c Cambridge University Press 2009
J. Fluid Mech., page 1 of 19 
doi:10.1017/S002211200999231X

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Experimental study on the kinetics of granulargases under microgravity

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S O I C H I T A T S U M I1 †, Y O S H I H I R O M U R A Y A M A,2
H I S A O H A Y A K A W A3 A N D M A S A K I S A N O1

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Department of Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan

Department of Applied Physics, Tokyo University of Agriculture and Technology, 2-24-16 Naka-cho,
Koganei-shi, Tokyo 184-8588, Japan
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Yukawa Institute for Theoretical Physics, Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan

9(Received 12 August 2008; revised 7 September 2009; accepted 20 September 2009)

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The kinetics of granular gases, including both freely cooling and steadily driven
systems, is studied experimentally in quasi-two-dimensional cells. Under microgravity
conditions achieved inside an aircraft ﬂying parabolic trajectories, the frictional force
is reduced. In both the freely cooling and steadily driven systems, we conﬁrm thatthe velocity distribution function has the form exp(−α|v|β ). The value of exponent β
is close to 1.5 for the driven system in a highly excited case, which is consistent withtheory derived under the assumption of the existence of the white-noise thermostat(van Noije & Ernst, Gran. Mat., vol. 1, 1998, p. 5764). In the freely cooling system,
the value of β evolves from 1.5 to 1 as the cooling proceeds, and the system’s energy
decays algebraically (Tg = T0 (1 + t/τ )−2 ), agreeing with Haﬀ’s law (Haﬀ, J. Fluid
Mech., vol. 134, 1983, p. 401430).

granular gas, kinetic theory, steady state, cooling state, microgravity,
experiments
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1. Introduction
In recent years, interest in the rheological properties of granular material assemblies
is rapidly growing. Granular systems exhibit solid, liquid or gas-like behaviours,
depending on the external condition (Jaeger, Nagel & Behringer 1996; Duran2000; Goldhirsch 2003). The diﬀerent behaviours are owing to the diﬀerence inthe dominant physical process of energy dissipation. In the gas-like state, so-called
granular gas, energy dissipation is governed by inelastic collisions between particles.
Many interesting behaviours such as cluster formations promoted by the instabilityof homogeneous state (Goldhirsch & Zanneti 1993; Kudrolli, Wolpert & Gollub
1997), chaotic behaviour (Baxter & Olafsen 2007) and the anomalous scaling of thepressure (Géminard & Laroche 2004; Falcon et al. 2006) are found. For granulargases, the dynamics can be understood as collections of binary collisions between
granules and can be studied using the methods of the kinetic theory of gases basedon the Boltzmann–Enscog equation.
In the granular gas system, since the kinetic energy of the system is dissipated byinelastic collisions, a non-equilibrium steady state is sustained by a continuous energy
injection; we call this state a ‘steady state’. When the energy injection is stopped, thesystem is brought into a freely evolving state. The system evolves from the steadystate towards the resting state, where we call this transient state a ‘cooling state’. If the† Email address for correspondence: sk tatsumi@issp.u-tokyo.ac.jp

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S. Tatsumi, Y. Murayama, H. Hayakawa and M. Sanosystem is spatially uniform, one can expect the existence of the scaling properties ofthe system during the ‘cooling state’. The typical feature of granular gases appears inthe statistical properties of the system, especially in the velocity distribution function(VDF), f (v, t). Unlike the ideal gas system, in which the VDF is Gaussian, several
experiments for granular gas systems exhibit non-Gaussian VDFs (Losert et al. 1999;
Olafsen & Urbach 1999; Rouyer & Menon 2000; Baxter & Olafsen 2003; Painter,
Dutt & Behringer 2003; van Zon et al. 2004; van Zon & MacKintosh 2005; Reis,
Ingale & Shattuck 2007; Maaß et al. 2008).
The granular gas systems are widely studied both experimentally (Olafsen & Urbach
1998; Blair & Kudrolli 2001; Aranson & Olafsen 2002) and numerically (Murayama &Sano 1998; Das & Puri 2003; Kawarada & Hayakawa 2004; Miller & Luding
2004; Moon, Swift & Swinney 2004; Herbst et al. 2005; Ahmad & Puri 2006,
2007; Brilliantov et al. 2007; Wang & Menon 2008). It is known that the inelastic
Boltzmann–Enskog equation is semi-quantitatively accurate to describe such granular
gas systems (Jenkins & Richman 1985; Santos et al. 1989; Goldshtein & Shapiro
1995; Esipov & Pöschel 1997; van Noije & Ernst 1998; van Noije, Ernst & Brito
1998; Aspelmeier, Huthmann & Zippelius 2001; Jenkins & Zhang 2002; Brilliantov &Pöschel 2004; Goldhirsch, Noskowicz & Bar-Lev 2005; Ernst, Trizac & Barrat 2006;
Mischeler, Mouhot & Richard 2006; Mischeler & Mouhot 2006; Pöschel, Brilliantov &Formella 2006; Villani 2006). The high energy tail of the VDF for the granular gas
is predicted to be exp(−α|v|β ), where β = 2. The value of the exponent β in thesteady state with the white-noise thermostat is β = 3/2, implying that random noise isapplied frequently between collisions. On the other hand, β is 1 in the cooling state or
the steady state with Gaussian thermostat, which is reduced to the velocity rescaling
thermostat for molecular dynamics simulation in the small amplitude limit (Santos2003). The non-Gaussian VDF of β = 3/2 is also derived in the dense system by aphenomenological approach based on the experimental results (Fiscina & Cáceres2007).
However, two problems still remain unanswered among these studies. One is thelack of the systematic experiments in the cooling state. Although the evolution of
granular temperature, the spatial correlation, the cluster formation and the VDF in
the cooling state is numerically investigated (McNamara & Young 1994, 1996; Nie,
Ben-Naim & Chen 2002; Kawahara & Nakanishi 2004; Hayakawa & Kawarada
2005), most experiments are performed in the steady state, except for a few studies(Losert et al. 1999; Painter et al. 2003; Maaß et al. 2008). Losert et al. (1999) haveobserved the VDF of β = 1 in the cooling state, where the velocity data are averaged
across the entire cooling process, even though the total energy has dramaticallychanged in the course of the process. Theories (Haﬀ 1983; Brilliantov & Pöschel
2000) predict that the kinetic energy of the system, called granular temperature Tg ,algebraically decays as the cooling proceeds:
Tg =

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T0
,
(1 + t/τ )γ(1.1)

where T0 is the initial granular temperature, t is the time after the cooling starts and τ isa characteristic decay time. From the theoretical point of view (Brilliantov & Pöschel
2004), γ = 2 for hard particles whose restitution coeﬃcient is independent of therelative velocity, while γ = 5/3 for viscoelastic particles whose restitution coeﬃcientdepends on the relative velocity. Quite recently, the time evolution of Tg with γ = 2 hasbeen observed in an experiment that uses a technique called the magnetic levitation(Maaß et al. 2008). A few tens of particles are trapped in a magnetic potential, andparticles are quickly forced to make clusters at the bottom of the potential. Therefore,

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the particles outside the cluster are regarded as gaseous particles, and then theirvelocities outside the cluster are used to test how the granular temperature decays
in the cooling state. Although their results agree with the theoretical prediction, theeﬀect of the external potential is unclear, and a large number of particles are required
to guarantee suﬃcient statistics. Another problem of previous studies is that thestatistical properties in the steady state depend on the magnitude of excitation. In thecase of high acceleration, the VDF of β = 3/2 has been observed (Losert et al. 1999;
Rouyer & Menon 2000), which is consistent with the theory with the white-noise
thermostat (van Noije & Ernst 1998; Santos 2003). On the other hand, in the caseof low acceleration, such relations remain obscure. Although several papers indicatedpossible origins of the deviation such as the friction of the wall (van Zon et al. 2004)or the way in which the system is excited (van Zon & MacKintosh 2005), we still do
not understand the matter at hand.
To check the validity of theoretical prediction and to obtain clear statistics on
granular gases, inelastic collisions between particles should be dominant, where
additional eﬀects such as frictional forces or an external potential should be reduced.
To this end, a part of our experiments is conducted under the microgravity condition.
Under the normal gravity condition, for particles with small velocities, the dominantdissipation is not caused by the inelastic collisions but by the friction between
particles and the wall of a container, since the collision rate is low. The microgravity
condition suppresses the eﬀect of friction against the wall, because the frictional
force is proportional to the normal force. Falcon et al. (2006) have studied a threedimensional granular gas under microgravity condition and succeeded in obtaining
the probability distribution function of the collision frequency between the particle
and the wall.
Under the normal gravity condition, we have used quasi-two-dimensional cellswith a rough top plate for the excitation of particles and a smooth bottom plate tofacilitate rolling motions of particles after the vertical vibration has stopped. Because
the rolling friction is much smaller than the sliding one, we can reduce frictional
force.
The microgravity condition allows us to create an ideal state for granular gases,in particular for the cooling state. There is good agreement between the theory andour experiments in both the energy decay and the shape of the VDF under the
microgravity condition, whereas the agreement is not clear under the normal gravity
condition.
The organization of this paper is as follows. The experimental setup and analytic
methods in our experiments are, respectively, explained in § § 2 and 3. The experimental
results for the steady state and the cooling state are shown in § § 4 and 5, respectively.
Finally, we discuss and conclude our report in § 6. In Appendix A, we estimate therole of the hydrodynamic interaction among particles. In Appendix B, we evaluate
the degree of g-jitter during the experiments under the microgravity condition.

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2. Experimental setup

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2.1. Materials
The experimental setups are presented in ﬁgure 1. Mono-dispersed zirconium beads(ZnO2 , diameter d = 1.00 mm ± 0.05 mm; Toray Industries, Inc.) are conﬁned in quasitwo-dimensional cells. The material constants of zirconium beads are shown in table 1.
The restitution coeﬃcient of a bead with a glass plate, εgl , and that with a zirconium
plate, εzir , are determined by measuring the speeds before and after a collision withthe plate. The restitution coeﬃcient determined by a particle–plate collision is usually

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S. Tatsumi, Y. Murayama, H. Hayakawa and M. Sanoρ ( g cm−3 )E (MPa)Y (MPa)εglεzir

Γzir ( mJ m−2 )

6.0

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780

0.981 (2)0.954 (27)

41∼45

ZrO2

Table 1. Material constants of working granules, Zirconium beads. ρ, E, Y , εgl , εzir , and Γzir are
density, Young’s modulus, yield stress, restitution coeﬃcient between glass plate and zirconium
bead, restitution coeﬃcient between zirconium plate and zirconium bead, and surface energyof zirconium, respectively. ρ, E, and Y are obtained from the datasheet or the informationfrom Toray Industries, Inc. We refer to the experimental study by Król & Król (2006) for thevalue of Γzir .

(b)

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Horizontal cell
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Cooling state

Position (mm)

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Vertical cell
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Figure 1. Setups for (a–c) the horizontal cell and (d–f ) the vertical cell. (a, d ) Schematic
diagram of the cells. (b, e) Top views of the cells. The grey zones in (b, e) are used for ouranalysis, namely the region of interest. (c, f ) The time-averaged area fractions of particles areplotted as a function of the distance from the centre for the horizontal cell and as a functionof the position along the vertical axis for the vertical cell.
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larger than that determined by a particle–particle collision. For aluminium and brassmaterials, Weir & Tallon (2005) report that restitution coeﬃcient of the particle–
particle collision is about 80 % of that of the particle–plate collision. Hence, we
regard the restitution coeﬃcient between zirconium beads as 0.8 εzir . A tangential
frictional coeﬃcient μ between particles is usually less than 0.2; Labous, Rosato &Dave (1997) showed a typical value of μ being 0.175. Therefore, we can use the theoryof Jenkins & Zhang (2002), in which the frictional eﬀect can be absorbed by the
suppression of the eﬀective restitution coeﬃcient εeﬀ = ε − π2 μ + 92 μ2 . The quantitative
validity of this theory has been extensively studied by Saitoh & Hayakawa (2007).

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Thus, we can estimate that εeﬀ for a binary collision of zirconium beads is 0.62,
assuming ε = 0.8 εzir and μ = 0.175. In addition, the lower limit of the impact speedof a particle to stick to the plate is 1.25 mm sec−1 estimated from a value of Γzir
(Thornton & Ning 1998), which is suﬃciently small for our experimental conditions.
Thus, (i) the frictional eﬀects can be absorbed in the eﬀective restitution coeﬃcient,and (ii) sticking force between the particle and the wall is negligible.

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2.2. Horizontal cell
We use two types of quasi-two-dimensional cells: a horizontal cell and a verticalcell. The horizontal cell (ﬁgure 1a) is set in a horizontal plane and is accelerated
vertically. The vertical cell (ﬁgure 1b) is set in a vertical plane and is accelerated
vertically. The horizontal cell is a cylinder with a diameter of 80 d and a depth of
2.5 d. The top plate and the sidewalls are made of aluminium, and the bottom plate
consists of a glass plate coated with electrically conductive indium tin oxide (ITO)
ﬁlm to prevent static electrical charge. To randomize the motion of the particles,
1 mm diameter glass beads are glued to the top plate. The minimum gap between
two plates is roughly 1.5 d, and the average gap is 1.8 d. The horizontal cell is used
under both the microgravity and the normal gravity conditions. For the microgravity
experiment, the number of particles conﬁned in the cell is 2000 ± 5, the correspondingarea fraction is 0.313 ± 0.001 (the error corresponds to an error bar in the weight
per each particle due to small dispersion). For the normal gravity experiment, mostof the experiments use the same number of particles, 2000 ± 5, except for several
data sets with 1000 ± 3 and 3200 ± 10 particles which are performed to test the eﬀectof area fraction. The cell is kept in vacuum condition (200–300 Pa at maximum) to
avoid hydrodynamic interaction between particles. Under this condition, we estimatethat the velocity reduction due to hydrodynamic drag between two collision eventsis roughly 6.6 × 10−2 mm sec−1 (see Appendix A). In both the steady state and thecooling state, the root mean square (rms) velocity of particles is suﬃciently higher
than this value. Therefore, the hydrodynamic eﬀect is negligible in our setup.
External force is applied by sinusoidal acceleration using an electromagnetic
vibration system (9514-AB/SD vibrator, EMIC Corp.). By the vertical vibrationof the cell, all the particles gain high vertical velocity. When the particles strike the
top plate, they are scattered and randomized by the glued beads. Because of thevertical vibration, the magnitude of the velocity of particles in the vertical direction ismuch larger than that in the horizontal direction. In our experiments, the horizontal
components of the velocity are measured.
Under the normal gravity condition, the steady state is realized by setting thefrequency f ranging from 70 to 200 Hz and the acceleration Aω2 ranging from to
10 to 200 m sec−2 , where A is an amplitude of the vibration and ω = 2πf is anangular frequency. Under the microgravity, the frequency is ﬁxed at 100 Hz, andthe acceleration is varied from 6.0 to 48 m sec−2 . Throughout the experiments, theacceleration is measured by a vibration accelerometer (VM-83, RION).
Moreover, to test the eﬀect of the area fraction on the statistics of the system, theexperiments on two other area fractions, 0.156 ± 0.001 (N = 1000 ± 3) and 0.50 ± 0.01
(N = 3200 ± 10), are conducted.
The cooling state is realized by the following procedure. First, we set the frequency
to 100 Hz and the acceleration to 48 m sec−2 for more than 10 sec, then suddenly turnoﬀ the vibration. In the cooling experiments under the normal gravity, the particles
roll on the bottom glass plate soon after the vertical vibration is stopped, indicatingthat the frictional eﬀect is suﬃciently reduced even under the normal gravity.

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Gravitational acceleration (m sec2)

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Gravitational acceleration (m sec2)

S. Tatsumi, Y. Murayama, H. Hayakawa and M. Sano

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0.15

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gx
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Figure 2. Time sequences of acceleration during a parabolic ﬂight. The level of microgravity
condition is maintained below 0.1 m sec−2 . The typical duration of the microgravity condition isabout 20 sec. The experiments are conducted within about 10 sec of each microgravity period.
The measurement time is 1.6 sec. Measurements are performed as the level of microgravity
became lower and more stable. In total, about 100 data sequences are taken.

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2.3. Vertical cell
As shown in ﬁgure 1(b), the vertical cell is a thin rectangular parallel-piped cell withan area of 50 d × 70 d and a thickness of 1.1 d. The entire cell wall is made of glass
plate coated with ITO ﬁlm. The vertical cell is only used under the microgravity
condition. The number of particles conﬁned in the cell is ﬁxed at 300, in which thecorresponding area fraction is 0.067. This setup is similar to that in the experiment
by Rouyer & Menon (2000) except for our use of the microgravity condition. The
hydrodynamic eﬀect is estimated in the same manner for the horizontal cell. Since
the vertical cell is not in vacuum, its hydrodynamic eﬀect is much larger than that of
the horizontal cell. The velocity decrease between consecutive collisions is estimatedas 4.2 × 10−1 mm sec−1 , which is suﬃciently low compared with the rms velocity ofparticles. Thus, the hydrodynamic eﬀect is also negligible in the vertical cell. Similar
to the horizontal cell, the external force is applied by sinusoidal acceleration. Owing
to the vertical acceleration, all the particles near the top and bottom boundaries gainhigh velocity towards the vertical direction. Then, particles reach the centre region
after 2–3 collisions. Because of the suﬃciently high acceleration and the microgravity
condition, the density proﬁle of the system becomes symmetric (see ﬁgure 1b). It is
worth noting that the system becomes uniform in the cooling state. The steady state
and the initial condition for the cooling state are realized with a ﬁxed frequency of
40 Hz and a ﬁxed acceleration of 250 m sec−2 .
2.4. Microgravity condition
A microgravity condition is achieved aboard a parabolic ﬂight of a Gulfstream II
jet aircraft (Diamond Air Service Co. and Japan Space Forum). Figure 2 shows anexample of the microgravity process.

The kinetics of granular gases under microgravity
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The ﬂuctuation of microgravity is called g-jitter. The g-jitter in the z-direction, δgz ,is kept below about 0.1 m sec−2 for approximately 20 sec, and the xy directions, δgx
and δgy , are kept under 0.03 m sec−2 . However, δgz does not aﬀect our measurements
because we analyse the motion of particles in the xy plane for the horizontal celland that in the y-direction for the vertical cell. In the cooling experiments, we further
choose a suitable condition in which the microgravity level becomes very low during
the ﬂights, and choose the appropriate data that include no average drift. Thus,
we estimate that δg is 0.01 m sec−2 in the cooling experiments. This is validated bymonitoring the mean velocity of particles in the cooling state; see Appendix B.
To estimate the eﬀect of g-jitter, we introduce a ratio R:
R = τ∗

δg
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vth

(2.1)

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where τ ∗ is a characteristic time for the particle–particle
 or the particle–plate collisions,and vth is the thermal velocity of particles, vth = vx2  + vy2 , where vx and vy arevelocity components and · denotes the average over all the particles. The estimationof R is discussed in the last part of § § 4 and 5.

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3. Analysis

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3.1. Particle tracking
Particle motion is captured by a high-speed camera (1024PCI, PHOTRON), at aframe rate of 1 kHz. The size of the image is 1024 × 1024 pixels, and the spatial
resolution is about 80 μm pixel−1 on the image. The particle size on the image isabout 12 pixels, and the centroid of each particle is obtained with a precision of
0.007 d (7 μm). A simple algorithm for particle tracking is performed by detecting thenearest particle within a distance d between consecutive images. Thus, the particles
with velocities less than 1 m sec−1 can be tracked. We calculate the velocity of eachparticle from the displacement in eight frames. This interval and the precision of
the centroid determine the lower limit, 1 mm sec−1 , in detecting velocity. For thehorizontal cell, this simple algorithm enables us to detect more than 99.8 % of theparticles. For the vertical cell, to detect particles with a high velocity, we ﬁrstly
perform the simple tracking and remove detected trajectories. Secondly, we perform
a more elaborated tracking algorithm by minimizing the sum of the displacements
for the remaining particles. For particles with velocities less than 1 mm sec−1 , thevelocities are recalculated after obtaining their trajectories. Because of the absence ofobstacles and a low area fraction in the vertical cell, collision events can be easily
detected; thus, the velocities are obtained from the linear trajectories between collision
events. Finally, we detect particles within the speed range, 0.2 mm sec−1 < v < 3 m sec−1 .
Typical trajectories in the horizontal cell under the microgravity are shown inﬁgure 3.
From the data on the position and velocity of all particles, we can calculate thediﬀusion constant, the granular temperature and VDF. Based on the measurements
error analysis by Xu, Reeves & Louge (2004), we estimate that the obtained granulartemperature could be 27 % lower than the actual one in the horizontal cell, which isdue to the scattering of particles by the obstacles within eight frames. However, inthe vertical cell, since most collisions are detected, there is no underestimation of thevelocity and the granular temperature.

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S. Tatsumi, Y. Murayama, H. Hayakawa and M. Sano

10 mm

Figure 3. Typical trajectories of particles in the horizontal cell under the microgravity(steady state). Seven particles during 1.6 sec are picked up and shown. The frequency andthe acceleration of external vibration are 100 Hz and 48 m sec−2 , respectively.
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3.2. Calculation of velocity distribution function
For the steady state, the velocity data for all particles across the measurement timeof 1.6 sec are used for the calculation of the VDF. We then scale the velocity data by
the rms velocity σ ; we call the normalized velocity c ≡ v/σ . For the cooling state, σdecreases with time. We therefore calculate σ (t) for every frame and obtain the VDF
by averaging over every 50 msec.
β
Most VDFs satisfy the form, f (c) = C exp(−α|c|), where C is the normalization
constant. From the normalization condition, f (c) dc = 1 and c2 f (c) dc = 1, we canwrite the shape of VDF as β/2 β Γ (3/β)1/2
Γ (3/β)f (c) =
exp −
cβ ,
(3.1)
2 Γ (1/β)3/2
Γ (1/β)271
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and consequently, only β is a ﬁtting parameter. From the ﬁtting of VDF by (3.1), we
obtain the value of β for each VDF.

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4. Steady state274

4.1. Granular temperature and diﬀusion constant
The temperature and the diﬀusion constant are important physical quantities for the
kinetic theory of molecular gas. In a weakly non-equilibrium time-dependent process,relaxation or dissipation is characterized by transport coeﬃcients, e.g. diﬀusivity,
viscosity and thermal conductivity. In the kinetic theory of collisional gases, all threeof these transport coeﬃcients are proportional to vth l, where vth is the thermal velocity
and l is the mean free path. In this section, we focus on the granular temperature (Tg )

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The kinetics of granular gases under microgravity
(a)

(b)
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D (cm2 sec–1)

Tg (cm2 sec–2)

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N = 2000
100 Hz (μg)
70 Hz
100 Hzf = 100 Hz
120 Hz150 HzN = 1000
N = 3200
200 Hz510
Vpl (cm sec–1)

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Figure 4. Dependence of (a) Tg , (b) D, (c) vth and (d ) D/Tg on vpl , respectively. The
legends in ﬁgure 4(d ) are common to all ﬁgures.

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and the diﬀusion constant (D) of particles in the steady state by an analogy between
the granular system and the molecular gas. The granular temperature is deﬁned as2
Tg = 12 vth, where we subtracted the centre of mass velocity of all particles, v̄, for the
calculation of vth to avoid mean drift eﬀect. In our analysis, we set m as unity forsimplicity. The diﬀusion constant is deﬁned by, D = dtd (x(t) − x(0))2 , where x(t) is
the position of a tracer particle at time t.
Figure 4 shows the dependence of various physical quantities on the maximum
plate speed vpl for the horizontal cell experiments. As shown in ﬁgures 4(a) and 4(c),the values of vth are almost independent of the external frequency for small vpl , whilethe values of vth strongly depend on the external frequency for large values of vpl .
Moreover, the slope of vth decreases as vpl increases. Even if we plot Tg as a functionof Aω2 , the curves of Tg strongly depend on the frequency (data not shown). As
shown in ﬁgure 4(b), D shows a monotonic increase with increasing vpl . However, thedependencies of D on vpl , Aω2 and Tg are not simple.
On the other hand, the value of D/Tg shows a good scaling relation on vpl except
for the lowest density case, N = 1000. This indicates that vpl is a good parameterto characterize the nature of the system at a high enough density. Moreover, D/Tg
seems constant in the case of excitation at a high degree (vpl is large). It meansthat if the particles experience random forcing from the plate continuously, they

10

S. Tatsumi, Y. Murayama, H. Hayakawa and M. Sano
(a)

10–1

ln(ln(–(f (c)/f (0))) + const.

Probability density [f (c)](b) 66 m sec–2
17 m sec–2
48 m sec–2

100

10–2
10–3
10–4
–6

4
2
0
–2
–4

–4

–2
0
2
4
Normalized velocity (c)

–4

6

–3

–2

–1
ln(|c|)

0

1

2

Figure 5. VDF obtained in the horizontal cell under the microgravity condition: f = 100 Hz.
(a) The semi-log plot. (b) The double-logarithmic plot for the same result. The three lines in(a) represent reference data of the VDF with β = 1, 1.5 and 2. The two solid lines in (b) serve
as a guide to the eyes for the slopes of 1.5 and 1. f (0) is chosen as the experimental value forc = 0. The legends are common to all ﬁgures.
(b)

Exponent

(a)
1.6

1.6

1.4

1.4

1.2

1.2

1.0

1.0

0.8

0

10

20
30
Tg (cm2 sec–2)

4050

N = 2000
100 Hz, μg
70 Hz
100 Hzf = 100 Hz
120 HzN = 1000 150 Hz
N = 3200 200 Hz

0.8
0

5

10

15

Vpl (cm sec–1)

Figure 6. Dependence of the exponent β for the VDF on Tg (a) and vpl (b). Dotted lines are
to guide the eyes, and their value is 1.5. The legends are common for plots in both (a) and (b).
300
301

behave like Brownian particles. In such a case, the diﬀusivity of the particle should
be proportional to its kinetic energy, i.e. D/Tg becomes a constant.

302

4.2. Velocity distribution function
Figures 5(a) and 5(b) show the VDFs under the microgravity condition in thehorizontal cell. All VDFs show the form as in (3.1) for most of the velocity regions,where the exponent β depends on the external acceleration. The value of β increases
from 1 to 1.5 with the increase of the external acceleration. This trend is clearlyobserved in ﬁgure 6(a) for both microgravity and normal gravity conditions, andamong diﬀerent densities.
These results are consistent with experimental studies on granular gases by Losert
et al. (1999). In ﬁgure 6(b), the same data in ﬁgure 6(a) are plotted for vpl . We notethat the data collapse to β = 1.55 ± 0.05 for high excitation cases (vpl > 4 cm sec−1 ),

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11

The kinetics of granular gases under microgravity(a) 100(b)ln(ln(–f (c)/f (c))) + const.

Probability density [f(c)]

2
10–1

10–2

10–3

10–4

Vertical Cell (250 m sec–2)–5 –4 –3 –2 –1 0 1 2 3Normalized velocity (c)

1
0
–1
–2
–3

4

5

–4

–2

0

2

ln(|c|)

Figure 7. VDF obtained in the vertical cell under microgravity condition (open circles).
f = 40 Hz and Aω2 = 250 m sec−2 . (a) The semi-log plot. (b) The double-logarithmic plot forthe same result. The solid line in (a) represents reference data of the VDF with β = 1.5. The
two solid lines in (b) serve as a guide to the eyes for the slopes of 1.5 and 1. f (0) is chosen as
the experimental value for c = 0.
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314
315

while the data are distributed in the range from 0.9 to 1.6 for low excitation cases(vpl < 4 cm sec−1 ). For high excitation cases, our results agree with the theory of vanNoije & Ernst (1998), in which the VDF with β = 1.5 is derived for homogeneousand dilute granular gas excited by the white-noise thermostat.

316

4.3. Results of the vertical cellIn the vertical cell, the VDF for the steady state is obtained at f = 40 Hz andAω2 = 250 m sec−2 under microgravity condition (ﬁgure 7). The VDF shows the formin (3.1), where the exponent β = 1.28 ± 0.04 is obtained by ﬁtting. The deviation fromβ = 1.5 can be caused by low excitation compared with the mean collision time.

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320

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327
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4.4. The eﬀect of g-jitterIn the steady state under the microgravity condition, the eﬀect of g-jitter is estimatedby R (see (2.1)) with δg = 0.1 m sec−2 . Under high enough excitation, τ ∗ is characterized
by the mean collision time between a particle and the plate, deﬁned as h/vpl , where
h is the distance between the top and bottom plates of the horizontal cell, and we
obtain R 10−2 for the acceleration, f = 100 Hz and Aω2 = 48 m sec−2 . On the other
hand, under low excitation, τ ∗ is characterized by the mean collision time between
particles, deﬁned as l/vth , where l is the mean free path of a particle, and we obtain
R 10−1 . Thus, we conclude that the eﬀect of g-jitter is negligible in the steady state.

330

5. Cooling state331

5.1. Energy decay
The cooling processes are observed about 1 sec after the external vibration is stopped.
The initial conditions for the cooling processes are steady states (see § 2). The snapshots
during the cooling processes are shown in ﬁgure 8. Because of the inelastic collisions
during the cooling process, the total energy of the system decreases. At the last stageof cooling, the velocity of particles becomes very small and the inhomogeneity of thesystem arises.

321

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12

S. Tatsumi, Y. Murayama, H. Hayakawa and M. Sano
(a)

1g||
cell

(b)

μg||
cell

(c)
μg
⊥cell

0 sec

0.2 sec

0.4 sec

0.6 sec

Figure 8. Snapshots during the cooling processes. (a) Horizontal cell under normal gravity
condition. (b) Horizontal cell under microgravity condition. (c) Vertical cell under microgravity
condition. The initial conditions are f = 100 Hz and Aω2 = 48 m sec−2 for the horizontal celland f = 40 Hz and Aω2 = 250 m sec−2 for the vertical cell. The times shown at bottom indicate
the elapsed time after the vibration is stopped.
338
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342
343
344
345
346
347
348
349
350
351

The time evolution of Tg is shown in ﬁgure 9(a). The decay of Tg under microgravity
condition and normal gravity condition is diﬀerent in the form. Under microgravity
condition, Tg decays with (1.1). From ﬁtting, τ = 36 ± 3 msec and γ = 2.0 ± 0.1 are
obtained for the horizontal cell, and τ = 38 ± 4 msec and γ = 2.1 ± 0.1 are obtained
for the vertical cell. This algebraic decay sustains until 0.6–0.8 sec for both cells. On
the contrary, under the normal gravity, it is hard to ﬁt with (1.1). Based on theexperimental results under microgravity condition, using the ﬁtting values of τ , therescaled temperature Tg /T0 (1 + t/τ )2 is plotted as a function of the elapsed time inﬁgure 9(b), where the rescaled temperature under the normal gravity condition isclearly diﬀerent from that of the microgravity condition. For microgravity condition,
rescaled temperatures are almost 1 until 0.6–0.8 sec, which means that the decay ofTg obeys Haﬀ’s law with the restitution coeﬃcient independent of the impact speed(γ = 2) (Haﬀ 1983). In contrast, the decay of Tg for the normal gravity deviates fromHaﬀ’s law soon after 0.2 sec.

352

5.2. Velocity distribution function
The VDF at the elapsed time t is obtained from the data set during t ± 25 msec. For
each VDF, the exponent β is obtained by ﬁtting with (3.1). The VDFs at t = 0 (steadystate), 100 and 300 msec for each condition are shown in ﬁgure 9(c–e). At t = 0, theexponent β is 1.55 for the horizontal cell under both the microgravity and normal
gravity conditions, and β is 1.28 for the vertical cell under microgravity condition. The
double-logarithmic plot of VDFs and time evolutions of β are shown in ﬁgure 10.
In the horizontal cell under normal gravity condition, β is ﬂuctuating around 1.5
for about 0.6 sec (ﬁgure 10b). Nonetheless, in the horizontal cell under microgravity

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13

The kinetics of granular gases under microgravity(b) 3.5

10–1
1g || cellμg || cellμg cell

10–2

10–3

10–4

2.5
2.0
1.5
1.0

10–5
0
100

Probability density [f(c)]3.0Rescaled temperature

Granular temperature [Tg (m2 sec–2)]

(a)

0.2

0.4
0.6
Time (sec)

0.8

0.5

1.0

0

0.2

(d)

(c)

0.4
0.6
Time (sec)

0.8

1.0

(e)

10–1
10–2
10–3
10–4
–5

0
c

Steady state
100 m sec
300 m sec

Steady state
100 m sec
300 m sec

Steady state
100 m sec
300 m sec
5 –5

0
c

5 –5

0
c

5

Figure 9. (a) Decay of Tg in the cooling state. (b) Time evolution of the rescaled temperature
Tg /T0 (1 + t/τ )2 . The legends are common for plots in both (a) and (b). (c–e) VDF at t = 0(steady state), 100 and 300 msec for each condition. (c) Horizontal cell under the normal
gravity. (d ) Horizontal cell under microgravity condition. (e) Vertical cell under microgravity
condition. The VDF in (d ) shows an asymmetric shape for 300 msec. It is caused by the eﬀectof g-jitter in the horizontal direction.
361
362
363
364
365
366
367
368
369
370
371
372

condition, β is ﬂuctuating around 1.5 for t < 0.2 sec, and decreases to 1 during
0.2 sec < t < 0.6 sec (ﬁgure 10d ). In the vertical cell under microgravity condition, βimmediately decreases from 1.28 to 1 and is ﬂuctuating around 1 (ﬁgure 10f ).
In the cooling state, the theory by van Noije & Ernst (1998) predicts β = 1 for thehigh energy tail. In our experiments under microgravity condition, the VDFs with
β = 1 are observed for both horizontal and vertical cells. However, under normal
gravity condition, no agreement with the theory is found throughout the decaying
process. In the horizontal cell under microgravity condition, β is almost 1.5 for theﬁrst 0.2 sec. In this setup, the particles with a large initial speed in the z-component
can remain for a short time, which randomizes the system because of the glued beads
on the top plate. The ﬁrst 0.2 sec might be understood as the relaxation time for the
randomization.

373

5.3. The eﬀect of g-jitterIn the cooling state under the microgravity condition, the eﬀect of g-jitter is estimatedby R with δg = 0.01 m sec−2 , and τ ∗ is characterized by τth at each moment. HereR becomes 1 at t = 0.5 and 1.0 sec for the horizontal and vertical cells, respectively.
Thus, after these time periods, the eﬀect of g-jitter cannot be neglected. The validityfor δg = 0.01 m sec−2 is shown in Appendix B.

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14

S. Tatsumi, Y. Murayama, H. Hayakawa and M. Sano3ln(ln(–f (c)/f (0))) + const.

(b)

4
2

Steady state
100 m sec
300 m sec

1.5

1Exponent

(a)

0
–1
–2

1.0
Cooling state
Steady state

0.5

–3
–4
–5
4

ln(ln(–f (c)/f (0))) + const.

3
2

(d)
Steady state
100 m sec
300 m sec

1.5

1Exponent

(c)

0

0
–1
–2

1.0

Cooling state
Steady state

0.5

–3
–4
–5

3
ln(–ln(–f (c)/f (0))) + const.

(f)

4
2

Steady state
100 m sec
300 m sec

1.5

1Exponent

(e)

0

0
–1
–2

1.0

0.5

–3

Cooling state
Steady state

–4
–5
–3

–2

–1

0

1

Normalized velocity (c)

2

0

0

0.2

0.4

0.6

0.8

1.0

Elapsed time (sec)Figure 10. (a, c, e) Double-logarithmic plot of VDFs in the steady state (open circles), att = 100 msec (asterisks) and 300 msec (crosses). The two solid lines guide the eyes for the slopesof 1 and 1.5. (b, d, f ) Time evolution of the exponent β obtained by ﬁtting. (a) and (b) Normal
gravity condition in the horizontal cell. (c) and (d ) Microgravity condition in the horizontal
cell. (e) and (f ) Microgravity condition in the vertical cell.

379

6. Discussion and conclusion

380

6.1. Discussion
We have conducted experiments on granular gas in detail, dominated by inelastic
collisions in both the steady state and the cooling state. Present experiments underthe microgravity condition allow us to reduce friction drastically.

381
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383

The kinetics of granular gases under microgravity
384
385
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393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409

15

The experiment for the steady state with suﬃciently high acceleration produces
similar results to those predicted by the theory of dilute granular gas with the whitenoise thermostat (van Noije & Ernst 1998). Moreover, we found the scaling relationin the value D/Tg with respect to vpl . When vpl is large enough, D/Tg is almost
independent of vpl and the exponent of the VDF is close to 1.5 except for the case of
low area fraction.
Strictly speaking, the theory that predicts VDF with the exponent of 3/2 is only
applicable to high energy particles. Indeed, theory predicts that the shape of VDF at
low energy is supposed to be Gaussian with a correction by the Sonine polynomials.
Reis et al. (2007) ﬁtted a deviation of VDF from Gaussian distribution by the onedimensional Sonine polynomials well, and we also succeeded. However, we shouldnote that the diﬀerence between the ﬁtting by Sonine polynomials and ﬁtting by the
exponent of 3/2 is very small. A similar result has been reported in previous studies(Rouyer & Menon 2000). We still do not understand the reason why we observe the
VDF with the exponent of 3/2 for the whole range of the velocity.
Theoretically, the 
relaxation time τ in (1.1) for the cooling process isφ−1
predicted as τ = 2 Tπ0 (1 − ε2 ) 1 −(1(7/16)φ
(Brilliantov & Pöschel 2004). Under− φ)2 d2
−2
microgravity condition, T0 = 27.4 cm sec and φ = 0.313 for the horizontal cell, andT0 = 1070 cm2 sec−2 and φ = 0.067 for the vertical cell. Using εeﬀ = 0.62, τ = 50 and
60 msec are obtained for the horizontal and the vertical cells, respectively. These
values are of the same order to the experimentally obtained values τ = 36 ± 3 msecand τ = 38 ± 4 msec for the horizontal and the vertical cells, respectively. We shouldnote that, using εzir = 0.95, τ = 300 and 400 msec are obtained for the horizontal
and the vertical cells, respectively. These results suggest that the tangential friction
between particles is not negligible, which is also found in numerical studies on thecooling state (Huthmann & Zippelius 1997; Luding et al. 1998).

411
412
413
414
415
416
417
418
419
420

6.2. Conclusion
In conclusion, under a condition of high enough acceleration, a universal form of the
VDF with an exponent of 3/2 is observed for a wide range of velocities. Moreover,in this range, D/Tg which shows scaling relation on vpl becomes constant.
The experiments in the vertical and the horizontal cells under microgravity condition
provide an ideal system to study the freely cooling state of granular gas. Under the
microgravity condition, the time evolution of the energy decay agrees with Haﬀ’slaw (Haﬀ 1983) by assuming a constant restitution coeﬃcient. However, under thenormal gravity condition, energy decay does not agree with Haﬀ’s law. The shape of
the VDF satisﬁes an exponential distribution under microgravity condition which isconsistent with the theory.

421
422
423
424
425
426
427
428

We thank Akinori Awazu, Igor Aranson, Harry Swinney and Danniel I. Goldman
for their fruitful discussions; Y. Takahashi and K. Takeuchi for their help in
microgravity experiments; and Ryo Suzuki for careful reading of the manuscript.
This research is fully supported by the Japan Society for the Promotion of Scienceand the Japan Space Forum. The experiment involving parabolic ﬂight was conductedwith the help of the Diamond Air Service Co., for which we express our gratitude.
S.T. thanks the Research Fellowships for Young Scientists from the Japan Society for
the Promotion of Science 17-11808.

410

16

S. Tatsumi, Y. Murayama, H. Hayakawa and M. Sano
Tg || cell
v–2/2 || cellTg ⊥ cellTg, v–2/2 (cm2 sec–2)

102

v–2/2 ⊥ cell
100

10–2

0

0.2

0.4
0.6
Elapsed time (sec)

0.8

1.0

Figure 11. Time evolution of the Tg (square) and v̄ 2 (circle) is shown. Open symbol andsolid symbol denote the results in the horizontal and vertical cells, respectively.

430
431
432
433

Appendix A. Hydrodynamic interaction
Hydrodynamic loss between all the collision events is calculated theoretically fromthe packing fraction, the density of the particles, the diameter of the particles and theviscous coeﬃcient of air as follows. First, we consider only the average loss of thesystems. We can write the hydrodynamic interaction with air for each particle as

434
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436
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438
439
440
441
442
443

dv
= −μv,(A 1)
dt
where m is the particle mass, v is the velocity of each particle and μ is the viscouscoeﬃcient by Stokes’ law, μ = 3πdη. Here d is the particle diameter, and η is theviscosity of air and is constant against air pressure. Note that this Stokes drag force
is applicable in the small range of Reynolds number (Re 6 1). In this case, Re is
estimated as Re = (U L)/(η/ρair ), where U and L are the typical velocity and thetypical length of the system. Note that U and L, in our case, are the velocity andthe diameter of each particle, and ρair is the density of air, which is 1.2 × 10−3 g cm−3
at 1 atm. Since the value of Re becomes Re = U (cm sec−1 ) × P (atm) × 6.7 × 10−5 , thetypical velocity of this experiment is much less than the limitation value of Stokes’
law. Then, the velocity decay due to viscosity is calculated as429

m

9π η 1 − φv = − √,2 2 ρd φ

(A 2)

444
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446
447
448
449

where φ is the packing fraction and ρ is the net density of the particles. In our setup,the diameter of the particle is 1.0 mm, the net density is 6.0 g cm−3 , and the viscouscoeﬃcient η is 1.81 × 10−4 g cms−1 . Finally, we obtain a typical velocity decay between
collisions at 6.6 × 10−2 mm sec−1 for the horizontal cell and 4.2 × 10−1 mm sec−1 forthe vertical cell. As mentioned in the main text, these values are much smaller than
the root mean square velocity of particles, and are therefore negligible.

450

Appendix B. Eﬀect of g-jitter
We assess the eﬀect of g-jitter in the cooling state quantitatively from the mean
velocity of particles v̄, where v̄ is calculated for all the particles at each moment. Origin

451
452

Q2

The kinetics of granular gases under microgravity
453
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459
460
461
462
463

17

of g-jitter in the parabolic ﬂight mainly comes from manoeuvering of the aircraft andexternal wind. The frequency power spectrum of g-jitter ﬂuctuations dominates at low
frequency (<2 Hz). Since all the particles experience the same ﬂuctuations of g-jitter,a ﬁnal outcome of the eﬀect should appear as a mean drift motion of particles,
which can be detected by monitoring the mean velocity of particles. Figure 11 showstime evolutions of v̄ 2 and Tg for the horizontal cell and the vertical cell under
microgravity. (Note that we subtracted the centre of mass velocity, v̄, in calculating
Tg throughout this paper.) When Tg is larger than v̄ 2 , one can judge that the eﬀect ofg-jitter is negligible. As shown in ﬁgure 11, this condition holds until t < 0.5 sec forthe horizontal cell, and until t < 0.8 sec for the vertical cell. These facts validate ourestimation of δg = 0.01 m sec−2 for cooling experiments given in § 5.

464

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