Pattern Formation in Granular Materials-Siegfried GrobMann

8/17/2019 Pattern Formation in Granular Materials-Siegfried GrobMann

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G era ld H R i s tow

P a t t e r n o r m a t i o n

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S p r i n g e r T r a c t s i n o d e r n P h y s ic s

S p r i n g e r T r a c ts i n M o d e r n P h y s i c s p r o v i d e s c o m p r e h e n s i v e a n d c r it ic a l r e v ie w s o f t o p ic s o f c u r r e n t

i n t e r e s t i n p h y s i c s . T h e f o l lo w i n g f i e ld s a r e e m p h a s i z e d : e l e m e n t a r y p a r ti c l e p h y s i c s , s o l i d - s t a t e

p h y s i c s , c o m p l e x s y s t e m s , a n d f u n d a m e n t a l a s t r o p h y s i c s .

S u i t a b l e r e v i e w s o f o t h e r f i el d s c a n a l s o b e a c c e p t e d . T h e e d i t o r s e n c o u r a g e p r o s p e c t i v e a u t h o r s t o

c o r r e s p o n d w i t h t h e m i n a d v a n c e o f s u b m i t t i n g a n a r t ic l e. F o r r e v i e w s o f t o p ic s b e l o n g i n g t o t h e

a b o v e m e n t i o n e d f ie ld s , t h e y s h o u l d a d d r e s s t h e r e s p o n s i b l e e d it o r, o t h e r w i s e t h e m a n a g i n g e d it o r.

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ISSN oo81-3869

ISBN 3-54o-667ol-6 Spr inger-Verlag Ber l in Heid e lbe rg N ew York

T h i s w o r k i s s u b je c t t o c o p y r i g h t . A l l r i g h t s a r e r e s e r v e d , w h e t h e r t h e w h o l e o r p a r t o f t h e m a t e r i a l i s c o n c e r n e d ,

spec i f i ca l l y t he r i g h t s o f t r ans l a t i on , r ep r i n t i ng , r euse o f i l l u s t r a t i ons , r ec i t a t i on , b r oadcas t i ng , r ep r odu c t i on on

m i c r o f i l m o r i n a n y o t h e r w a y, a n d s t o r a g e i n d a t a b a n k s . D u p l i c a t io n o f t h i s p u b l i c a t io n o r p a r t s t h e r e o f i s

p e r m i t t e d o n l y u n d e r t h e p r o v i s i o n s o f th e G e r m a n C o p y r i g h t L a w o f S e p t e m b e r 9 ,1 9 65 , i n i t s c u r r e n t v e r s i o n , a n d

p e r m i s s i o n f o r u s e m u s t a l w a y s b e o b t a i n e d f r o m S p r in g e r -V e r la g . V i o la t i o n s a r e l i a b l e fo r p r o s e c u t i o n u n d e r t h e

G e r m a n C o p y r i g h t L aw .

© Sp r i nge r - V er l ag B e r l i n H e i de l b e r g 2ooo

P r i n t e d i n G e r m a n y

T h e u s e o f g e n e r a l d e s c r i p t i v e n a m e s , r e g i s t e r e d n a m e s , t r a d e m a r k s , e t c . i n t h i s p u b l i c a t i o n d o e s n o t i m p l y, e v e n i n

t h e a b s e n c e o f a s p e c i fi c s t a te m e n t , t h a t s u c h n a m e s a r e e x e m p t f r o m t h e r e l e v a n t p r o te c t i v e l a w s a n d r e g u l a t i o n s

and t he r e f o r e f r ee f o r gene r a l u se .

T ypese t t i ng : C am er a - r ead y copy by t h e a u t h or u s i ng a S pr i nge r LATEXm a c r o p a c k a g e

C o v e r d e s i g n :

design 6 production

G m b H , H e i d e l b e r g

Pr in ted on ac id- f r ee pa pe r SPIN: 1o746o98 56/3144/t r 5 4 3 21 o


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F o r C l a u d i a S


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o r ew o r d

I t i s h i g h l y e x c i t i n g t o o b s e r v e h o w a n e w f i el d o f s c i e n c e d e v e l o p s . A r e c e n t

d i s t i n c t a n d l iv e l y e x a m p l e i s t h e f ie ld o f g r a n u l a r m a t e r i a l s d y n a m i c s .

T y p i c a l l y , t h e b i r t h o f a n e w f ie l d p r o c e e d s a s f o ll o w s : i n i ti a l ly , s o m e

p a p e r s a p p e a r o n q u i t e u n c o n v e n t i o n a l p h e n o m e n a , m a t e r i a l s o r m e t h o d s ,

p o i n t i n g i n n e w d i r e c t i o n s a c r o s s h i t h e r t o a g r e e d b o r d e r s , a l b e i t v a g u e l y .

C o m m o n f e a t u r e s o f o r i g i n a l l y i s o l a t e d w o r k e n h a n c e t h e i n t e r e s t , a ( st il l

s m a l l) c o m m u n i t y j o i n s f o r c es , i n c r e a s i n g l y m o r e p a p e r s a p p e a r , t h e o r ig i -

n a l o n e s n o w b e c o m i n g p i o n e e r i n g r e f e r e n c e s . H i s t o r i c a l r o o t s a r e g e n e r a t e d ,

a g r e e m e n t e m e r g e s o n t h e n a m e o f t h e n e w f ie ld , an d i ts i d e n t i t y b e c o m e s

e s ta b l is h e d . F i r s t a t t e m p t s a r e m a d e t o g iv e a s y s t e m a t i c e x p l a n a t i o n , c h a r a c -

t e r is t ic t o o l s a n d m e t h o d s e m a n a t e . S o o n , a n e x p o n e n t i a l i n cr e a se o f p u b l ic a -

t io n s i n d ic a t e s t h a t a n e w c o m m u n i t y h a s g a t h e r e d , a t t r a c t e d b y a f a s c in a t i n g

n e w s u b j e c t . T h e s e p e o p l e m e e t a t w o r k s h o p s a n d c o n f er e n c es , p r o c e e d i n g s

i n d i c a t e a v i v a c i o u s , o r i g i n a l d e v e l o p m e n t , m o r e a n d m o r e d e t a i l s a r e c o n -

t r i b u t e d , s u b - b r a n c h e s a p p e a r . A t t h i s s t a g e , t h e e x p l o s i o n o f d e t a i ls m a k e s

i t in c r e a s i n g l y d if f ic u l t t o k e e p t r a c k o f t h e m a i n s t r u c t u r e s a n d t h e l e a d i n g

i d e a s .

T h e r e i s t h e n a n u r g e n t n e e d f o r a c o m p r e h e n s i v e o rd e r i n g o f t h e c o n c e p t s

a n d n o t i o n s, a s u m m a r i z i n g o f t h e s t r u c t u r e s t h a t a r e p r es e n t i n t h e m a s s

o f d e t a i ls , a n d f o r a n i d e n t i f i c a ti o n o f t h e i m p o r t a n t q u e s t i o n s t o b e t a c k l e d .

T h i s i s t h e t i m e a t w h i c h a m o n o g r a p h o n t h e s u b j e c t is n e e d e d t o fu lf il t h e s e

r e q u i r e m e n t s .

E x a c t l y t h i s s i t u a t i o n n o w p e r t a i n s i n t h e f ie ld o f g r a n u l a r m a t e r i a l s d y -

n a m i c s . G e r a l d R i s t o w s b o o k a d d r e s s e s t h e s i t u a t i o n a n d w i ll g i v e f u r t h e r

m o m e n t u m t o t h e d e v e l o p m e n t o f t h e f a s c in a t i n g y o u n g f ie ld o f g r a n u l a r m e -

d i a a n d t h e i r d y n a m i c s . I t p r o v i d e s n u m e r o u s r e s u l t s a n d d a t a a s w e l l a s a n

a t t e m p t t o f o r m u l a t e u n i fy i n g c o n c e p t s w i t h w h i ch to u n d e r s t a n d t h e m . A n d

o n r e a d i n g i t o n e b e c o m e s a w a r e o f f u t u r e a im s a n d p r o b a b l e d i r e ct i o n s o f f u r-

t h e r r e s e a rc h . O n e c a n h a r d l y o v e r e m p h a s i z e t h e m e r i ts a n d t h e i m p o r t a n c e

o f p u b l i s h i n g t h i s m a n u s c r i p t .

T h e c u l t u r e o f m o n o g r a p h s i s o f u t m o s t i m p o r t a n c e i n t h e d e v e l o p m e n t

o f o u r s c ie n c es . E v e r y d a y r e s e a r c h i n te r e s t s , t h e n e e d t o w r i t e p r o p o s a l s o f

l i m i t e d s c o p e , a n d t o p r o v i d e s u f f i ci e n tl y m a n y n e w e n t r i e s in o n e s l is t o f

p u b l i c a ti o n s a b s o r b m o s t p e o p l e s t i m e c o m p l e t e ly a n d l e ad t o a r u s h o f


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VIII Foreword

original papers. But we also need the overview, the extraction of the essential

elements from the details, we need to build the house from the many bricks .

This is one such rare and valuable monograph, even rarer within the

young but expanding fields of granular systems. Its part icular emphasis is on

elucidating the various characteristic patterns in these systems in terms of a

unique but ubiquitous dynamics.

The reader will feel the author's enthusiasm for this fascinating field. A

new world of surprising observations opens, often unexpected, even counterin-

tuitive, and it is a big challenge to explain all this, and to use it. The flavour of

the book is interdisciplinary and it thereby reveals the evident high potential

of granular materials for many applications in science, industry, agriculture

and beyond. Granular media are real world systems. To understand their

dynamics one must go beyond the common tendency to over-simplify.

Important scientific progress in two areas now facilitates the study of gran-

ular dynamics: on the one hand, we have learned to deal with open nonequi-

librium systems, and, in addition, computational methods have reached suffi-

cient maturity. This monograph presents an impressive body of dat a obtained

by a mixture of real world experiment and computer simulation which inti-

mately complement one another.

This monograph will be of interest both to the specialist, who finds an

invaluable compilation of facts, graphs, data and theoretical background,

useful for his ongoing research, and also to the newcomer, who is introduced

effectively but enthusiastically to the granular world of sand, rocks, marbles,

pills and more. I am sure that the reader cannot fail to be impressed. This

monograph will be equally useful for research, for teaching and for learning.

I wish this book deserved success, a strong impact and an enthusiastic

reception, for the sake of the field, granular dynamics and pattern formation,

and as a contribution to its further growth and prosperity.

Marburg, October 3, 1999 Siegfried roJ~mann


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r e f a c e

G r a n u l a r m a t e r i a l s a r e a n i n t e g r a l p a r t o f o u r e v e r y d a y lif e: j u s t i m a g i n e

y o u r s e l f s t r o l l in g a l o n g t h e b e a c h o r f u m b l i n g w i t h y o u r m u l t i g r a i n m i ie s li in

t h e m o r n i n g b e i n g o n l y h a l f a w a k e . I n t h e l a t t e r c a s e h a v e y o u e ve r w o n d e r e d

w h y t h e l a r g e r o r l ig h t e r g r a i n s a r e a l w a y s f o u n d o n t o p o f t h e b o x r e g a r d l e s s

o f h o w w e l l i t i s s h a k e n ? T h i s i s t h e f a m o u s razil nuts e f f ec t w h i c h a l r e a d y

p o i n t s t o a m a j o r m i x i n g p r o b l e m e n c o u n t e r e d i n in d u s t r i a l p ro c e s si n g w h e n

h a n d l i n g p a r t i c l e s o f d i f f e r e n t s iz e d e n s i t y o r e v e n s h a p e . T h i s is o n e o f t h e

q u e s t i o n s a d d r e s s e d i n t h i s b o o k .

T h e t e r m

granular material

j u s t s t a t e s t h a t t h e m a t e r i a l in m i n d i s m a d e

u p o f s m a l l p a rt i c le s w h i c h m i g h t b e g r a i n s b u t i t c a n a ls o r e f er t o r o c k s

s a n d o r p i ll s. T h i s m a k e s i t a v e r y g e n e r a l t e r m a n d c o n s e q u e n t l y a m u l t i -

d i s c i p l i n a r y f i el d o f r e s e a r c h w h e r e s u c h m a t e r i a l s a r e s t u d i e d b y s u c h d i ff e r-

e n t c o m m u n i t i e s a s b i o lo g i st s e n g i n e e r s g e o l o g is t s m a t e r i a l s c i e n ti s ts a n d

l a s t b u t n o t l e a s t p h y s i c i st s . H o w e v e r s in c e t h e t h e o r e t i c a l d e s c r i p t i o n o f

g r a n u l a r m a t e r i a l s i s s ti ll i n i t s in f a n c y t h i s f ie l d h a s a l s o a t t r a c t e d m a n y

m a t h e m a t i c i a n s a n d c o m p u t e r s c i e nt is ts l e a v in g a m p l e r o o m f o r n e w t h e o -

r ie s a n d l a r g e - s ca l e c o m p u t e r s i m u l a t i o n s t o m o d e l a n d d e s c r i b e t h e d i f f e r e n t

p h e n o m e n a .

S i n c e i t is s u c h a l a rg e a n d a c t i v e fi el d o f r e s e a r c h u n f o r t u n a t e l y n o t a ll

i n t e r e s t i n g t o p i c s c o u l d b e c o v e r e d i n t h i s b o o k a n d a s e l e c t i o n h a d t o b e

m a d e . K n o w i n g t h a t e v e r y se l e c ti o n m u s t b e b i a s e d I a p o l o g i z e f o r t h e g a p s

I h a d t o l ea v e h o p i n g t h a t t h e r e s t o f t h e m a t e r i a l p r e s e n t e d w i ll b r i d g e m o s t

o f t h e m .

T h e m a t e r i a l c h o s e n f o r t h i s b o o k c o n c e r n s collective e f f ec t s i n v o l v i n g

m a n y p a r t ic l e s . D u e t o t h e h i g h l y d i s s ip a t i v e n a t u r e o f t h e p a r t i c l e c o ll is io n s

e n e r g y i n p u t is n e e d e d i n o r d e r t o m o b i l i z e t h e g r a i n s . T h i s i n t e r p l a y o f di ss i-

p a t i o n a n d e x c i t a t io n l e ad s t o a w i d e v a r ie t y o f p a t t e r n - f o r m a t i o n p r o ce s se s

w h i c h w i l l b e a d d r e s s e d i n t h i s b o o k . T h e r e a d e r i s i n t r o d u c e d t o t h i s w i d e

f ie l d b y f i rs t a d e s c r i p t i o n o f t h e m a t e r i a l p r o p e r t i e s o f g r a n u l a r m a t e r i a l s

u n d e r d i ff e r en t e x p e r i m e n t a l c o n d i t io n s w h i c h a r e i m p o r t a n t i n c o n n e c t io n

w i t h t h e p a t t e r n - f o r m a t i o n d y n a m i c s a n d s e c o nd f u r t h e r d e ta i ls w h e n d e-

s c r i b in g t h e s p e c if ic s y s t e m . I n g e n e r a l t h e o b s e r v e d p a t t e r n s c a n i n v o lv e t h e

s a m e k i n d o f p a r t i c l e s l e a d i n g f o r e x a m p l e t o c o n v e c t i o n r o ll s a n d s t a n d i n g


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X Preface

waves under vibrations or they can involve different kinds of particles, giving

rise, for example, to stratification patterns and segregation.

My work in this field was initiated a few years ago when I had the oppor-

tuni ty to work at the HLRZ in Jfilich. A very stimulating atmosphere created

by young scientists from all over the world led to lively discussions and new

ideas, especially during the short-notice coffee seminars. I have continued to

work in this field since then and have included some parts of my own research

results in this book.

I welcome this opportunity to thank all the people that have supported

and helped me in pursuing this project with their comments and ideas. I

am especially grateful to H. J. Herrmann for introducing me to this fasci-

nating and active field of research and to S. Grot~mann for his many most

valuable and critical comments during our long discussions. I would be lost

in parameter space) without the detailed experimental information from my

coworkers D. Bideau, F. Lebec, J. L. Moss, M. Nakagawa, I. Rehberg and G.

Stragburger. I also have to thank A. Schmiegel, Helga and Tony Noice for

valuable comments and W. Zimmermann for critically reading parts of the

manuscript and for giving me the freedom to finish this work. However, most

of all, I would like to thank all of my family for their love and understanding

throughout my entire life and for their continuing belief in me.

Saarbr/icken, August 1999 Gerald H Ristow


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o n t e n t s

2

I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

S o m e E x p e r im e n ta l P h e n o m e n a o f G r a n u l a r M a te r i a l s . . . . 5

2.1 S h ea r F low . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 D i l a ta n c y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 S o l i d F lu id T r an s it i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 .4 Convec t ion Roll s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 .5 F ree Sur face F low . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 .6 Inc l inat ion Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 .7 Dens ity and S t re ss F luc tua t ions . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 .8 Commonl y Used Ma te r ia l s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

V e r t i c a l S h a k i n g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.1 H e ap F o r m a t ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2 C o n ve c ti v e M o t ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 .3 S ur f ac e P a t t e r n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 .3 .1 Surface Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3 .3.2 S t a t i on a r y P a t t e r n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3 .4 Compac t i f i c a t ion and Clus te r ing . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.4.1 Compac t i f i c at ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.4 .2 C lu st e r in g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.5 S e gr e ga t io n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4 H o r i z o n t a l S h a k i n g 37

4.1 So l id F lu id T ransi t ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4 .2 Cr i t i c a l Po in t Exponen ts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4 .3 C r y st a l li z at i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4 .4 C o n ve c ti o n R ol ls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4 .5 S ur fa ce P a t t e r n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.6 Lif t ing the Hyste res is by Gas Flow . . . . . . . . . . . . . . . . . . . . . . . 48

4 .7 Inve r ted Funnel F low in Hoppe rs . . . . . . . . . . . . . . . . . . . . . . . . . 49


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XII Contents

S t r a t i f i c a t i o n

5.1

5.2

1

Exper imenta l F indings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Discre te Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.2.1 Model Based on Angle of Repose . . . . . . . . . . . . . . . . . . . 57

5.2.2 Model Based on Energy Dissipation . . . . . . . . . . . . . . . . 58

5.3 Cont inuum Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.3.1 Model Descr ipt ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.3.2 Steady State Profi les . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.3.3 S tabi l i ty Analysi s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.3.4 Thin Flow Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.4 Dependence on Geomet ry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

6 . C o n i c a l H o p p e r . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . 67

6.1 Segregation During Fi l ling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

6.2 S tat ic Wal l S tresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

6.3 Out flow Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

6.3.1 Dependence on Orifice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6.3.2 Dependence on Silo Geometry . . . . . . . . . . . . . . . . . . . . . 70

6.4 F low Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

6.5 Segregation During Outflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

6.6 Dens i ty Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6 .7 Dynamic Wal l S tresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.8 Silo Design to Decrease the Stress Fluctua tions . . . . . . . . . . . . . 79

.

R o t a t i n g D r u m

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

7.1 Different Flow Regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

7.1.1 Avalanches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

7.1.2 Continuous Surface Flow . . . . . . . . . . . . . . . . . . . . . . . . . . 85

7.1.3 Centr ifugal Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

7.2 Segrega tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

7.2.1 Radial Size Segregation . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

7.2.2 Radial Density Segregation . . . . . . . . . . . . . . . . . . . . . . . . 96

7.2.3 Interplay of Size and Density Segregation . . . . . . . . . . . . 99

7.2.4 Fr ic t ion Induced Segregat ion . . . . . . . . . . . . . . . . . . . . . . . 99

7.3

7.4

7.5

7.2.5 End Longi tudinal Segregat ion . . . . . . . . . . . . . . . . . . . . . . 101

7.2.6 Axial Segregat ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

Axial Band and Wave Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 105

Competit ion of Mixing and Radial Segregation . . . . . . . . . . . . . 107

Front Propagat ion and Radial Segregat ion . . . . . . . . . . . . . . . . . 111

7.5.1 Experimental Setup and Studies . . . . . . . . . . . . . . . . . . . . 112

7.5.2 Approx imatio n Through Diffusion Process . . . . . . . . . . . 115

7.5.3 Conce ntra tion Depe nden t Diffusion Coefficient . . . . . . . 116

7.5.4 Calcula tion of Diffusion Coefficients . . . . . . . . . . . . . . . . 119

7.5.5 Front Propa gati on with and With out Segregation . . . . 123


14/172

Contents XIII

8 . C onc l ud i n g R e m a r k s a nd O u t l o ok . . . . . . . . . . . . . . . . . . . . . . . . 125

A . N u m e r i c a l M e t h o d s U s e d t o S t u d y r a n u la r M a t e r i a ls . . . 129

A.1 Monte Ca rlo Me thod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

A.2 Dif fusing-Void Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

A.3 Method of S teepes t Descent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

A .4 C e ll ul a r A u t om a t a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

A.5 Event -Dr iven S imula t ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

A.6 Time-Dr iven Simula t ions Molecular Dynamics) . . . . . . . . . . . . 133

A.6 .1 T ime- In tegra t ion Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 133

A.6.2 Forces Dur ing Col lis ions . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

A.6 .3 Numerica l S tab i l ity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

A.6 .4 Compar i son wi th Expe r iments . . . . . . . . . . . . . . . . . . . . . 141

A .7 Sum m a r y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

R e f e r e n c e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

I nde x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159


15/172

1 I n t r o d u c t i o n

J e d e N a t u r w i s s e n s c h a f t w ~ r e w e r t l o s , d e r e n B e h a u p t u n g e n n i c h t i n

d e r N a t u r b e o b a c h t e n d n a c h g e p r i i f t w e r d e n k S n n t e n ; j e d e K u n s t

w s w er t l o s , d i e d i e M en s ch en n i ch t m eh r zu b ew eg en , i h n en d en

S in n d es D as e in s n i ch t m eh r zu e r h e l ten v e r m Sc h te . 1

Werner Heisenberg

T h e s e c on d l aw o f t h e r m o d y n a m i c s s t a t e s t h a t t h e e n t r o p y o f a n isolated

s y s t em can o n ly i n c r eas e i n t im e o r s t ay co n s t an t f o r a s y s t em in eq u i l i b r i u m .

T h i s s u g g es t s t h a t t h e m o s t l i k e ly s t a t e f o r s u ch s y s t em s i s a f u l l y d i s o r d e r ed

o n e , ex h ib i t i n g no s p a t i a l s t r u c tu r e .

T h i s n o l o n g e r h o ld s f o r

dissipative

s y s t e m s w h i c h e x c h a n g e e n e r g y i n

t h e m o s t g e n e r a l s e n s e w i t h a n e x t e r n a l s o u r c e . S o m e c o m m o n , e v e r y d a y

e x a m p l e s a r e t h e f a s c i n a t in g g r o w i n g p a t t e r n s f o u n d i n s n o w fl a ke s o r b a c t e r i a

c o lo n ie s a n d t h e c o m p l e x t u r b u l e n t f lo w p a t t e r n s f o u n d i n t h e a t m o s p h e r e .

T h r e e i n g r e d i e n t s a r e m o s t l y f o u n d i n s u c h s y s t e m s t h a t s h o w a c o m p l e x

p a t t e r n - f o r m a t i o n p r o c e s s :

exchange of energy i n o r d e r t o d r i v e t h e s y s t em

ins tabi l i ty

i n o r d e r t o s t a r t t h e p a t t e r n

non l inear i t y i n o r d e r t o ch o o s e t h e p a t t e r n .

M a t e r i a l s t h a t a r e c o m p o s e d o f v e r y m a n y l o o se l y p a c k e d i n d i v id u a l p a r -

t i c l e s , o r g r a in s , a r e ca l l ed g r an u l a r m a te r i a l s . N u m er o u s d i f f e r en t t y p es o f

p a r t ic l e s c a n c o n s t i t u t e a g r a n u l a r m a t e r i a l , f o r e x a m p l e s a n d , p e b b l e s o r

r o ck s . H o w ev er , a l t h o u g h w e h av e a f ee li n g f o r s o m e o f t h e p r o p e r t i e s o f

g r an u l a r m a te r i a l s , e . g . i t i s k n o w n th a t an i n t e r s t i t i a l f l u id can g r ea t l y i n -

c r e a se t h e a g g r e g a t i o n o f s a n d g r a i n s t o e n a b l e s a n d c a s t l e s t o b e b u i l t w i t h

ta l l towers , ver t ica l wal l s and wide b r idges , i t i s s t i l l no t poss ib le to p red ic t

a c c u r a t e l y m a n y o f t h e p h e n o m e n a a s s o c i a t e d w i t h t h e m , e .g . t h e o c c u r r e n c e

an d im p a c t o f r o ck av a l an ch es o r m u d s li d es .

1 Any n atural science whose assu mptions can t be ver ified through observat ions

in nature would be worthless; any ar t tha t isn t ca pable of moving people or of

enl ightening the m as to the mean ing of their exis tence would also be wor thless .


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2 1. Introduction

In the engineering community, granular materials have been the subject

of a very active field of research over many decades due to their importance

in industrial processing: the majority of materials used there are at some

stage in granular form, e.g. pills, grains, stones or plastics, and even though

carefully designed, large silos used for storage can completely block due to

the formation of arches and sometimes collapse due to unforeseen stress fluc-

tuations.

Traditionally, continuum theory was used to describe granular materials

and, more recently, numerical calculations starting from a discrete-particle

description entered the field. Most measurement techniques cannot record

properties inside the bulk of a granular material and specially designed meth-

ods have to be used in order to obtain a full three-dimensional picture of the

particle dynamics. I will present two such specialized experimental methods,

as described below.

i )

ii)

For vertical hoppers, one can place numbered particles at well-chosen ini-

tial positions and reconstruct the flow dynamics from the time of outflow

of all the numbered particles [223].

To test the mixing rate in rotating cylinders, the ro tation can be stopped

and samples taken from small holes that are drilled into the granular ma-

terial in order to take height samples of the particle concentrations [122].

This will disturb the mixture considerably and in order to get other sam-

ples for different times, the experiment has to be repeated many times.

With nuclear magnetic resonance a very powerful non-invasive measurement

technique was recently applied to the study of granular materials. This makes

use of the spin echo of protons, usually tuned to the frequency of the hydro-

gen nucleus, and nowadays it is commonly referred to as magnetic resonance

imaging MRI). The original experiment was designed to measure the flow

properties and velocity and density profiles in half-filled rotating drums [213],

and since then this method has also been used to investigate granular con-

vection in vibrated systems [88, 155].

The fascinating and also puzzling field of granular materials received in-

creasing attention from the physics community only in the last decade. The

first studies were numerical investigations on size segregation under verti-

cal vibrations [258], and the first experiments were on avalanche detection

and their statistics [91], which is especially interesting in conjunction with

the concept of self-organized criticality [11]. By now, the field of research has

broadened to include such different areas as vibrations, hoppers, piles, surface

flow and segregation, to name but a few. Review articles about granular ma-

terials can be found in popular science magazines like Na tu re [185,203, 293],

Physics Today [130] and Science [88, 129], and even newspaper articles on

the newest research topics in granular materials are not uncommon.

Since granular materials are intrinsically dissipative in nature due to the

energy losses in collisions, the dynamics of pattern formation are mostly


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1. Introduction 3

studied by supplying energy in the form of

gravity

e.g. pouring),

acceleration

e.g. shaking) or a combination of both e.g. in drum mixers).

In this book, some of the most fascinating pattern-formation processes

found in granular materials are reviewed and common features outlined and

highlighted. Theoretical explanations are given where available but are usu-

ally sparse since a satisfactory theoretical description of granular materials is

still in its infancy. The list of pattern-forming granular systems in this book is

by no means complete and should be regarded as a starting point to explore

this relatively new and still-emerging field of research.

The presented material is organized in the following fashion:

In Chap. 2, some of the peculiar properties of granular materials found

in different experiments and numerical simulations will be given, which will

also provide the reader with the necessary background for the later chapters.

The perhaps most-widely studied granular system with respect to pat-

tern formation is a shaken one. Vertical vibrations are considered in Chap. 3

whereas horizontal vibrations are considered in Chap. 4. Both vibration types

often show a similar behaviour and the connection to patt erns found in fluids

is striking.

A common handling technique of granular materials is pouring or dis-

charging them, e.g. to form heaps. This can lead to stratification patterns

or a strong segregation, depending on the materia l parameters, which is pre-

sented in detail in Chap. 5.

The most common storage devices for granular materials are silos and

hoppers. Chapter 6 is devoted to their study, where the flow properties of

granular matter in different geometries in addit ion to density and stress fluc-

tuations are discussed. These fluctuations can lead to waves and unwanted

early hopper failure.

Mixing of particles is often done in a rotating drum, but such an attempt

can lead to unwanted segregation demixing), which is shown in Chap. 7. In

addition, the different flow regimes and the evolution of an initially segregated

configuration are presented.

Some concluding remarks are given in Chap. 8 as well as speculations

on future research devoted to granular materials and on novel numerical

techniques and approaches.

In the appendix, the most commonly used numerical methods to study

granular materials are reviewed with a clear emphasis on my favourite tech-

nique, the discrete-element method which I will also refer to as molecular

dynamics MD) simulations, given in Appendix A.6.


18/172

2 S o m e E x p e r im e n t a l P h e n o m e n a

o f G r a n u l a r M a t e r i a l s

A few surprising or puzzling experiments and corresponding simulations from

the fascinating world of granular materials will be presented in this chapter in

order to introduce the reader to the subject. They are important for a better

understanding of the results and findings given in the remainder of this book.

The selection is by no means exhaustive and should be regarded merely as

a st arting point for further exploration. Additional experimental results and

possible explanations in some cases can be found in the literature [27, 76,

129, 130, 194].

2 1 S h e a r F l o w

To understand the flow properties of granular mater ials, simple shear exper-

iments, most ly in a circular geometry, have been conducted for more than 40

years. A two-dimensional cross section of a simple shear experiment using two

parallel plates is shown in Fig. 2.1, in which the bottom plate is at rest and

the top plate moves with a constant velocity U to the right. If a Newtonian

fluid is put between the two plates, the shear stress T on the bounding walls

is proportional to the shear rate in the laminar-flow regime, i.e.

du

T = rl~yy , 2.1)

with ~? being the temperature-dependent fluid viscosity [301].

For granular materials this is only approximately true for very low shear

rates [114]. A quadrat ic dependence is found for higher shear rates, i.e. for the

so-called

grain inertia

regime, since the momentum transferred per collision

and the frequency of granular collisions are both proportional to the mean

shear rate [10, 114]. The same two dependencies on the shear rate were found

for the normal stress where the thickness of the shearing layer was 5 to

15 grain diameters. The origin of the quadratic dependence of the shear

and normal stresses on the shear rate at high shear rates can be explained

by writing down continuum equations for the mass conservation and the

linear-momentum balance, and by applying a kinetic theory which explicitly

considers the fluctuating component of velocity [133, 180].


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6 2. Some Experim ental Phe nom ena of Granular Materials

a

Fig . 2.1. Sketch of a simple shear-flow expe rime nt and correspond ing velocity

profile for a Ne wto nian fluid in the la min ar regime ; the lower wall is at rest a nd

the upper wall moves with a constant velocity U to the right

The expe r i m en t a l s e t up shown i n F i g . 2 .1 was s t ud i ed num er i ca l l y u s i ng

a ha rd - sphe re m ode l by Cam pbe l l and Brennen [42 ] . Pa r t i c l e -wa l l co l l i s i ons

were t re a te d in two d i f feren t ways : e i ther a fu l ly rou gh w al l o r a no-s l ip condi -

t i on . Af t e r t he s t eady s t a t e was r eached i n t he l a t t e r ca se, wh i ch co r re sp onde d

t o t h e a b o v e - c it e d e x p e r im e n t s , t h e e x p e r i m e n t a l l y o b s e rv e d q u a d r a t i c d e p e n -

dence o f t he she a r and no rm a l s t r e ss es on t he m ean shea r r a t e was found . In

an independent s imula t ion [299] , a s imi lar resu l t was ob ta ined .

In a m ore - r ecen t s i m u l a t i on by Thom pson and Gres t [285 ] , so f t pa r t i c l e s

wi t h equa l r ad i i were u sed . S t i ck - s l i p m o t i on was obse rved , t he t h i cknes s o f

t he shea r l aye r be i ng be t ween 6 and 12 pa r t i c l e d i am e t e r s . A m os t su rp r i s ing

resu l t a ro se i n t ha t t he shea r s t r e s s d i d not s h o w a q u a d r a t i c d e p e n d e n c e

o n t h e m e a n s h e a r r a t e , b u t r a t h e r s tur ted fo r l a rge shea r r a t e s . Th i s was

e x p l a i ne d b y t h e dil t ncy i n t he s t e ady - s t a t e r eg i m e s ee a lso nex t s ec ti on ) .

2 2 D i l a t an c y

Befo re s ingle g ra ins in p i les o f g ran ular ma ter ia l s can m ove, the loca l dens i ty

has t o dec rease . Th i s de c rease in dens i t y i s known as

dil t ncy

and i s a un ique

fea t u re o f g ranu l a r m a t e r i a l s [242]. D i l a t ancy can bes t be un de r s t oo d by

l ook i ng aga in a t t he t wo-d i m ens i ona l shea r - f low exam pl e ske t ched i n F i g . 2 .1.

Im ag i ne t ha t t he spac i ng be t ween t he t wo pa ra l l e l p la t e s i s dense l y fi ll ed w i t h

m on od i spe r s e sphe res o f r ad i u s R i n t he fo rm o f a t r i angu l a r l a t t i ce . Wh en

t he shea r i ng m o t i on i s s t a r t ed , no pa r t i c l e m o t i on w i l l s e t i n a s l ong a s t he


20/172

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Yo ~ ~

~ [ I l I l I i l X I [ I ~ i W ~

I I I l ] f l I q [ I ] l l ] l •

~ 2 * I * * ' ~ A sin w t

2 .3 Sol id -F lu id Tran s i t ion 7

1

P0.8

0.6

0.4

0.2

0

0

1 2 3 4

H e i g h t ( c m )

F i g 2 2 S o l i d - f lu i d t r a n s i t i o n i n a g r a n u l a r m e d i u m u n d e rg o i n g v e r t i c a l v i b r a t i o n :

( a ) s e t u p w i t h 3 00 p a r t ic l e s a n d (b ) d e n s i ty p ro fi le m e a s u re d a l o n g t h e d o t t e d l i n e

of pa r t (a ) , acco rd ing to [54]

t o p l a y e r d o e s n o t m o v e u p w a r d s b y ( 2 - v ~ ) R ~ 0 .2 6 8 R a n d t h u s a l lo w s

f o r w h o l e p a r t i c l e la y e r s t o s l id e o v e r o n e a n o t h e r . T h i s m o t i o n l e a d s t o a

l o c a l d e n s i t y d e c r e a s e a n d r e d u c e d f r i c ti o n . C a m p b e l l [4 1] s t a t e d t h a t s u c h

a m i c r o - s t r u c t u r e i n d u c e s a p r e f e r e n c e f o r a p a r t i c l e t o c o l l id e w i t h p a r t i c l e s

w i t h i n i t s o w n l a y e r a l o n g l i n e s p a r a l l e l t o t h e l a y e r a n d w i t h p a r t i c l e s i n

t h e i m m e d i a t e l y n e i g h b o u r i n g l a y er s a t a n g l e s r o u g h l y p e r p e n d i c u l a r t o t h e

d i r e c t io n o f f lo w . T h i s is th e p h e n o m e n o n o f d i l t n c y a n d c a n b e f o u n d

w h e n e v e r g r a in s s t a r t t o m o v e . M o s t s t r ik i n g l y , t h i s is o b s e r v e d a t t h e b e a c h

w h e n o n e s ta r t s t o s h e a r w e t sa n d . T h e s u r f a c e a p p e a r s d r y w h e n m o t i o n

s t a r t s , s i n c e th e i n t e r s t i t i a l f lu i d o c c u p i e s t h e a d d i t i o n a l v o i d s c r e a t e d i n t h e

s a n d b e l o w t h e s u r f a c e .

2 .3 S o l i d F l u i d T r a n s i t i o n

W h e n e v e r a g r a n u l a r m a t e r i a l m o v e s , t h e d e n s i t y h a s t o b e b e l ow t h e b u l k

d e n s i t y o f t h e m a t e r i a l a t r e s t . T h i s w a s d e s c r i b e d i n t h e p r e c e d i n g s e c t i o n

f o r s h e a r f lo w , a n d is fi rs t v is i b le a t t h e f r e e s u r f a c e w h e n v i b r a t e d s y s t e m s

a r e c o n s i d e r e d . I t i s v i s ib l e i n v e r t i c a l ly a n d h o r i z o n t a l l y v i b r a t e d s y s t e m s

a n d c a n b e v i e w e d a s a s o l id - f l u id - l ik e p h a s e t r a n s i t i o n .

A v e r t i c a l l y e x c i t e d , q u a s i - t w o - d i m e n s i o n a l s y s t e m c o n s i s t in g o f 3 0 0 s t e e l

s p h e r e s w i t h d i a m e t e r 2 .9 9 m m w a s in v e s t i g a te d b y C l e m e n t a n d R a j c h e n b a c h

[5 4]. T h e s y s t e m , s k e t c h e d i n F i g . 2 .2 a , w a s e x c i t e d w i t h a l o u d s p e a k e r w i t h

a n o u t p u t o f t h e f o r m

. 4 ( t ) = A s i n w t ,

w h e r e A i s t h e a m p h t u d e o f t h e v i b r a t io n s , t a k e n t o b e 2 . 5 r a m , a n d w is th e

a n g u l a r f r e q u e n c y o f t h e v i b r a t i o n s , t a k e n t o b e 2 0 H z . T h e s i d e w a ll s w e r e


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8 2. Some Experimental Phenomena of Granular Materials

built at an angle of 30 ~ to the vertical to give a fully compacted medium

when the system is at rest. The positions and corresponding density fields

of all particles were recorded for different phases of the external excitation,

and it was found that the density profile was independent of the phase of

the external excitation. The universal form of the density profile, measured

along the dotted line shown in Fig. 2.2a, is drawn schematically in Fig. 2.2b,

where p represents density and the height was measured along the dotted

line. The average density for a triangular packing is 0.91 and the density

decreases rapidly in a narrow region close to the free surface. No convection

was found for this set of parameters, but particles above the surface were

seen to undergo ballistic flights.

To characterize the strength of the external vibration, one introduces the

dimensionless acceleration F given by

Aw 2

F_=

g

where g represents the gravitational constan t./~ measures the maximum ac-

celeration given to a particle in contact with the vibrating plate and for

/ < 1, the bo ttom layer will always be in contact with the plate. It was

found experimentally [93] that the threshold value for the solid-fluid tran-

sition is /'c = 1.2 4-0.05, which is in agreement with molecular dynamics

simulations [278].

2 .4 C o n v e c t i o n R o l l s

Above the fluidization threshold, Fc, one finds convection rolls in vertically

vibrating cells, as reported by different experimental [92, 95, 148] and nu-

merical [101, 158, 278] works. Since the motion of the individual particles

can be monitored in detail in the numerical simulations, a brief review of

the numerical results is given here with a more detailed analysis following in

Chap. 3.

The rolls are found to sta rt in the middle of the cell and they will fill the

whole cell for higher excitations. The sense of rotation of the rolls depends on

the ratio of the particle-particle friction, fp, and the particle-wall friction,

fw [158, 278]. If fp < fw, the particles will move upwards in the middle region

of the cell, as shown in Fig. 2.3a, whereas when fp > fw, the particles will

move downwards in the middle, as shown in Fig. 2.3b.

Recently it was argued by Luding et al. [175] that the onset of motion

and the pat terns in convection cells found in molecular dynamics simulations

may be partly due to questionable microscopic interactions and unphysically

large contact times. Another interesting result in this context is the finding of

turbulent flow in powders and a suggested

k 5/3

scaling as found in [280,281].

However, the error bars given in these works indicate that further research

needs to be done in this area.


22/172

a)

2 5

Free -Sur f ace F low

b )

j O

Fig . 2 .3 . Sense of ro ta t ion o f conv ec t ion rol ls : a ) wa ll shea r f r i c tion is g r ea te r than

in te r -pa r t i c le fr i c t ion and b) wa ll shea r f ri c t ion i s l es s tha n in te r -pa r t i c le f r i c t ion

2 .5 F r e e - S u r f a c e F l o w

P l a s t i c s p h e r e s w e r e p l a c e d i n a n i n c l i n e d g l a s s - w a l l e d c h u t e i n s u c h a w a y

t h a t t h e y c o u l d m o v e f re e l y u n d e r t h e i n f lu e n c e o f g r a v i t y , a s s k e t c h e d i n

F i g . 2 . 4 a . B y u s i n g a h i g h - s p e e d c a m e r a , i t w a s p o s s i b l e t o i d e n t i f y t w o d i f -

f e r e n t f lo w r e g i o n s , n a m e l y a f r i c t i o n a l r e g i o n c l o s e t o t h e b e d a n d a c o ll i-

s i o n a l r e g i o n a b o v e i t [7 1]. T h e s e t w o f lo w r e g i o n s c o u l d b e s u b d i v i d e d e v e n

f u r t h e r , w h e r e t h e d i v i s i o n t h i c k n e s s e s d e p e n d e d o n t h e c h o s e n e x p e r i m e n -

t a l c o n d i t i o n s . D i f f e r e n t c l u s t e rs o f s p h e r e s w e r e i d e n ti f ie d a n d m o n i t o r e d ,

a n d t w o d i f f e re n t t y p e s o f b e d s w e r e st u d i e d : s m o o t h b e d s m a d e o f p o l i s h e d

a l u m i n i u m o r h i g h - fr i c t io n r u b b e r a n d r o u g h b e d s w h e r e s p h e r e s o f d i f fe r e n t

r a d i i w e r e g l u e d a t r a n d o m p o s i t i o n s o n t o t h e b e d s u r f ac e . T h e m o s t s t r i k i n g

d i ff e re n c es b e t w e e n t h e t w o a f o r e m e n t i o n e d b e d s w e r e th e b u l k d e n s i ty a n d

t h e v e l o c i t y o f t h e p a r t i c l e s m o v i n g d o w n t h e c h u t e a s a f u n c t i o n o f h e i g h t

a b o v e t h e b e d , t h e l a t t e r b e i n g s h o w n s c h e m a t i c a l l y i n F ig . 2 .4 b . T h e s t e e p

i n c r e a s e in v e l o c i t y in t h e c a s e o f t h e s m o o t h b e d f o r s m a l l v e l o c i ti e s c a n b e

e x p l a i n e d b y b l o c k s o f s p h e r e s s li d in g d o w n w a r d s . B y u s i n g m o l e c u l a r d y -

n a m i c s s i m u l a t i o n s , t h e v e l o c i t y a n d d e n s i t y p r o fi le s f o r t h e s m o o t h a n d t h e

r o u g h b e d s c o u l d b e r e p r o d u c e d , w i t h t w o d i f f er e n t r e g i m e s in t h e c a s e o f t h e

s m o o t h b e d , c o r r e s p o n d i n g t o l o w a n d h i g h k i n e t i c e n e rg i e s, w h i c h w e r e a l so

o b s e r v e d in t h e s a m e s i m u l a t i o n s [2 31 ].

T h e i m p o r t a n c e o f t h e b e d r o u g h n e s s o n t h e p a r t ic l e m o t i o n i s m o s t

c l e a r l y s e e n b y c o n s i d e r i n g s i n g l e p a r t i c l e s r o l l i n g o r s l i d i n g d o w n a n i n c l i n e d

p l a n e . I n t h i s c a s e , t h r e e d i f fe r e n t r e g i m e s a r e o b s e r v e d a n d t h e p a r t i c l e m o -

t i o n i s f o u n d t o d e p e n d o n t h e a n g l e o f i n c l i n a t i o n o f t h e p l a n e . F o r l ow

i n c l i n a t i o n a n g l e s , a p a r t i c l e w i l l c o m e t o r e s t a f t e r h a v i n g t r a v e l l e d o n l y a

s h o r t p a t h , w h e r e a s f o r v e r y h i g h i n c l i n a t i o n a n g l es , a p a r t i c l e a c c e l e r a t e s

c o n s t a n t l y u n t i l i t fa l ls o ff t h e p l a n e [2 43 ]. T h e r e a l s o e x i s t s a n i n t e r m e -


23/172

1 0 2 . S o m e E x p e r i m e n t a l P h e n o m e n a o f Gr a n u l a r M a t e r i a ls

O o

,0% o .

~o~ o

W ~ ~ o o

S m o oth b e ~ o O o O

I r i

( a ) h~6 ( b ) x

12

y / / / / / / /

~ s

~2opO

%g,%o o

- 2o o

- ~ O o O O O 4 [ -. / / Ro ug h be d . .. .. .. .. .

R o u g h b ; _ _ _ 0

1 2

Ve l o c it y ( m / s )

F i g . 2 . 4 . P a r t ic l e s o n a n i n cl in e d c h u t e : ( a ) s k e t c h o f t h e s e t u p a n d ( b ) p a r t i c le

ve loc i ty a s func t ion of bed he igh t h* ( in pa r t i c le d iamete r s ) : th e d o t te d l ine denotes

a r o u g h b e d a n d t h e s o l id li n e a s m o o t h b e d ( s c h em a t i c )

d i a t e r e g i m e , w h e r e a f t e r a s h o r t t i m e a p a r t i c l e w i l l t r a v e l w i t h a c o n s t a n t

d o w n w a r d s v e l o ci ty . I t h a s b e e n s h o w n u s i n g m o l e c u l a r d y n a m i c s s i m u l a t i o n s

t h a t t h e e x i s t e n c e a n d t h e r a n g e o f i n c l in a t io n a n g le s o f t h e c o n s t a n t - v e l o c i t y

r e g i m e d e p e n d c r u c i a l l y o n t h e r o u g h n e s s o f t h e p l a n e [2 55 ].

2 . 6 I n c l i n a t i o n A n g l e

I m a g i n e y o u r s e l f b u i l d i n g a s a n d p i l e b y s l o w ly p o u r i n g d r y g r a i n s c o n s t a n t l y

a t t h e s a m e s p o t a s d e p i c t e d i n F i g . 2 . 5 a . D u e t o local r a i n r e a r r a n g e m e n t s ,

t h e p i le w i ll b e c o m e s t e e p e r a n d s t e e p e r i n t i m e u n t i l a c r i ti c a l s u r fa c e s lo p e

i s r e a c h e d , c a l l e d t h e

a n g l eo f m arginal ta b il ity

O m . W h e n t h i s a n g l e i s

r e a c h e d , a global r a i n m o t i o n s e t s i n , n a m e l y a n a v a l a n c h e d e t a c h e s , w h i c h

c a n t r a n s p o r t g r a i n s a ll t h e w a y d o w n t h e s l o p e o f t h e p i l e, a s s h o w n i n

F i g . 2 .5 b . W h e n t h e g r a i n m o t i o n h a s s t o p p e d , t h e s l o p e o f t h e p i le h a s

r e a c h e d t h e ang l eo f repose f t h e m a t e r i a l , O r w h i c h i s a r o u n d 5 d e g r e e s

l o w e r t h a n t h e a n g l e o f m a r g i n a l s t a b il i ty .

A s i m i l a r s i t u a t i o n a r is e s in r o t a t i n g d r u m s o r c y l i n d e r s . A ll p a r ti c l e s a r e

a t r e s t a t t h e s t a r t o f t h e r o t a t i o n a n d t h e s u r f a c e a n g l e i n c r e a se s in t im e .

W h e n t h e a n g l e o f m a r g i n a l s t a b i l i t y i s r e a c h e d , a n a v a l a n c h e e v e n t o c c u r s .

D e p e n d i n g o n t h e r o t a t i o n r a t e o f t h e d r u m , i n d iv i d u a l a v a la n c h e s c a n b e

v i s i b l e o r c o n t i n u o u s a v a l a n c h e s c a n o c c u r , w h i c h w i l l b e d i s c u s s e d i n m o r e

d e t a i l i n C h a p . 7 .


24/172

a)

2.8 Com mo nly Used Mate rials 11

c)

Fig. 2.5. Surface angles: (a) building a pile through pouring, (b) pile right before

an

avalanche at the

angle of marginal stability,

Om and (c) pile right after an

avalanche at the angle of repose of the m aterial, 0r

2 7 D e n s i t y a n d S t re s s F l u c t u a t i o n s

Since the individual particles , e.g. grains or spheres, forming a specific gran-

u la r m ate r ia l , exper ience no , o r on ly very weak , a t t r ac t ive fo rces, the dens i ty

can very eas i ly be decreased f rom the bu lk dens i ty by a cons iderab le am ount .

This g ives r ise to dens i ty waves due to the form at io n of arches , as found, for

exa mple , in ho ppe r f lows [21] an d pipe f lows [237], which som etim es leads

to a com plete blocking of the f low. Large den s i ty f luctuat ions , w hich follow

a power - law d is t r ibu t ion o f the fo rm

P w) = Po w -~

where w denotes f re-

quency, can be observed, e .g . in ver t ical p ipes with a pos i t ion- independent

expo nent of a ~ 1 .5 th at c lass if ies the p ar t ic ular f low as a

self-organized

process [125].

Much theore t ica l and exper imenta l work has been devo ted to the s ta t i c

and dy nam ic s t ress d is t r ibu t ions in hoppers , as descr ibed in for exam ple [131,

303]. Not only the averag e value of these dis t r ib ut ion s seems im po r ta nt , but

large power- law f luctuat io ns with varyin g expo nent are a lso observed [22],

which migh t l ead to f a i lu re and co l lapse wi th in the hoppe r s t ruc tu res . L arge

s tress f luctu at ions are in add i t ion observed whe n glass bead s are cont in uously

sheared [205] . In th is case , for large values of w, the ra te- inv ar iant power

spec trum var ies as w -2 , indica t ing a rand om process , whereas an e xpon ent

of 0.6 is found for smaller values of w.

2 . 8 C o m m o n l y

U s e d M a t e r i a l s

Many granu la r mate r ia l s a re met in indus t r ia l and pharmaceu t ica l app l ica -

t ions and the i r par t i c le - s ize d i s t r ibu t ions l ead to par t i c le numbers tha t a re

of ten beyond the scope o f toda y s numer ica l s imula t ions, which a re based on

the concept of d iscrete par t ic les . However , in recent years , ma ny la bor ato ry


25/172

12 2 . S o m e E x p e r i m e n t a l P h e n o m e n a o f G r a n u l a r M a t e r ia l s

T a b l e 2 . 1 . M a t e r ia l p r o p e r t i e s o f c o m m o n l y u s e d g r a n u la r m a t e r ia l s in e x p e r i-

m e n t s : d e n s i t y p , P o i s s o n s r a t i o a , Y o u n g s m o d u l u s Y a n d n o r m a l r e s t i t u t i o n

coefficient en

M ate r i a l p a Y en

( g / c m 3 ) ( N / m 2)

V i ta m in- E pi l ls 1 .1 ? 5 .4 x 10s 0.89

M us tard seeds 1.3 ? ? 0 .75

A ce ta te be ad s 1 .319 0.28 3.2 x 109 0.87

G lass be ad s 2.5 0.22 7.1 x 10 l~ 0.97

Alu m in ium bea ds 2 .7 0 .33 6 .9

X 1 1

0.8

Steel be ad s 7.83 0.28 1.91 x 1011 0.6

e x p e r i m e n t s h a v e b e e n c o n d u c t e d in o r d e r t o g et a b e t t e r i n s i g h t in t o t h e p a r -

t ic l e d y n a m i c s o f g r a n u l a r s y s t e m s . F o r c o n v e n i e n ce , T a b l e 2.1 s u m m a r i z e s

t h e r e l e v a n t m a t e r i a l p r o p e r t i e s w h i c h w i l l e n t e r n u m e r i c a l s i m u l a t i o n s , f o r

t h e s ix m o s t c o m m o n l y u s e d g r a n u l a r m a t e r i a ls . M o s t l y m a t e r i a l s w e r e c h o -

s e n t h a t c o n s i s t o f s p h e r i c a l p a r t ic l e s . H e r e a d e n o t e s

Poisson s ra tio

w h i c h i s

d e fi n ed a s t h e r a t i o o f t h e t r a n s v e r s e c o m p r e s s io n t o t h e l o n g i t u d in a l e x t e n -

s i o n , Y d e n o t e s t h e

m o d u l u s o f e x t en s io n

a l s o c a l l e d

Yo u n g s m o d u l u s

w h i c h

m e a s u r e s t h e s t if f n es s o f t h e m a t e r i a l a n d e n s t a n d s f o r t h e r e s t i t u t i o n c o e f-

f ic ie n t i n t h e n o r m a l d i r e c t io n . T h e l a t t e r q u a n t i t y i s d e f i n e d a s t h e a b s o l u t e

v a l u e o f t h e r a t i o o f t h e r e l a t iv e p a r t i c l e v e l o c i ti e s r i g h t b e f o r e a n d r i g h t a f t e r

a co l l i s i on .

R e f e r r i n g t o T a b l e 2 . 1 , t h e v i t a m i n - E p i l l s h a v e a l i q u i d c o r e a n d t h e

m u s t a r d s e ed s c o n t a i n a c o n s i d e r a b l e a m o u n t o f w a t e r . B o t h o f t h e a f o r e -

m e n t i o n e d p r o p e r t i e s a l lo w a n o n - in v a s i v e m a g n e t i c r e s o n a n c e i m a g i n g i n-

v e s t i g a t i o n , w h i c h is o f t e n u s e d i n c o n n e c t i o n w i t h r o t a t i n g d r u m s [1 19 , 2 13 ].

I t s h o u ld b e n o t e d t h a t t h e m u s t a r d s e e ds a re n o t a s p e r fe c t l y r o u n d a s t h e

o t h e r m a t e r i a l s l i s t e d i n T a b l e 2 . 1 , w h i c h m a y l e a d t o a h i g h e r s u p p r e s s i o n

o f r o t a t i o n s .


26/172

3 Vert ical Sha king

P a r t i c l e m o t i o n i n g r a n u la r m a t t e r s t a r t s w h e n t h e e n e r g y i n p u t o v e r c o m e s

t h e s t ro n g en e rg y d i s s i p a t i o n cau s ed b y co l l i s i o n s . P ro b ab l y t h e b es t - s t u d i ed

s y s t em o f t h i s k i n d is en e rg y i n p u t t h ro u g h v e r t i ca l v i b ra t i o n s o f e i t h e r t h e

w h o le c o n t a i n e r o r j u s t t h e b o t t o m p l a te .

Fo r ex c i t a t i o n s h i g h e r t h an a ce r t a i n t h re s h o l d acce l e ra t i o n , t h e i n i t i a l l y

f la t su r face becomes uns tab le and forms a heap , which wi l l be d i scussed in

Sec t . 3.1. Wi t h i n t h e h ea p , p a r t i c l e s u n d e rg o a co n v e c t i v e m o t i o n , wh ere t h e

s t r u c t u r e o f t h e p a t t e r n d e p e n d s o n t h e a c c e l e r a t io n a n d o t h e r p a r a m e t e r s ,

e .g . p a r t i c l e -p a r t i c l e f r i c t i o n an d g as -p res s u re e f fec ts , wh i ch w i ll b e p re s en t ed

i n Sec t. 3 .2 . Th e co n v ec t i v e m o t i o n can l ead t o s u r f ace wav es an d s t a t i o n a ry

pa t te r ns l ike square s and s t r ipes , as d i scussed in Sect . 3 .3 . I f the e ner gy supply

i s red uce d to a sequen ce of s ingle shakes o r taps t h e g r a n u l a r m a t e r i a l c a n

be compact i f i ed , which wi l l be shown in Sect . 3 .4 . In the same sec t ion , the

ine las t i c co l lapse in a mo nola yer o f par t i c les i s d i scussed . Las t b u t no t l eas t ,

p ro b ab l y t h e l a rg es t i n d u s t r i a l ap p l i ca t i o n fo r v e r ti ca l v i b ra t i o n s is t h e s o r t i n g

of g ra ins v ia seg regat io n ou t l ine d in Sect . 3 .5 .

3 . 1 H e a p F o r m a t i o n

T h e e n e r g y i n p u t i n m o s t v e r t i c a l - v i b r a ti o n e x p e r im e n t s c o m e s t h r o u g h a

s i n u so i d a l ex c i t a t i o n o f e i t h e r t h e wh o l e co n t a i n e r o r t h e b as e p l a t e , w h i ch

c a n b e w r i t t e n a s

A t ) = A s in 27rf t ) , 3 .1)

w h e r e A a n d f d e n o t e t h e a m p l i t u d e a n d t h e f r e q u e n c y o f t h e v i b r a t io n ,

respect ive ly . I f a sys tem i s p repared wi th an in i t i a l ly f l a t su r face , th i s sur face

remains f l a t fo r low exci ta t ions , i .e . smal l ampl i tudes and smal l f requencies .

Th e p a r t i c l e s cl o se t o t h e s u r f ace can m o v e m o re f r ee l y t h a n t h e p a r t i c l e s in

the bu lk .

Wi t h i n c reas i n g ex c i t a t i o n , p a r t i c l e m o t i o n c l o s e t o t h e s u r f ace b eco m es

vis ib le , which ca n be v iewed as a f lu id iza t ion of the g ranu lar ma ter ia l . In a

n a r ro w reg i o n c lo s e t o t h e f r ee su r f ace , t h e d en s i t y d ec reas es r ap i d l y f ro m i ts

bu lk va lue to zero [54]; see a l so Fig . 2 .2. At a cer ta in p ar am ete r c om bina t ion of

f r eq u en cy an d am p l i t u d e , t h e f l a t s u r f ace b eco m es u n s t ab l e an d t h e g ran u l a r


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a)

14 3. Vertical Shak ing

~

0 1

0.01

10

0.2 mm o

100

Frequ ency f (Hz)

Fig . 3 .1 . (a) The free surface becomes unstable beyond a threshold accelerat ion

with the surface inclined at 0 and a flow of particles as indicated by arrows; (b)

thresho ld a mplitu de to g enerate the surface instabil it_y as function o f frequen cy for

different bea d d iameters, showing a scaling of A ,,- ] - , according to [92]

m a t e r i a l fo rm s a h eap a s s k e t ch ed in F i g . 3 .1a . In l a rg e -as p ec t - r a t i o s y s t em s ,

t h e s i t u a t i o n is s o m e wh a t d i ff e r en t an d i t was r ep o r t ed t h a t a t t h e i n s tab i l i t y

t h r e s h o l d t h e d i s t u r b a n c e s a p p e a r a t t h e l a t e r a l b o u n d a r i e s , a n d t h e n t h e

p a r t i c l e s m i g ra t e t o wa rd t h e cen t e r o f t h e ce l l an d fo rm a m o u n d [161].

Pa r t i c l e s f lo w d o wn t h e p i le l ead in g t o a p e rm an en t s u r f ace cu r r e n t J s,

a n d a r e t r a n s p o r t e d b a c k t o t h e s u r fa c e t h r o u g h c o n v e c ti v e m o t i o n f r o m t h e

b o t t o m t o t h e t o p t h ro u g h t h e p i l e ; s ee t h e a r ro ws s h o wn i n F i g . 3 .1 a . Th e

s l o pe o f t h e p i le was fo u n d t o b e a l way s s m a l l e r t h a n t h e an g l e o f r ep o s e o f

the ma ter ia l [92]; see Sect . 2 .6 fo r a d ef in i t ion of the ang les .

By u s i n g d i f f e ren t g l as s b ead s w i t h d i am e t e r s o f 0.2, 0 .4 an d 1 .0 m m ,

Ev es q u e an d Ra j ch en b ach [92 ] i n v es t i g a t ed t h e am p l i t u d e an d f r e q u en cy re l a -

t i o n o f t h e i n s t a b i li t y t h r e sh o l d . T h e d a t a f o r t h e o n s e t o f t h e h e a p f o r m a t i o n

a re s h o wn i n F i g . 3 .1 b i n a d o u b l e - l o g a r i t h m i c p l o t . Th e s h ak i n g am p l i t u d e

was v a r ied o v e r n ea r l y t wo o rd e r s o f m a g n i t u d e an d i t was fo u n d t h a t t h e

re l ev an t p a ram e t e r t o d es c r i b e t h e i n s t ab i l i t y t h re s h o l d i s t h e

acceleration

o f

t h e g ra i n s . Th i s m i g h t b e u n d e r s t o o d b y r ea l i z i n g t h a t t h e l o ca l acce l e ra t i o n

h as t o exceed g rav i ty , g, i n o rd e r fo r g ra i n s t o m o v e r e l a t i v e t o each o t h e r . Th e

i n s t ab i l i t y t h re s h o l d Fc , wr i t t en a s a d i m en s i o n l e s s acce l e ra t i o n , was fo u n d

t o b e i n d e p en d en t o f t h e g ra i n d i am e t e r i n t h e ex p e r i m en t s g i v en i n [92]. Th e

fo ll o win g r e l a t i o n co u l d b e ex t r ac t ed f ro m t h e d a t a s h o wn i n F ig . 3 .1 b

Acw c

Fc -= - 1.27 + 0 .1 0 , (3.2)

g

wh ere w _= 2 ~rf d en o t e s t h e an g u l a r f r eq u en cy. Th e accu racy o f t h e ex p o n e n t

for the f requency was 2 .0 + 0 .1 .


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3.1 Heap Form ation 15

a )

3O

~

2

;~ 10

I

-

Exper iment

Parabol ic f i t - -

b)

i O

0

I I I I I

I 3

Dimensionless accelerat ion/ '

Fig. 3 .2 . (a) Sketch of the glass tes t tub e experim ent and ( b) invading thickness

as function of dimensionless acceleration, according to [93]

T h e a b o v e - c i te d m e a s u r e m e n t s w e r e r e p e a t e d i n v a c u u m a n d

no

di f ference

was found fo r t he va l ue o f t he i n s t ab i l i ty t h re sho l d [92]. However , La roche

e t al . [161] pe r fo rm e d a si m i la r expe r i m en t and t hey found t ha t i n vacuu m

(10 -5 t o r r ) , t he convec t i ve m o t i on d i s appea r s and t he l aye r f r ee su r f ace r e -

m a i ns f l a t , excep t c l o se t o t he l a t e r a l boundar i e s . A heap fo rm a t i on can s t i l l

be observed on ly a t h igh enough f requencies in very nar row ce l l s , bu t th i s i s

due t o l a t e r a l bo un da ry e f f ec ts . Th i s con t rove r sy has r a i s ed a l o t o f d i scus -

s ion [89, 162, 240] but i t seems th at the gas effect i s no t negl igible, especia l ly

i f d _< 1 m m [283], which wi ll be f ur t he r d i scussed be low. Ple ase a l so no te tha t

fo r l a rge r acce l e ra t i on va l ues (F > 4 ) , a pe r i od -doub l i ng i n s t ab i l i t y cou l d be

observed [69].

An o t he r exp e r i m e n t t o de t e rm i ne t he onse t o f pa r t i c l e fl u i d iza t i on fo r

ve r t i ca l shak i ng was p e r fo rm ed by Evesque e t a l . [ 93] and t he i n it i a l s e t up

is ske tch ed in Fig . 3 .2a. T he lower pa r t o f the g lass tub e cons i s ted o f two

ver t i ca l and coax i a l t ubes hav i ng d i am e t e r s o f 5 and 10 cm , r e spec ti ve l y . Th e

i nves t i ga t ed m a t e r i a l was s and m a de up o f g ra i n s w i t h a d i am e t e r r ang i ng

f rom 200 to 600 ~m . Af t e r t he shak i ng was s t a r t e d and i f t he acce l e ra t i on

exceeded t he f l u i d i za t i on t h re sho l d , t he l oosened s and m oved up be t ween

t he i nne r and ou t e r t ubes , s ee F i g . 3 .2a . Whe n r each i ng a s t eady s t a t e , t he

inv d ing th ickness

was measured and i t s va lues are shown in Fig . 3 .2b as

a func t i on o f t he d i m ens i on l e ss acce l e ra t i on F . Th e au t ho r s co nc l uded f rom

tha t f igure a c r i t i ca l va lue of Fc = 1 .2 • 0 .05 for the f lu id iza t ion th resh old ,

even thou gh the ad ded para bol ic f i t in Fig . 3 .2b sugges t s a s ligh t ly lower

value for Fc .

Th i s i s cons i s t en t w i t h (3 .2 ) and ano t he r i ndependen t expe r i m en t [224 ]

whe re a va lue of Fc - - 1 .17 • 0 .05 was found , w hich was a l so ind epe nde nt o f


29/172

16 3. Vertical Shaking

h

a)

hi

r

100

10

~X,x ' ' ' Exper'im|ent

(b ) hX, xo

I i o o o o ol

0.1 1

Part icle diameter d (mm)

Fig. 3.3. Gas-pressure effect: (a) sketch of app ara tus a nd (b) scaled satur ation

height as func tion of particle dia mete r for glass spheres in air at 1 atm; solid l ine

is a power-law fit with an exponent of -1.63, according to [225]

t he m a t e r i a l u sed. A l l t hese r e su l t s sugges t t ha t t he onse t o f f l u id i za t ion and

f low i s r a t h e r un i ve r s a l and occu r s a ro und Fc = 1.2 fo r g ra i n d i am e t e r s o f 0 .2

t o 2 m m .

Dif feren t phys ica l mechanisms have been iden t i f i ed as poss ib le causes for

t he above -d i s cus sed heap i ng i n s tab i l it y . C l em en t e t a l. [ 53 ] i nves t i ga t ed t he

i m p o r t a n c e

of frict ion

by u s i ng a quas i - two-d i m ens i ona l s e t up ope ra t i ng c l o se

t o t he f l u id i za t ion t h re sho l d i n t he l ow - f r equency r ange 8 - 30 Hz . If t he g ra i n -

gra in f r i c t ion was too low no heaping was found , which was the case for f resh

s t ee l and a l um i n i um beads . F o r ox id i zed beads wh i ch have a s i gn if i cant l y

h i ghe r f r i c t i on coe f fi c ien t , t hey ob t a i n ed a hea p wh i ch l ooked s i m il a r t o t he

t h ree -d i m ens i ona l expe r i m en t and wh i ch was i n i t i a t ed by t wo ro l l s s t a r t i ng

f rom t he s i de wa l l s . Wi t hou t s i de wa l l s no heap was found . The ex t en t o f

t he ro l l s was cons t an t du r i ng t he heap i ng p roces s and i t i nc reased l i nea r l y

wi t h acce l e ra t i on . Each ro l l l ed t o a peak whose d i s t ance f rom t he l a t e r a l

b o u n d a r y i n c r e a s e d p r o p o r t i o n a l t o t h e l o g a r i t h m o f t im e .

T h e i m p o r t a n c e o f th e gas-pressure effect was add res sed i n m ore de t a i l

by Pa k e t a l . [225] who a l so wante d to reso lve the d i scuss ion ab ou t a h eap

being ab le to fo rm in vacuum ra i sed in [89 , 92 , 161 , 162] . The exper imenta l

app a ra t u s cons i s t ed o f annu l a r ce ll s w i t h a na r row gap o f 1 cm , a s ske t ched i n

F i g . 3.3a. O ne o f t he m ea su red quan t i t i e s was t he ve r t i ca l d is t ance b e t ween

t he t op a nd t he bo t t o m o f t he i nc l ined su r f ace H m ea su red a t F - - 1 .3 a s a

f u n c t io n o f t h e p a r t i c l e d i a m e t e r d , t h e m e a n d e p t h o f t h e g r a n u l a r m a t e r i a l

h and the p re s su re P . Th e au t ho r s found , i n ag reem en t w i t h [161], t ha t

hea p ing e i the r ceases o r i s s ign i f i can t ly red uce d for P = 0 [225]. T he y a lso

found t h a t gas e f fec ts depe nd s t rong l y on t he s ize o f t he g ra i n s , by m easu r i ng


30/172

3.2 Co nvectiv e Mo tion 17

Fig. 3 .4 . Schematic diagram of the

experimental setup; the arrows indi-

cate the long-time convective parti-

cle motion of the vertically vibrating

granular bed; please note that the bot-

tom plate is fixed at the side walls, ac-

cording to [266]

t he s ca l ed s a t u ra t i on he i gh t , H/d as a func t i on o f pa r t ic l e d i am e t e r d , a s

shown i n F i g . 3 .3b . The da t a l ed t o t he r e l a t i on

H

-., d -1 63 (3.3 )

d

wh i ch s t a t e s t ha t t he ve r t i ca l d i s t ance H i nc reases w i t h dec reas i ng pa r t i c l e

d ia me ter d . No pressure ef fec t was v is ib le fo r d = 0 .1 3c m a t h = 10cm . I t

was a l so con f i rm ed t ha t heap i ng i s supp res sed i f h/d is small .

W hen i nves t i ga t ing c i r cu l a r heaps m a de o f i r r egu l a r l y shaped a l um i n a

g ra in s , F a l con e t a l. [ 94 ] found a s a t h re sho l d fo r the heap fo rm a t i on a va l ue

o f Fc = 1 .17 +0 .0 6 i n t he f r equen cy r ange 20 < f < 120 Hz , bu t t he t h re sho l d

decr ease d wi th incre as ing f req uen cy g iv ing a va lue of Fc = 0 .74 + 0 .03 a t

160 Hz.

3 2 C o n v e c t i v e M o t i o n

An e a r l y s t udy o f t he convec t i ve m o t i o n o f g ranu l a r m a t e r i a l s i n ve r t ica l l y

v ibra ted ce l l s was done by Savage [266] , and the appara tus i s ske tched in

F i g. 3 .4 . The s ide wa ll s were f i xed and t he am p l i t ude o f t he base v i b ra t i ons

was spa t i a l l y non -un i fo rm as shown i n t he f i gu re . Even t hough t he ce l l was

t h ree -d i m ens i ona l , t h e f low t ha t deve l oped was r easonab l y t wo-d i m ens i ona l

as ske tched in Fig . 3 .4 . The shak ing f re que ncy was var ied f rom 20 to 45 Hz

and t he shak i ng am p l i t ude be t ween 2 .5 and 5 .0m m , wh i ch gave va l ues fo r

t he d i m ens i on l e s s acce l e ra t i on F we l l above 1 . Due t o t he l a rge r v i b ra t i on

am p l i t ude a t t h e cen t r e o f t he ba se t he f r ee su r face a lso showed one cen t r a l

heap . T he m a i n r e su l t s were t ha t ( i) t he s t r eam i ng ve l oc i t y a l ong the a r rows

shown i n F i g . 3 .4 f i r s t i nc reased w i t h f r equency , r eached a m ax i m um and

t hen dec reased fo r h i ghe r f r equenc i e s and ( ii) w i t h i nc reas i ng am p l i t ude , t he

peak i n t he s t r e am i n g ve l oc i t y fo r a g i ven am p l i t ude sh i f ted t o l ower f r e-

quencies . The explanat ion goes as fo l lows [266] : the granular mater ia l can

be t r ea t ed a s a com pres s i b l e f l u i d and t he v i b ra t i ons s end ( acous t i c ) waves


31/172

18 3. Vertical Shaking

upwards which are attenuated on their way up. A significant contribution

to the pressure comes from particle-particle collisions and the particles are

in contact with the vibrating bed over the

complete cycle

of the vibration.

With increasing acceleration the streaming velocity increases since more col-

lisions occur. However above a critical parameter combinat ion the granular

bed loses collisional contact with the particles over some par t of the cycle and

the vibrations become less effective in inducing pressure waves. This explains

the maxima and the fact that the streaming velocity decreases for higher

accelerations.

The material properties can be varied easily in numerical simulations

which can greatly help in studying the dependence of the convective mo-

tion on a single parameter. This was done independently by Taguchi [278]

and Gallas et al. [101] using molecular dynamics simulations; the simula-

tion technique is described in more detail in Appendix A.6. The lat ter group

also investigated the effect of a spat ially modulated ampli tude as sketched in

Fig. 3.4 and found the strongest convection at a vibration frequency of about

60 Hz. The maximum of the convection current increased with decreasing

dissipation coefficient i.e. for decreasing energy loss during collisions. Ac-

cording to Walker [298] the explanation for the existence of the maximum

of the convective motion at around 60 Hz can be found in the work by the

British petroleum engineer R. A. Bagnold.

Taguchi [278] calculated the onset of convection using two different quan-

tities namely the cell-to-cell flow of particles and the vertical radius of

the convection roll. He found that a critical dimensionless acceleration of

Fc -- 0 . 9 - 1.2 is needed to start the convection which is in perfect agree-

ment with the experimental results given in Sect. 3.1 above. In the same

paper it was argued th at the fluidization threshold and the convection insta-

bility should have the same value of Fc since the convection which is induced

by the elastic interaction between particles was found to be the origin of the

heaping [278]. However the last argument only works in vessels having a fi-

nite width and thus cannot be responsible for all the experimentally observed

convection patterns. The convection rolls start ed at the fluidized surface and

their vertical extent increased with increasing excitation. The orientation of

the rolls depended on the ratio of the particle-particle friction ]p and the

particle-wall friction fw as already discussed in Sect. 2.4. If fp < fw the

particles will move upwards in the middle region of the cell whereas when

fp > fw the particles will move downwards in the middle see Fig. 2.3.

When investigating the convective motion of 2 mm glass beads in a 35 mm-

wide Pyrex cylinder at vibrational accelerations of 3 to 7 g Knight et al. [148]

found tha t the sense of rota tion of the convection rolls could be reversed i.e.

particles moved upw rds along the walls and downw rds in the middle when

a container with outwardly slanted walls was used i.e. a container which

became wider with height.


32/172

3.2 Convective Motion 19

The dynamic phase transitions in the roll pattern were studied experi-

mental ly by Aoki et al. [5] using a 30 mm thick and 200 mm high glass vessel

having a variable length of 100, 150 or 200 mm. Nine sizes of glass beads were

used ranging in diameter from 0.10 to 1.29mm. The dimensionless accelera-

tion, F , could be varied up to 10.5 and the dependence of the roll patterns

and their transitions on the frequency, the container length, the bed height

and the bead diameter was invest igated. The observed pat terns were classified

through the number of visible rolls and the direction of the particle motion

close to the vertical side walls. As before, no convective motion occurred as

long as F < 1, and a downw rds convection along the side walls showing two

rolls set in above an acceleration of Fc > 1.

The direction of this convection pattern is due to the shear force at the

walls, which was nicely illustrated through computations by Lee [165]. The

shear force at the vertical walls opposes gravity during the upwards and

the downwards motions of the pile. However, since the pile is more densely

packed when moving upwards, the shear force is then larger, resulting in a net

shear force pointing downwards after a whole vibration cycle. This mechanism

breaks down when F becomes too large, since then the time span when the

bed is in flight increases, which leads to alternating shear directions during

the upwards and downwards motions of the pile. Furthermore, the downwards

mode, denoted by the letter D, was found to be not very stable, which led to

a breakdown of the symmetry of the heap resulting in an inclined slope; see

Fig. 3.1a.

When F is further increased, a transition in the convection pat tern oc-

curred, leading to a rather stable upw rds motion of particles along the side

walls. In contrast to the downwards mode,

multiple

pairs of convection rolls

can appear in the upwards mode, leading to a patt ern similar to the Rayleigh-

B~nard convection [87, 126] in fluids. The pattern in the upwards mode was

denoted by the letter U followed by the number of rolls, e.g. U2 or U4. For

slowly increasing F, the system exhibited the sequence of patterns D -+ U2

--4 U

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Pattern Formation in Granular Materials-Siegfried GrobMann

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