Recurrent clogging and density waves in granular material flowing through a narrow pipe

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Recurrent clogging and density waves in granular material flowing through a narrow pipe

Thorsten Pöschel

To cite this version:

Thorsten Pöschel. Recurrent clogging and density waves in granular material flowing through a narrow

pipe. Journal de Physique I, EDP Sciences, 1994, 4 (4), pp.499-506. �10.1051/jp1:1994155�. �jpa-

00246925�

Classification Physics Abstracts

05.60 47.25 46.10 02.60

Recurrent clogging and density

_waves

ⁱⁿ granular material flowing through

_{a narrow}

pipe

Thorsten Pôschel

HLRZ, KFA Jülich, Postfach 1913, D-52425 Jülich, Germany and

Humboldt-Universitàt _zu Berlin, FB Physik, Institut für Theoretische Physik, Invalidenstrafle 110, D-10115 Berlin, Germany

(Received

29 November1993, received in final form l7December 1993, accepted 7

January1994)

Abstract. We report _on density _{waves m} granular material, investigated both experimen- tally and numerically. When granular material faits through _a long _narrow pipe _one observes

recurrent cloggmg. The kinetic _energy of the

falling

partiales increases _up to _a characteristic

threshold corresponding to the onset of recurrent clogging and density _waves of _no definite _wave- length. The distances between _regions of high density depend strongly _on the initial conditions.

They _vary irregularly without _any characteristic time and length scale. The partiale-flow _was investigated using 2D Molecular Dynamics simulations. Experimental investigations lead to

equivalent results.

The

astonishing

^{elfects observed} in

moving granular

material bave been

subjects

^of interest for _at least 200 years.

Examples

of such eflects _are trie

angle

of _repose

il,

_2] size

segregation [3-5], heap

^formation on

vibrating

media

[6-9], density

_{waves [10]} and

granular

fiow down in-

dined surfaces

[11, 12].

^Trie

origin

of these eflects _is that

granular

material _can behave like _a solid _or like _a

liquid, depending

_on its

density

_[13].

Particularly,

in recent times

dynarnic

_as

well _as static behaviour of

granular

mater1al bas been

investigated by

_many authors

experi- mentally

and

theoretically

with various

techniques

either

analytical,

such _as

thermodynamic

[14] ^and

hydrodynamic

_[15]

approaches,

_or numericallike cellular automata [16], Monte-Carlo

simulation _[4] and molecular

dynamics

_{[3, 8,}

17].

Even random walk

approaches

bave been

proposed

_[18].

Trie atm of this Letter is to report trie eflect that

granular

mater1al like sand that falls

through

_a vertical _narrow

pipe

_or

capillary

with _a diameter of

only

^few

partiale-diameters changes

^from a

homogeneous

to _an

inhomogeneous

flow when trie kinetic _energy

Ek

of trie

falling grains

reaches _a characteristic threshold. In this _{state one} observes _recurrent

dogging

of trie

partide

flow

(stick slip motion)

and

density

_waves. We simulated this behaviour

using

molecular

dynamics

and demonstrate that trie results _agree with _our

experimenta1observations.

500 JOURNAL DE PHYSIQUE I N°4

The

experiments

were carried out in two vertical fixed

glass-pipes

^of diameters Dl ₌ 2 _mm

and D2 _" 4 _mm and

length

L

= 1.4 _m. An _upper funnel contained

enough granular

material _to

ensure

time-independent

initial conditions.

During

trie experiments _very

fine-grained

sand of

typical partiale

diameter

Dg

₌ ^0.18 mm

begins

to flow

homogeneously

with trie

velocity

_{u =} 0 at trie _upper end of trie pipe. After _a

typical

distance of

approximately

20 _cm trie uniform flow becomes unstable and _tutus into _a fiow of _recurrent

clogging.

In this

regime density

waves can be observed.

Figure

1 shows trie

pipes

with

typical density

distributions. Trie

distances between dark

high-density regions

_vary

irregularly.

There is _no definite

wavelength.

Trie distance between trie top ^{of trie tubes} and trie _onset of trie stick

slip

motion _as well _as for trie distances between trie

plugs depends sensitively

_on ^trie

humidity

of trie air and varies

significantly

from _one

experiment

to trie next _one, 1-e-

depend

_on initial conditions. Similar

experiments

for trie fiow of sand

through hourglasses

^bave been described in [19]. ^As m trie _case of _our numerical simulations

they

found

clogging,

trie distribution of trie distances between trie

high-density

_regions

obeys

_{a power} law.

in fact _we bave doue trie simulations first and trie

reported

experiments were intended to check trie accuracy of trie results

predicted by

trie numerical simulations.

According

to trie

experiments reported

above _we

investigated numerically

the flow of

sphencal- shaped grains

with rotational

degrees

of freedom

falling

under

gravity through

_a pipe of diam- eter D

(Fig. 2)

_using 2D molecular

dynamics

simulations. Trie

length

of trie

pipe

considered

was much

larger

trier its diameter L _» D. We

investigated

systems of

partiales

of

equal

radii r~ = R _as well _as of

partiales

with _a Gaussian

probability

distribution for trie radii with _mean

value R

(Fig. 3).

For trie fiow _we assumed

periodic boundary

conditions. Trie inner surface of trie

pipe

is build of smaller

spheres

with _a diameter _rs

=

)

R of trie _same materai _as trie _grains to simulate _a

rougir

surface. Since _our model mcludes elastic and inelastic interaction between tl~e _grains eacl~ otl~er and between trie

grains

and tl~e watt _we simulated tl~e watt

exactly

_in tl~e

same mariner as trie

grains

but tl~e

partiales

tl~e

pipe

is build of _were neither allowed _{to move}

nor to rotate.

Trie

grains

with _{mass m~ were} accelerated due to

gravity (g

₌ -9.81

m/s~,

their

density

~~~Î~3

~ ~~~ ~° ~~~~~~'

Fz _{= -mi g}

Ii)

where Fz _is tl~e force

acting

_upon trie itl~

partiale

in trie direction of trie axis of trie

pipe.

Starting

with

given

initial

positions

of trie

particles

and

randomly

chosen small initial hnear and

angular

velocities trie

dynamics

of tl~e system was determined

by mtegrating

Newtons

equation

of motion

numerically

for eacl~

particle

for ail furtl~er _time steps. For tl~e

integration

we used _a sixtl~ order

predictor-corrector

method for trie

partiale positions

and _a fourtl~ order for tl~e calculation of tl~e

partiales' angular

movement [20].

In _our model

partiales

are allowed to penetrate eacl~ otl~er

sligl~tly.

Tl~e force acting ^between

two

partiales

and j is assumed to be _{zero m case} tl~e

partiales

do _net toucl~ eacl~

otl~er,

1-e- when trie distance between their centres _is

larger

then trie _sum of trie

partiale

radii _{r~ +} r~.

Otherwise trie force separates into normal and shear force

(Eq. (2)).

~ ₌

FN ii

_{+ FS}

_Il

_&~)

ii

_if

_{(xi x~l}

< ri + r~

" ~~j

0 otherwise

with

FN _" kN (r~ + r~ (x~ x~ ()~.~ ~iN

m~~(k~ k~) (3)

I

j

i

'

'

' i f

'

j <

~ ~

j

(

502 JOURNAL DE PHYSIQUE I N°4

acceleration due to

gravity

_- D

L

Fig.

2. The _pipe. The rough surface consists of spheres with diameter _rs

=

j

R of the _same

material

as the _grains.

Fig.

^3. Emargement of _a small part ^{of the} pipe filled with grains of dilferent radii. Gravity acts

horizontally from left to right.

and

FS =

mm(-'fs

_{me~ urei P' (FN}

Î)

_~~~

where

meoE "

~~

₍₆₎

m~ +Î~~

The

erms

itl~ partiale.

equation

(3) includes tl~e Hertzian contact force wl~icl~ rises

witl~

tl~e _power

I.à

of

trie

penetration

depth r~ + r~ _- (x~ - x~( of two _partiales. It

of tl~e grains

[21].

This nsatz

for force

ggested in [22j

and

ligl~tly

dified in _[3].

quation 4 _takes

into

_account, that trie aximum

momentum two

partiales

are able ^to

transmit

~vhile

is

determmed by trie

Coulomb friction law iii.

p

_is trie _of

trie

partiales.

To

discuss

lengtl~ of tl~e pipe

was

L = ₆₆₆ . R _and

ils widtl~

D

=

was

~is " 3 x 10~ s~~, trie _riction oefficient ~iN " 3

x 0~ tl~e material constant

kN = lob )

and

p

=

o-S- For trie integration

lime ^{step as hosen} lit

=

_10~~ s. For

value ^{trie alculation}

is

_umencally

xact, 1-e- trie _results do

non

depend on _lit.

After _filling trie

artiales

into

_trie pipe they start oving ai

rest.

_{While falling}

kinetic

energy

due

to

trie _constant

acceleration.

To

avoid infinitely _rising velocity of

trie

E kin E_roll 2.5e10

2e10

1.5e10)

5e09

les 1.5e5 2e5 time steps

Fig.

4. Evolution of the kinetic energy of the transversal partiale movement and the rotational energy. Since the rotational _{energy is very} small trie _curve

is almost indistinguishable of the time-

axis. While flowing accelerated, the system becomes unstable ai 35.000 lime steps approximately and

transits into another flow regime ^where ii reaches _a state of relative steady kinetic energy

(E

G3 6 _x 10~

for this

simulation).

partiales

without

colliding they

_were initialized with _{a very} small random

velocity perpendicular

to the axis of trie

pipe.

At _a certain energy trie flow does net get faster anymore but trie

grains

are decelerated _very

intensively.

From _{now on} trie system moves with

approximately

constant kinetic _energy.

Figure

4 shows trie evolution of trie kinetic _energy of trie transversal

particle

movement Ek and trie energy of trie

partiale

rotation Er.

In trie _case of trie low

velocity regime

^trie

granular

materai _moves

through

trie

pipe

with almost

homogeneous partiale density. Figure

5

displays snapshots

^of trie pipe each 1,000 time steps. ^{Time increases}

upwards

while

horizontally

_{one sees} trie evolution of trie

density

_wave from trie left to trie

right. Gravity

acts from left to

right.

In trie

early

times trie

partiale-density

is

approximately homogeneous

due _to trie initial conditions.

Dunng

trie evolution ii becomes

more and _more

inhomogeneous

and

density

_{waves are} observed. As visible _at later times trie flow is unstable: trie

regions

^of

high density

_can

diverge

_as well _{as converge} while trie _average

velocity

remains

approximately

trie _{same as} shown in

figure

4.

After trie transition into trie

recurrent-plug-flow regime,

where trie

velocity profile

does not vary with trie distance from trie centre on trie pipe, trie system ^reaches a

dynamic

state of

nearly

constant kinetic _energy due _to trie

equilibrium

between constant acceleration and _energy

dissipation by

friction. Nevertheless _{one can} observe small fluctuations of trie energy which

are

approximately

due to trie

genesis

and

vanishing

of

regions

of

higher density

with lime, _as visible

comparing

trie evolution of trie _energy

(Fig. 4)

with trie

corresponding snapshot

ai trie

same lime

(Fig. 5).

This behaviour _proves that trie

inhomogeneous

flow is unstable and ii could be _an indication of

coexisting

metastable states but _we dia trot check ibis

possibility

yen.

Our simulations show further that there _{are no}

significant

diflerences between trie behaviour of

a system ^with

partiales

of

equal

and

randomly

distributed radii.

Simulating

trie system ^with

slightly

^dilferent initial conditions _we got _very ^similar results for trie _energy evolution and for trie structure of trie

density

_wave but trie concrete

density

wave as it _cari be observed _as dark and

bright remous

in

figure

5 _varies

significantly. Very

small dilferences in trie initial conditions lead to very dilferent macroscopic behaviour. In ibis _{sense we} call trie flow

chaotically.

There _are essent1al dilferences between trie behaviour of _a

liquid

and _a fluidized

granular

material: for trie _case of _an

incompressible

fluid _{one never} finds

density

_waves. Nevertheless there _are similanties too: The evolution of trie energy of trie system is similar to trie evolution

504 JOURNAL DE PHYSIQUE I N°4

t=26

t=24

t=22

t=20

t=18

t ₌ iô

t= 14

t=12

t = io

t = 8

t = 6

t = 4

t = 2

t ₌ 0

Fig.

5. Snapshots of the pipe each 1,000 time-steps. The time _is given in lo~ _time steps. The

density fluctuations

are net equidistant and _net stable. They _{con converge as} well _as diverge. Dark regions correspond to

higher

densities. The partiales of which the wall _{consists of}

are non drawn.

of trie

velocity

of _a

liquid flowing through

_{a pipe [23] while} trie _pressure is increased

gently.

Ai low flow

velocity

_an

incompressible

viscous fluid forms _a laminar flow

(Poisseuille-flow)

which gels faster with

rising

_energy. If trie

velocity

reaches _a certain value _itc trie Poiseuille-flow

becomes unstable and _an

inhomogeneous

flow

regime

becomes stable. For trie _case of _a fixed pipe geometry and given fluid parameters trie critical

velocity

of trie flow _itc is described

by

trie

cntical

Reynolds-number

_{[24] Rec}

=

fi

m 2, 300 where _~lk is trie kinematic

viscosity

of trie fluid. Provided there _are critical fluctuationsk trie system tums into _a non-laminar

regime.

If trie pressure is increased _very

gently

and trie system is net disturbed in another way,

however,

it is

possible

to maintain trie Poiseuille-flow _at velocities far _over Re~ _m trie unstable

regime.

Trier _a small fluctuation _causes trie system to turn to trie stable state, 1-e- it

suddenly

lowers trie

velocity

of trie flow and

dissipates

trie energy. As shown for trie fiow of _a

granular

material

there exist dilferent fiow

regimes

too and trie transition between them is due _{to a} sudden _energy

dissipation.

Hence trie transition from low energy ^to

high

_energy

regime

of

granular

flow in _a pipe is similar to trie transition of _a laminar fiuid-fiow into _a non-laminar _one.

Experimental

and recent numerical

investigations

of trailic fiows [25] lead to similar results

as those

presented

here. At _a

given

_average

car-velocity

trie

homogeneous

^{trailic flow} becomes unstable and traffic

jamming

occurs

provided

tl~ere _are random

perturbations

^{of tl~e}

velocity

of tl~e _cars whicl~

correspond

to tl~e collisions of trie

grains

witl~ trie wall in _{our case.}

Concluding

we state that

granular mater1alfalling through

_a narrow pipe _appears elfects far from trie behaviour of _a laminar fluid. At _a certain _energy trie

granular

flow transits from trie

homogeneous

to an

inhomogeneous flow-regime

where

clogging

and

density

_{waves can} be observed. Trie regions ^of

high partiale density

_{can converge as} well _as

diverge,

their distances vary

irregularly.

Trie results of trie

experiment

shown in trie

beginmng

_agree with trie numencal

simulations.

Acknowledgments.

Tl~e author tl~anks J.

Gallas,

H.

Herrmann,

G. Ristow and S. Sokolowski for fruitful and

stimulating

discussions and H. Pul~l for

experimental

assistance.

References

iii

Coulomb C. A., Acam. R. Sa. Mem. Math. Phys. _par Dmers Savants 7

(1773)

343.

[2] Jaeger H. M., Lin C. and Nagel S., Phys. Reu. Lent. 62, 40

(1989);

Lee J. and Herrmann H. J., J. Phys. A 26

(1993)

373.

[3] Half P. K. and Werner B. T., Powder Technol. 48

(1986)

239.

[4] ^Devillard P., J. Phys. France 51

(1990)

369.

[Si ^{Williams J.} C., Powder Techn. 15

(1976)

245;

Rosato A., Strandburg K. J., Prinz F. and Swendsen R. H., Phys. Reu. Lent. 58

(1987)

1038;

Ohtsuki T., Takemoto Y. and Hayashi A., prepnnt.

[6] Faraday M., Phil. Trans. R. Soc. Land. 52

(1831)

299.

[7] Dinkelacker F., Hübler A. and Lüscher E., Biot. Cybern. 56

(1987)

51;

Walker J., Soi. Am. 247

(1982)

167.

[8] Gallas J. A. C., Herrmann H. J. and Sokolowski S., Phys. Reu. Lent 69

(1992)

1371; J. Phys.

France 2

(1992)

1389;

Sokolowski S. and Herrmann H. J. E~rophys. Lent. 18

(1992)

415;

Taguchi Y. H., Phys. Reu. Lent. 69

(1992)

1367;

Taguchi Y. H., report titcmt-1992-7

(1992);

Herrmann H. J., Physica A191 (1992 263.

[9] Evesque P. and Rajchenbach J., Phys. Reu. Lent. 62

(1989)

44;

Rajchenbach J., E~rophys. Lent. 16

(1991)

149.

[10] ^{Baxter G.} W., Behringer R. P., Fagert T. and Johnson G. A., Phys. Reu. Lent. 62

(1989)

2825.

[Il]

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

(1990)

8681; J. Fhid Mech. 225

(1991)

121.

[12] Savage S. B. and Hutter K., J. Fhid Mech. 199

(1989)

177;

Pôschel T., J. Phys. France 3

(1993)

27.

[13] Reynolds O., Philos. Mag. 20

(1885)

469.

[14] ^{Edwards S. F. and Oakeshott R. B.} S., Physica Amsterdam157A

(1989)

1080;

Edwards S. F., J. Star. Phys. 62

(1991)

889;

506 JOURNAL DE PHYSIQUE I N°4

Herrmann H. J., J. Phys. France 3

(1993)

427;

Mehta A. Ed., Granular Matter

(Springer,

New York,

1993).

[15] Savage S. B., J. Fhid Mech. 92

(1979)

53.

[16] ^{Baxter G. W. and} Behringer R. P., Physica D 51

(1991)

465; Phys. Reu. A 42

(1990)

1017.

[17] Thompson P. A. and Grest G. S., Phys. Reu. Lent. 67

(1991)

1751;

Hong D. C. and McLennan J. A., preprint;

Ristow G. H., J. Phys. France 2

(1992)

649.

[18] ^{Caram H. and} Hong D. C., Phys. Reu. Lent. 67

(1991)

828.

[19] ^{Schick K. L. and Verveen A.} A., Nat~re 251

(1974)

599.

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

(Clarendon

Press, Oxford,

1987);

Walton O. R., Braun R. L., J. Rheol. 30

(1986)

949.

[21] ^{Landau L. D. and Lifschitz E.} M., Theory of Elasticity

(Pergamon

Press, London, 1959).

[22] ^{Cundall P. and Strack O. D.} L., Géotechmpe 29

(1979)

47.

[23] ^{Landau L. D. and Lifshitz E.} M., Fluid Mechanics,

(Pergamon

Press, London, 1959).

[24] ^O. Reynolds, Phil. Trans.

(1883);

Froc.Roy.Soc.

(1894).

[25] ^{Herman R. and} Prigogme I., Science 204

(1979);

Nagel K. and Schreckenberg M., J. Phys. France 2

(1992)

2221;

Nagel K. and Herrmann H. J., Physica A 199

(1993)

254;

Treiterer J., Technical Report PB 246 094, Ohio

(1975).