes

of the flow patterns. After the _{onset of} the

s

flow

(t

>

1000At)

_we find in the first

regime (up

to

10000At)

that the vertical component ^of the

velocity (<

_vy

>)

is

nearly

constant _as found

experimentally

_{[2] and in} other simulations [4].

The fluctuations of the horizontal

velocity

_component

(<

_v~

>)

_are rather small

indicating

that the

particles

_come from

a central flow

regime.

After _a

sharp

transition at to @ 10000At, < vy ^>

takes _on another constant value whereas the fluctuations in _{< v~ >} become rather

high

but

are of the _same order in the

positive

and

negative

z-direction. This behaviour characterizes

a flow similar to the _one sketched in

figs.

lb and 2b when the central

region

is

already

empty and

particles

fall down from the inclines of the side _cones. This

explains

the constant value of _< vy ^> and the

higher

fluctuations in _{< v~ >.} In

figure

4 _we show the value of _< vy ^{> in}

the first

regime

_as a funtion of the hole size D and _we note that it

only

varies

slowly

which

might mostly

be due to statistical fluctuations of the individual

samples

which _were of the order of10i~. The

discharge R, meaning

the number of

already

outflown

particles

from the upper

chamber, during

a

typical

simulation is

given

in

figure

5. Schwedes [2]

reported

that the

discharge

rate

~~

in three dimensions scales

roughly

_as

(D Do)" (a

₌ 2.

2.5)

^where Do is _a reduction

fd~or

_{of the order of}

two to four times _a which takes into account the efsect of the

edges

of the hole. Since _our simulations _were done in two dimensions _we expect a value of

a = I. 1.5

[14].

^The

reported

results

by

Schwedes _were obtained for _a flow field in which the

0 5 lo 15 20 25 30 35 40

oz

04

~

off

_****~

-la

x

-

14

~

- 16

k _~

* a)

03

~

Da ~

#

~~ ^{# ~ ~}

~~kk~x~~~*~~~~****#~

~*

^~

~

~

~ *~

~ #

-01 #

~

oz

x 03

0 5 lo 15 20 25 30 35 4O ~~

Fig.3.

Averaged outflow velocities

((a)

_{< vz >,}

(b)

< vy

>)

right below the hole of size D

= 10a

for 900 spheres and ₇

" 30.

central

region

_was not empty

(number

of

particles

N _-

cc)

_{so we}

compared

_our results of the

discharge

rate for different hole sizes D

by

linear

interpolation

towards t ₌ 0 with them. In

figure

6 _we show _our results for different values of the friction coefficient 7. The dotted _curves

were obtained

by

_a ^{linear least} _square ^fit and the

slopes

are

given

in table I.

One _sees that _our first model agrees

qualitatively

with the available data from

experiments especially

the behaviour of the

velocity components right

below the hole. Also the

properties

^of

the

discharge

rate _are well included and the behaviour of different materials _can be simulated

by varying

either the

prefactor

_or the exponent ^{of the}

single particle potential

_as

given by

equation

I _or the friction coefficient _{7 as} indicated

by

the values

given

in table 1.

654 JOURNAL DE PHYSIQU,E I N°5

-o.05

~ o

-0.I _{o o o}

o

o o o o

o o

o o

o.15

o

-0.2

0 2 4 6 8 10 12 14

FigA.

Outflow velocity < vy ^> as function of the hole size D for 900 spheres and ₇

= 30.

aoo

700

800 , ,

a a ? '

, n ?

~ o °

~ o

5©O ~ n °

a o

°

o

°

MO

~

°

o o o o a o o 200

~

°

o a o a o o

o z 4 6 a to La m 16 La zo zz m za Za 30 3z 34 36 3a 4o

Fig.5.

Number of outflown pacticles N _as function of time for 900 spheres and ₇

# 30 for _a hole

size of D =10a.

Table I.

Discharge

rate

~~

for dilLerent values of the friction coe~licient _7.

At

fi

30 79.1

60 42.3

90 28.8

1000 j

7 " 30

o o

7 = 60 _*

150 7 " 9°

° °.

8

~

500 ^°

~

*

> * _.

250 *

o ^"

o

1

0

~

0 2 4 6 8 10 12 14

Fig.6. Discharge

rate

~~

for different friction coeficient _7.

at

3. Model with rotation and

varying sphere

diameters.

The

underlying

model for _our second

approach

dates back to our

knowledge

to _a iuethod used

by

Cundall and Strack [6] to describe the force distribution of _an

asseiubly

of

spheres

under strain. It has been used in

slightly

modified form to describe the mechanical

sorting

of

grains (Hafl

and Werner [8]) ^{and the} shear-induced

phase boundary

between static and

flowing-states (Thompson

and Grest

[9]).

^{In this}

model,

normal and shear forces between

interacting spheres

are included. Whenever two

spheres

_are closer than the _sum of their radii

(d

_{= vi} +i>j, ai =

2ij)

the ith

particle

feels _a contact force of

Fij

=

(kN(d (xi

_xi

)fi)~'~ 7Nm~«(ki kj )fi)fi

+

(2) +min(-7Sm~fiVr~i, Sign(-75

_''llefl

Vrel)P(FijU()S

where

Vre< " (11

k~)I

+

~ioi

+

~joj

lllilll;

'~~"

lily + ill;

Here _xi denotes the

position

vector of the ith

particle,

fi _:=

fi

is the unit _vector

pointing

from the centre of

particle

I to the centre of

particle j

and

j is'a

_unit

vector

perpendicular

to fi

turning

^clockwise. fly stands for the

angular velocity

of the ith

particle

where clockwise rotation _was chosen _as

being positive

and _p denotes the static friction coefficient.

Since in _our model the

particles

_can penetrate each other

slightly

_we used _a Hertzian contact force to take into _account the deformations of the

spheres

which leads to an exponent of1.5 in the first _term of the normal force component [10]. ^{We considered the force law for three} dimensions

making

^it easier to extent _our simulations in _a later

stage.

^One can think of _a setup where the

spheres

_can

only

_move between two

glass plates

which _are

separated by only

_a little _more than the

sphere

diameter. Such _a setup is feasible and _our results could be checked

experimentally ill].

The

prefactor

kN _can be calculated from material

properties

and is of the order of10~ 10~

~~.

All

lengths

_are measured with respect to the

averaged sphere

m

656 JOURNAL DE PHYSIQUE I N°5

f-., ~

~ ' ~

~

Fig.7.

Equipotential lines for the summed translational energies during the first 50000 iterations for 2500 spheres and the parameters _{7N =} ^{300 and} 7s = loo

(D

= 12).

Fig.8.

Equipotential lines for the summed rotational energies during the first soooo iterations and the _same parameters _as in figure 7.

diameter _{ro "< r>} and all _{masses mi} have the _same

density

and _are measured in units of _a

spherical particle

with radius _ro.

The parameters _TN and _{7s are} the normal _resp. shear

~amping (friction)

coefficients where 7s controls the _energy transfer between the translational and rotational

degree

of freedom.

To fulfill the Coulomb relation between the normal and shear force components [12] we put

an upper bound _on the shear force and used _p

= 0.5

throughout

_our simulations. The term

proportional

to the relative surface

velocity

_(v~~i) in the shear force component ^{reflects the} fact that _we consider

particles

that slide _upon each other and do _not stick. The

newly

in- troduced rotational

degree

of freedom of _an individual

sphere

_was

incorporated by

_a third- order

predictor-corrector algorithm

in _our second model. As time step we

mostly

used At

=

0.0001 _s which had to be reduced when

higher

values of the

damping

coefficients _TN and _7s

were considered.

We will

present

simulations for different parameters _7s to

investigate

^{the influence} of the shear force which converts translation into rotational _energy and vice _versa. The

dependence

on the

damping

coefficient _TN and the interaction coefficient kN is also discussed.

First _we present some

general

flow

properties

of _our second model. In

figure

7 the summed translational _energy

during

the onset of the

flow,

similar _to

figure 16,

is shown for _a

typical

simulation.

Figure

8 shows the summed rotational energy

during

the _same iteration and in

both _cases the _upper chamber is half _way filled with

spheres initially.

^One notes, that there

is _no motion in the _corners of the _upper chamber _as mentioned

by

Schwedes [2]

showing

the

2

o o

O o ~

'~~ _o

~°o o ~ ~

° O Oo o o o ° O O

~~ ~ O o ~ ~

0 ^{~ ~ ~}

O °

o °

~

~ O

O

~

O ~

~ o

° o o o~

O '2

~ o

°

0 100000 200000

Fig.9.

Outflow velocity < _vz >

during

_a typical iteration for 2500 spheres and _7N

= 300 and

7s = 100

(D

= 16).

6

o ~

o o o

o o o o

~

o o o °

o ° o

o o

~

o

O

io _o OO °

~ O

o O

~

o

o o

o

14 °° °

0 100000 200000

Fig.10.

Outflow velocity < vy ^> during _a typical iteration for 2500 spheres and _7N

= 300 and

7s = 100

(D

= 12).

existence of _a central flow

region.

This is

emphasized by figures

9 and 10 where the horizontal and vertical

averaged

velocities

right

below the hole

(as

in

Fig. 3)

are shown for the _same simulation but, for _a

larger

observation time. Two

regimes

are

clearly

visible _as in

figure

3.

In the

former,

the variations in < v~ > are rather small

stressing again

the existence of _a central flow

region

and < vy > shows _a transition to a maximum value with _a

plateau.

When

compared

to

figure

3 and other simulations [4] one expects a constant value in this whole

regime

for < vy ^{>. This} difference

might

be due _to the fact that _one has _to increase the number of

particles

for the second model in order to get ^{similar results indicated}

by

_an

already

small

plateau

in

figure

10 which could be _seen in simulations for other parameter values _as well. In the latter

regime,

the variations in _{< v~} > increase but _are of the _same

magnitude

in both

directions

indicating

_a flow

along

the sides of the two side _cones and _< vy ^> shows

a slow

transition to _a second _constant value.

Interesting

in itself and also in

comparison

to

figure

4

we show the values of _< vy ^> in the first

regime

_{as a} function of the hole diameter D in

figure

Il. For _our second

model,

the fluctuations with the hole size _are much

larger

than in the first model for the considered parameter values and

they

have _an

opposite tendency

with

respect

to

increasing

hole size.

658 JOURNAL DE PHYSIQUE I N°5

~

l

12

.

18

o to lo Jo Jo

Fig.ll.

Outflow velocity _{< vy > as a} function of the hole size D for 2000 spheres and _7N

= 300

and _{7s =} I00.

o.ooos

*

~

o o

* o

*

o

0.001

~ o

*

*

° o ^° *

*

* * D=8 _o

* * * D=19 _w

0.0015

42 46 50 54 58

Fig.12.

Outflow velocity < v~ > as a function of the z-coordinate for 2 different hole sizes D and

7N = 300 and _7s

" Ioo

(2500 spheres).

In

figure

12 _we present the

velocity profile right

below the hole

averaged

_over the time when the flow _comes from the central flow

region (first regime)

for two different hole diameters. The

diagram

indicates _a

parabolic profile

with _some side effects due _to the

boundary

conditions _at the

edges.

The middle of the hole h at _z

= 50 and the units _are in

sphere

diameters which is considered to be the _same for all

particles.

To check this statement, we

plot

in

figure

13 both branches of the _{same curves as a} function of

(z zo)~

^where _zo ^stands for the z-coordiante of the middle of the hole. One _sees that _an almost fluid like

profile

is obtained.

An

important quantity

in industrial

applications

is the measurement of the amount of out-

flowing particles

_per time

(discharge

rate

fl).

_{Since in real industrial bins and bunkers}

t one

looks at the

steady

state _we extract the value

graphically

from the _curve

R(t),

which _we show in

figure 14,

after the onset of the flow in the first

regime

where R grows linear with time.

Doing

_so for different parameter values and

plotting

^~~ _{as a} function of the hole size _we found that _an almost

straight

line could be drawn

throughi(e

_data

_{points suggesting}

a law like

fl=c.(D-Do) ₍₃₎

o.ooos

o

* o

*

o ~

0.001

~ ^o

*

o *

* 4 _D=8

O

* D=12 *

0.0015

0 10 20 .?0 40

Fig.13,

Outflow velocity < vy > as a function of the

(z zo)~-<oordinate

for 2 different hole sizes

D and _7N

= 300 and _7s

" loo

(2500 spheres).

1600

1200

800

o°o

400 o°°

o°

o°

o°

0

0 100000 200000

Fig.14.

Outflow R _{as a} function of time for _a hole size of D

= 12 and _7N

= 300 and _7s

= loo

(2500 spheres).

The parameter c reflects the material

properties

and

Do

takes into _account

arching

effects which will block the flow for values of D <

Do (see

_e.g. ^{lteisner and} von Eisenhardt

[3]).'An example

of such

a least _square fit to

~~

as a function of D is shown in

figure

15. In table II,

a collection of the values _c and Do

fd~different

_{parameter values}

are

given

for _a simulation with

just

_one

sphere

diameter. To check the accuracy of _our method _we

averaged

^~~ _over five

different hole

positions

for

varying

^initial

configurations

and found _a

change

of

ti~

discharge

rate of less than 3i~.

In table III _we present _some simulations for _an

assembly

of _a Gaussian distribution of

sphere

diameters

(~ ~r )~

rj # roe ^~("~)

JOURNAL DE PHYS>QUSI -T 2, N'5, MAY <W2 26

660 JOURNAL DE PHYSIQUE I N°S

600

o

400

o

»

~

0 '

0

10

- _rate fl

t

7s

Table II. Values of _c and Do for different parameter values and _an

assembly

of

spheres

with

equal

diameters.

N kN _c Do

2000 10~ 99 297 13.385 3.711

300 0 15.574 3.809

600 0 13.942 2.998

3000 0 14A13 3.250

0 300 13.855 4.285

300 100 14.665 4.402

600 100 13.768 5.754

Table III. Valuer of _c and Do for dilLerent parameter values and _an

assembly ofspheres

with dilLerent diameters.

N kN _c

1000 99 297 8.733 2.447

10~ 300 0 8.807 1.164

10~ ₀ 300 10.lsl 4.101

10~ _{300 100 8.727 2.405}

10?

_{300 100 8.677 1.814}

10~ 300 100 9.022 2.673

1500 10~ _{99 297 11.707 3.697}

where _xi is _a random variable

fulfilling

the relation _{r~~ < xi < r~_} and Ar is _a quarter of the considered interval

[r~~,

_r~~~]. We have chosen _{r~;~ =}

0.6ro,

_{r~_}

= 1.4ro and _ro

" 0.5 _as

in the simulations with constant

sphere

^diameters. The results

suggest

^{that the}

investigated properties depend only slightly

_on ^{the chosen} parameter ^values

kN,

_TN and _7s but rather _on the

geometry

^{of the whole}

problem.

The main difference is

taking spheres

with

equal

_or

varying

diameters which leads to _a

drop

of the outflow rate of 20it. Similar observations _were observed

by looking

for

density

_waves in the central flow

regime (Baxter

and

Behringer [13]). Any

other

systematic

behaviour could not be found for the considered parameter ^{values and number of}

particles.

4. ConcIusions.

We

presented

two models to simulate the behaviour of

granular

flow. To be able to compare our

results to

experimental

data _we had chosen _as setup _a

simplified

bin

resembling

_an

hourglass.

We looked at the summed kinetic

energies

of

particles being

in different

regions,

the

averaged

outflow velocities of

single particles

and the

particle discharge

for difserent hole diameters.

Our first model

only

considered the normal force component

during

interactions and _we introduced _a

damping

coefficient _to simulate difserent materials. We found the flow pattern and the outflow velocities _to be in

qualitative good

_agreement with

experiment indicating

_a

central flow

regime

in which the

particle discharge

increased linear with time. The

discharge

rate showed _a linear

dependence

_on the reduced hole diameter which _was

justified

for _our two dimensional setup.

Our second model also considered shear forces and had the

ability

to choose

spheres

with

varying

diameters. We considered _a Hertzian contact force which seemed _more realistic for

slightly penetrating

^soft

^spheres.

^{The results}

agreeded qualitatively

with the _ones from model I but

they

did _not

depend

that

strongly

_on the parameter values. This

suggested

that the flow behaviour of

granular

media for this

particular

_setup is dominated

by

the geometry ^{of the}

problem

which is further

emphasized by

^{the fact} that the outflow

properties changed drastically

when

spheres

with

varying

diameters _were considered.

Some aspects of

granular

flow

(e.g. arching efsects)

_were described _more

accurately by

_our second model but rotational

degrees

of freedom

might play

_a crucial role when

phenomena

like surface fluidization _or instabilities _are considered where

investigations

_are

under»lay.

Acknowledgements..

I would like to thank Hans Herrmann for

introducing

_me to this

topic,

_many fruitful and

clarifying

discussions and _a critical

reading

of the

manuscript.

Also I have _to thank Stefan

Sokolowsky

for

explaining

the secrects of Molecular

Dynamics

to _me and

discussing

the details

of vector

programming

with _me.

Note added in

proof

_:

Thanks to Eric Clement who mentioned reference [14] to _us. We tested _our second model for the

following discharge

rate

fl

= c~

(D D[)~/~

and found it in

good

_agreement with _our data for small hole diameters. To

clarify

this

point

one has _{to use a}

larger

^box to reduce the side efsects

by

the walls where

investigations

_are

underway.

References

ii]

Clement E. and Rajchenbach J., Europhys. Lett. 16

(I991)

133.

[2] ^Schwedes J., Flieflverhalten _non Schittgfitern in Bunkem

(Verlag

Chemie, Weinheim,

1968).

[3] ^{Reisner W. and} von Eisenhart-Rothe M., Silos und Bunker fir die Schfittgutspeicherung

(Trans

Tech Publ., Clausthal-Zellerfeld,

1971).

[4] Hong D.C. and McLennan J-A-, preprint.

is]

Baxter G.W. and Behringer R-P-, Phys. Rev. A 42

(1990)

1017.

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[6] ^{Cundall P. and Strack} O.Dl., Gdotechnique 29

(1979)

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[7] ^{Allen M.P, and} Tildesley D.J., Computer Simulations of Liquids

(Clarendon

Press, Oxford,

1987).

[8] ^{Half P.K.} and Werner B-T-, Powder Technol. ₄₈

(1986)

239.

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175I.

[lo]

Landau L.D. and Lifschitz E-M-, Elastizitdtstheorie

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Berlin,

1989) [english

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ill]

Clement E., private communication.

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[13] ^Baxter G.W, Behringer R.P., Fagert T. and Johnson G.A., Phys. Rev. Lett. 62

(1989)

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