Density waves in granular flow: a kinetic wave approach

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Density waves in granular flow: a kinetic wave approach

Jysoo Lee, Michael Leibig

To cite this version:

Jysoo Lee, Michael Leibig. Density waves in granular flow: a kinetic wave approach. Journal de

Physique I, EDP Sciences, 1994, 4 (4), pp.507-514. �10.1051/jp1:1994156�. �jpa-00246926�

Classification Physics Abstracts

05.40 46.10 62.20

Density

_waves

in granular ^flow:

_a

^kinetic

_wave

approach

Jysoo Lee(*)

and Michael

Leibig(**)

HLRZ-KFA Jiilich, Postfach 1913, W-5170 Jiilich, Germany

(Received

15 November 1993, received in final form 16 December 1993, accepted 3 January 1994

)

Abstract. Ii _was recently observed that Sand flowmg down _a vertical tube sometimes forms

a traveling density pattern in which

a number of regions with high density _are separated from each other by regions of low density. In this work, _we consider this behavior from the point of view of kinetic

wave theory. Similar density patterns _are found in molecular dynamic simulations of the system, and _a well defined relationship _is observed between local flux and local density

a strong mdicator of the presence of kinetic _waves. The equations of motion for this system are

also presented, and they allow kinetic _wave solutions. Finally, the pattem formation _{process is} investigated _{using a} simple model of interacting kinetic _waves.

Systems

of

granular partiales (e.g. sand)

exhibit _many

interesting phenomena,

such _{as segre-}

gation

^{under vibration} or

shear, density

_waves in trie outfiow

through

tubes and

hoppers,

and

probably

most

strikingly,

trie formation of

heap

and convection cell under vibration

il,

_{2, 3, 4].}

In

granular

flows

through

_{a narrow} vertical

tube,

Pôschel found _[Si that trie

partiales

do trot flow

uniformly,

but form

high density regions

which travel _as coherent structures with _{a ve-}

locity

dilferent from trie _center of _mass

velocity.

He also

reproduced

these

density

_waves

using

molecular

dynamics (MD)

simulations [Si.

However,

trie motion of these

high density regions

and trie mechamsm which is

responsible

^{for their} formation _are non

fully

understood.

In this paper, we present ^{numencal and} theoretical evidence that these

density

_waves are of _a kinetic nature [6].

Using

MD

simulations,

_{we measure} trie

dependence

of trie

partiale

flux _on trie

density.

We find _a well-defined

flux-density

relation _an indication that _a kinetic _wave

theory

descnbes trie behavior. A direct measurement of trie

velocity

of these

high density regions

shows

a

dependence

_on trie _mean

density

which is in

good

_agreement with trie

predictions

from kinetic

wave

theory.

On trie theoretical

side,

_we consider _one dimensional

equations

of motion for trie

density

and trie

velocity

fields in trie tube. These

equations, together

with

Bagnold's

law for

friction [7], ^{allow kinetic}

density

_wave solutions.

(*) Present address: Levich Institute, Stemman Hall, ^140~~ St and convent av., New York, NY 10040, U-S-A-

(**)Present

address: Institute of Theoretical Physics, University of Califomia, Santa Barbara, CA 93106, U-S-A-

508 JOURNAL DE PHYSIQUE I N°4

In order to understand the formation of these

high density regions,

_we consider trie

general problem

of

interacting

kinetic _waves. We first show

numerically

that _a system with _an

initially

random

density

field evolves to _a

configuration

in which

neighboring regions

bave _a

high density

contrast. At trie

early

stage ^of

development,

_{we can} show

analytically

that trie

density

contrast between

nearby regions

increases

linearly

with time.

We first discuss trie MD simulations of trie system, and

begin

with _a brief

description

of trie

interparticle

force laws that _were used in _our calculations. Trie

partiales

mteract with each other

(or

with _a

wall) only

if

they

_are in contact. Trie force that _{acts on} partide1 due to

partide j

_cari be divided into two components. Trie

first, F)~~,

^is

parallel

to trie _{vector r e} R~

R~,

where

R~

and

R~

are trie coordinates of trie centers of

particles

and

j respectively.

We refer _to this _as trie normal component. Trie second component,

orthogonal

to r, is trie shear comportent

(~~~.

^{Trie normal} component ^is

given by

~Î-

"

~n(~i

+ ~J lTll~~~ 7nme

)) ^{~, (la)}

where a~(a~ ^{is trie}

radius,

and _m~ (m~ ^trie mass of

partiale1 (j). Also,

_m~ is trie effective _mass m~m~

/(m~

+

m~),

and _{v +}

dr/dt.

Trie first _term in

equation (la)

is trie Hertzian elastic

force,

where kn is _a material

dependent

elastic constant. Trie second term is _a

velocity dependent

friction term, where _~in is _a normal

damping

coefficient. Trie shear component is

given

_as

lÇ~i

_"

~'lsLle)jj~, ⁽¹⁶⁾

where _{s is} defined

by rotating

_r clockwise

by ~/2.

Trie shear

force, equation (16),

is

simply

_a

velocity dependent

friction term similar to trie second _term in trie normal component.

Finally,

we must

specify

trie interaction between _a

partiale

and _a wall. Trie force _on

partiale1,

in _contact with _a

wall,

is

given by equations (1)

_{with a~}

= cc and m~ = cc. Trie choice of trie interactions defined

by equations (1)

is rather

typica1in

trie MD simulations of

granular

material [8]. ^A detailed

explanation

of trie interaction _is

given

^elsewhere _[9].

For

simplicity,

_we

study granular

fiows in 2 dimensions and _{use a} fifth order

predictor-

corrector scheme to mtegrate trie

equations

of motion,

calculating

both trie

positions

and

velocity

of each

particle

ai all limes. Trie tube _is modeled

by

two vertical sidewalls of

length

L with _a

separation W,

and _we

apply

_a

periodic boundary

^condition m trie vertical direction.

Between trie

sidewalls, partiales

of radii o-1 _are

mit1ally

^{filled with} _a ^uniform

density

of _p~

(throughout

ibis _paper, numerica1values _are

given

in CGS

unit).

Trie

particles begin

to _move under trie influence of

gravity,

and _soon reach _a

steady

_state, where trie

gravitational

force is

balanced

by

trie frictional force from trie interactions with trie sidewalls.

in

figure

1, we show trie time evolution of trie

density

and the

velocity

fields for L

= 15 and

W ₌ 1, measured _{at every} 5 _ms. At _a

given

time, we divide trie tube into 15 vertical

regions

of

equal length,

and _measure trie

density

and trie average

velocity

in each

region.

These fields

are

displayed

_{as a} vertical

strip

of _square

boxes,

where each box

corresponds

to _a

region

in trie tube. Trie

grayscale

of trie box is

proportional

to trie value of trie field in that

region.

Trie

parameters we used in this simulation _are kn

= I.o _x 10~,~in _{= ~is =} S-o _x 10~, with trie time step ^S-o x 10~5. Trie initial

density

_{p~ is} 25

particles

_per unit _area. In trie

figure,

we find

(1)

a

region

^of

large density

fluctuations _is formed _out of trie

init1ally

uniform system,

(2)

trie fluctuations _seem to travel witl~ almost constant

velocity (dilferent

from tl~e center of _mass

velocity),

and

(3)

tl~ere _seems to be strong correlation between tl~e

density

and tl~e

velocity

fields. These

findings

_remam true for tl~e simulations _we bave

performed

with dilferent values of _~i, kn and _p~, except when _p~ is very

small,

where _a

steady

state is non reached. These

traveling density

_{pattems were} first observed in trie simulations

by

Pôschel [Si.

Time

Space

(a)

(b)

Fig.

1. Time evolution of

(a)

density and

(b)

velocity fields. These simulations _were doue with _a tube,of width W

= and length L

= 15. Fields ai _a given lime

are shown

as a vertical line of small boxes. The grayscale of each box is proportional to the value of the density _or velocity _m that region of trie tube. Regions of high density _are formed, and travel with almost constant velocity.

In order to

quantitatively study

trie correlation between trie

density

and trie

velocity fields,

we measure trie local

partiale

flux _{as a} function of tl~e local

density

in tl~e

following

_manner.

Once tl~e system ^{bas reached} a

steady

state, _we measure trie _mean

velocity

_u~ and trie

density

p~ in

region

1. Trie flux

j(p)

for _a

given density

_{p is} then taken _to be _p

(u(p)),

where () is _a time average over ail

regions

which had _a

particular density

_p. Trie

flux-density

_curve,

obtained

by averaging

_over 10,000

iterations,

_are shown in

figure

2.

Here,

trie parameters _are trie _{same as} those of

figure

1. Trie fact that _a well-defined

flux-density

_curve exists suggest that trie

density

_waves

(traveling density fluctuations)

_are kinetic _m nature.

Furtl~ermore,

tl~e

flux-density

_curve for tl~e

granular

^fiow resembles tl~at of _a trailic

fiow,

wl~icl~ is considered _as

a prime

example

of trie systems which shows kinetic _{waves [GI.}

One additional

piece

of evidence that trie

density

_{waves are} of _a kinetic _nature is their

dependence

_on trie initial

density

_p~. Trie

theory

of kinetic _waves

predicts

_[6] ^{that small}

density

fluctuations in _a uniform

density background

_p~ travel with _a

velocity

U(Po)

=

fl

lp=p~, 12)

which is trie

slope

of trie

flux-density

_curve at tl~e _mean

density.

We tl~us expect a

large negative velocity

^for small _{p~, a} decrease _{to zero}

velocity

at pa m

15,

^witl~ _an

increasingly large positive velocity

_{as p~} is increased furtl~er. To cl~eck

tl~is,

_{we measure} trie _wave velocities for several values of _p~

(keeping

all otl~er parameters fixed _as

above). Writing

tl~e _mean

density

_p~ and tl~e measured

velocity U(p~)

_as

_(p~, U(p~)),

_we find

(10.o,

-41+

2), (15.0,

5

+9), (18.7,12 +11)

and

(22.5,113.o

+

4),

wl~icl~ _are ail consistent witl~ tl~e above

prediction.

We _now consider trie theoretical aspect of trie

density

_waves. Consider trie

equations

of

510 JOURNAL DE PHYSIQUE I N°4

1400

1000 _."

,,

~~~~

_.,

..

800

~~~

j

>,

600 ^,

~

400

~,,

~

l'~'j"~

j'

0 5 10

lj

2fl 25 30 35

nUm ero palicles

Fig.

2. Local flux _{as a} function of local particle density. This _{curve was} found for _a tube with width W

= 1 and length L

= 15, obtained by time averagmg. The parabolic shaped _curve resembles the flux-density relation _in traflic flows.

motion wl~icl~ _govern tl~e time evolution of tl~e

density p(z, t)

and tl~e

velocity vii, t)

fields for

a

granular

flow _{in a} vertical tube. Tl~e first

equation

is tl~at of _mass conservation

(P

₊

_)lPu)

= 0>

13a)

and tl~e second is _a momentum conservation

equation

p~u

+

pu)u

₌

F(x, t), (3b)

wl~ere

F(x,t)dx

is the total amount of force

acting

_on the

particles

in _a

region [x,x

₊

dx].

Tl~e force

F(x, t)

bas tl~ree contributions

gravity,

internai pressure, and friction from tl~e sidewalls. Tl~e exact form of tl~e internai _pressure and tl~e friction _is trot known.

Here,

_{we use}

Bagnold's law,

wl~icl~ is believed to be valid in tl~e

grain

inertia

regime

_[7].

Therefore,

trie force

F(x, t)

is

Flx, t)

=

-PgW SignlU)PBf~ylP)U~ Dl

lPBf~~lP)U~l

14)

Here,

_{g is}

gravitational acceleration,

_pB trie

density

of trie material which forms trie

particles,

_p

trie

packing

fraction

(p

₌

pBp)

and D is trie diameter of trie

partiales.

We _assume tl~e thickness of trie shear

layer

to be of order of D.

Also, f~~

^and

f~y

_are

geometry dependent functions,

which contain trie information about trie

density dependence

of trie forces.

Trie uniform

density

solution of

equation (3

with trie force

given by

equation

(4)

is

P(x,t)

= pBp~

u(x,t)

₌

-~ ₍₅₎

If _we add small

density

fluctuation _p

= p~ +

dp

in trie uniform

density fiow,

trie fluctuation

travels with _a

velocity

UiPo)

_~

-l~

^~~~~~~°~

j~iill~~~°~/~~,

¹⁶⁾

which is

given by equation (2). Equations (5)

and

(6)

_are

exactly

what _one expects if trie kinetic _wave

theory

is to

apply

uniform fiow _{is a} solution to trie

equations

of motion, and

density

fluctuations travel with _a

density dependent velocity.

Thus,

it is clear then that trie motion of trie

density

pattern _can be understood

by applying

trie ideas of kinetic _wave

theory. However,

this basic formalism

only

describes trie motion of

a

pre-existing density

pattern. It does _not

explain

trie observation that

regions

^with

large density

contrasts _are

being

formed _out of trie uniform

background.

Our simulations show that trie

large

scale

density

pattern

begins

_{as a} collection of small fluctuations in trie

density.

These small fluctuations _grow in time and _a pattern _emerges in which

large density

contrasts exist between

neighboring regions.

^Trie evolution to such _a state _can be understood

by considering

trie system _as set of

interacting

kinetic _waves. A detailed treatment of trie

general problem

of

interacting

kinetic _{waves can} be found elsewhere [11] m this _{paper, we} present

only

trie results from _a

simple

^{model for trie} pattern ^evolution _process in sand

fiowing

down _a tube.

Consider trie

early

stages of trie fiow in which trie

density

of sand is

nearly

uniform at p ^m p~.

Because of trie

roughness

of trie grains, trie

roughness

of trie walls _or from trie stochastic nature of trie inelastic

collisions,

small

density

fluctuations _appear in trie system. In trie

interacting density

wave

approach,

_we treat trie fluctuations _{as a} set of distinct

density regions

with interfaces whose velocities _are determined

by

_a discrete form of

equation (2).

In this _case, trie interface

separating

_a

region

of

density

_pi from _a

region

with

density

_{p2 moves} with _a

velocity, U(1, 2), given by

U(1, ₂₎

=

~~~~~

~~~~~,

(7)

Pl P2

which is trie kinetic _wave

theory

result for interfacial velocities

involving

^finite

density

dilfer-

ences [GI. In trie _case that trie dilference between _pi and _{p2 is very}

small,

trie

equation

becomes

equation (2).

Trie evolution of trie system ^is determined

by

trie motion of trie interfaces

and,

as shall be shown, trie nature of their interactions leads to a final state m which

large density

contrasts _occur.

In trie

computational

and

analytic

results that

follow,

_we choose _a

specific

form for trie

density

fluctuations _m trie system. ^In our

model,

trie initial

positions

of trie interfaces _are taken _{as a} set of

No points placed randomly

on trie interval [o,

L],

^with

regions

between successive interfaces

being assigned

_a

density randomly

in trie _range [p~

W,

_{p~ +}

W].

Trie

principal

virtues of this model _are its

simplicity

and trie fact that there _{are no} correlations in trie initial state which

might

influence trie final structure. A _more realistic model for trie fluctuations of trie system would require _a

microscopic understanding

of each

specific

_source of noise.

It is also necessary ^to choose _a form for trie flux _curve

j(p).

We bave taken trie

parabolic

form

j(p)

= J~ ^~

(1-

^~

,

(8)

where R is trie

density

at which _no flow _occurs, and J~ is _one quarter ^{of trie} maximum flux of trie system. ^This curve was chosen for several _reason. Trie first is that its

simplicity

_{eases some} of trie

hardships

of

analytical

calculations. Trie second _{reason is} that for

density

fluctuations

over a

sufficiently

small _range, trie _true flux _{response can} be

approximated by

this form

(with

R and J~

being fitting parameters).

And

finally,

it is _a first

approximation

to trie form observed for trie

j(p)

observed in

figure

2.

512 JOURNAL DE PHYSIQUE I N°4

(ai

(b)

Fig.

3. Evolution of interacting ^kinetic _waves. Both strips _{use a} linear grayscale ^with white

mdicating _{po =} 0 and black pa = R, the jamming density.

(a)

shows the initial configuration of 400 interfaces with densities in the _range

[0.3R, 0.8R]. (b)

shows the configuration when only 33 interfaces

remain, and illustrates the tendency for

alternating

high and low density regions.

Numerical simulation of this system is _a very

straight-forward

exercise. Trie values of trie densities in two

adjacent regions

determine trie

velocity

of trie associated interface. Consider three successive

density

_regions

A,

B and C.

During

trie _course of trie

simulation,

trie interface A-B may ^encounter trie interface B-C. This indicates that ail of trie _mass that _was inside

region

B bas been

completely

"swallowed up"

by

trie

regions

^{A and C.} In this _case, trie interfaces A-B and B-C _are

replaced by

_a

single

A-C interface. Trie

velocity

of this interface _cari be calculated from trie densities _in

regions

A and C.

Thus,

it is _a matter of

tracking

all of trie interfaces,

checking

for

collisions,

and when

they

_occur,

replacing

trie two old interface with _a

single

_new

one.

Therefore,

this

technique

does trot allow for any

density

values other than those

initially

present, and trie number of interfaces is

always decreasing.

For convenience, trie simulations

were done

using periodic boundary

^condition.

Trie first set of results shown below _are from _a simulation _m which there _are 400 interfaces

initially placed randomly

in trie interval [o,

ii

(1.e. ^L ₌

1).

We also choose trie values J~

= 1

and R

= 1. Trie densities _are chosen at random from trie interval

[o.3,o.8]

(1.e. _{p~ =} o.55R, W =

o.25R). Figure

3a shows trie initial

density configuration,

while

figure

3b shows trie system ^alter a time t ₌ o.486

(where

time is measured in trie units of

RL/J~),

and there _are

only

33 interfaces which remain

along

trie interval. Trie system bas evolved to _a state in which trie

density

contrast is very

high

^between

neighboring regions,

and this behavior _was observed for all values of _p~ and W.

This _increase in trie

density

contrast _can be characterized

quantitatively

in trie

following

way. Let trie

density

of each

region

be _p~, with1

indexing

trie dilferent

regions,

and

N(t)

be trie number of

regions

at time t. Define trie

quantity

N(t)

M(t)

₊

q ~

_(Pi

Pi+il,

₁₉₎

where pN(t)+i ^e pi Trie

larger

^trie value of

Mit)

trie

larger

trie

density

contrast between

neighboring regions. Figure

4 shows trie

quantity M(t) M(o) averaged

_over 10 simulations

with

No

_" 10,ooo interfaces. At

early times,

there is _a linear increase in

M(t)

with _{a crossover}

to a

nearly

constant value at late times.

At

early limes,

it is

possible

to calculate

Mit) analytically

and trie results _are shown as trie dotted line in

figure

4. In this

regime,

trie

changes

_m

Mit)

_are dominated

by

trie interaction of interfaces whose movements are determined

by

trie initial

configuration

of trie system. ^Trie

1-où

'~

É

CJ

~Î

1

~# -

Ù

fl _~

~f

-5.oo

-

,'

-6.oo

-8.oo logjo(t)

Fig.

4. Contrast,

M(t),

as function of time, t. The sohd line shows trie results from averaging

over 10 simulations with

N(0)

= 10,000, and densities chosen in the range

[0.3,0.8].

Ai early limes, the increase in contrast is linear and at long limes ii becomes _a constant. The dashed line shows the

results from _an analytical calculation of trie short lime behavior.

calculation _{averages over} all

possible configurations

of trie initial random densities and inter-

faces,

determines trie time ai which each interface collision _occurs and how much that collision

changes

trie value of

Mit).

In this

regime,

^trie agreement with trie simulation is

good.

It is also

possible

to show

exactly

that

M(o)

₌

2W/3.

At later

times,

^after there bave been _many

collisions between

interfaces,

trie structure of trie system

depends

_on tl~e nature of tl~e earlier evolution.

Tl~us,

finis

long

lime bel~avior is much _more difficult _to calculate. Trie results from trie calculation described above break down in finis

regime

because trie distribution of

density regions

is _no

longer

that of trie initial random distribution.

At

long

limes

Mit)

_m 2W.

Thus,

trie

density

contrast at

long

times

is,

_{on average, as}

large

_as trie

largest density

contrasts present in trie initial

configuration.

lt turns out that trie

interacting

kinetic _waves do non create

large

contrasts.

Rallier,

trie interfaces from trie initial distribution which survive _are those that bave _a very

large density

contrast [11]. Thus, while trie noise in tl~e system _may

provide

_a

variety

of sucl~ contrasts, tl~e

interacting

kinetic _waves will

destroy

all but tl~e _very

largest.

This paper outlines _a kinetic _wave

approacl~

to

understanding

trie

density

patterns observed in sand flow

along

_a vertical tube

(many

of trie details omitted here _con be found in references [10,

11]). ^However,

^{these ideas}

certainly

do not constitute _a

complete theory

for trie pattems observed in trie

experimental

system. Trie role that trie flow of air

plays

in this _process

[12],

as well _as trie _sources of noise in trie system, are

certainly

not well understood. Further

experimental investigation

of tl~ese _issues would be most

enligl~tening.

From _a tl~eoretical point ^of

view,

_it is not dear wl~etl~er tl~e frictiona1force _at tl~e watt and tl~e intemal _pressure

obey Bagnold's

law. While this form bas been observed in trie sheer cell geometry [7,

13],

tl~ere bas been _no direct measurement of tl~e frictiona1force for

gravity

driven flow.

Finally,

it is known that trie interface between two

regions

^of

diflering

densities _may not be _a stable

514 JOURNAL DE PHYSIQUE I N°4

structure [6], and that diflusive eflects _may

strongly

influence trie

long

time behavior of _a system ^of

interacting

kinetic _waves.

Acknowledgments.

Trie authors would like to thank trie members of trie HLRZ

Many Body Group

for

stimulating

discussions

throughout

this work.

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