Different characteristics of the motion of a single particle on a bumpy inclined line

HAL Id: jpa-00246976

https://hal.archives-ouvertes.fr/jpa-00246976

Submitted on 1 Jan 1994

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Different characteristics of the motion of a single particle on a bumpy inclined line

Gerald Ristow, François-Xavier Riguidel, Daniel Bideau

To cite this version:

Gerald Ristow, François-Xavier Riguidel, Daniel Bideau. Different characteristics of the motion of a

single particle on a bumpy inclined line. Journal de Physique I, EDP Sciences, 1994, 4 (8), pp.1161-

1172. �10.1051/jp1:1994246�. �jpa-00246976�

J. Phys. I £Fance 4

(1994)

1161-l172 AUGUST 1994, PAGE l161

Classification Pllysics Abstracts

64.60C 05.60 46.30

Dilllerent characteristics of the motion of

_a

single particle

_{on a}

bumpy inclined fine

Gerald H. Ristow

(*), François-Xavier Riguidel

and Daniel Bideau

Groupe Matière Condensée et Matériaux (**), Université de Rennes1, 35042 Rennes Cedex,

France

(Received

18 January1994, revised 24 Marcll 1994, accepted 15 April 1994)

Abstract. In order to better understand the rote of friction and disorder in segregation

mechanism in surface flow, _we investigate the motion of _a single bail of radius R _{on a one-} dimensional bumpy fine made of balls of radius _r. The smoothness is then controlled by

R/r,

the ratio of radii. In addition to experimental results for different ratios, _we perform numerical simulations where the bumpiness disorder in spacing between the balls forming the fine is aise controlled. Different kinds of behavior _are observed. In particular, _we obtain

a regime where the bail quickly reaches

a constant _mean velocity. In this regime, we investigate the friction forces during contacts and identify in _a generalized model the regions which _are dommated by solid friction _or by loss due to collisions. We show that smaller balls get trapped _more easily _{on a} rough surface which might explain the origin of the segregation _process found in rotating drums and sandpiles. We aise study this system for different values of the bumpiness of the fine and find conditions under which the constant velocity _regime disappears completely.

1. Introduction.

Surface

phenomena

are of great importance in

rapid granular

flow but

despite

their wide _use in industrial

applications

and expenments, a detailed

understanding

of the relevant parameters ^is not yet ^{achieved. It} was first studied

by Bagnold iii

who defined two flow

regimes (macroviscous

and

grain inertia)

in the _case of _a mixture of

grains

and fluid. The

grain

inertia

regime

in the

case of

high partiale

velocities is often described

by

a kinetic

theory (see

_{e-g- [2]} but for denser systems or low

velocities,

the above theories have to be

greatly

modified to account for the

Coulomb-type

friction forces

along

^surface contacts. This dualism of

granular

materials

(they

can behave like solids _or like

fluids)

leads to instabilities which often have the form of

density

waves, observed in experiments _[3] and numerical simulations

([4,

5])

(*) present ^address: Fachbereich Physik, Philipps-Universitàt Marburg, Renthof 6, 35032 Marburg, Germany.

(**) URA CNRS 804.

Dne of the most important mechanisms of

segregation

in

granular

media is surface flow [6].

When

partiales

of diiferent _{sizes move} down _an inclined bed _{or are} part of _an avalanche the

bigger

ones are found _more

likely

_on top at the end of the _process. This

happens

aise _m

a

rotating

drum _or while

building

a

sandpile

where the

underlying partiales

net

being

part of _an avalanche _can be

regarded

_as

being

fixed

(solid phase)

in _a first approximation. ^If

two-dimensional

experiments

_are

performed,

these

partides

form _a one-dimensional

rough

fine where the scale of the

roughness depends

_on the

grain

size distribution. This

roughness

aise

depends

_on the size and the

shape

of the

rolling grains:

the

larger

the

grain,

the less stable its

equilibrium

when it is _at rest, and the

longer

its _mean free

path along

the

slope

when it flows.

Recent

experiments

in 2D drums and numerical simulations

using

diiferent models illustrate this

phenomenon

_very well

iii.

In this context, we have

performed

experiments and numerical simulations ta get a better

understanding

of the individual behavior of _a

single

^bail

moving

down _an inclined

bumpy

fine with controlled

roughness.

We define the smoothness _as the ratio of the radius of the

single

bail

moving

down the fine _ta the radius of the balls

forming

the fine

(RIT).

The motion is net easy ^ta understand and _even for this

simple

_case the energy balance between

gravity,

collision and friction forces leads ta

interesting

behavior

([8, 9]).

In _a

previous

_paper

[10],

_we have

given

_a

phase diagram showing

the diiferent

regimes

^of

motion observed

experimentally

in _a two-dimensional system. Here, _{we use} the

hypothesis proposed by

Jan et al. _[8] which _was

developed

for the _constant

velocity regime

and find that it can

only

^be

applied

in _{a narrow} range of the control parameter

RIT.

We then extend the

model

proposed by Jaeger

et al. [9] ^to describe the

general

behavior of the bail

moving

down and discuss the importance of the different forces _{as a} function of the inclination

angle le),

the surface smoothness

(RIT)

and the

bumpiness

of the fine

le).

We present ^{conditions under} which the constant

velocity regime disappears completely

and for

particular

values of

angle

and size

ratio,

chaotic behavior _can be reached, _very similar to that obtained in the _case of _a

bouncing

bail

iii].

In the next section, we

explain

the experimental and numerical

techniques

_we used. The results _are

presented

and discussed in section 3.. In section 4., we present diiferent theoretical

approaches

to understand and

classify

_our results and _we conclude

by giving

_an outlook _on the

applicability

of _our results to other

phenomena,

1-e- the

segregation

_process.

2.

Experimental

and numerical system.

2.1 EXPERIMENTAL _SETUP. Dur

experimental

setup is

slightly

dilferent from the _one used in reference [8]: ^{it is} made _up of _a 2 _m

long

mclined fine of 400 identical steel

spheres

of

diameter 5 _mm. Each

sphere

is

touching

its

neighbors

and

they

_are confined

by

_a

L-shaped

flume sketched in

figure

1. The two sides X and Y _can have _a diiferent inclination

angle

_ax

and _ay with the horizontal. In ail _our

experiments,

_we have chosen _ax

= 50° and _ay

= 40°.

We then

study

the motion of _a

single

steel

sphere

of variable size

moving

down

along

either

one of the two sides of the flume. In order to start the motion, we

give

it a small well defined initial

velocity.

2.2 NUMERICAL MODEL.

Concurrently

to these experiments, we have

performed

numerical

simulations _on the motion of _a bail _{on a}

rough

inclined fine formed

by

balls

placed

in _a disordered _way. The distance between

neighbors

_is

randomly

chosen between 2R and

2R(1+e),

where the disorder _can be controlled

by

the parameter e

le

= 0 refers to the ordered _case of

touching spheres).

A

snapshot

of the simulation for _e

= o-à is shown in

figure

2. Periodic conditions _are

imposed

to the fine _so that the system can be considered _as infinite.

N°8 MOTION OF A PARTICLE ON A BUMPY LINE 1163

x

y

Fig. 1. L-shaped flume used in the experiments with approximately 400 steel spheres.

~Î~

Ô

Fig. 2. ID motion of _one bail _{on an} inclined fine. Roughness of the hne _is created by placing

balls randomly _or net _on it. In the random _case, the distance between neighbours _is randomly chosen between 2R

(touching spheres)

and

2R(1+ e).

The base of this simulation is the Molecular

Dynamics

method [12]. ^In

1979,

Cundall and Strack [13]

proposed

_a model to describe the forces

acting

_on

partiales

in _an

assembly

of

spheres

_{on a}

grain

size level which

they

called the distinct element method. Since then this

model has been extended and

slightly

modified to describe _among others the mechanical

sorting

of

grains [14],

the shear-induced

phase

boundaries [15] ^the

origin

of convection cells [16] ^and the outflow from _a

hopper ([17, 4]).

In Molecular

Dynamics simulation,

_one solves Newton's equations of motion for each

particle

for successive times. The time axis is discrete and the order of the time step ^{àt is crucial and} has to be checked for numerical

stability.

Whenever two

particles

_are doser than the _sum of their

radii,

1-e-

they

have been

deformed, they

interact via normal and shear forces and the force _on

partiale1,

caused

by partiale j

is given

by

Fij

_" Fné+FSE

Fn ₌ -kn

(ri

_{+ rj}

(ri

_rj

)é)~ inme~(éi

_Fg

)é Il

Fs ₌

-isme~((éi fg )§

+ rioi +

rjoj

where

mimj

~~~

mi + mg

Here _ri denotes the

position

vector of the ith

partiale,

il is the unit vector

pomting

from

partiale

to

j

and § is _a unit vector

perpendicular

to

ilrotating

dockwise. The constant

kn

stands

for the

Young modulus,

_{in is} the

damping

coefficient in the normal

direction,

_is accounts for the transfer of translational _energy into rotational _energy and _me~ denotes the effective

(reduced)

_mass. For _a

= 1, _{we are} left with Hooke's law and for _a

= 1-à _{we can}

investigate

the Hertzian contact force due to

shghtly

deformable discs.

Normally,

the shear force Fs _can

have two

regimes

where the transition is

given by

the friction coefficient _~t due to Coulomb's criterion which _we

neglected

in _{our case} (~t ~

cc)

since it introduces another parameter ^and it _was found to

give

_no

qualitative change

for the

investigated

setup.

We used cgs units

throughout

_our simulations where the diarneter of the watt

partiales

_was

1 _cm. The

Young

modulus kn is lo~

gs~~cm~~~,

_in

= 1000 Hz, _i~

= 100 Hz and _we used _a

time step ^{of àt} = o-à x lo~~

s which assured numerical

stability.

The

relationship

between

our

damping

pararneter _in ^{and the}

expenmentally

_more

commonly

used restitution coefficient er can be calculated for Hooke's law

la

=

1) directly

from

equation Il) by comparing

the velocities

right

before and alter _a collision. It is of the form

'In '~

~~~~(

[2)

1r + in

Ier)

For _o

=

1-à,

_one gets a very weak

velocity dependence

^of _in but

equation (2)

_can still be used

as a

good approximation

over a wide

velocity

_range which has been verified for _our model

by

one of _us

iii].

The chosen value of _in in ail _our simulations is

approximately equivalent

to _a restitution coefficient of 0.7.

3.

Interpretation

of results.

3.1 PHASE DIAGRAMS. For _a

general understanding

of the motion of the

single baÎÎ,

_we

show _m

figure

3 the

phase diagram

obtained in _one dimension which is very similar to the

phase diagrams given

in

[loi

for _one and two dimensions. As central parameters _we used the smoothness

RIT

and the inclination

angle

8. In _our expenments, we followed the motion of the

single

bail with _a vider _camera and calculated the

velocity

for diiferent

positions

_on the fine. For _our numerical

simulations,

_we get the bail

velocity

_every time step ^and monitor the

over 600 time steps

averaged

value. The phase boundanes for the diiferent bail motions

(which

will be

explained

in more detail in the next

paragraphs)

is

given.

We show the data from experiments and numerical simulations in the _same

plot

for direct comparison.

For small inclination

angles

8 with the

horizontal,

the bail _{comes to a}

complete

_stop

(called regime A) regardless

of its initial

velocity.

The

gain

in

potential

_energy, <

Ep

>m

2rmg sine,

is toc small

compared

to the

dissipation

due _to friction _or collisional fosses.

When the inclination

angle

_is

increased,

_one reaches _a

regime

^{where the behavior of the bail}

depends

_on its initial

velocity.

For small values, the bail gets

trapped

in _one of the next

valleys

formed

by

two balls. If the initial velocity is

high enough

to climb _over at least the next bail,

one finds _{a regime} of _{constant mean}

velocity,

this

velocity being

defined at the top ^{of each} bail

forming

the fine

(called

_regime

B) ([8, 9]).

The

phase boundary

for the _case of _a

high

initial

velocity

with the

regime

where the bail stops ^{is shown in}

figure

3 and for

RIT

₌ 1, we get an

angle

^of

7°(exp.)

and

9.5°(sim.), respectively.

By

increasing the inclination

angle further,

_we reach the regime where the bail _starts to

jump,

1-e- the bail

rolling

down

might

not be in contact with _{any one} of the balls

forming

the fine

(called regime

C). Such _a situation _is shown _m

figure

2 and the

length

of the

jumps

_seems to increase with time for small times. In this _case, the

phase boundary

also

depends

_on the initial

velocity

since _a

triangular

lattice is stable _up to _an

angle

of 30°

(and

8 is less thon 30°

).

For _a

high

initial

velocity

and for

RIT

= we get an

angle

of

20°(exp.)

and

17°(sim.).

On the base of numencal simulations, we found that this is

only

_a transition

regime:

for _very

long

times the

single

bail reaches _a constant

velocity

which _was two orders of

magnitude larger

than that obtained in

regime

B.

Unfortunately,

we do not see any chance to

verify

this

experimentally

smce fines

containing

_a few thousand balls would be

required

and _we will _naine it the

regime

N°8 MOTION OF A PARTICLE ON A BUMPY LINE l165

experimental data

20 C numerical data

8

B

io

A

0

0.5 1.5 2 2 5

R/r

Fig. 3. Comparison of the phase diagram for the experiment

(thin hnes)

and the numerical simula- tion

(thick fines).

A: decelerated _regime, B: constant velocity regime, C: accelerated regime

[umps).

of

increasing velocity (motivated by experimentally

accessible time behavior and to

distinguish

it from

regime B).

Figure

3 also shows that when the ratio

RIT

_~ 0, ^{the width} of the constant

velocity regime

B decreases since the size of the

valley

between two balls becomes

bigger expressed

in bail diameters 2R of the bail

rolling

down. This _means that the

trapping probability

increases for

RIT

_~ 0. On the other

hand,

for

RIT

> 1 _we found that the width of the constant

velocity regime

decreases when

RIT

increases.

The

experimental

points are connected

by

_a thin fine and the numerical

points by

_a thick fine _to

guide

the _eye. Both capture ^the essential features and for the

phase boundary

between

regimes

A and B the dilference is less thon 10$l for values of

RIT

> 1.2. In _our numerical

simulations,

the

bump

in the upper curve for

RIT

= 1.2 is not

significant.

Since the type of motion for the

single

bail

depends strongly

_on the initial

velocity

this

might

_{serve as} an

indication for the

magnitude

of the numerical _error.

Nevertheless,

_we believe that _{an even} better agreement con be achieved if _more accurate values for the simulation parameters

kn,

_in and _is could be obtained from

experiments.

In _our numerical

simulations,

_{we were} able _to

change

the

bumpiness

of the fine

by placing

the balls _a random distance apart ^from each other. We used 70 balls with

periodic boundary

conditions and the distance between _two of them _was chosen from _a uniform distribution

between 2R and

2R(1+ e).

We could therefore control the

bumpiness by

the parameter e

and _e

= 0 reflects the ordered _case discussed above. We find that the width of the constant

velocity regime

B also decreases when _e increases. This _is shown in

figure

4 where the dotted fine _was obtained

by

_a least square fit and for _{e >} 0.2 _we do not _see this regime any more.

When _{e increases,} the average hole _size between balls gets

larger

and the

trapping probability

is increased for fixed

RIT.

This leads to the fact that for

high enough

values of _{e, we} do not find _{a constant}

velocity

_regime before the bail gets

trapped.

This outlines the great ^influence of the disorder in this _case.

B ,*

14 1

*

12

io

0.00 0.04 0 os 0 12 0 16 0 20

Fig. 4. Phase diagram for _our numerical simulations for varying disorder of the fine. The dotted fines _{are a} least square fit and show that for

a finite disorder _{ec Cz} 0.2 the _constant velocity regime B

disappears completely.

3.2 CONSTANT vELociTIEs IN REGIME B. As _a first

comparison,

_we have

performed

_ex-

periments comparable

to those of reference _[8] but _we used _up to 400 balls. Moreover the radius of the balls

forming

the fine

(r)

and the radius of the

single

bail

rolling

down

(R)

_was

not

necessarily

the _sonne. We

investigated

the average

velocity

of the

single

bail in the constant

velocity regime

B for diiferent values of the inclination

angle

8. The measured results for four diiferent values of

RIT

_are shown in

figure

5. The

change

in

slope

of _{a curve} for _a

given

value of

RIT

for

higher

^{velocities indicates} the

beginning

^{of the} next

regime

C where the

partiale

starts to

jump.

This transition _seems to be smoother for smaller values of

RIT, making

the

detection of the

phase

boundaries _more difficult. It is also visible in

figure

7 at the end of the measured _curves. One also notes in

figure

5 that the

slope

of _{a curve} for _a fixed value of

RIT

becomes steeper ^when the

angle

decreases.

In

figure

6, _we show the data obtained

by

_our numencal simulations for the _same values of

RIT

and the parameters _{in =} 1000, _{i~ =} 100 and _a

= 1.5

Ii.e. using

_a Hertzian contact

force).

The

position

of the _{curve can} be fine tuned

by adjusting

the parameters kn, _in and _i~. The data _we present ⁱⁿ this and the next section _use values of the

right

order of

magnitude

since

exact values of the collision laws _are hard to get

experimentally. However,

m our numerical

model _we

neglected

^the

slight dependence

^of kn _on the _mass of trie

partiale. Keeping

_m mind that

by using

real

physical

units _{m our} numerical simulations _we do _not have _a free parameter

to

adjust

_our

length

and time scales the agreement ^between

figures

5 and 6 is remarkable. The

starting point

in sine of each _{curve agrees} within 15-20$l close _to

RIT

= and the diiference

m the initial

velocity

_is less than 15$l for ail _curves. For

higher

constant velocities _vm, trie differences become

bigger

but _we believe that _more accurate values for kn, _in and _i~ and the

right dependence

of kn _on the _mass of the balls will lead to _{an even} better agreement. One should also note that due _{to our}

expenmental

setup, ^{where the bail is forced} to roll _{on one} side of the

I<shaped

flume

(see Fig. l).

This constant

rubbing

leads _{to an} additional friction force not considered in _our numencal simulations. We tried _to determine the influence of this

N°8 MOTION OF A PARTICLE ON A BUMPY LINE 1167

20

R/r=0.S - R/r=1.0 +- 16 R/r=1.6 é- R/r=2.0 -

12

Vm

s

4

0

0 o-1 0.2 0.3 0.4

s1n8

Fig. 5. Variation of _{vm in}

cm/s

_{as a} function of sine for _our expenments ^{for different values of}

R/r.

30

R/r=0 8 -

R/r=1.0 +-

R/r=1.6 é- R/r=2.0 _-

20

Vm

io

0

0 01 0 2 0 3 0 4

sin 8

Fig. 6. Same _as figure 5 but _now for _our numerical simulations with _~tn

= 1000 and _~ts

= 100.

additional friction

by

_varymg the inclination

angles

_ox and _ay.

Unfortunately,

this could

only

be done _{m a} very narrow range ^to avoid that the bail would

jump

_over the borders in

regime

C and does not lead _to diiferent results. We also

thought

about _{using a}

U-shaped

flume similar to that in reference [8] but the additional collisions with the two vertical watts _seem to

be _{even worse.} We feel that _our model with better

adjusted

_{parameters can even} be used for

quantitative

_companson with

expenmental

data and believe that the contact forces _are well described

by

the

proposed

model.

4. Characterization of trie

general

motion.

4.1 MODEL _oF JAN ET AL.. In _a recent

experiment,

Jan et al. _[8] studied the motion of _a

single

bail _{on an} ordered fine

le

₌

0)

made of balls of the _same kind for three types ^{of balls and}

a fixed

roughness (RIT

₌

1). They

found the three

regimes

described above and

proposed

_a

theoretical

description

for the constant

velocity regime

B to find the

velocity

_{vm as a} function of the

experimental

pararneters. ^The

velocity

_{vm was} obtained

by writing

^{the balance} between the

potential

_energy diiference and the energy lost in inelastic collisions and

by

friction _on the

plane, along

_an

elementary path

ôx

= 2T defined _as the distance between two successive

collisions,

_or between two

neighboring

balls [8]:

mg2T

sin 8

=

kmv$

₊

_~tmg2T

_cos 8

,

(3)

where _m stands for the mass of the

bail,

_g the acceleration of

gravity,

k is the collision coefficient

(related

to coefficient of

restitution),

and _{~t a} friction coefficient. This equation leads to a constant

velocity

of the form

vm =

)~ (sin

8 ~t cos

8) (4)

By dividing

the

right

hand side

by

the left hand side in

equation (3)

_one is left with

g211Ù8

^~

^~~°~~~

^~~~

The first term _on the

right-hand

^side measures the collision part ^{and the second} term the friction ratio of the energy

dissipation.

Using

their

experimental

determination of the diiferent parameters in

equation là),

the authors found that each of these parts are

roughly equivalent,

the friction part

being larger

at small

angles,

the collision part at

large angles.

If _we want to compare these results with _our

experiments

_or numerical

simulations,

_we have to take into _account three facts

. the friction _{is a}

priori

_more considerable in the above descnbed

experimental

setup due to the two walls of the flume

. the _size of the system ^{is infinite in} our numencal simulations, and of the order of 400

sphere

diameters

(compared

_to 220 in Ref. [8]) ⁱⁿ _our

experiments

. our

numencally

^studied systems _can be disoTdeTed

The above fine of

thoughts, equations (3)-(5),

is also valid for

RIT #

1 but R does not enter the formulae. We therefore expect that

they

_are

only

valid close _to

RIT

= 1 and tested this

ansatz for the

velocity

_vm obtained in _our experiments ^and numencal simulations. We found

that it

only

worked well in the _range <

RIT

< 2. For smaller values of

RIT,

the theoretical

curve did not capture ^the

sharp

_increase in the

slope

when

decreasing

the inclination

angle

8.

Also for values of

RIT

greater ^than 2 the

predicted slope

was too steep.

Since we are interested in _a

general description

which captures ^{all of the observed} phenomena,

we have to extend the above model.

4.2 MODEL _oF JAEGER ET AL.. In 1990,

Jaeger

et ai. [9]

investigated

the forces

acting

_on

a

single grain moving

_{on a} ^{surface made}

by

other

grains.

No

assumption

about the

underlying

surface _was made and the

general

form of the forces

acting

_on the _grain is

mù=a-F, (6)

N°8 MOTION OF A PARTICLE ON A BUMPY LINE l169

where _m and _v stand for the _mass and the

velocity

of the

grain.

On the

flght

hand

side,

the first term a stands for the

driving force, usually gravity,

^{and the second} term is _a

general

friction force which accounts for ail forces

during grain-grain

contacts. The latter _is of the form

~

l

/flV~ ^~'~~~

' ~~~

where _a,

fl,i

_are material

dependent

parameters. ^For

high velocities,

_{we are} left with F _c~

v~ which _was

already

studied

by Bagnold iii

and for low velocities _one reaches trie solid friction

regime. By adjusting

the three parameters, _{one con}

smoothly interpolate

between

these

regimes.

If

~~

> l,

equation Ii)

has _a minimum at vo "

~~ l)Ifl

and _one gets

'f 'f

an unstable

region

for velocities 0 < v < vo This is the situation _we observe in _our

experiments

and numerical simulations. There _are two stable solutions for the

velocity

_vm and the final state

depends

_on the initial

velocity

of trie bail.

4.3 GENERALIzED _MODEL. In _our case, the

driving

force is

given by

a = mg sin8 and

we are

looking

at the constant

velocity regime

where ù

= 0. If _we consider the ordered _case

le

=

0),

_{we can} write _a

generalized

form of

equation (6) by looking

at the losses due to collisions and friction:

àp

a = mg sin 8

= F

=

j

àp

AN

"

W $

AN Al

"

~?~ S

_àt '

j8)

where N stands for the number of

collisions,

for the travelled distance and

àpi

is the

averaged

loss between two collisions

(àp

for AN

collisions).

One _sees

immediately

that

allât

= v and

in _{our case} for the ordered line

àNlàl

is constant since _we have _one collision _on each bail

forming

the fine in the stable

regime

^B

(which

_was checked

by

numencal

simulations).

Thus the

only

^variable which

might depend

_on the

driving

force _is

àpi.

The

problem

is _more

complex

in two dimensions: the term

àNlàl

_may

depend

_on

8,

the inclination

angle,

and also _on the smoothness

RIT.

We have found in 2D

[loi

that the friction coefficient _<

f

> = < F

>Im

_c~

(RIT)~~.~

but _{we were} unable to

explain

the exponent of -1.5

theoretically.

Is it connected with the

specific

contact laws

during collisions,

to the

disorder,

etc.?

In

figure

7, _we show

àpi

_{as a} function of the _constant

velocity

_vm for _two different ratios

RIT.

The

general

behavior is _as

expected

and for

RIT

= 1-fi, _{one sees}

clearly

the

beginmng

of the _regime which is

proportional

to _v

giving

a force F _c~ v~ since the loss due _to friction _is

negligible.

The

change

_m

slope

at the end of the _{curves is} due to the

beginning

of

regime

C.

In

figure

8, we show the _same quantities as above but for _our numerical simulations _using four values of the

roughness RIT.

For small velocities, _we could

verify by

_a

log-log plot

that

àpi

scales like

1Iv

and thus leads _{to a} behavior

resembling

solid friction in the limit _{v ~} 0.

This is

only

due to the geometry ^{of the}

problem

since _we do not account for static friction in

our force laws

(Eq. Il )).

For

high velocity,

the _same

procedure

reveals _a

regime proportional

to

v which is visible _m the

plots

for

RIT

= 1.6 and

RIT

= 2.0. For certain parameter _ranges

le-g- RIT

₌ 1-o in _our numerical

simulations),

_one finds _a wide region where

àp,

is

nearly

constant

which leads to _a

general

^friction force of F _{c~ v.} This _is the law _we found in _our expenments on a two-dimensional inclined

plane

with controlled

roughness ([10, 18])

and _we also showed that the friction force varies _as

(RIT)~~

^~ This law could not be derived from the geometry of the

JOURNAL DE PHYSIQUE T 4 N'8 AUGUST 19Q4 42

0.030

R/r=10 _+- R/r=1.6 ^é-

0 029

0 015

0.028

AP> APT

0.027

0.013 0 026

0.025

4 S 12 14 4 8 12

vm vm

Fig. 7. Averaged loss Ap~

lin

arbitrary

units)

between two collisions

(experiments).

0 042 0 028

R/r=0 8 - R/r=1.0 _+-

0.038

0.024 0.034

àp~ àp~

0.030

0.020

0.026

0.022 0.016

5 15 25 5 15 25

vm _vm

0.015 0.014

R/r=1.6 é- R/r=2.0 -

0.013 Ô.ol?

APT APT

o.oit o,oie

0.009 0.008

5 15 25 5 15 25

vm vm

Fig. 8. Same _as figure 7 but for data obtained by numerical simulations.

N°8 MOTION OF A PARTICLE ON A BUMPY LINE l171

problem

and _we believe that the contact forces

during partiale

collisions _are

responsible

for it.

To test this

hypothesis,

_we

investigated

with _our numerical simulations the _average loss

àpi

as a function of the

velocity

_vm

(see Figs.

7 and

8)

for diiferent values of the surface

roughness

in the

region RIT

E

[0.8,

2]. ^The

procedure

_{goes as} follows: _we

plot

ail four parts ^of

figure

8

on one

graph

and read off the values of

àpj

for _a

given

_vm for the various ratios

RIT.

We then make _a

log-log plot

of

àp,

_{as a} function of

RIT by averaging

_over different values of _vm and

obtain for _a Hertzian contact force

la

=

1.5)

a

scaling

of

àpi

_c~

(R/T)~~.~~°.~, regardless

of the parameters in and _is. For Hooke's law

la

₌

1),

_we obtain

àpi

_c~

(R/T)~~.°".~

which is

significantly

different. Therefore _we condude that the friction law

proportional

_to

(R/T)~~.~

found in

experiments

and numerical simulations is

essentially

caused

by

the exact form of the contact forces

during

collisions in _one and _two dimensions.

The

proposed writing

of the

generalized

friction force in

equation (8)

has the

advantage

^that

it _can

easily

be extended to the disordered _case

le

>

o)

_or to two dimensions. The

only quantity

one has to _measure or estimate is AN

/àl

to obtain the

averaged

loss of momentum between

two successive collisions which _can then be

compared

with the one-dimensional ordered _case studied above.

5. Conclusions.

We

investigated by

_means of

experiments

and numerical simulations the movement of _a

single

ball _{on a} one-dimensional line with controlled

roughness

and

possible

disorder in the _arrange- ment of the balls

forming

the line. We extended the

phase diagram

_given in

(loi

and found that in the disordered _case, the constant

velocity regime disappears

for _e > ec Cz 0.2. In _a

generalized model,

_we could

give

_an

expression

^{for the} _average ^loss

àpi

between two collisions which is valid in the ordered _as well _as in the disordered _case and in any dimension. We found

àpi

_c~

(R/T)~~.~

and the exponent

depends

_on the _contact forces

during

collisions.

Concerning

the

trapping

and

segregation

mechanism _on the surface of avalanches in two

dimensions,

it is clear from _our

investigations

_on this

simple

model that smaller

grains

have

a shorter _mean free

path

in _an avalanche and therefore get

trapped

in the middle

region

of

a

rotating

drum in the

long

_run. From the

phase diagram

in

figure

3, one sees that the stable

region

^of no motion extends to

higher

values of 8 for smaller

grains.

This is _{even more}

pronounced

if _a disordered surface _is considered

(see Fig. 4).

The _reason is _seen in

figure

8 where the motion in the constant

velocity regime

_{is very} close to the static friction limit for smaller ratios of

RIT (Figs. la)

and

16)) leading

to a

scaling

of

àpi

_c~

llv,

_as

opposed

to

figures (c)

and

Id).

Acknowledgments.

We would like to thank _our

colleagues

of the GMCM and

especially

Alex Hansen for _very inter-

esting

discussions about this work. We also

acknowledge

_support from the ATP CNRS Matéri-

a~~

Hétérogènes

^from the

Groupement

de Recherches CNRS

Physiq~e

des Milie~~

Hétérogènes Compie~es.

References

Ill

Bagnold R-A-, Froc. R. Soc. London A 225

(1954)

49.

[2] Jenkins J-T- and Savage S-B-, J. Fluid Mech. 130

(1983)

186;

Lun C.K.K., Savage S-B-, Jeffrey D.J. and Chepumiy N., J. Fluid Mech. 140

(1984)

223.

[3] ^Baxter G-W-, Behringer R-P-, Fagert T. and Johnson G-A-, Phys. Rev. Lent. 62

(1989)

2825;

Drake T.G., J. Geophys. Res. 95

(1990)

8681.

[4] ^{Ristow G-H-} and Herrmann H-J-, Density Patterns in Granular Media, HLM preprint 1993.

[Si Pôschel T., J. Phys. I £Fonce 4

(1994)

499;

Lee J., Phys. Rev. E 49

(1994)

281.

[fil Jullien R. and Meakin P., Nature 344

(1990)

425-427;

Williams J-C-, Ghem. Process. Suppl.

(1965);

Savage

S., Interparticle percolation and segregation in

granular

materials, Developments in En- gineering Mechanics, A.P.S. Selvadurai Ed.

(Elsevier

Science Pub. B-V- Amsterdam, 1987) [7] Ristow G-H-, Cantelaube-Lebec F. and Bideau D., Avalanches

in a two-dimensional rotating drum: experiment and simulation, preprint 1994;

Clément E., Duran J. and Rajchenbach J., Mélangeage d'un Matériau Granulaire dans _un Tam- bour rotatif, Congrès de Mécanique, Lille

(sept.

1993);

Baumann G., Jobs E. and Wolf D.E., Fractals 1

(1993)

767.

[8] Jan C.D., Shen H-W-, Ling C.H. and Chen C.I., Proc of the Ninth Conf of Eng. Mech

(1992)

pp. 768-771.

[9] Jaeger H-M-, Liu C.-H., Nagel S-R- and Witten T.A., Europhys. Lent. Il

(1990)

619.

[loi

Riguidel F-X-, Jullien R., Ristow G., Hansen A. and Bideau D., J. Phys. I France 4

(1994)

261.

Ill]

Tullifaro N-B- and Albano A.M., Am. J. Phys. 54

(1986)

939;

Duran J., Europhys. Lent. 17

(1992)

679.

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

(Clarendon

Press, Oxford, 1987) [13] ^{Cundall P. and Strack} O.D.L.,

Géotechnique

29

(1979)

47.

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

(1986)

239.

[15] Thompson P-A- and Grest G-S-, Phys. Rev. Lent. 67

(1991)

1751.

[16] Taguchi Y.-h., Phys. Re~. Lent. 69

(1992)

1367;

Gaffas J-A-C-, Herrmann H-J- and Sokolowski S., Phys. Rev. Lent. 69

(1992)

1371.

[17] ^Ristow G-H-, J. Phys. I France 2

(1992)

649; Int. J. Mod. Phys. G 3

(1992)1281.

[18] Riguidel F-X-, ^Hansen A. and Bideau D., Gravity-driven Motion of _a Particle _{on an} inclined Plane with controlled Roughness, prepnnt 1993.