Free Cooling of a Granular Gas of Rodlike Particles in Microgravity

Granular gases as dilute ensembles of particles in random motion are at the basis of elementary structure-forming processes in the Universe, involved in many industrial and natural phenomena, and also excellent models to study fundamental statistical dynamics. The essential difference to molecular gases is the energy dissipation in particle collisions. Its most striking manifestation is the so-called granular cooling, the gradual loss of mechanical energy $E(t)$ in the absence of external excitation. We report an experimental study of homogeneous cooling of three-dimensional granular gases in microgravity. The asymptotic scaling $E(t)\propto t^{-2}$ obtained by Haff’s minimal model [J. Fluid Mech. 134, 401 (1983)] proves to be robust, despite the violation of several of its central assumptions. The shape anisotropy of the grains influences the characteristic time of energy loss quantitatively but not qualitatively. We compare kinetic energies in the individual degrees of freedom and find a slight predominance of translational motions. In addition, we observe a preferred rod alignment in the flight direction, as known from active matter or animal flocks.

Figure 1

(a) Sketch of the experimental setup and definition of the coordinates. Two side walls can be vibrated mechanically; the top and front walls are transparent. (b),(c) Typical frames of the top and front videos. Colored particles are tracked, and black rods provide thermal background. (d),(e) Overlay of eight frames of the top view, at the beginning of cooling [(d) homogeneous state] and at the end of the experiment [(e) slight clustering].

Figure 2

(a) Decay of the kinetic energy per particle in the individual degrees of freedom: After $\approx 2.5\mathrm{s}$, the energy partition is steady within statistical fluctuations (see Supplemental Material [46]), and the decay proceeds as $t^{-2}$ as predicted by Eq. (1) (see logarithmic plot in the inset). (b) Energies in the different types of d.o.f., all translational d.o.f. equilibrate within statistical fluctuations, rotations about the short rod axes are systematically less excited by about 10%–20%, and the rotations about the long axis were not considered here.

Figure 3

The cooling characteristics matches Haff’s model. (a) Decay of the average kinetic energy per d.o.f. and particle: After $\approx 1.25\mathrm{s}$, the curve is in good agreement with Haff’s model [41], Eq. (1) (logarithmic fit). (b) Comparison of the cooling rate and the measured mean velocities. The slope is $\xi (1-{ϵ}^2)/\lambda$. Solid symbols represent the data included in the fits; open symbols correspond to the initial 1.25 s when Haff’s equation is not yet applicable.

Figure 4

Cumulated number of collisions per particle determined from the rate of collision of traced (colored) particles in the videos. Only the dark data points were fitted.

Figure 5

Angle between the rod symmetry axis and $\overrightarrow{v}$ during the initial $0.5\mathrm{s}$ of cooling and during the last 0.5 s of the experiment. The pictures below illustrate the distribution $p(\theta )$ by bright dots projected on a unit sphere, looking in the flight direction, left: isotropically distributed rod axes, middle: experiment at $t\approx 6\mathrm{s}$, and right: best possible alignment.