Heap corrugation and hexagon formation of powder under vertical vibrations
E
´
ric Falcon,
1
Krishna Kumar,
2
Kapil M. S. Bajaj,
3
and Jayanta K. Bhattacharjee
4
1
Laboratoire de Physique Statistique, E
´
cole Normale Supe ´rieure, 24 rue Lhomond, 75005 Paris, France
2
Physics and Applied Mathematics Unit, ISI, 203 B T Road, Calcutta 700 035, India
3
Department of Physics and Centre for Nonlinear Science, University of California at Santa Barbara, Santa Barbara, California 93106
4
Department of Theoretical Physics, IACS, Calcutta 700 032, India
Received 21 October 1998
We report free-surface instabilities in a deep bed of fine granular material of irregular shape under vertical
vibrations. At low frequency of vibration, the conical heap due to convective flow becomes unstable above a
critical amplitude of vibration and acquires an azimuthal dependence which makes the heap surface corrugated.
At even higher amplitude, the heap is no longer stable and splits into small heaps on a hexagonal lattice. At
high frequency, we observe standing waves stripes at the same frequency as the driving one. The main
mechanism of these instabilities can be traced back to the presence of the surrounding gas, since they vanish
under vacuum conditions. S1063-651X9907605-9
PACS numbers: 83.70.Fn, 45.70.-n, 47.20.-k, 47.54.+r
Vibrated granular media display various fluidlike proper-
ties 1: roll convection 2, heaping 3–8, bubbling 9,
free-surface excitation 10–16. When the vessel accelera-
tion is greater than gravity, various patterns appear on the
free-surface of the grains. Pattern selection depends in par-
ticular on the dimensionless layer thickness N=h / d where
h is the layer thickness and d the typical size of particles.
Typically when N is greater than 10 to 20 a heap forms
spontaneously 1,15,16. The appearance of the heap has
been attributed to convective flow due either to grain friction
with the container walls 8,16 or to the presence of the
interstitial gas 6,16,17. Although two-dimensional 2D
simulations account for convective motions 18 and heap
formation 19 due to friction with the container sidewalls,
very few simulations take the effect of air into account 20.
However, the ambient gas effect is prominent for a relatively
deep bed ( N1) of small particles ( d 1 mm) 17, and
leads to a number of instabilities, as is reported here. In
addition to the convection patterns in the bulk of the granular
materials, its flat free-surface can exhibit different wave phe-
nomena: parametric extended 1,10–14 or localized 21
standing surface waves for N10-20), and period dou-
bling instabilities 1,13–16any N). For all these standing
waves, convection due to air and to friction on the edges are
negligible.
In this paper, we report qualitative observations of surface
instabilities in a deep (50N200) bed of fine granular
materials under vertical sinusoidal vibrations. We have used
noncohesive irregularly shaped fine grains from 40 to
120 m) to have large intergrain friction in order to inves-
tigate the role of large dissipation on granular material under
vibration. The small size of the grains makes the role of air
prominent 17. The choice of irregular shapes enhances dis-
sipation due to inelastic collisions as well as solid friction
22. Recently, both the effects of size and shape distribu-
tions have been shown to be linked to spontaneous stratifi-
cation 22,23. In such highly dissipative granular media, we
report secondary instabilities: first, the heap corrugation
which goes beyond the previous observations of heap forma-
tion see Ref. 24 for an overview and traveling waves on
its surface 25, second, the formation of hexagons, and
third, standing wave patterns stripes at the driving fre-
quency. Those last two patterns are not caused by parametric
instability, as was the case in previous observations on shal-
low layers of particles 12,13. All these phenomena are in-
dependent of the presence of the container sidewalls and
come from both the effects due to the interstitial gas and to
the irregular shapes of the grains.
The experimental setup consists of a vertically vibrating
vessel containing a powder alumina powder, silica powder,
or Ganga sand with grains of irregular shape see Fig. 1.
An electromagnetic vibration exciter BK4808 drives the
container sinusoidaly, with frequency accuracy better than
0.6%. The vertical acceleration amplitude a is measured
by means of a piezoelectric accelerometer BK4374 glued
on the top of the vessel. The horizontal component of the
acceleration is measured to be less than 1.3% of the vertical
one. The whole setup is fixed on a table with adjustable legs
in order to ensure that it is horizontal. Our control parameters
are the driving frequency f and the dimensionless accelera-
tion amplitude =a / g ranging from 0 to 12, where g is the
acceleration due to gravity. The experiments are conducted
by increasing or decreasing the acceleration at a fixed value
of the frequency in the range 10–300 Hz. Several containers
of circular shapes from 80 to 150 mm in diameter or square
FIG. 1. Image from a reflection microscope of alumina grains
used in our experiments.
PHYSICAL REVIEW E MAY 1999 VOLUME 59, NUMBER 5
PRE 59 1063-651X/99/595/57165/$15.00 5716 ©1999 The American Physical Society