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
in ranular M ater ia ls
W i th a Fo re w o rd b y S i eg f r ie d G ro f l m a n n
an d 83 -r igures
Springer
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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.
S e e a l s o h t tp : / / w w w . s p r i n g e r . d e / p h y s / b o o k s / s t m p . h t m l
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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
Vol u me 64
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R i s t ow , G e r a l d H . : Pa t t e r n f o r m a t i on i n g r an u l a r m a t e r i a l s / G e r a l d H . R is tow . - B e r l i n ; H e i de l be r g ; N ew Y ork ;
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ISSN oo81-3869
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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
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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
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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.
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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
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8/17/2019 Pattern Formation in Granular Materials-Siegfried GrobMann
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o a) /
\ ooooo
Voo Oooogoooooo o O o
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.
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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 -
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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 .
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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
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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 .
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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
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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
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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
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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.
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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