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P H Y S I C A
ELSEVIER Physica A 238 (19 97 ) 129-148
Time-dependent behav ior o f granular mater ia l
in a v ibrat ing box
Jysoo Lee
Benjamin Levich Institute and Department of Physics, City College of the City University of New York,
New York, NY 10031, USA
Received 17 May 1996
A b s t r a c t
Using numerical and analyt ic methods, we s tudy the t ime-dependent behavior of granular
materia l in a vibrat ing box. We f ind, by molecular dynamics s imulat ion, that the temporal f luc-
tuations o f the pressure and the height expan sion scale in A f , where A (f) is the ampli tude
(frequ ency ) of the vibrat ion. On the other hand, the f luctuations o f the veloci ty and the granular
temperature do not scale in any s imple com binat ion of A and f . Using the kinetic theo ry of
Haft, we s tudy the temporal behav iors o f the hydro dyn am ic quanti ties by perturbing about their
tim e-av erag ed values in the quasi-incompressible limit. The results o f the kinetic the ory disagree
with the num erical simulations. The kinetic t heo ry predicts that the w hole material oscillates
roug hly as a s ingle block. H owe ver, the numerical s imulations show that the region of act ive
part ic le movement is local ized and moves with t ime, behavior very s imilar to the propagation
of a sound wave .
Keywords: Granular media; Vibration; Wave; Scaling; Kinetic theory
1 . I n t r o d u c t i o n
S y s t e m s o f g r a n u l a r p a r t ic l e s ( e .g . s a n d ) e x h i b i t m a n y i n t e re s t i n g p h e n o m e n a , s u c h
a s s e g r e g a t i o n u n d e r v i b r a t i o n o r s h e a r , d e n s i t y w a v e s i n t h e o u t f l o w t h r o u g h a h o p p e r
a n d a t u b e , a n d t h e f o r m a t i o n o f h e a p s a n d c o n v e c t i o n c e l l s u n d e r v i b r a t io n [ 1 - 5 ] .
T h e s e p h e n o m e n a a r e c o n s e q u e n c e s o f th e u n u s u a l d y n a m i c a l r e s p o n s e o f t h e s y st e m s ,
a n d a r e f o r t h e m o s t p a r t s t i l l p o o r l y u n d e r s t o o d .
W e f o c u s o n t h e v e rt i c a l v i b r a t i o n o f a b o x c o n t a i n i n g g r a n u l a r p a r ti c l es . T h e r e
a r e m a n y i n t e r es t in g p h e n o m e n a a s s o c i a te d w i t h th i s s y s t e m , s u c h a s c o n v e c t i o n c e ll s
[ 6 - 1 1 ] , h e a p f o r m a t i o n [ 1 2 - 1 7 ] , s u b - h a r m o n i c i n st a bi li ty [ 1 8] , s u r fa c e w a v e s
[ 1 9 , 2 0 ] a n d e v e n t u r b u l e n t f l o w s [ 2 1 ] . T h e b a s i s f o r u n d e r s t a n d i n g t h e s e d i v e r s e
0378-4371/97/$17.00 Copyright ~ 1997 Elsevier Science B.V. All rights reserved
PH S 0 3 7 8 -4 3 7 1 ( 9 6 ) 0 0 2 8 1 - 6
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p h e n o m e n a i s t h e s t a te o f g ra n u l a r m e d i a u n d e r v i b r a t io n . T h e s t at e i s c h a r a c t e r i z e d
b y t h e h y d r o d y n a m i c f ie l d s o f th e s y s t e m , s u c h a s t h e d e n s i ty , v e l o c i t y a n d g r a n u l a r
t emp era tu re f i e ld s .
T h ere h av e b een sev era l s tu d ies o n th e s t a t e o f g ran u la r p a r t i c l e s u n d er v ib ra t io n .
T h o m a s e t a l. s t u d ie d t h e s y s t e m i n th r e e d i m e n s i o n s , m a i n l y f o c u s i n g o n t h e b e h a v i o r
o f sh a l lo w b ed s [22 ]. C 1 6 men t an d R a j ch en b a ch ex p er im en ta l ly measu red th e d en s i ty ,
v e l o c i t y a n d t e m p e r a t u r e f ie l ds o f a t w o - d i m e n s i o n a l v e r t i c a l p a c k i n g o f b e a d s [ 2 3] .
T h e y f o u n d t h a t t he t e m p e r a t u r e i n c r e a se s m o n o t o n i c a l l y w i th t h e d i s t a n c e f r o m t h e
b o t t o m p l a t e . T h e s a m e s y s t e m w a s s t u d i e d b y m o l e c u l a r d y n a m i c s ( M D ) s i m u l a t i o n
w i th s im i l a r r e su l t s [2 4 ] . In a se r i e s o f s imu la t io n s an d ex p er imen t s , L u d in g e t a l .
s t u d ie d t h e b e h a v i o r o f t h e o n e - a nd t w o - d i m e n s i o n a l s y s t e m s [ 2 5 - 2 7 ] . T h e y f o u n d
th a t t h e h e ig h t ex p an s io n , w h ich i s t h e r i se o f t h e cen te r o f mass d u e to th e v ib ra t io n ,
scales in the var iab le x = A f . H e r e , A a n d f a r e th 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 th e v ib ra t io n . War r e t a l. ex p e r im en ta l ly co n f i rm ed th e sca l in g , an d th ey a l so g av e
an a rg u m en t fo r i ts o r ig in [2 8] . In r ecen t MD s imu la t io n s o f t h e th ree -d im en s io n a l
s y s t e m , L a n a n d R o s a t o m e a s u r e d t h e d e n s i t y a n d t e m p e r a t u r e f i e l d s [ 2 9 ] . T h e y c o m -
p ared th e r esu l t s w i th th e th eo re t i ca l p red ic t io n s b y R ich man an d Mar t in [3 0 ] , an d
f o u n d g o o d a g r e e m e n t . A l s o , a n a p p r o x i m a t e t h e o r y w a s d e v e l o p e d f o r t h e s y s t e m i n
o n e d imen s io n , w h ich ag rees w i th s imu la t io n s in th e w eak an d th e s t ro n g d i s s ip a t iv e
reg im es [31 ] .
In a p rev io u s p ap er , w e s tu d ied th e
t i m e - a v e r a g e d
b e h a v i o r o f t h e t w o - d i m e n s i o n a l
s y s t e m u s i n g n u m e r i c a l a n d a n a l y t i c m e t h o d s [ 3 2 ] . U s i n g M D s i m u l a t i o n , w e f o u n d
t h a t th e t i m e - a v e r a g e d v a l u e o f n o t o n l y th e e x p a n s i o n b u t a l s o t h e d e n s i t y a n d t h e
g ran u la r t em p era tu re f i e ld s sca l e in x. W e a l so u se d th e k in e t i c t h eo ry o f H af t [3 3 ]
t o d e t e r m i n e t h e t i m e - a v e r a g e d h y d r o d y n a m i c f i e l d s i n t h e q u a s i - i n c o m p r e s s i b l e l i m i t .
T h e resu l t s a re , i n g en era l , co n s i s t en t w i th th e n u mer ica l d a t a , an d in p a r t i cu la r sh o w
sca l in g b eh av io r in th e v a r i ab le x . W e fo u n d th a t th e o r ig in o f t h e sca l in g can b e
u n d e r s t o o d w i t h i n t h e f r a m e w o r k o f t he t h e o r y .
In th e p resen t p ap er , w e ex ten d o u r s tu d y to th e t i m e - d e p e n d e n t b e h a v i o r o f t h e
s y s t e m , w h o s e u n d e r s t a n d i n g i s n o t o n l y e s s e n t i a l i n s t u d y i n g v a r i o u s t i m e - d e p e n d e n t
p h e n o m e n a , b u t a l s o n e c e s s a r y t o u n d e r st a n d t h e m e c h a n i s m s o f c e r ta i n s t e a d y - st a t e
p h e n o m e n a . F o r e x a m p l e , m a n y o f th e a r g u m e n t s f o r th e m e c h a n i s m o f t he c o n v e c -
t i o n i n v o l v e t h e v a r i a ti o n s o f c e r ta i n h y d r o d y n a m i c q u a n t it i e s o v e r a v i b r a ti o n c y c l e
[7 ,8 ,10 ,11 ,16] .
W e f in d , b y M D s imu la t io n , t h a t t h e t emp o ra l f lu c tu a t io n s o f t h e p ressu re an d th e
h e i g h t e x p a n s i o n s c a l e i n x , w h i l e t h e v e l o c i t y a nd t h e g r a n u l a r t e m p e r a t u r e d o n o t
s c a le i n a n y s i m p l e c o m b i n a t i o n o f A a n d f . W e a l s o s t u d y th e s y s t e m u s i n g t h e k in e t i c
t h e o r y o f H a f t , w h e r e t h e t e m p o r a l b e h a v i o r i s s t u d i e d b y p e r t u r b i n g a b o u t t h e t i m e
av erag e in th e q u as i - in co mp ress ib l e l im i t. T h e resu l t s o f th e k in e t i c t h eo ry d i sag ree
w i t h t h e n u m e r i c a l d a ta . T h e k i n e t i c t h e o r y p r e d i c t s t ha t t h e w h o l e s y s t e m o f p a r t ic l e s
m o v e s a s o n e e f f e c ti v e " b l o c k " . T h e n u m e r i c a l s i m u l a t io n s , o n t h e o t h e r h a n d , s h o w
t h a t th e r e g i o n o f a c t iv e p a r t ic l e m o v e m e n t i s l o c a l iz e d , a n d m o v e s w i t h t im e . T h e
p r e s e n c e o f t h e w a v e i s p a r tl y r e s p o n s ib l e f o r t h e d i s c r e p a n c y .
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T h e p a p e r i s o r g a n i z e d a s f o l l o w s . I n S e c t i o n 2, w e s p e c i f y th e i n t e r a c t i o n o f t h e
p a r t i c le s u s e d i n t h e M D s i m u l a ti o n s. W e t h e n p r e s e n t t h e t e m p o r a l f l u c tu a t i o n s o f t h e
e x p a n s i o n a n d t h e f ie l d s o b t a i n e d b y t h e s i m u l a t io n s . A n a l y t i c r e s ul t s w i l l b e d i s c u s s e d
i n S e c t i o n 3 . T h e c o n t i n u u m e q u a t i o n s f o r g r a n u l a r m a t e r i a l w i l l b e g i v e n , a n d p e r t u r -
b a t i o n e q u a t i o n s a r e d e r i v e d . W e p r e s e n t t h e s o l u t io n o f th e e q u a t i o n s , a n d c o m p a r e
w i t h t h e n u m e r i c a l d a ta . I n S e c t i o n 4 , w e c h e c k t h e a s s u m p t i o n s u s e d i n o b t a i n i n g
t h e s o l u t io n . W e a l s o d i sc u s s v a r i o u s p r o p e r t i e s o f th e w a v e s . C o n c l u s i o n s a r e g i v e n
in S ec t io n 5 .
2 . N u m e r i c a l s i m u l a t i o n
W e s t a r t b y d e s c r i b i n g t h e i n t e r a c t i o n s u s e d i n t h e M D s i m u l a t i o n s . T h e s i m u l a t i o n s
a r e d o n e i n t w o d i m e n s i o n s w i t h d i s k - s h a p e d p a r t i c l e s . T h e i n t e r a c t i o n b e t w e e n t h e
p ar t i c l e s i s t h a t o f C u n d a l l an d S t rack [3 4 ] , w h ich a l lo w s th e p a r t i c l e s to ro t a t e a s w e l l
a s t ra n s l a te . P a r t ic l e s i n t e r a c t o n l y b y c o n t a c t , a n d t h e f o r c e b e t w e e n t w o s u c h p a r t i c le s
i a n d j i s t h e f o l l o w i n g . L e t th e c o o r d i n a t e o f th e c e n t e r o f p a r ti c l e i ( j ) b e R i ( R j ) ,
a n d r = R i - R j . W e u s e a n e w c o o r d i n a t e s y st e m d e f i n e d b y t w o v e c t o rs h ( n o r m a l )
a n d g ( s h e a r ) . H e r e , h = r / I r L , a n d g i s o b t a i n e d b y r o t a t i n g h c l o c k w i s e b y ~ / 2 . T h e
n o r m a l c o m p o n e n t F j n i o f t h e f o r c e a c ti n g o n p a r ti c l e i f r o m p a r ti c l e j i s
Fjn~i = kn(ai 4- aj -
I r l )
- 7nme( v h) , ( 1 )
where
a i ( a j )
i s t h e r a d iu s o f p a r t i c le i ( j ) , a n d
v = d r / d t .
T h e f i r s t t e rm i s t h e l i n ea r
e l a s t i c f o r c e , w h e r e kn i s t h e e l as t i c co n s tan t o f t h e m ate r i a l . T h e co n s tan t Vn o f th e
s e c o n d t e r m i s th e f r i c ti o n c o e f f ic i e n t o f t h e v e l o c i t y - d e p e n d e n t d a m p i n g f o r c e , a n d
m e i s t h e e f fec t iv e mass, mim j / (m i J r m j ) . T h e s h e a r c o m p o n e n t F j S i i s g i v e n b y
F j S i = - s i g n ( 5 s ) m i n ( k s lf S l , # l F j ~ i l ) . ( 2 )
T h e t e r m r e p r e s e n t s s t a t i c f r i c t i o n , w h i c h r e q u i r e s a f i n i t e a m o u n t o f f o r c e ( # F j n i )
to b reak a co n tac t . H e re , p i s t h e f r i c t io n co ef f i c i en t , 6 s th e t o t a l s h e a r d i s p l a c e m e n t
d u r in g a co n tac t , an d ks t h e e l as t i c co n s tan t o f a v i r tu a l t an g en t i a l sp r in g .
T h e s h e a r f o r c e a l s o a f f e c ts t h e r o t a t i o n o f t h e p a r ti c le s . T h e t o r q u e a c t in g o n
p ar t i c l e i d u e to p a r t i c l e j i s
~ . ~ i = ,-~ x ~ F j S ~ , 3 )
w h e r e rc i s t h e v e c t o r f r o m t h e c e n t e r o f p a r t i c l e i to t h e p o i n t w h e r e p a r t i c l e s i a n d j
o v er l ap . S in ce th e p a r t i c l e s u sed in th e s imu la t io n s a re v e r y s t if f ( l a rg e kn) , t h e a rea
o f t h e o v e r l a p i s v e r y s m a l l . I t is t h u s a g o o d a p p r o x i m a t i o n t o u s e - a i h as rc.
A p ar t i c l e can a l so in t e rac t w i th a w a l l . T h e fo rce an d to rq u e o n p a r t i c l e i , i n
c o n t a c t w i t h a w a l l , a r e g i v e n b y ( 1 ) - ( 3 ) w i t h aj - - 0 an d m e = m i . A w al l i s
assu med to b e r ig id , i . e . , i t i s n o t mo v ed b y co l l i s io n s w i th p a r t i c l e s . A l so , t h e sy s t em
i s u n d e r a g r a v i ta t i o n a l f ie l d 9 . A m o r e d e t a i l e d e x p l a n a t i o n o f t h e i n t e r a c t i o n is g i v e n
e l sew h ere [3 5 ] .
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T h e m o v e m e n t s o f th e p a r t i c le s a r e c a l c u l a t e d u s i n g a f i f th - o r d e r p r e d i c t o r - c o r re c t o r
m e t h o d . We u s e t w o V e r l e t t a b l e s . O n e i s a u s u a l t a b l e w i t h f i n i t e s k i n t h i c k n e s s .
T h e o t h e r t a b l e i s a l is t o f p a i r s o f ac tua l l y i n t e r a c t i n g p a r t i c l e s , w h i c h i s n e e d e d t o
c a l c u l a t e t h e s h e a r f o r c e . T h e i n t e r a c t i o n p a r a m e t e r s u s e d i n t h i s s t u d y a r e f i x e d a s
f o l l o w s , u n l e s s o t h e r w i s e s p e c i fi e d : kn = 1 0 5 , k s = 1 0 5 , 7n = 2 x 1 02 a n d # = 0 . 2 . T h e
t i m e s t e p i s t a k e n t o b e 5 x 10 - 6 . T h i s s m a l l t i m e s t e p i s n e c e s s a r y f o r t h e l a r g e e l a s t ic
c o n s t a n t u s e d i n th e s i m u l a t i o n s . Fo r t o o s m a l l v a l u e s o f t h e e l a s t i c c o n s t a n t , th e
s y s t e m l o s e s th e c h a r a c t e r o f a s y s t e m o f d i s ti n c t p a r ti c le s , a n d b e h a v e s l i k e a v i s c o u s
m a t e r ia l . I n o r d e r to a v o i d a r t i fa c t s o f a m o n o d i s p e r s e s y s t e m ( e . g ., h e x a g o n a l p a c k i n g ) ,
w e c h o o s e t h e r a d i u s o f t h e p a r t ic l e s f r o m a G a u s s i a n d i s t r i b u t i o n w i t h t h e m e a n 0 .1
a n d t h e w i d t h 0 . 02 . T h e d e n s i t y o f th e p a r t i c l e s is 0 .1 . T h r o u g h o u t t h is p a p e r , C G S
uni t s a re impl ied .
W e p u t t h e p a r t i c l e s i n a t w o - d i m e n s i o n a l r e c t a n g u l a r b o x . T h e b o x c o n s i s t s o f
t w o h o r i z o n t a l ( t o p a n d b o t t o m ) p l a t e s w h i c h o s c i l l a t e s i n u s o i d a l l y a l o n g t h e v e r t i c a l
d i r e c t i o n w i t h g i v e n a m p l i t u d e A a n d f r e q u e n c y f . T h e s e p a r a t i o n b e t w e e n t h e t w o
p l a t e s H i s c h o s e n t o b e m u c h l a r g e r (1 05 t i m e s ) t h a n t h e a v e r a g e r a d i u s o f th e
p a r t i c l e s , s o t h e p a r t i c l e s d o n o t i n t e r a c t w i t h t h e t o p p l a t e f o r a l l c a s e s s t u d i e d h e r e .
W e a p p l y a p e r i o d i c b o u n d a r y c o n d i t i o n i n t h e h o r i z o n t a l d i r e c ti o n . T h e w i d t h o f t h e
b o x i s W = 1 . We a l s o t r y d i ff e r e n t v a l u e s o f W, a n d f i n d n o e s s e n t i a l d i f f e re n c e i n
t h e f o l l o w i n g r e s u l t s .
We s t a r t t h e s i m u l a t i o n b y i n s e r t i n g t h e p a r t i c l e s a t r a n d o m p o s i t i o n s i n t h e b o x .
W e l e t t h e m f a l l b y g r a v i t y a n d w a i t w h i l e t h e y l o s e e n e r g y b y c o l l i s i o n s . W e w a i t f o r
1 0 5 i t e r a t i o n s f o r t h e p a r t i c l e s t o r e l a x , a n d d u r i n g t h i s p e r i o d w e k e e p t h e p l a t e s
f ix e d . T h e t y p i c a l v e l o c i t y a t t h e e n d o f t h e r e l a x a t i o n i s o f o r d e r 1 0 - 2 . A f t e r t h e
r e l a x a t i o n , w e v i b r a t e t h e p l a t e s f o r 5 0 c y c l e s b e f o r e t a k i n g m e a s u r e m e n t s i n o r -
d e r t o e l i m i n a t e a n y t r a n s i e n t e f f e c t . M e a s u r e m e n t s a r e m a d e d u r i n g t h e n e x t 2 0 0
c y c l e s .
W e m e a s u r e h y d r o d y n a m i c q u a n t it ie s - d e n s i ty , v e l o c i t y a n d g r a n u l a r t e m p e r a t u r e
w h i c h c h a r a c t e ri z e t h e s t a te o f t h e s y s t e m . T h e m o s t d e t a il e d i n f o r m a t i o n i s c o n -
t a i n e d i n t h e t i m e s e r i e s o f th e i r f i e ld s ( e .g . , d e n s i t y f i e ld ) , w h i c h w i l l b e d i s c u s s e d
l a te r . H e r e , w e w a n t t o s ta r t w i t h s o m e t h i n g s i m p l e a n d r e p r e s e n t a t i v e o f t h e
s y s t e m .
T h e c e n t e r o f m a s s o f th e p a r t i c le s c a n b e l o o s e l y r e l a t e d t o th e d e n s i t y . L e t
y ( t )
b e t h e v e r t i c a l c o o r d i n a t e o f t h e c e n t e r o f m a s s a t t i m e t . T h e m e a n d e n s i t y i s r e l a te d
t o t h e m e a n i n t e r p a r t i c l e d i s t a n c e , w h i c h i s a l s o r e l a t e d t o y ( t ) . S i n c e y ( t ) i s a sca la r
w h i c h a l s o c a n b e e a s i l y m e a s u r e d , w e s t u d y i t a s a r e p r e s e n t a t i v e o f t h e d e n s i t y .
I n th e s a m e s p i r it , w e s t u d y t h e s p a c e - a v e r a g e d v e r t ic a l v e l o c i t y V y ( t ) a n d g r a n u l a r
t e m p e r a t u r e r ( t ) i n s t e a d o f t h e c o m p l e t e l o c a l f i el d s . W e d e f i n e t h e s p a t ia l a v e r a g e
o f A ( x, y , t ) as
(A(t)) = f f A(x, y,t)p(x, y, t)d xd y
f f p(x, y, t)dx dy
( 4 )
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,L Lee /Phys iea A 238 (1997) 129-148
1 3 3
w h e r e p(x, y , t) i s th e d en s i ty f i e ld a t p o s i t i o n (x , y ) . I t t h u s fo l lo w s th a t V y ( t )= (Vy(t))
a n d z ( t ) = (T ( t ) ) , w h e r e Vy ( X , y , t ) a n d T ( x , y , t ) a r e t h e v e r t i c a l v e l o c i t y a n d t h e
g r a n u l a r t e m p e r a t u r e f i e l d , r e s p e c t i v e l y . A n o t h e r q u a n t i t y o f i n t e r e s t i s t h e p r e s s u r e
f ie ld
p(x, y , t) .
Fo r in fo rmat io n o n th i s q u an t i ty , w e w i l l s tu d y th e to t a l p ressu re a t t h e
b o t t o m ,
po(t) = f p(x, y = O, t)p(x, y = O, t)d x
f p(x,y= O ,t)dx (5)
W e s t u d ie d t h e t i m e - a v e r a g e d b e h a v i o r o f th e s y s t e m i n o u r p r e v i o u s p a p e r [3 2] .
H e r e , w e s t u d y th e t e m p o r a l b e h a v i o r , e s p e c i a l l y f o c u s i n g o n t h e v a r i a t i o n s o f t h e f i e ld s
w i th in a v ib ra t io n cy c le . We f i r s t measu re th e t emp o ra l f lu c tu a t io n , w h ich i s d e f in ed as
t h e s t a n d a r d d e v i a t i o n o f a te m p o r a l s e q u e n c e . L e t (A)t b e t h e t i m e a v e r a g e o f A ( t ) ;
t h e n , w e u s e
f i e x p = ( Y e x p ) t , # e x p = v / ( y 2 x p ) , - ( Y e x p )t ,
(6)
l
~0 = (p0 ),, ap0 = V/(P20), - (p0)~,
w h e r e t h e e x p a n s i o n Y e x p ( t) i s d e f i n e d a s t h e d i f fe r e n c e b e t w e e n
y ( t )
d u r i n g a n d b e -
f o r e th e v i b r a t i o n . W e m e a s u r e t h e s e q u a n t i t ie s a t e v e r y 1 / 1 0 0 o f a p e r i o d f o r 2 0 0
cy c les .
In o u r p rev io u s p ap er , i t w as sh o w n th a t )S ex sca l es in
A f ,
i n a g r e e m e n t w i t h e a r -
l i e r s im u l a t i o n s a n d e x p e r i m e n t s [ 2 5 - 2 8 ] . W e f i nd th a t t h e r o t a t io n o f th e p a r t i c l e s
in c lu d ed in th e p resen t s imu la t io n d o es n o t ch an g e th i s sca l in g , b u t d o es s ig n i f i can t ly
d ecr ease th e v a lu e o f )Sex . T h e d e c reas e i s p ro b ab ly d u e to th e f ac t t h a t t h e av e rag e
t r a n s la t i o n a l e n e r g y b e c o m e s s m a l le r , s i n c e s o m e o f th e e n e r g y i s t r a n s f e r r e d t o t h e
r o t at i o n. W e t h e n c o n s i d e r t h e f l u c t u a ti o n o f th e e x p a n s i o n
6Yexp.
I n F i g . l ( a ) , w e s h o w
6Yexp f o r s e v e r a l v a l u e s o f A a n d f . T h e q u a l i ty o f t h e s c a li n g i s n o t v e r y g o o d , b u t
th e d a ta a re s t i l l co n s i s t en t w i th
A f
sca l in g , e sp ec ia l ly w i th o u t t h e p e r s i s t en t d ev ia t io n
a t th e lo w A p ar t o f th e f = 2 0 d a ta . T h i s d e v ia t io n i s a l so p rese n t i n th e sca l in g
of fexp"
T h e b e h a v i o r o f
z ( t )
i s v e ry s imi la r . I t w as sh o w n th a t ~ sca l es in
A f
[32] . Again ,
w e f in d th a t t h e ro t a t io n d o es n o t ch an g e th e sca l in g o f ~ , b u t d o es ch an g e i t s v a lu e .
T h e s i tu a t io n b e c o m e s a l i tt le d i f f e re n t w h e n w e c o n s i d e r t h e f l u c t u a t io n
6 z
a s s h o w n
in F ig . l (b ) . I t i s c l ea r f ro m th e f ig u re th a t b e d o es n o t sca l e in A f . In f ac t , i t d o es
n o t s c a l e in a n y o f th e s i m p l e c o m b i n a t i o n s o f A a n d f w e h a v e t r ie d .
In th e p rev io u s p ap er , t h e b eh av io r s o f V y ( t ) an d
po(t )
w ere n o t d i scu ssed in d e ta i l ,
s in ce th e i r t im e-av erag ed v a lu es a re t r iv i a l . S in ce th e sy s t em i s i n a s t ead y s ta t e , ~ , i s
z e r o , w h i c h i s a l s o c o n f i r m e d b y t h e s i m u l a ti o n s . S i n c e t h e p r e s s u r e
po(t )
i s c a u s e d
b y t h e w e i g h t o f th e p a r t ic l e s , o n e m i g h t g u e s s /3o is s im p ly th e to t a l w e ig h t o f t h e
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134 J. Lee/Physica A 238 (1997) 129-148
(a)
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0
f = 2 0 ~ - - -
f = 5 0 + ~ -
f = l O 0
5 10 15
A f
/ "
/ , '
. /
, /
. Z
/
I
20 25
25
20
15
10
f = 2 0
f = 50 . . . . .
f = 100 ..D-
/ , / /
/ / / / /
. - " " . E2
0 r I I
0 5 10 15 20 25
(b) A f
Fig. 1. Scaling behaviors o f the hydrodynam ic qua ntities. Each datum is averaged ove r at least three sam ples,
where 20 000 m easureme nts are mad e in a sample. (a ) Th e fluctuation o f the expansion 6Yexp seems to scale
in A , where (b) the fluctuation of the temperature fiz does not scale. (c) The fluctuation of the velocity
6Vy does not scale, where (d) the fluctuation of the pressure
6po
seems to scale in
A f .
p a r t i c l e s d i v i d e d b y t h e a r e a o f th e b o t t o m ( W = 1 ). W e f i n d th a t / 30 i s i n d e e d a
c o n s t a n t i n d e p e n d e n t o f A a n d f , w h o s e v a l u e i s c o n s i s t e n t w i t h t h e t o ta l w e i g h t . O n
t h e o t h e r h a n d , t h e b e h a v i o r o f th e i r f l u c tu a t i o n s i s fa r f r o m t r iv i a l. T h e f l u c tu a t i o n o f
t h e v e l o c i t y 6V y f o r s e v e r a l v a l u e s o f A a n d f i s s h o w n i n F i g . l ( c ) . I t i s a p p a r e n t
t h a t 6 V y d o e s n o t s c a l e i n A f . A l s o , 6 Vy d o e s n o t s c a l e i n a n y s i m p l e c o m b i n a t i o n o f
A a n d f . T h e d a t a f o r t h e f lu c t u a t io n o f t h e p r e s s u r e 6po a r e c o n s i s t e n t w i t h s c a l i n g
7/23/2019 129-148
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J . Le e / P h y s i c a A 2 3 8 ( 1 9 9 7 ) 1 2 9 - 1 4 8 135
20
(c)
18
16
14
12
10
8
6
4
2
0
f= 20 o
. . . . . 4 / / / / / / / "
+ . +~ . ~ _ 4. + ++ . . . . . . . . . . ~ . . G
," .C~'"
' .~3 . .B - ' ~"
- t~12 E ] ~ I I I I
5 10 15 20
A f
25
(d)
1 0 0 0
900
800
7 0 0
6 0 0
500
4 0 0
300
200
0
/ / / / / "
D . Z 3 / / /
. . . r
~ " / /
o . ? 5 . . . . .
o
~ " f = 5 0 . . . .
J " f = 1 0 0 a
o
I I I I
5 10 15 20
A f
25
F i g . 1 . C o n t i n u e d .
in A f a s s h o w n i n F ig . l ( d ) . T h e q u a l i t y o f c o l l a p s e i s a g a i n n o t v e r y g o o d , e s p e c i a l l y
f o r t h e l o w A p a r t o f t h e f - - - - 2 0 d a t a .
T h e b eh a v i o r o f t h e t i m e- a v er a g ed q u a n t i t i e s i s e i th er t r iv i a l ( l ? y a n d / 3 0 ) , o r s c a l e s
in A f (35exp a n d ~ ) . T h e r ea s o n f o r th e t r i v ia l b eh a v i o r h a s b een d i s c u s s e d , a n d t h e
s c a l i n g i n A f c a n b e u n d er s t o o d f ro m a k i n e t i c t h eo ry o f g ra n u la r p a r t i c le s [ 3 2 ] . T h e
b e h a v i o r o f t h e f lu c t u a t io n s i s , h o w e v e r , n o t e a s y t o u n d e rs t an d . F o r e x a m p l e , o n e m i g h t
n a i v e l y e x p e c t t h a t 6V y b e h a v e s a s A t h e v e l o c i t y f l u c t u a t i o n o f t h e b o t t o m p l a t e ;
b u t t h e n u m e r i c a l d at a s u g g e s t s t h i s is n o t s o . A l s o , o n e m i g h t g u e s s
6po
b e h a v e s
a s A f 2 t h e v a r i a t i o n o f t h e e f f ec t i v e g ra v i t y ; b u t t h e n u m er i c a l s i m u l a t i o n s s u g g es t
s c a l i n g i n A f . T h u s , t h e b e h a v i o r s o f t h e t e m p o r a l f l u c tu a t i on s s e e m t o b e i n c o n s i s t e n t
7/23/2019 129-148
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136 J . LeeIPhys ica A 238 (1997) 129 148
w i th an in tu i t i v e p i c tu re . In th e n ex t sec t io n , w e d i scu ss an a t t emp t to u n d er s t an d th i s
b eh av io r .
3. Kinetic theory
I n th i s s e c t io n , w e s t u d y th e t i m e - d e p e n d e n t b e h a v i o r o f t h e s y s t e m u s i n g a k i n e t i c
t h e o r y o f g r a n u la r m a t e r ia l . W e u s e t h e f o r m a l i s m b y H a f t [ 3 3 ], w h i c h w a s s u c c e s s f u l l y
a p p l ie d t o t h e t i m e - a v e r a g e d b e h a v i o r o f t h e s y s t e m [ 3 2] . F o r m o r e d e t a i ls o n t h e
fo rm al i sm an d o th e r k in e t i c t h eo r i es o f g ran u la r ma te r i a l , s ee R ef . [3 2] an d re fe ren c es
th e re in .
H a W s f o r m u l a t i o n c o n s i st s o f e q u a t i o n s o f m o t i o n f o r m a s s , m o m e n t u m a n d e n e r g y
c o n s e r v a t i o n . T h e m a s s c o n s e r v a t i o n e q u a t i o n i s
~ p + I 7 . ( p v ) = 0 , ( 7)
w h ere p an d v a re th e d e n s i ty an d th e v e lo c i ty f ie ld s , r e sp ec t iv e ly . N ex t i s t h e i t h
c o m p o n e n t o f t h e m o m e n t u m c o n s e rv a t io n e q u a ti o n,
~ 3 [ ~ V j ~ V i~
p vi+p(v. 17)v,= \ a x + ax j JJ +po t , 81
w h ere s u m ma t io n o v e r in d ex j i s imp l i ed . T h e co e f f i c i en ts 2 an d r / a re v i sco s i t i e s
w h ich w i l l b e d e te rmin ed l a t e r . A l so , p i s t h e in t e rn a l p ressu re , an d g ~ i s t h e i t h co m-
p o n e n t o f t h e g r a v i ta t io n a l f ie l d. A l t h o u g h ( 8 ) r e s e m b l e s t h e N a v i e r - S t o k e s e q u a t i o n ,
th e co ef f i c i en t s a s w e l l a s t h e in t e rn a l p ressu re a re n o w fu n c t io n s o f t h e f i e ld s in s t ead
o f b e i n g c o n s t a n t. T h e l a s t o f t h e e q u a t i o n s o f m o t i o n i s e n e r g y c o n s e r v a t i o n ,
at
+ + G , \oxj + x,j
+ pvigi + ~xi
H e re , T i s t h e g ran u la r t emp era tu re f i e ld , K i s t h e " th e rm al co n d u c t iv i ty " , I i s t h e
ra t e o f t h e d i s s ip a t io n d u e to in e l as t ic co l l i s io n s , an d su m ma t io n s o v er in d ices i an d j
a r e i m p l ie d . A l t h o u g h t h e f o r m o f ( 9 ) is s o m e w h a t d i ff e r e nt f r o m t h a t o f t h e N a v i e r -
S to k es eq u a t io n s , t h e eq u a t io n can s t i ll b e eas i ly u n d er s to o d . T h e l e f t -h an d s id e o f (9 ) i s
s imp ly th e ma te r i a l d e r iv a t iv e o f th e to t a l k in e t ic en e rg y , w h ere th e to t a l k in e t i c en e rg y
i s d iv id ed in to th e co n v ec t iv e p a r t ( i n v o lv in g v ) an d th e f lu c tu a t in g p a r t ( i n v o lv in g T ) .
O n th e r ig h t -h an d s id e o f t h e eq u a t io n , t h e f ir s t t h ree l i n es a re s im p ly th e r a t e o f w o rk
d o n e b y t h e i n t e r n a l p r e s s u r e , v i s c o s i t y a n d g r a v i t y , r e s p e c t i v e l y . T h e t e r m i n v o l v i n g
7/23/2019 129-148
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J . Le e / P h y s i c a A 2 3 8 ( 1 9 9 7 ) 1 2 9 - 1 4 8
137
K i s t h e r a te o f e n e r g y t r a n s p o r t e d b y " t h e r m a l c o n d u c t i o n " . T h e d i s s ip a t i o n t e r m I ,
w h i c h i s a c o n s e q u e n c e o f t he i n e l a s t ic i t y o f t h e p a r ti c le s , i s r e s p o n s i b l e f o r m a n y o f
t h e u n i q u e 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 .
W e n o w d i s c u s s t h e c o e f fi c ie n t s w h i c h a r e y e t t o b e d e t e r m i n e d . D e r i v a t io n o f t h e
r e l a t i o n s o f t h e s e c o e f f i c ie n t s t o t h e f i e ld s i s b a s e d o n i n t u it i v e a r g u m e n t s [ 3 3 ]. A l s o ,
t h e d e r i v a t i o n a s s u m e s t h a t t h e d e n s i t y i s n o t s i g n i f i c a n t l y s m a l l e r t h a n t h e c l o s e -
p a c k e d d e n s i t y , i . e . , t h e s y s t e m i s a l m o s t i n c o m p r e s s i b l e . T h e r e l a t i o n f o r t h e i n t e r n a l
p r e s s u r e i s
T
p = t d p -- , ( 1 0 )
S
w h e r e t i s a n u n d e t e r m i n e d c o n s t a n t , a n d d i s t h e a v e r a g e d i a m e t e r o f t h e p a r ti c l e s . T h e
v a r i a b l e s , w h i c h i s r o u g h l y t h e g a p b e t w e e n t h e p a r t i c l e s , i s r e l a t e d t o t h e d e n s i t y b y
m
p ~ ( d + s ) 3 , ( 1 1)
w h e r e m i s t h e a v e r a g e m a s s o f t h e p a r t i c le s . T h e n , t h e v i s c o s i t y r/ i s g i v e n a s
r l= q d 2 p v @
(12)
S
w h e r e q i s a n u n d e t e r m i n e d c o n s ta n t . I n a s i m i l a r w a y , t h e th e r m a l c o n d u c t i v i t y i s
f o u n d t o b e
K = r d 2 ~ (13)
S
H e r e a g a i n , r i s a n u n d e t e r m i n e d c o n s t a n t . F i n a l l y , t h e ra t e o f d i s s ip a t i o n i s
T3/2
1 = 7 P - - , ( 14 )
S
w h e r e 7 i s a n u n d e t e r m i n e d c o n s t a n t . T h e v i s c o s i t y 2 i s l e f t u n d e t e r m i n e d , d u e t o t h e
f a c t t h a t , i n t h e r a n g e w h e r e t h e s e r e l a t i o n s a r e v a l i d , t h e t e r m c o n t a i n i n g 2 i s n e g l i g i b l e
a n d i s d r o p p e d f r o m t h e c a l c u l a t i o n .
W e i m p o s e t w o c o n s t r a i n t s i n o r d e r t o m a k e t h e e q u a t i o n s a n a l y t i c a l l y t r a c t a b l e . T h e
f i r s t i s t h e h o r i z o n t a l p e r i o d i c b o u n d a r y c o n d i t i o n . Du e t o t h e b o u n d a r y c o n d i t i o n , t h e r e
a r e n o s i g n i f i c a n t v a r i a t i o n s o f t h e f i e l d s a l o n g t h e h o r i z o n t a l d i r e c t io n . T h u s , w e o n l y
h a v e t o d e a l w i t h a o n e - d i m e n s i o n a l e q u a t i o n i n s t e a d o f a t w o o r th r e e - d i m e n s i o n a l
o n e . T h e o t h e r c o n s t r a i n t i s i n c o m p r e s s i b i l i t y , w h i c h i s a l i t t l e t r i c k y . I n c o m p r e s s i b i l i t y
i m p l i e s , s t r i c t l y s p e a k i n g , t h a t t h e d e n s i t y p i s c o n s t a n t . Du e t o t h e r e l a t i o n b e t w e e n
p a n d s ( 1 1 ) , s a l s o h a s t o b e c o n s t a n t . H e r e , w e a r e i n t e r e s t e d i n t h e s i t u a t i o n w h e r e
s i s m u c h s m a l l e r t h a n d , b u t s t i ll n o n - z e r o . I n s u c h a c a s e , th e v a r i a t i o n o f t h e d e n s i t y
c a n b e i g n o r e d , b u t n o t t h e v a r i a t io n o f a v a r i a b le t h a t d e p e n d s d i r e c tl y o n s . W e c a l l
t h i s c o n d i t i o n q u a s i - i n c o m p r e s s i b i l i t y .
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138 J. Lee/Physica A 238 (1997) 129-148
U n d e r t h e s e c o n d i t i o n s , w e s o l v e d t h e e q u a t i o n s f o r t h e t i m e - a v e r a g e d f i e l d s ,
T ( ) ( y ) ~ _ B 2 v 2 Y e x p ( - 2 y / ( ) ,
W y 0 - y
t Z d Z P O B Z v 2 y o
s( ) (Y)
~ g w ( Y o - y ) 2
e x p ( - 2 y / ( ) ,
v { )( y) = 0 ,
H e r e ,
an d
an d
( 1 5 )
p ( O ) ( y ) = P o g ( Y o - Y )
V w = 2 n A f t h e m a x i m u m v e l o c i t y o f t h e b o t to m , y 0 th e h e i g h t o f t h e f r e e s u r f ac e ,
= v / ~ d t h e d i s si p a ti o n l e ng t h. A l s o , P 0 i s t h e d e n s i ty o f t h e m a x i m u m p a c k i n g ,
w h e r e ew i s th e co e f f i c i en t o f r e s t i tu t io n o f co l l i s io n s b e tw een a p a r t i c l e an d a w a l l [32 ].
W e s t u d y t h e t i m e - d e p e n d e n t b e h a v i o r o f a q u a n t i t y b y p e r t u r b i n g i t f r o m i t s t i m e -
a v e r a g e d v a l ue . W e a s s u m e t h a t t h e p e r t u r b a ti o n t e r m o s c i ll a t e s w i t h f - t h e f r e q u e n c y
o f th e v ib ra t io n . In g en era l , t h e assu m p t io n i s n o t v a l id , s in ce o n e h as to co n s id e r a l l
t h e m o d e s w i t h d i f fe r e n t f r e q u e n c i e s . H o w e v e r , w h e n t h e a m p l i tu d e o f t h e v i b r a t i o n is
s m a l l e n o u g h , i n m a n y c a s e s, th e m o d e w i t h f r e q u e n c y f d o m i n a t e s t h e t i m e - d e p e n d e n t
b e h a v i o r . W e e x p e c t t h e r e i s a r a n g e o f A in w h i c h t h e a s s u m p t i o n i s v a li d , w h i c h w i ll
b e d e t e r m i n e d l a t e r b y t h e n u m e r i c a l s i m u l a t i o n s . W e t h u s u s e
T ( y , t ) = T ( ) ( y ) + T ( I ) ( y ) e x p ( i o ~ t ) ,
s ( y , t ) = s ( ) ( y ) + s ( l ) ( y ) e x p ( i t n t ) ,
( 1 7 )
(0) v~ ) ( y ) exp( /~o t)
y ( y , t ) = V v ( Y ) +
p ( y , t ) = p (O ) ( y) + p ( l ) ( y ) e x p ( i ~ o t ) .
S u b s t i tu t in g (1 7 ) in to th e mass co n se rv a t io n co n d i t io n (7 ) an d u s in g th e q u as i -
i n c o m p r e s s i b i l i t y c o n d i t i o n , w e o b t a i n
d v ~ O ( y ) / d y
= 0 . H ere an d in th e r es t o f t h e
ca lcu la t io n , w e co n s id e r t e rms u p to th e f i r s t o rd e r in th e ex p an s io n . S in ce V y ( y = 0 , t )
s h o u l d b e t h e v e l o c i t y o f th e b o t t o m p l a t e ,
V ( y l ) ( y ) = i V w , ( 1 8 )
w h i c h is a c o n s e q u e n c e o f q u a s i- i n c o m p r e s s i b i li t y a n d t h e o n e - d i m e n s i o n a l n a t u re o f
t h e s y s t e m . A l s o , m o m e n t u m c o n s e r v a t i o n e q u a t i o n ( 8 ) , c o m b i n e d w i t h ( 1 7 ) , b e c o m e s
p O ) ( y ) = p o A ~ o Z ( y o - y ) , ( 1 9 )
w h ich can b e eas i ly u n d er s to o d . T h e p ressu re a t t h e b o t to m i s p ro p o r t io n a l t o th e to t a l
w e ig h t o f t h e p a r ti c l e s . T h u s , t h e f lu c tu a t io n o f
p o ( t )
i s t h e to t a l mass
P o Y o
t imes th e
B2 _- (1 +
e w ) 2
( 1 6 )
1 - e2w - (2 r d /a ( ) (1 - ( / 2 y 0 ) '
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f luc tua t i on o f t he e f f ec t i ve g rav i t y / / o9 2 . F ina l l y , we cons ide r the ene rgy co nse rva t i on
e q u a t i o n ( 9 ) . C o m b i n e d w i t h ( 1 7 ) , i t b e c o m e s
io9po U ()
U (1) +
ipov(y U () d u(O)
dy
rd a (p(O) d_ u(1) + U(0) )
t d y P ~ l d - - -
dy dy
~d(U {1}p {) + U {)p{~ )) ,
(20)
where we i n t roduce t he va r i ab l e U ( y , t ) = ~ . O ne h a s to so lv e (2 0 ) fo r T (y , t ) .
W e can not , unfor tuna t e ly , ge t an a na ly t i c so lu t ion o f t he r esu l t ing non- l i nea r
di f ferent ia l equat ion. S ince
s(y, t)
h a s t o b e c a l c u l a t e d f r o m t h e r e l a t i o n ( 1 0 ) b e t w e e n
T( y , t ) , s ( y , t )
a n d
p( y , t ) ,
w e a l s o c a n n o t g e t a n e x p r e s s i o n f o r s ( l ) ( y ) .
W e c o m p a r e t h e r e s u l t s f r o m t h e k i n e t i c t h e o r y w i t h t h e n u m e r i c a l s i m u l a t i o n s
(Fig. 1) . F ir s t, we cons ide r
Vy (y, t).
Since t he t empora l f l uc tua t i on o f
vy(y, t)
i s propor-
t ional to Vy( l) (y) , the kinet ic the ory p redic t s
6Vy ,.~ A f .
The da t a f rom the s imula t i ons ,
however , a r e i ncons i s t en t w i th t h i s sca l i ng . The s i t ua t i on i s s imi l a r fo r
p( y , t ) .
T h e
kine t i c t heory p red i c t s
bpo ~ A f 2,
whi l e t he numer i ca l da t a sca l e i n
A f .
W e d o
not have ana ly t i c express ions fo r
T (y , t )
a n d
s(y , t )
t o c o m p a r e w i t h t h e s i m u l a t i o n
data .
The f a i l u re o f t he k ine t i c t heory , when app l i ed t o t he t ime-dependent behav ior , i s
i n sha rp con t r as t w i th i ts success i n s t udy ing t he t ime-ave raged behav ior . The k ine t i c
t h e o r y c o r r e c t l y p r e d i c t s t h e s c a l i n g b e h a v i o r o f a l l t h e t i m e - a v e r a g e d h y d r o d y n a m i c
quan t i t i e s . We suspec t t he f a i l u re i s a s soc i a t ed wi th t he b reakdown of a key as sump-
t i o n ( s ) u s e d i n t h e t h e o r y . T h r e e k e y a s s u m p t i o n s , b e s i d e s t h e o n e s e m p l o y e d i n t h e
form ula t i on o f t he k ine t i c t heory , a r e m ade t o ob t a in a s imple an a ly t i c so lu t i on fo r
the t ime-dependent behav ior . The f i r s t i s quas i - i ncompress ib i l i t y , which i s shown to
be va l i d fo r
A f ~ l . 5
f r o m t h e s t u d y o f t h e t i m e - a v e r a g e d b e h a v i o r [ 3 2] . H o w e v e r ,
t h e s c a l in g b e h a v i o r o f th e t im e - a v e r a g e d q u a n t it ie s r e m a i n s u n c h a n g e d e v e n i n t h e
c o m p r e s s i b le r e g i m e . T h e s e c o n d a s s u m p t io n i s t h a t t h e t im e d e p e n d e n c e o f a q u a n -
t i ty i s a s i n u s o id a l o s c il la t io n w i t h f r e q u e n c y f , w h i c h w e e x p e c t to b e v a l id f o r
smal l A . The l a s t a s sum pt ion i s t ha t t he t im e-dep end ent t e rm in (17) i s m uch smal l e r
t han i t s t ime-averaged va lue i n o rde r fo r t he pe r tu rba t i on t o be va l i d . In t he nex t
s e c t i o n , w e c h e c k t h e v a l i d i t y o f t h e s e a s s u m p t i o n s b y c o m p a r i n g w i t h t h e n u m e r i c a l
s imula t i ons .
4. V alidity of assum ptions
F i r s t , w e c h e c k t h e a s s u m p t i o n t h a t t h e m o d e w i t h t h e d r i v i n g f r e q u e n c y d o m i n a t e s
t h e t im e - d e p e n d e n t b e h a v i o r o f th e h y d r o d y n a m i c q u a n t it ie s . W e o b t a i n a t i m e s e ri e s
of yexp( t ) by measur ing i t a t eve ry 1 /100 of a pe r iod fo r 200 cyc l es . We then ca l cu l a t e
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0.06
Z Lee /Phy sica A 238 (1997) 129-148
(a)
0.05
0.04
0.03
0.02
0.01
f = 1 0 0, A = 0 . 0 1 - -
0 I
0 0.008 0.01.002 0.004 0.006
f (x 104 Hz)
0.25
0.2
f = 100 , A = 0 .03 - -
0.15
0
0.05
0 ^~- ^ _. t . , I
0 0.002 0.004 0.006 0.008 0.01
(b) f (x 104 Hz )
Fig. 2. Power spectrum of
yeo(t)
with f = 100 and (a) A 0.0 1, (b) A =0.0 3. The expansion is measured
at every 1 0- 4s for 200 cycles. The mode w ith f = 100 is dominant at A =0.0 1, but loses i ts dominance at
A = 0.03.
t h e p o w e r s p e c t r u m o f y e x p ( t ) - )~exp u s i n g a F F T r o u t i n e i n th e N A G l i b ra r y ( c 0 6 g b f ) .
T h e r e s u l t s w i t h f = 1 0 0 a r e s h o w n i n F i g . 2 . T h e m o d e w i t h f = 1 00 is d o m i n a n t
f o r A = 0 .0 1 ( F i g . 2 ( a ) ) . W h e n A i s i n c r e a s e d f u r t h e r to 0 . 0 3 , h o w e v e r , t h e f _ ~ 2 0
m o d e b e c o m e s d o m i n a n t (F i g . 2 ( b ) ) . T h e re s u lt s w i t h f = 2 0 a r e e n t i r e l y s im i l ar . T h e
f = 2 0 m o d e l o s e s it s d o m i n a n c e w h e n A i s i n c r e a s e d t o a b o u t 0 .5 . I n b o t h c a s e s , t h e
m o d e w i t h f r e q u e n c y f i s d o m i n a n t u n t il F ~ 1 0 , w h e r e
F = A o 9 2 / g .
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141
>
(a)
2.5
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
-2.5
L . i i . .. .. i
/ / ' ,~
, ~ - f ' - - "~ "% A = 0 .0 0 5
" , ' ~ ~ : : ~ , , A =0 .0 1 . . .. . .
" ' , , " ' - - - - J .
i i i i i i i [ " " 1
0 10 20 30 40 50 60 70 80 90 100
ohase (x 2~ / 10 0)
450
400
350
300
250
200
150
100
50
0
(b)
_-o0o5 -
\ ' , , , , h = 0 . 0 1 . . . . . . , , /
~
' , i " '
A = 0 . 0 3 / . . t /
0 10 20 30 40 50 60 70 80 90 100
phase (x 2 ~ / 100)
Fig. 3 . Time evolution o f (a) Vy(t), (b)
po( t)
in one cycle, where f = 100 and the data are averaged over
200 c ycles, Deviation from a sinusoidal is apparent for A =0 .03 .
T h e m e a s u r e m e n t s o f V y (t) a n d
p o ( t )
a l so s u p p o r t t h e a b o v e o b s e r v a t i o n s . I n
F i g . 3 ( a ) , w e s h o w t h e v a r i a t i o n o f V y (t) i n o n e c y c l e , w h e r e f = 1 0 0 a n d t h e d a ta
a r e a v e r a g e d o v e r 2 0 0 p e r i o d s . T h e c u r v e w i t h A = 0 .0 1 i s n e a r l y s i n u s o i d a l , w h i l e
c l e a r d e v i a t i o n i s s e e n f o r A = 0 .0 3 . A n i m p o r t a n t p o i n t t o n o t e i s th e b e h a v i o r o f th e
m a x i m u m v a l u e o f V y (t) . T h e k i n e t i c t h e o r y p re d i c t s , in ( 1 8 ) , t h e m a x i m u m v a l u e to
b e p r o p o r t i o n a l t o A , w h i c h i s c l e a r l y n o t c o n s i s t e n t w i t h t h e da t a. T h e m e a s u r e d v a l u e
i s q u i t e s m a l l e r t h a n w h a t i s p r e d i c t e d . F o r e x a m p l e , f o r A = 0 . 0 3 , t h e p r e d i c t e d v a l u e i s
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142 J. Lee/Physica A 238 (1997) 129 148
V y (
10
5
0
-5
- 1 0
(a)
100
90
(X 2 ~ / 100)
T(y , t )
0 1 ' q
0 . 6 ~ 0 1 0 0
0 : 4
0 .5 ~ . ~ 7 40 ph ase (x 2 rc / 100)
1 ~ - - - - - . . . . . ~ ~ f 2 0 3 0
(b) y 1.5 10
Fig . 4 . T ime evo lu t ion o f ( a )
Vy(y,t),
( b )
T ( y , t )
a n d ( c )
p ( y , t )
f ie l ds , w h e r e f = 1 0 0, A = 0 . 0 1 a n d t h e
da t a a re ave raged ove r 200 cyc l e s . The dens i t y f i e ld i s norma l i zed t o be t he vo lume f rac t i on .
6n , w hi le the m easured va lue i s about 2 .4 . I n F ig . 3 (b) , w e show po(t) in one period,
where again f = 100 and the data are averaged over 200 cycles . The curve seems to
deviate from the sinusoidal even for small A, and it is diff icult to determine whether
the mod e with f = 100 dominates . Again, the predic ted beha vior of the maxim um
value of
po(t)
i s not cons is tent wi th the measurement . The maximum value of
po(t)
i s predic ted to increase l inear ly with A, as in (19) , which is c lear ly not cons is tent wi th
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p :
0 . 8
0 . 6
0 . 4
(c)
Fig. 4. Continued.
1 0 0
) 0
(x 2 r t / 1 0 0 )
t h e d a t a . T h e o b s e r v e d m a x i m u m v a l u e ( ~ 4 0 0 ) i s q u i t e s m a l l e r t h a n t h e p r e d i c t e d
v a l u e ( ~ 4 0 0 0 ) .
T h e m o d e w i t h th e d r i v i n g f r e q u e n c y d o m i n a t e s th e t i m e - d e p e n d e n t b e h a v i o r f o r
s m a l l v a l u e s o f A , w h e r e a r o u g h c r i t e ri o n f o r th e d o m i n a n c e i s F , ~ 10 . A l so , i t w a s
s h o w n t h a t q u a s i - i n c o m p r e s s i b i l i t y i s v a l i d f o r
A f
< 1 .5 [3 2 ] . T h e v a l id i ty o f t h e
l in ea r p e r tu rb a t io n ap p ro x imat io n (1 7 ) i s a l i t t l e t r i ck y . We req u i re th a t t h e p e r tu rb a -
t i o n t e r m s a r e s m a l l e r t h a n t h e t i m e - a v e r a g e d t e r m s , w h i c h i s v a l i d f o r t h e e x p a n s i o n
a n d t h e t e m p e r a t u r e . T h e t w o t e r m s a r e , h o w e v e r , c o m p a r a b l e f o r t h e p r e s s u re e v e n
a t a s m a l l v a l u e o f A = 0 . 05 . T h e c o n s e q u e n c e o f th e l a r g e p r e s s u r e f l u c t u a ti o n o n
th e v a l id i ty o f t h e p e r tu rb a t io n i s u n c lea r . Fo r l a rg e A , al l o f t h e ab o v e assu m p t io n s
a r e n o t v a l id , w h i c h c o m p l i c a t e s t he a n a l y s i s o f t h e s y s t e m . F o r e x a m p l e , i n o r d e r to
s t u d y t h e t i m e - d e p e n d e n t b e h a v i o r , o n e h a s t o c o n s i d e r a d d i t i o n a l m o d e s w i t h d i f f e r e n t
f r e q u e n c i e s .
T h e s u r p ri s e i s th a t t h e p r e d i c ti o n s f o r t h e m a x i m u m v a l u e s o f t h e v e r t i c a l v e l o c i t y
a n d t h e p r e s s u r e a r e n o t c o r r e c t e v e n w h e n a ll t h e a s s u m p t i o n s s e e m t o b e v a l i d ( e . g .,
A = 0.01 an d f = 1 0 0 ) . W e in sp ec t ag a in th e p red ic t io n s o f t h e k in e t i c th eo ry . A s g iv en
in ( l 8 ) , t h e v e lo c i ty f i e ld is u n i fo rm , an d i t s v a lu e i s Vw t h e v e l o c i t y o f t h e b o t t o m .
T h e r e f o r e , t h e s o l u t io n o f t h e p e r t u r b a ti o n e x p a n s i o n s u g g e s t s t h a t th e w h o l e s y s t e m o f
p a r t i c le s i s m o v i n g a s a s i n gl e " b l o c k " a t t a c h e d t o t h e b o t to m . T h e s p a ti a l a n d t e m p o r a l
v a r i a ti o n s o f th e o t h e r f i el d s d o n o t c h a n g e t h e s i n g l e b l o c k p i c t u r e , b u t r a t h e r d e s c r i b e
t h e s t r u c t u r e o f th e b l o c k . A c o n s e q u e n c e o f t h e p i c t u r e is t h a t th e p r e s s u r e a t th e
b o t t o m i s p r o p o r t io n a l t o t h e e f f e c t i ve g r a v i ty w h i c h r e a c h e s i t s m a x i m u m ( F + 1 ) 9 a t
p h ase 3 z r/2 o f th e v ib ra t io n , w h ich i s ex ac t ly (1 9 ) .
H o w e v e r , t h e m e a s u r e m e n t s o f V y ( t) a n d p o ( t ) are n o t co n s i s t en t w i th th e p i c tu re .
T h e m e a s u r e d m a x i m u m v a l u e o f V y ( t ) i s m u c h s m a l l e r t h a n w h a t i s p r e d ic t e d , w h i c h
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su g g es t s t h a t o n ly a smal l f r ac t io n o f t h e p a r t i c l e s mo v e to g e th e r a t a g iv en t ime . A l so ,
t h e f a c t t h at t he m e a s u r e d m a x i m u m v a l u e o f p o ( t ) i s smal l e r t h an th e p red ic t io n a l so
su p p o r t s t h i s o b se rv a t io n . Fu r th e rmo re , t h e p h ase a t w h ich p o ( t ) r e a c h e s t h e m a x i m u m
is ab o u t 0 , i n co n t ras t t o 3 n / 2 s u g g e s t e d b y t h e s i n g l e b l o c k p i c t u r e . T h e d i s c r e p a n c y
c a n b e u n d e r s t o o d a s f o ll o w s . S i n c e t h e m a x i m u m a c c e l e r a t i o n o f th e b o t t o m i s la r g e r
th an th a t o f g rav i ty , p a r t i c l e s in i t i a l ly ly in g o n th e b o t to m w i l l b e " l a u n c h ed " a t a
c e r t a in p h a s e o f t h e v i b ra t i o n. T h e m a x i m u m p r e s s u r e a t t h e b o t t o m w i l l o c c u r w h e n
m o s t o f th e l a u n c h e d p a r t i c le s c o m e b a c k a n d c o l l id e w i t h t h e b o t to m , w h i c h o c c u r s
a ro u n d q S = 0 .
T h e d i r e c t e v i d e n c e a g a i n s t t h e s i n g le b l o c k p i c t u r e i s th e t i m e e v o l u t i o n o f t h e
w h o l e v e l o c i t y f ie l d s h o w n i n F ig . 4 ( a ) . H e r e , w e u s e f = 1 00 , A - - 0 . 0 1 , a t w h i c h t h e
assu m p t io n s u s ed to d e r iv e th e p red ic t io n s o f t h e k in e t i c t h eo ry seem to b e v a l id . I t
i s c l ea r f ro m th e f ig u re th a t t h e r eg io n o f s ig n i f i can t mo t io n i s l o ca l i zed , an d t r av e l s
l ik e a w a v e . T h e p r o p a g a t i o n o f t h e d i s tu r b a n c e s e e m s t o b e v e r y s i m i l a r t o t h a t o f
so u n d w av es in a g as . A l so , t h e max imu m v e lo c i ty i s ab o u t 6 , c lo se to th e p red ic -
t io n 2 n . T h e lo ca l i za t io n o f th e p a r t i c l e mo t io n can a l so b e seen in th e t ime ev o lu -
t io n o f t h e g ran u la r t emp era tu re f i e ld sh o w n in F ig . 4 (b ) . A g a in , t h e r eg io n o f h ig h
tem p era tu re i s l o ca l i zed , an d t rav e l s u p w a rd s [3 6 ] . Fu r th e rmo re , t h e lo ca t io n o f t h e
h i g h - t e m p e r a t u r e r e g i o n c o i n c i d e s w i t h t h a t o f t h e l a r g e v e l o c i t y re g i o n . T h e d e n s i t y
f i e ld , o n th e o th e r h an d , d o es n o t v a ry s ig n i f i can t ly as sh o w n in F ig . 4 (c ) , w h ich
a g r e e s w i t h t h e p r e v i o u s e x p e r i m e n t [ 23 ]. I t is c l e a r th a t t h e p r e s e n c e o f th e " w a v e s "
ch an g es th e b eh av io r s o f t h e f i e ld s , an d p o ss ib ly th e i r sca l in g p ro p er t i e s . T h e a b sen c e
o f th e w av es in th e s in g le b lo ck so lu t io n i s, a t l eas t, p a r t ly r e sp o n s ib le fo r t h e f a i l -
u re o f th e k in e t i c t h eo ry . T h e ab sen ce i s d u e to q u as i - in co m p ress ib i l i t y . In f ac t , it is
easy to d e r iv e th e s in g le b lo ck p ic tu re o n ly f ro m th e q u as i - in co mp ress ib i l i t y co n d i -
t io n an d o n e-d ime n s io n a l n a tu re o f t h e eq u a t io n . I t is t h u s n eces sa ry to co n s id e r th e
g e n e r a l c a s e o f th e k i n e t i c t h e o r y o f a c o m p r e s s i b l e g a s, w h i c h u n f o r t u n a t e l y is q u i t e
c o m p l i c a t e d .
W e w an t to f in i sh th i s sec t io n b y d i scu ss in g so me p ro p er t i e s o f t h e w a v es . W e f i r st
c o n s i d e r th e m o t i o n o f t h e m a x i m u m d i s tu r b a n c e . I n F ig . 5 ( a ) , w e s h o w t h e p h a s e
q~max(Y) at
w h i c h t ~ ( y , t ) r e a c h e s a m a x i m u m w i th f = 1 00 a n d s e v e ra l v a l ue s o f
A . I n o t h e r w o r d s , w e p l o t t h e p o s i t io n o f t h e m a x i m u m v e l o c i t y in t h e y - q ~ p l a n e .
T h e v e l o c i t y o f th e w a v e , i n v e r s e l y p r o p o r t io n a l t o t h e s l o p e o f th e
q~max(Y)
cu rv e ,
i s a b it s m a l l. T h e t i m e n e e d e d f o r t h e w a v e t o p r o p a g a t e f r o m t h e b o t t o m t o t h e
t o p o f th e p i le i s o f th e o r d e r o f t h e p e r i o d o f th e v i b r a t io n . T h e s i m u l a t io n s w i t h
. / = 5 0 s h o w t h a t, f o r th e s a m e v a l u e s o f A , th e v e l o c i t y o f t h e w a v e d o e s n o t c h a n g e
s ig n i f i can t ly , su g g es t in g th a t t h e re i s a f ix ed t ime sca le fo r t h e w av e p ro p ag a t io n . A l so ,
i t can b e seen f ro m th e f ig u re th a t t h e v e lo c i ty d ec rea ses w h en e i th e r y o r A in c reases
w i t h t h e o t h e r p a r a m e t e r s f i x e d . T h e d e c r e a s e p r o b a b l y r e s u l t s f r o m t h e d e c r e a s e o f
t h e c o l l i s io n f r e q u e n c y b e t w e e n p a r ti c le s , d u e t o t h e d e c r e a s e o f th e d e n s i t y . A l s o ,
t h e l o c a t io n o f t h e m a x i m u m g r a n u l a r t e m p e r a t u r e i n th e y - q 5 p l a n e i s s h o w n i n
F ig . 5 (b ) , w h e re o n e can see th e c lo se co r re l a t io n w i th F ig . 5 (a ) . In f ac t , t h e max im u m
t e m p e r a t u r e a l w a y s o c c u r s j u s t a b o v e t h e m a x i m u m v e l o c i t y a t a g i v e n p h a s e , w h i c h
7/23/2019 129-148
17/20
Z Lee/Physica A 238 (1997) 129-148 145
(a)
90
80
70
60
50
40
30
20
10
0
i i f t i , i i , ~
,':,".',()"A
A = 0 0 1 , ( ,. ' / ~
A = 0 .05 . . . . . . . . . . . = ::9 /
I I I I I I I I I
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
Y
- e -
(b)
90
80
70
60
50
4 0
30
20
10
0
i i i i r I I I
A = 0.01 . i : i '~
