Behaviour of a sphere on a rough inclined plane

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Behaviour of a sphere on a rough inclined plane

François-Xavier Riguidel, Remi Jullien, Gerald Ristow, Alex Hansen, Daniel Bideau

To cite this version:

François-Xavier Riguidel, Remi Jullien, Gerald Ristow, Alex Hansen, Daniel Bideau. Behaviour of a sphere on a rough inclined plane. Journal de Physique I, EDP Sciences, 1994, 4 (2), pp.261-272.

�10.1051/jp1:1994106�. �jpa-00246903�

Classification

Physics

AbstJ.acts

46.30 64.60 68.22

Behaviour of

_a

sphere

_{on a}

rough inclined plane

François-Xavier Riguidel (~),

Remi Jullien

(2),

Gerald H. Ristow

(1),

Alex Hansen

(')

and Daniel Bideau

(1)

(') Groupe

Matière Condensée et Matériaux

(*),

Université de Rennes I,

Campus

de Beaulieu, 35042 Rennes Cedex, France

(2) Laboratoire des Sciences des Matériaux Vitreux (~*). Université de Montpellier II, Case 069, Place

Eugène

Bataillon, 34095 Montpellier Cedex 5, France

(Received ii Jane J993, revised 7 September J993,

accepted

J2 October J993)

Abstract. The tlifferent states of _a bail _{on a} rough indmed plane are analyzed ^from

experimental

results and numencal simulations. The conditions of _static

equilibrium

are given. The control parameters are the inclination angle of the plane antl the surface roughness. Then, _{we propose a}

«phase

tliagram

» for the motion of the bail. These results _are expectetl to be useful _m

untlerstantling

the

ongin

of the

segregatîon

effects on the surface of _a

sandpile.

This paper is devoted to the

study

^of the different regimes goveming the motion of _one

particle

_{on an} indmed surface of controlled

roughness (or smoothness).

The

goal

of this work

is to reach _a better

understanding

of various

phenomena

_occurrmg at the surface of _a

flowing granular

medium,

especially segregation

_:

generally

if _one observes the surface of _a

granular

medium

(e.g,

a sand

pile),

the

Iarger grains

_are at the lower part, ^{while the smaller}

grains

_are at the top.

Une of the most

important

mechanisms of

segregatîon

is the surface flow

il -3]

: the grains

near the surface _{move on a}

rough

surface which is the

loyer below,

and the

roughness

_seen

by

a

grain depends

_on its own size

compared

_to the size of

bumps

^{beneath it. The}

forger

the

grain,

the less stable its

equilibrium position

when _not

moving,

and the

longer

^its _« mean free

path

_»

L

along

the

slope

when moving. This

explains why

one

always

finds the

Iargest grains

at the bottom of

piles. Experiments

_{using a} two-dimensional drum

[4]

confirrn that surface flow _{is an} important ^{mechanism of}

segregation.

The

equilibrium

of _a

sandpile

_is

generally

defined from the

angle of

_{repose 0~,} which is the

slope

of _a

pile

built

dynamically [5, 6]. However, starting

from this

angle,

such _{as in}

experiments

on

avalanches,

_{one can} define the _maximum

angle ofstability 0~ [5] (or angle of

(~) URA CNRS _n 804.

(.~*) URA CNRS n° II19.

JOLRNAL D~ PHYSIQUE T 4, N' 2 FEBRUARY <994

niaJ.gmctl stability [6])

_as the maximum

angle

for which the

pile

is Table. For

examp[e,

this

angle

_may be measured

by tilting

the

pile

until _an avalanche _occurs.

However,

according

to the initial conditions and the

angle

o between the inclined

plane

and the horizontal, the motion and the

equilibrium

of _a

particle

_{on a}

rough

inclined surface i~ not a

simple problem

to describe

[7, 8].

Dur

simplified

model pennits a formai and

relati~ely complete investigation

of this

problem (roughness

and inclination

angle

_are both control

parameters),

_even if _a direct

analogy

with the surface of the

sandpile

_is non very strong. ^If one i~

only

concemed

by

the static behaviour, 1-e-

by studying

the motion of the bail without _any initial energy, we can define _a

stability

threshold 0~. ^For _~

o,,

the hall _is stable,

trapped

in a

well _at the surface.

Agam

with respect to

equihbrium

of

piles, 0,

is a

locally

defined

angle

of

maximum

stability, depending mamly

on the geometry of the grains ^{and that} of the ~y;tem ^and

to a fesser

degree

on the friction coefficient.

If _one gwes an initial kinetic energy ^to the hall, the

problem

becomes different. The hall

moves on the surface _{m a} sort of directed random walk.

Starting

with _an initial energy, the

motion is first decelerated and, for small

enough

H, it stops. ^For

larger

value~ of 0, the hall reache~ _{a constant}

speed,

such _{as m} the sedimentation of _one

partide

submitted _to Stokes

drag

force. When the

angle

increases above _a threshold value

(,

^{the hall}

jumps

on the

plane,

and the scale of the motion is

completely changed.

This

regime

_seems to be _a transient one : the numefical simulations show thon after _a very

long

lime, a new stationary _regime i~ reached,

with _{a constant}

speed

and small

jumps.

The paper is

orgamzed

in the

following

_{manner in} the first _{section we} pre~ent ^{the resu[t~ of} the

experimenta[

and the numencal methods _we used in this

study.

The second ~ection

describes _our results concerning static behaviour, whereas we present ⁱⁿ section three _a

<~

phase diagram

_{» conceming} the different kinds of behaviour observed in the

dynarnic

_ca~e.

1.

Experimental

and numencal system.

We _{want to} model the surface of _{a «}

sandpile

_)>, ^but with control parameters which _{are the}

slope

of the

pile,

and the

roughness

of it~ surface. In order to do this, _{we use an} inc[ined

glas~

plane

P (one meter

long,

70 _cm wide, and _cm

thick),

which _can have _a variable

angle

i&.ith the horizontal. The

roughness

of the

plane

_is obtained

by ~ticking

smooth

glas~ spheres

(of radius _1± 0.1 1) _on contact paper fixed _on the

plane. Figure

shows _a part ^of a

rough

~urface

Fig.

l Photograph of _a pan ^of a rough ^{surface obtainetl} by sticking gla,~ ^beatl~ (tliametei 0.5 _{mm on}

a

plane.

The obtainetl

packmg

fraction _is C 0.7.

obtained

by

this method. The two-dimensional

(2D) packing

is

homogeneous,

with _a

packing

fraction C 0.70-0.75. This is far from the usual

packing

fraction for dense disordered 2D

systems (C 0.82) [9],

^but seems to

give

_a

roughness comparable

to that observed _at the surface of _a

pile.

Note that this 2D

packing

_{is a} disordered _one. The radius of the

spheres moving

down the

plane

P is

R,

and _we define here the smoothness 4l of the

plane

as the ratio Rli

(the roughness

in this

particular

case

being

the ratio

ilR).

In one-dimensional

ID)

numerical

simulations,

_we build an indmed _row of

equal spheres,

whose distance between

neighbours

_con be

randomly

fixed between 2 _r

(touching spheres)

and

2>.(1+ e):

_see

figure

2. This mimics the disorder observed in 2D

experiments

and simulations. The motion of _one ball _on this inclined _row is studied

by

_using molecular

dynamics

simulations

developed by

_one of _us

[loi.

The hall _{can rotate} and _we consider

dissipation

and

dynamic

friction

during

collisions. As _an

integration scheme,

_{we use} a third- order

predictor-corrector algorithm [1Il.

Periodic

boundary

conditions _are

imposed,

in such _a way that

long

distance _{motion con} be studied.

8

Fig. 2. ID _motion of _one ball _{on an} inclined line. The roughness of the litre is createtl by placing balls

at rantlom _or equally tlistanced positions. In the rantlom _case, the distance between neighbours _is

randomly

chosen between 2 _r

(touching

spheres) and 4 _r.

From the 2D numerical

point

^of _view, the ensemble of

equal

sized small

spheres

stuck _on the

plane

P is modeled

using

the random

sequential adsorption

model m two dimensions

(RSA) just

^below its

jamming

threshold, which is at a

packing

^{fraction of} _{C~ =} 0,547

[12].

In

figure

3 _we show an

example

of such _a

configuration,

with _a

packing

^{fraction of}

C

=

0.52,

built _{on a} square of

edge length

L

=

180 _1, where _r is the radius of the small

sphere,

with

periodic boundary

conditions at the

edges

^of the square. The

geometrical properties

of such RSA

packing

near the

jamming

threshold have been

extensively

studied in the past

[12].

We have

only reproduced

here the

computations

that _are necessary for the purpose of _our

study.

We simulate the motion of _a

larger sphere by

_assuming that it

always

_stays in contact with at least _two smaller

spheres dunng

^its entire

trajectory (this

_assumes that the

angle

0

should not be too

large

m order _to prevent ^the

sphere

from

jumping).

In this scheme, the details

of the _contact force between the moving

sphere

and the surface

spheres

are

unimportant.

However, one can assume

sufficiently large

friction to make sure that the

rolhng sphere strictly

follows the

edges

^{of the} Voronoi

polygons,

m

particular

when it reaches _{a vertex} where it has to

change

its direction quasi

instantaneously. Consequently,

_m

projection

on P, the

trajectory

of the center of the

larger sphere

is made of _successive segments ^which are all

edges

of

Voronoi cells

[12]

of the RSA

configuration

of the small

spheres. Figure

4 shows the Voronoi cells

corresponding

to the

configuration

of

figure

3.

2. Static results _:

stability

threshold.

We studied 0~, ^the «

stability

_» threshold first _: it defines the limit of

stability

of _a grain on the

surface,

and

depends

_on the disorder of the 2D

packing

stuck on the

plane

and _on the

smoothness 4l.

Fig. ^3.

Fig.

4.

Fig. 3. A typical RSA

configuration

obtainetl by numerical simulations with C

= 0.52 (L

= 180 _J

Fig. 4. Set of Voronoi ce[ls correspontling to the RSA

configuration

~hown _m figure 3.

Consider first the

problem

from _a theoretical and numencal point of view _in two dimensions.

At each vertex, i-e- at each

point

_i common to three Voronoi cells, where the

langer sphere

i~ in

contact with three smaller

spheres

of centers A, B, C (see

Fig. 5),

the

large ~phere

has to

choo~e its way

according

to steepest ^{descent rules. This} can be determined

by ca[cu[ating,

at

the _vertex point, ^the

respective slopes

of the

possible

further

trajectofies

of the

large

ball

center. Let _us calculate this

slope

for the motion in _contact with

spheres

A and B. We call _x the

unit _vector

defimng

the steepest ^descent on P, _y the unit vector

defining

the

in-plane

horizontal

direction, and _z

= x x y

perpendicular

to P. In

projection

_on P, the direction of _{motion i~}

defined

by

_a unit _{vector u,}

perpendicular

to the segment ^{AB. The} unit vector t tangent, at the

vertex point, ^{the further}

trajectory

of the

large

bail _{center, is} given

by

t ₌ Cos qJu + sin ~Rz

z

~ t

tan~'zj

rll+©1

L ~ C

ti

Fig.

where the

angle

~R, indicated in

figure 5,

is

given by

~~~ ~

N/(l

+

~)2

r~ p~

~~

where h is the distance between the _vertex point ^{and the} segment

AB,

and _p is the radius of the circle which is circumscribed

by

ABC. As _a function of _{x, y} and _{z, t} is given

by

t ₌ u~ Cos ~Rx + u~ ^{cas ~Ry +} sin _~Rz

(2)

and,

using

now the _«

laboratory

_» set of unit vectors X, _y and Z

(X being

horizontal and Z

vertical),

it is

given by

t ₌

(u~

Cos _~R Cos à + sm _~R sin 0

)

X ₊ uy Cos ~Ry ⁺

(-

_u~ Cos _~R sin à ₊ sin

~R Cos 0 Z

(3)

Consequently

the

slope

_a~ of t is

given by

u~ cos ~R sin 0 + sin

~R Cos 0

a~ =

(4)

u~ cos _~R cos 0 + sin q~ sm 0

Une _can write

tx~ = tan

(&

à

) (5)

with

tan &

= tan _~R

(6)

u~

8

might

be

mterpreted

_{as a} local threshold

angle.

If the actual inclination ois

larger

than

8,

the

rolhng sphere

_can roll _over AB.

Using

_equation

(I)

and h

=

iu,

where 1 is the

algebraic length

of iL

(see Fig. 5),

one gets

tan &

~/~i

+

~~

~.~

~~

~7)

If it is released at the _vertex

point

without kinetic _energy, the

Iarger sphere

is able _to roll _over

AB if _a~ SO, 1-e- if MS 0. At each vertex _i, we define _{r~ as}

being

the minimum of

tan

Mi,

tan

8~,

tan

8~,

where

Mi, 8~, 8~

are the three

possible angles

⁸

(rolhng

over AB, BC

or

CA).

_{If r~ is} smaller than tan 0, _a

large sphere

released _{at vertex} without kinetic energy will roll

naturally

_on the

couple

of small

spheres

^which

correspond

to the minimum

angle

^{8. If} r~ is

Iarger

than tan 0, the vertex i

corresponds

to a stable

position

of the

large sphere.

In

figure 6,

_we have

represented

all the

steepest

^descent

trajectories

_starting from all the vertices

(they

_are the true

trajectories

for tan 0 _m max~ r~. For smaller 0 values,

only

some of these

trajectories

_are

actually

used _smce the

rolling

hall _can be

trapped

at some

vertices). They

look like _a network of rivers and tributaries with main _avers closed _{on to} themselves due to

penodic boundary

conditions.

In _our numerical

calculations, starting

from _an initial RSA

configuration

_we have

determined the

r,'s corresponding

to all vertices of the

corresponding

Voronoi network. Then

we have calculated the

quantity p(0 )

which

corresponds

to the fraction of r~ values smaller than _a given tan 0. When comparing with _an experiment,

p(0

also represents ^the fraction of balls which _are able to move when

they

_are released

anywhere

on P without initial kinetic

energy. It is easy to define

stability

thresholds from these distributions _{: it is} the

angle

for which

p(0)

=

0.5,

_{as m} the experiment. ^{The results} are given in

figure

7. Une _notices that

p(0

^does net staff from p =

0 for 0

=

0. This _is due to the existence of _some vertices which

Fig.

6. Ail ~teepe,t descent trajectories on Voronoi cell

edge~

for the RSA configuration ,hown _in figure 3.

° lt/r=2

~

. lt/r=4 Ù,4

,

° lt/r=16

0,2

0 10 20 30 40 50 60

8

Fig. 7. Curves giving p(0) _{as a} function of à.

are «

naturally

_» un~table, _even for

= 0 such vertices are located outside their

carre,ponding

ABC

triangles.

All the results that are

reported

here have been obtained with L

=

1801.

However, we have

performed

other calculation~ with different L values and _we ha,,e checketl that for such

large

~ystems, ^{the numefical results} are quite

mdependent

_on the size _as fat _{a~ we} stay

sufficiently

close _to the

jamming

threshold.

Experimentally,

H, is determmed _on average as follows _{: at each} experiment, ⁵⁰ stee[

spheres

(with diameters very well defined) are

placed

on the top ^of the (horizontal)

plane,

each of them

being

in stable

equilibrium

m a _« cavity _» ^formed

by

three smal[

glass sphere~

in

contact. This

explams

that _{is is not}

possible

to

explore expenmentally

the non-stable _situations

descfibed

just

above _: we must start with 50 halls _in stable

equflibrium.

From this initial situation, ois

progressively increased,

until _one

sphere

_moves, and falls down

along

^the

plane,

and _so on until all the 50

spheres

are down, the expenment is done again.

Finally, by

_summing

six expenments (300

halls),

_we have _a

(Gaussian)

distribution of limit

equi[ibrium angles.

Figure

8

gives

this distribution for

4l=4,

and r=0.25 _mm the _mean value defines ô~. ^This distribution becomes wider antl wider when the smoothness

decreases,

as shown _in

figure

9 which gwes the standard deviation of these distributions _versus the smoothness

4l,

in _a

log-log plot,

for r ₌ 0.25 _{mm, r}

=

0.5 _mm and _r

= mm.

N _' r=lmm

° Î ° r=0,smm

é'oà

° r=0~25mm

. @

O

~

0 10 20 30 40 1 10 ioo

O

Fig.

8.

Fig.

9.

Fig.

8. Distribution of the

stability angle

(~P ₌ 4, antl _r

= 0.25 mm) as the resu[t of six elementary experiments (300 balls).

Fig.

9. Variations of the standartl-tleviation of the distribution of the static angle with the smoothness

~P.

Figure

10 shows _{m a}

log-log plot

the variation of _H~

(normalized by

its value for 4l

=

2)

versus the smoothness 4l.

Experimental

and numencal values _are well fitted

by

_{a unique curve}

corresponding

to a power law _{H~ oz} 4l

~, with _an effective exponent x =

0. 8. When companng

simulations with the expenments, ^the most

important approximation

_{appears to} be the

~~i

° Exp (r ₌ 0,25 mm)

' Exp (r ₌ 0,5 mm)

- ^. Exp(r=lmm)

C~

,

. simulations (C

= 0,52)

~°' _~~o

a

° simulations (C ₌ 0,59)

~ o ,

ce

j

W ^° _o

à $

o .

o

lo~

1o° 1o~ o~

m

Fig.

10.

Log-Log

plot ^{of the} variations of the stability thresholtl _H~ (normahzetl by its value for ~fi 21 for tlifferent _sizes of stuck balls (expenments) or

packing

fraction C on the

plane

(2D numencal

simulations). All the results coalesce _{on a} smgle _curve well fittetl by a power law _{H~ cc} ~P~~ with _an exponent x = 0.8 _± 0.05.

replacement

of the

rough plane by

an RSA

configuration.

When

devising

the

rough plane,

the

sticking

_process is

certainly

more

complicated

thon the

simple

rules of the RSA model. In

particular

due to the

unphysical

constraints of the RSA model, _{we are} limited to _a

quite

low

concentration

(lower

thon

0.547)

while

expefimental

concentration _is of order of 0.7. We have

also

perforrned

similar calculations

using

_{a more} realistic RSA model. in which ~nme

restructuring

is allowed

[1?].

With such _a model _{one con} reach

higher

densities of small

spheres

since the

jamming

concentration _is

larger

(C _m 0.61). The values obtained _are

only slightly

srnaller than the previous ones, but

collapse

_on the _{Sartre curve} when normalized

by H,(2)

as can be _seen in

figure

10. Une _can also believe that

finding

0.8 and _{not is} due to the fact that _{we are} in the _zone in which corrections to

~caling

are important (and this _zone grows

when the

packing

fraction increases), see

equation

(7).

Another

interesting

^result concerns the _maximum value _z~

= tg _H~, ^of the

T,'s,

to which

corresponds H~

such that pi ₌ I for

m

H~.

Ii _can be calculated

exactly

at the

jamming configuration

in the asymptotic ^{hmit of} an

infinitely large plane.

This should

corre~pond

to a

vertex with the

largest

_p value, which _{is p}

=

2 _r at

jamming,

and with _a

configuration

a; that

represented

in

figure

11, i&.here the

large sphere

should roll _{over a} ~mal[

sphere ju~t

_in front of it. Thus

=

2 _r and _one finds

T~ = tg

H~

₌ ^~ j8)

, (4S -1)(4l + 3)

à

A

r r

x

c

Fig.

Il Sketch of _a

configuration

correspontling to ₇

= 7~.

Numerical simulations agree very well with this formula, but there _{are some} differences with the experiment, ^{whose results} are

presented

in

figure

12

compared

to theoretical values. A

linear variation _is also observed

expenmental[y,

but _tan

H~

is not zero at the

ongin

if _we

extrapolate

the

expenmental

^{results. This} _is due _to the contribution of friction, and

eventually

of _attractive interactions between the ball and the stuck

glass spheres (capillary

forces..

these forces _ensure the

stability

of the hall _on non-horizontal

plane

(smal[ _H) even for infinite smoothness.

3.

Dynamical

behaviour,

phase diagram.

In the above section, we have defined _a

stability

threshold H~. which is

completely

^defined

by

static conditions,

mamly geometncal

_ones. We _are here concemed with _a

global analysis

of the different

dynamica[

_regimes, and _on its

dependence

_on the geometry, i-e- on the smoothness of the

plane.

In these expenments ^{and in the lD numerica[} simulations, the bail is thrown on the

3

R/r

' D

~im.ÎÎ

, ^BIC

° . Exn(r

m 0.25 mm)

° Exn(r_mO.Smm)

. . Exp (rm mm)

~

a N1Im.olm. ~

o 50 ioo iso o ~ ~~ ~~ ~~

~'~~"

~

Fig.

12.

Fig,

13.

Fig. 12, Variations of tg H~ versus 2/

,~

_according

to

equation

(8). The

straight

fine is the theoretical

prediction.

Fig. 13. ID phase tliagram obtainetl from numencal simulations (for _s 0).

plane

with _a small initial kînetic energy : we use a

drawing

straw

weakly

inclined with respect

to the

plane

as

launching

_ramp. The

length

necessary to

forget

this initial kinetic energy is found

experimentally

to be very small

compared

to the

plane length

(a few

centimeters).

We have to compare here the static threshold _H~ with the

dynamic

^threshold _H~, ^{which is defined in} this section. But _our

goal

is _to give a

phase diagram

first

descnbing

the different

dynamical

behaviour of the ball _{on a}

rough

surface _or

hne,

with control parameters ^which are the

angle

and the smoothness of the surface 4l. The ID

(numencal)

and 2D

phase diagram

are obtained.

Figure

13 shows the 1D

phase diagram

for _no disorder

(e

=

0).

The _contact conditions in this

example

are the

following

_: the coefficient of restitution _in this simulation is 0.5. For small values of 4~

(~ 0.8),

the behaviour of the ball is

dependent

on its initial

velocity.

^{For 4~} _~ O.8, the

plane

is divided in three parts,

according

_to the values of H. For small values of the inclination

angle (region A),

the _{motion is} decelerated. For

angles roughly

between 5° and

15°,

a constant

velocity

motion _is observed

(region B),

whereas for

larger angles (region D),

the ball

velocity

_increases

leading

to motions _in which steps are

larger

than the minimum _one

defined

by

two

neighbounng

balls thus

leading

to

jumps.

The situation observed in 2D

experiments

is very

comparable. Figure

14

gives

the

phase

diagram

obtained, also with 4~ and _as control parameters. ^{In the A} region, the motion _is decelerated _: the initial kinetic energy is

quickly

lost. In the B region, a constant

speed

is observed, but the _mean free

path (see

the

quantitative

definition

below)

is smaller than the size of the table

(2 m)

_: the ball stops on the

plane.

In the C region, the

speed

is constant, but the

mean free

path

is

larger

than the table

length.

In the D region,

comparable

to its

equivalent

in the lD

diagram,

the bail

jumps,

^and the scale of the motion _is

completely changed.

We are

essentially

concerned with the constant

speed

_regime. A

complete analysis

^{of the}

dynamics

of this

problem, including

statistical aspects, is

given

elsewhere

[7]. Briefly,

the motion _is the result of _a balance between the gravity ^{force and} a « viscous » force integrating

«

microscopic

_» collisions _{: so,} the differential

equation descnbing

the average

(1.e. neglecting fluctuations)

_motion of the ball is identical to that of the Stokes sedimentation of _a

sphere

m a

fluid. It is clear

expenmentally

that this stable motion _is

suddenly stopped

(not

decelerated) by

a

large

^hole m the stuck

packing. According

to this observation, if _one wants to descnbe the behaviour of the ball

completely,

we must add to the constant motion

description

_a

probabihty

e~a~

fllr 2,8 ~"a~

D @~a~~

a~

a~

z ~~am~

c ea~

w a~~

© @~

m~~

A

~

0 4 a 2 16 20 40 60 80 100

8 1*

Fig.

14.

Fig.

15.

Fig. 14. 2D phase tliagram obtained from experiments.

Fig.

15.

Semilog

plot of the cumulatetl distribution N

(1

m 1* j of the stoppmg distance (1.e. number of spheres whose stoppmg ^distance Î is

larger

than _a given value

Î")

obtamed under the followmg

conditions R

= mm, 1 = 0.25 _mm,

= 4.7°, total number of spheres : 300.

for the ball _to be

trapped

in _a well. This

probabihty

is the _same

everywhere

on the

plane

since it

is

supposed

to be

homogeneous,

and _{as a} consequence, the distribution of the stoppmg distance

1 (projection

of the distance covered

by

the ball before

stopping

_on the

largest slope line)

for

given

values of and 4l _must be

exponential.

This is what we observe, as shown in

figure

15. which is a

semilog plot

of the cumulated distribution of

1.

The model descnbed

just

above allows us to make _some

quantitative predictions

_concemmg

the

dynamical

parameters

[7].

The _mean

velocity

is

given by

~~

~~(ÎÎ

^{~ ~~~}

where

f)

is the _mean value of the

(fluctuating)

Stokes constant, and _m the _mass of the

rolhng

ball. We find that

(f)

varies with 4l _as 4~+~~~.

Combining

the observed Stokes motion of the ball, and the _use of _extreme statistics _to

analyse

the effects of traps on

1,

_{we propose}

[7]

the

following expression

^for the mean free

path

L _on the

plane,

defined _as the average

characteristic value

of1

_{determined from} distributions similar to that shown _m

figure15 (L

~

(i)

L

(H,

^4l ₌

Lo

^e~~~ =

Lo e~'

^~~~~~~~

(10)

Table I

gives

different values of this _mean free

path

L obtained from different values of and of the smoothness 4l. It shows in

particular

the very strong

dependence

^{of L} on the

angle

^and the

smoothness 4l: half _a

degree

_is sufficient _to

multiply

L

by

⁵

Starting

from the distributions of the stoppmg ^distance

1,

_we also define the

dynamical

threshold H~ as

being

the

angle

for which half of the balls have _a stoppmg ^distance

larger

than the

plane length. Figure16

shows the variations of H~

(norrnahzed by

its value for 4l

=

2)

»eisus the smoothness 4l, for three different values of _r. The three curves coalesce _{m a} well

defined power law H~ =

4l~",

where _y

=

1.55 _± 0.05.

Finally,

it is clear that _{H~ is}

always

larger

than H~.

Table I.

Dijfierent

values

of

the _mean

free path

obtained

f>.om dijfierent

values

of

and

of

the smaothness 4l,

expi"essed

in _cm. The table

Îength

is 200 _cm.

4l

(°)

4 5 6 8 lO

31

1,3 29 95

1,7 ¹³⁶ _~ ²⁰⁰

2 45

~ 200

2,5 18 139

3 81

~ 200

3,5

10 ~200

4 54

4,5 143

5 ~ 200

,~i

o

,o° o ^" ~*o,25mm

~

° r₌ O,5 mm

. * rm mm

m .

'

10'

~ o

4l

Fig. ^16.

Log-log

plot _givmg the variations of the dynamic threshold (normalized by its value for

~P

= 2) _ver.sus the smoothness for three values of _r.

4. Conclusion.

Surface

phenomena

are of great

importance

_m

rapid granular

flow. This paper is a contribution to basic

understanding

^of what _are the relevant parameters in the _motion of

partiales

_on the

surface of _{a «}

sandpile

_», with _a

particular

attention to segregation ^effects

by

surface flow. The

results obtained here outline the fact that _{on a}

rough surface,

and

especially

_on the surface of _a

sandpile,

the

equîhbrium

and the motion of _one

grain

is very

dependent

on its size

compared

to the surface

roughness.

The extreme

dependence

of the different thresholds and of the _mean free

path

on the size of the

partiale

is at the

origin

of surface

segregation

: even if _our results _are obtained for mdividual

partiales, they explain why

the

large grains

are at the bottom of the

pile (not trapped by roughness),

_as observed

by

numerical simulations

by

_one of _us

[2].

These

results _must be

completed by

_{a more} accurate

analysis

of the

dynamical

behaviour of

spheres

on a

rough

surface, and of collective

effects,

_{now m progress at} Rennes.

Acknowledgements.

We would like to

acknowledge

_our

colleagues

of the

laboratory,

and S. Roux for very useful discussions. This work _was

supported

in part

by

the CNRS ATP _« Matériaux

Hétérogènes

_»,

and

by

^the

Groupement

de Recherche CNRS

«

Physique

des Milieux

Hétérogènes Complex-

es »

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