数理解析研究所講究録
1184 巻 2001 年 189-200189
鉛直に加振した粒状体薄層における準結晶的パターン
Pattern formation of thin granular layer

due to vertical vibration
農工大工 鵜川 亜希子 (Akiko Ugawa)
農工大工鈴木勝博 (Katsuhiro Suzuki)
農工大工 佐野 理 (Osamu Sano)
Abstract

Experimental study of the pattern formation on the thin horizontal granular layer,
which oscillated vertically with frequency under atmospheric pressure, was made. Patterns were observed by a high-speed video camera and classified by means of two-dimensional
Fourier transform, as well as a direct measurement of the images. In addition to spots,
square cells, polygons and stripes with frequency $f/2$ , we observed squares, polygons andquasi-crystal patterns with frequency $f/4$ . The dependence of oscillation amplitude andfrequency on the pattern size was analyzed.
$f$

1INTRODUCTION

Pattern formation of thin granular layer due to vertical vibration has
been extensively studied since the pioneering work by M. Faraday
et al. 1989, Melo et al.
. Recently many experimental(1994, 1995, Goldshtein et al. 1995, Umbanhower et al. 1996, Metcalf
et al. 1997, Sano &Suzuki 1998, Sano et al. 1999) and numeriet al. 1996, Clement et al. 1996, Lan&Rosato 1997, Bizonet al. 1998) investigations have revealed wavy motions on this layer.
In spite of industrial importance and academic interest, however, thebasic understanding of the physical mechanisms underlying the collective behaviors of this material is still inadequate (Jaeger&Nagel1992, 1996, Lubkin 1995, Jaeger et al. 1996, etc.). In this paper weshall elucidate the detailed pattern diagram and dispersion relation
to understand the fundamental processes of pattern formation.
$(1831\backslash )$$\mathrm{c}\mathrm{a}\mathrm{l}(\mathrm{A}\mathrm{o}\mathrm{k}\mathrm{i}$$\mathrm{D}\mathrm{o}\mathrm{u}\mathrm{a}\mathrm{d}\mathrm{y}$190

2EXPERIMENTAL APPARATUS

An experimental study of the pattern formation on thin granular materials in a vessel, which is vibrated vertically under atmospheric pressure, was made. A circular cylindrical vessel of a diameter
anda height 31mm was mounted with its principal axis parallel to gravity
on an electro-mechanical vibration generator. The block diagram ofour experimental apparatus is shown in Figure 1.
$106\mathrm{m}\mathrm{m}$Figure 1. Experimental Apparatus.

The vertical oscillation of the container was given by $z=a\sin(2\pi ft)$ ,where the frequency and amplitude were given by a function
synthesizer and an amplifier. Noof the bottom of thecontainer was recognized. The observation was lnade by means of ahigh-speed video camera to which close-up lens is mounted. Theseimages were later reproduced and theFourier image,in addition to direct observation of their planar form, were analyzed.
$f$

$a$

$\mathrm{d}\mathrm{e}\mathrm{f}\mathrm{o}\Gamma \mathrm{l}\mathrm{n}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$$\mathrm{t}_{\mathrm{W}\mathrm{e}\succ}\mathrm{d}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{S}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{a}1$

191

3

RESULTS

3.1

Planar form

We filled the glass beads of a diameter $d=0.13\pm 0.05\mathrm{m}\mathrm{m}$ in a vessel
to a depth, and observed the planar form of thegranular layer under vertical vibrations. Examples of the patterns areshown in Figure 2. The typical sequence of pattern changes, which
we found under frequency forcing of fixed amplitude of medium
values is as follows: initiallysurface disrupted into spot-like pat) for frequencies
terns (Fig.
, which diffused into sheet
around spots forto square patterns (Fig.
)
for(sometimes(Fig.
) wereobserved), and stripe patterns (Fig.
) with some dislocations for, which sometimes evolved into spirals (Fig. )patterns repeated with $2T$ (Fig. 3), where $T(=1/f)$ is the period ofexternal forcing. When we increased the frequency, patterns disappeared (Fig. ) for, and cellular patterns of approximately
hexagonal shape reappeared for, which repeatedwith period $4T$ . Between the patterns (e) and (g), we observed almost
flat surface, although constituent particles of the latter were not atrest. These findings agree with the previous works. We also observed
quasi-crystal pattern(1974, Mackay 1981, Shechtman et al.
1984, Levine&Steinhardt 1984, Christiansen et al. 1992, Edwards&Fauve 1994) with 8-fold symmetry (Fig. ). Typical Fourier patterns
corresponding to Figures
andare shown in Figure 4. Figure 5 is the pattern diagram in normalized frequency $f^{*}(=f\sqrt{h}/g)$against normalized acceleration amplitude $\Gamma(=4\pi^{2}f^{2}a/g)$ , where is
the acceleration of gravity. In contrast to the previous works whichwere made under evacuated circumstance, we have wider region ofconvection regime as well as the region of quasi-crystal patterns in
the phase diagram.
$h=0.7\mathrm{m}\mathrm{m}\sim 1.2\mathrm{m}\mathrm{m}$$a$$\mathrm{f}\mathrm{l}\mathrm{a}\mathrm{t}_{l}$$2\mathrm{a}$$f\approx 20\mathrm{H}\mathrm{z}$$2\mathrm{b}$$f\approx 25\mathrm{H}\mathrm{z},\mathrm{d}\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{d}$$f=25\sim 35\mathrm{H}\mathrm{z}$$\mathrm{h}\mathrm{e}\mathrm{x}\mathrm{a}\mathrm{g}\mathrm{o}\mathrm{n}\mathrm{s}/\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{l}\mathrm{e}\mathrm{S}$$2\mathrm{c}$$2\mathrm{d}$$f=40\sim 45\mathrm{H}\mathrm{z}$$2\mathrm{f}$$2\mathrm{e}$$.\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{s}\mathrm{e}$$f\approx 50\mathrm{H}\mathrm{z}$$f\approx 60\mathrm{H}\mathrm{z}(\mathrm{F}\mathrm{i}\mathrm{g}.2\mathrm{g})$$\mathrm{p}\mathrm{e}\mathrm{n}\mathrm{r}\mathrm{o}\mathrm{S}\mathrm{e}$$2\mathrm{h}$$2\mathrm{b},$$2\mathrm{c}$$2\mathrm{h}$

$g$

192

(a)

(b)

(d)

(e)

(g)

(h)

(f)

Figure 2. Typical patterns observed on vertically vibrated thin
granular layer:,(a) spots,(b) squares,(c) polygons(d) stripes $(f/2)(f=45\mathrm{H}\mathrm{Z}, a=1.00\mathrm{m}\mathrm{m})$ ,,(e) spirals,(f) flat(g) hexagons $(f/4)(f=60\mathrm{H}\mathrm{z}, a=0.60\mathrm{m}\mathrm{m})$ ,(h) quasi-crystal pattern$(f=20\mathrm{H}_{\mathrm{Z},a}=0.80\mathrm{m}\mathrm{m})$$(f/2)(f=25\mathrm{H}_{\mathrm{Z}}, a=1.20_{\mathrm{m}}\mathrm{m})$$(f/2)(f=30\mathrm{H}_{\mathrm{Z},a}=2.18\mathrm{m}\mathrm{m})$$(f/2)(f=45\mathrm{H}\mathrm{z}, a=1.2\mathrm{o}\mathrm{m}\mathrm{m})$$(f=45\mathrm{H}\mathrm{Z},$$a=1.60_{\mathrm{m}\mathrm{m})}$$(f/4)(f=75\mathrm{H}_{\mathrm{Z},a}=1.0\mathrm{o}\mathrm{m}\mathrm{m})$

.

193

(a)

(b)

$\mathrm{t}=0$(c)$\mathrm{t}=\mathrm{T}$$\mathrm{t}=2\mathrm{T}$Figure 3. Time sequence of the square pattern; (b) and (c),respectively,
are the patterns at a tilne and $2T$ after (a), where is the periodof external vibration. Crosses indicate the same position.
$T$.
$-\mathrm{I}+_{\mathrm{I}}^{\mathrm{h}}$$-$$-\tau$

(b)

$T$

...
(c)

(h)

Figure 4. Fourier patterns of Figures
6-fold and 8-fold symmetries, respectively.
$2\mathrm{b},$$2\mathrm{c}$and$2\mathrm{h}$, showing 4-fold,

194$\sim\sim \mathrm{b}0$$\mathrm{Q}\mathrm{o}^{\mathrm{g}}\vee$$\mathrm{b}^{\mathrm{I}}7$$f^{*}=f(\mathrm{h}/\mathrm{g})1/2$

Figure 5. Phase diagram.
: squares $(f/2),$ : hexagons and triangles $(f/2),$ $/$ : stripes$(f/2),$: squares $(f/4),$ : hexagons and triangles $(f/4)$ , and: octagons $(f/4)$ .
$\square$$\otimes$$\Phi$$\mathrm{O}$$\mathrm{A}$

3.2

Size of the cell

The size of the cell depends on $f,$ and . We plot the wavelengthof the patternnormalized by the depth (i.e.
) against
normalized frequency
in Figure 6. Data estimated from previousworks (Melo et al. 1994; Metcalf et al. 1997) are also plotted. By thisscaling, each family of data belonging to $f/2$ and $f/4$ is well fitted by. Their intercepts in log-log plot differ by about
with
0.3. Note that in the case ofwaves of the water, thedispersion relation leads to $\alpha=-2\sim-1$ for lower frequency, while$\alpha=-2/3\sim-1/2$ for higher frequency (Miles &Henderson 1990,
$a$

$h$

$\lambda^{*}$$\lambda^{*}=\lambda/h$$f^{*}$$\lambda^{*}\propto f^{*\alpha}$$\alpha\approx-1$$\mathrm{c}\mathrm{a}\mathrm{p}\mathrm{i}\mathrm{l}\mathrm{l}\mathrm{a}\mathrm{r}\mathrm{y}/\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{V}\mathrm{i}\mathrm{t}\mathrm{y}$195
Tufillaro et al. 1989, Milner 1991, M\"uller 1993, Edwards &) Fauve$\mathrm{i}994)$.

$\log(\mathrm{f}(\mathrm{h}/\mathrm{g})^{1}\prime \mathrm{z})$Figure 6. Dispersion relation.
present result,: squares,: triangles and hexagons,/: stripes, $\#:f/4$ patterns.
Melo. et al. (1994) $(\Gamma=3.5)$ ,$h/d=7,$,, :,.
Metcalf. et al. (1997) $(\Gamma=3.0)$ ,$h/d=6,$,$d=0.13\mathrm{m}\mathrm{m},$$h/d=4\sim 13,$$\square$$P=\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{m}$$\mathrm{O}$$P=\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{m}$$\mathrm{A}:d=0.2\mathrm{m}\mathrm{m}$$\bullet$$d=0.3\mathrm{m}\mathrm{m}$$\blacksquare:d=0.41\mathrm{n}\mathrm{m}$$P=10\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{r}$$0:d=0.5\mathrm{I}\mathrm{n}\mathrm{m},$$\square :d=0.5\mathrm{I}\mathrm{n}\mathrm{m},$$h/d=8,$$P=0.04\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{r}$.

196$L$$\cup$$0$

Time$0$$(d\mathrm{T})$Figure 7. Schematic picture of the orbit of a granular particle:-pattern, and (b) $f/4$-pattern.
$(\mathrm{a})f/2$

4

DISCUSSION

Our results are explained as follows: Granular particles pushed upby the upward motion of the vessel obtain upward momentum andmake a free flight during the time interval in which the upward acceleration due to external forcing exceeds that of the gravity. When
granular particles fall down and touch the bottom of the vessel, they
are reflected and make successive free flights with given positions andvelocities as new ‘initial conditions’. These conditions, however, arein general random, so that no collective motion is observed. On theother hand, if particles make ‘soft-landing’ so that they have the same
velocities as the wall velocity at the time of collision, they are convected by the downward motion of the vessel, which are duly releasedin the same phase of the next elevation as they did previously, whichforms the periodic and correlated motion of the granular particles, .
$\mathrm{i}.\mathrm{e}$‘patterns’.

The above-mentioned process is possible for a particular combination of $f,$
and , whose minimum time interval is $2T$ (subharmonicoscillation). The container may oscillate even-number
during
$a,$

$h$

$g$

$\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{l}\mathrm{e}\mathrm{s}$197
the free flight, so that similar patterns reappear for a period $4T$ , or
$6T,$ or(Fig. 7), although the observation becomes increasingly
difficult.
Finally we shall consider the experimentally observed dispersion
relation. To this end we assume$\cdots$(assumption 1)$2\lambda_{1}=\lambda_{2}$$c$$f_{2}=f_{1}/2$,,
is a constant. From assumption 1, we have(assumption 2)

where

,

$\lambda=cf^{\alpha}$$\log\lambda_{2}=\log 2+\log\lambda 1=0.301+\log\lambda_{1}$,which explains the difference 0.3 of the intercepts. From assumptions
1 and 2, we have
$2cf_{1}^{\alpha}=cf_{2)}^{\alpha}$or 2$f_{1}^{\alpha}=(f_{1}/2)^{\alpha}$

which explains the exponent $\alpha=-1$ , i.e.

5$\lambda=cf^{-1}$).

CONCLUSIONSPattern formation on vertically oscillating thin granular layer underatmospheric pressure was investigated by means of a high-speed video

camera.
Fourier
1. Pattern diagram is obtained by means of direct
images, and the sequence of transitions are elucidated.
2. Pattern formation process, temporal periodicities, translationalrotational symmetries of the patterns are checked.
, etc
3. In addition to regular patterns(of subharmonic type), we found quasi-crystal pattern of 8-fold symmetry with frequency $f/4$ , as the external forcing frequency
are increased.
$\mathrm{a}\mathrm{n}\mathrm{d}/\mathrm{o}\mathrm{r}$$\mathrm{a}\mathrm{n}\mathrm{d}/\mathrm{o}\mathrm{r}$$\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{l}\mathrm{e}\mathrm{S}/\mathrm{h}\mathrm{e}\mathrm{x}\mathrm{a}\mathrm{g}\mathrm{o}\mathrm{n}\mathrm{s}$$\mathrm{S}\mathrm{q}\mathrm{u}\mathrm{a}\Gamma \mathrm{e}\mathrm{S},$$f^{*}\mathrm{a}\mathrm{n}\mathrm{d}/\mathrm{o}\mathrm{r}$$\Gamma$

198

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