Mesoscopic Theory of Granular Fluids

Abstract

Using fluctuating hydrodynamics we describe the slow buildup of long range spatial correlations in a freely evolving fluid of inelastic hard spheres. In the incompressible limit, the behavior of spatial velocity correlations (including r 2d behavior) is governed by vorticity fluctuations only and agrees well with two-dimensional simulations up to 50 to 100 collisions per particle. The incompressibility assumption breaks down beyond a distance that diverges in the elastic limit. [S0031-9007(97) In the characterization of granular matter as an unusual solid, fluid, or gas by Jaeger et al. [1], this Letter addresses the granular gas regime, controlled by inelasticity, clustering [2], and collapse [3]. Clustering is a long wavelength, low frequency (hydrodynamic) phenomenon and inelastic collapse a short wavelength, high frequency (kinetic) phenomenon. In the granular gas regime, also called rapid granular flows, the dynamics is dominated by inelastic collisions. Here the methods of nonequilibrium statistical mechanics, molecular dynamics, kinetic theory, and hydrodynamics are most suitable for describing the observed average macroscopic behavior [2][3][4][5][6][7] and the fluctuations around it.The lack of energy conservation makes the granular gas, whether driven or freely evolving, behave very differently from molecular fluids. The essential physical processes and detailed dynamics are described in [2,3] and references therein: the similarities and differences with molecular fluids; lack of separation of microscales and macroscales, not only because the grains themselves are macroscopic, but also because of the existence of intermediate intrinsic scales which are controlled by the inelasticity and are only well separated when the system is nearly elastic. A simple model which incorporates the inelasticity of the granular collisions consists of inelastic hard spheres (IHS), taken here of unit mass and diameter, with momentum conserving dynamics. The energy loss in a collision is proportional to the inelasticity parameter e 1 2 a 2 where a is the coefficient of normal restitution.For an understanding of what follows, we recall two important properties of the undriven granular gas: (i) the existence of a homogeneous cooling state (HCS) and (ii) its instability against spatial fluctuations. The hydrodynamic equations for an IHS fluid, started in a uniform equilibrium state with temperature T 0 , admit an HCS solution (see, e.g., [2,3,7]) with a homogeneous temperature T ͑t͒, described by ≠ t T 22g 0 vT. Here the collision frequency is v͑T͒ ϳ p T͞l 0 with a mean free path l 0 ,given by the Enskog theory [8] for a dense system of hard disks or spheres (d 2, 3) and g 0 e͞2d. Then T ͑t͒ T 0 ͓͞1 1 g 0 v͑T 0 ͒t͔ 2 T 0 exp͑22g 0 t͒, where t is the average number of collisions suffered per particle within a time t. It is found by integrating dt v͑T ͑t͒͒dt. Moreover, this HCS solution is linearly unstable once the linear extent L of the system exceeds some dynamic correlation length, which increases with decreasing e, and is ...

Role of Molecular Chaos in Granular Fluctuating Hydrodynamics

“…The HCS is unstable against spatial fluctuations: this instability appears at scales larger than a critical length L c which depends on α and on the mean free path, therefore it can be avoided by taking the linear size of the system L < L c [19]. It is possible to analyze the effects of spatial fluctuations by deriving mesoscopic equations through a linearization around the HCS [35]. The resulting equations are generally coupled, but in the Fourier representation the transverse velocity field results decoupled from the other modes.…”

“…where ν(t) is the kinematic viscosity which, in the HCS, is proportional to T g (see [4] for definitions), v th (t) ≡ 2T g /m and c is expected to be 1 [21,35,3]. The last term in the Eq.…”

“…This question is usually more complicate than the analogous problem at equilibrium, because less symmetries are available, the most important being the invariance under time-reversal. Models of granular gases [30,29] lend themselves to be analyzed under this aspect: they are usually far from equilibrium and exhibit peculiar properties, such as: breakdown of energy equipartition [16,25,28], spontaneous symmetry breaking (vortices and clustering) [35,18], "Maxwell demon" effects [15], "ratcheting" effects [12], and lot more. At the same time granular experiments usually concern small samples, or the order of 10 3 ± 10 4 grains, and therefore extensive observables are affected by large (easily detectable) fluctuations [18].…”

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“…In general, the collision frequency ω(T ) is proportional to the root mean square velocity v 0 = 2T /m, and its explicit form for hard sphere fluids can be found in Refs. [24,25]. When the system is prepared initially in a homogeneous equilibrium state, it evolves at short times in a spatially homogeneous cooling state (HCS) with a time dependent temperature.…”

“…The search for the proper macroscopic description of unstable granular fluids has been pursued by many authors [1][2][3][4][5][6][7][8][9][10][11][12][13]18,20,[22][23][24][25][26][27][28][29][30][31][32]. Recently, two new points of view have been presented, namely by Ben-Naim et al [11] and by Soto et al [12].…”