A universal scaling law for the evolution of granular gases

Abstract

Dry, freely evolving granular materials in a dilute gaseous state coalesce into dense clusters only due to dissipative interactions. This clustering transition is important for a number of problems ranging from geophysics to cosmology. Here we show that the evolution of a dilute, freely cooling granular gas is determined in a universal way by the ratio of inertial flow and thermal velocities, that is, the Mach number. Theoretical calculations and direct numerical simulations of the granular Navier-Stokes equations show that irrespective of the coefficient of restitution, density or initial velocity distribution, the density fluctuations follow a universal quadratic dependence on the system's Mach number. We find that the clustering exhibits a scale-free dynamics but the clustered state becomes observable when the Mach number is approximately of O(1). Our results provide a method to determine the age of a granular gas and predict the macroscopic appearance of clusters.Geophysical processes [1], the solar corona [2], the asteroid belt between Mars and Jupiter, planetary rings [3,4], protoplanetary disks [5], and the formation of cosmological structures [6] are systems where granular clustering processes are at play. Even a small degree of dissipation in the kinetics of granular particles produces spatial correlations and structures in a dilute, homogeneous gas [7]. Hydrodynamic treatments suggest that a shear instability initiates this transition [7]. However, what exactly initiates this process is not known. The equations of granular hydrodynamics [8] predict a linear instability of the transverse mode [9] when the wavevectorand ε is the coefficient of restitution. This instability leads to the formation of vortices. In regions where the particle velocities are correlated the temperature drops, which, in turn, creates a region of low pressure. These are the seeds for a second instability if the system size is larger than k * −1 [13,15,17].Experimental studies [18][19][20][21][22] of granular gases are rather scarce because of formidable challenges in preparing a system free of external forces. Microgravity experiments on parabolic flights or drop towers potentially offer good conditions, but it is very difficult to remove the influence of confining potentials or surrounding walls on the granular system. The necessity of studying large system sizes and of characterizing fluctuations in regions of sharp gradients in temperature and density, developing into supersonic flow, without ambiguities motivates us to tackle the continuous hydrodynamic equations beyond perturbative schemes. Hydrodynamic fields can be rigorously defined in a manner similar to molecular fluids by means of a coarse-graining of the microscopic kinetic equations. First, one considers the one-particle distribution function f ( r, w, t), which obeys the Boltzmann equation and represents the number of particles within a volume d r centered at r and with velocity w within the interval d w. Then, the transport equations for inelastic systems are deri...

Electrification in granular gases leads to constrained fractal growth

“…Secondly, granular gases are characterized by the emergence of an advective flow 27,36 which we find, in the present case, induces the non-monotonicity in the temporal evolution of the number density [Fig. 3(c)].…”

Aggregation and pattern formation in charged granular gases

“…Secondly, granular gases are characterized by the emergence of a convective flow (Hummel et al , 2016;Brilliantov & Pöschel, 2010) which we find (see Methods and Excluding the two above mechanisms explains the slight deviation of our quasimonodisperse Boltzmann theory from the non-monotonic behavior of B(t) after the crossover to aggregative collapse.…”

Velocity Distribution of a Homogeneously Cooling Granular Gas

“…For the homogeneous cooling state, the validity of Eq. (1) has been shown both in numerical simulations [31,[71][72][73] and experiments [30,60,62]. For the exponents α characterizing the tail of the velocity distribution values in the range 0.6 to 1.5 have been reported for boundarydriven, two-dimensional systems [49].…”

“…However, these differences to molecular gases also signify the chance to "reinvent statistical mechanics in a new context" [2]. Properties of granular gases not known from their molecular counterpart include: non-Fourier heat flow [3][4][5], correlations [6][7][8], breaking of time-reversal symmetry [9,10], segregation [11][12][13][14][15][16][17][18], non-equipartition [6,[18][19][20][21][22], and clustering [23][24][25][26][27][28][29][30][31][32]. Sometimes granular gases unexpectedly behave even like equilibrium systems [33].…”

“…P (c) ∼ exp(−kc). The HCS ends with the onset of particle clustering which destroys its homogeneity [31].…”