Free cooling and inelastic collapse of granular gases in high dimensions

The connection between granular gases and sticky gases has recently been considered, leading to the conjecture that inelastic collapse is avoided for space dimensions higher than 4. We report Molecular Dynamics simulations of hard inelastic spheres in dimensions 4, 5 and 6. The evolution of the granular medium is monitored throughout the cooling process. The behaviour is found to be very similar to that of a two-dimensional system, with a shearing-like instability of the velocity field and inelastic collapse when collisions are inelastic enough, showing that the connection with sticky gases needs to be revised. PACS. 47.50.+d Non-Newtonian fluid flows – 05.20.Dd Kinetic theory – 51.10.+y Kinetic and transport theory of gases An important difference between a molecular fluid and a gas of mesoscopic or macroscopic grains is the possibility for the latter, associated with the inelastic nature of the collisions, to exhibit clustering and collapse [1–3]. A large and rapidly growing body of theoretical work is devoted to clustering, which consists in a long-wavelength low-frequency hydrodynamic phenomenon, and refers to the formation of density inhomogeneities. On the other hand, the phenomenon of inelastic collapse, which is a short-wavelength and high-frequency singularity inherent to the inelastic hard sphere (IHS) model, seems much less understood, except in one dimension [4]. In the IHS model, grains are modeled as smooth hard spheres undergoing binary, inelastic and momentumconserving collisions, which dissipate a constant fraction (1 − r) of the component of the relative velocity v12 along the center-to-center direction σ̂. Noting with primes the post-collision velocities, this translates into v 12 · σ̂ = −r v12 · σ̂, while the tangential relative velocity (perpendicular to σ̂) is conserved. In an interesting article, Ben-Naim et al. proposed that a freely evolving inelastic gas belongs asymptotically to the universality class of the sticky gas [5], for which v ′ 12 = 0 after each collision. Noticing that the temperature T of an inelastic gas is a monotonically increasing function of the restitution coefficient r and therefore bounded from below by the totally inelastic case (r = 0), these authors invoked a mapping onto Burgers’ equation to conjecture that the inelastic collapse is avoided for space dimensions d > dc = 4, and that the standard Haff’s cooling law a e-mail: [email protected] b e-mail: [email protected] c Unité Mixte de Recherche UMR 8627 du CNRS. T ∝ (ǫt), where ǫ = (1 − r)/(2d), holds indefinitely above this critical dimension dc. Velocity fluctuations and scaling exponents in one dimension can indeed be described by the inviscid Burgers equation [5]. However, in higher dimensions, the completely inelastic version of the IHS model, with r = 0, does not strictly correspond to the sticky gas limit because the tangential relative velocity is not dissipated in a binary encounter (v 12 · σ̂ = 0, but a priori v 12 = 0), so that it is interesting to test the validity of the above-mentioned predictions. In this article, we report Molecular Dynamics (MD) simulations of IHS gases for d = 4, 5 and 6. For each space dimension, the relevant parameters φ (packing fraction) and r (normal restitution) are varied for systems consisting typically of N = 5·10 to 5 · 10 particles. The code was first successfully tested by comparing for various densities the MD equation of state (or, equivalently, the pair correlation function at contact) with the analytical approximation of Song et al. [6]. For all investigated dimensions and for high enough dissipation (r ≤ 0.2 for d = 5 and r ≤ 0.1 for d = 6), the system exhibits the finite-time singularity characteristic of the inelastic collapse, in contradistinction to the conjecture of [5], with a situation closely reminiscent to its two-dimensional counterpart [2]: the (hyper)spheres collide infinitely often in a finite time along their joint line of centers. Following references [2] we probed this multiparticle process occurring through the accumulation of an infinite sequence of binary collisions by a contact criterion: after each collision, the relative distance d between the next two colliding partners is monitored; if this interparticle spacing normalized by the diameter σ has decreased and becomes of the order of machine precision, a three body interaction has occurred, corresponding to an inelastic collapse. The results of a typical run (d = 5) are shown 292 The European Physical Journal E 0 100 200 300 400 500 Number of collisions 10 10 10 10 -6 d* /σ 0 100 10 10 10 Fig. 1. Normalized inter-particle separation (see text for definition) as a function of the number of collisions since an arbitrary time origin, for d = 5, N = 16807, φ = 0.08 and r = 0.1. Each circle corresponds to a collision and the data were obtained specifying a “quadruple precision” computation (with reals coded on 16 bytes). Each decrease (from 10 to 10) corresponds to repeated collisions between a small number of particles, typically three particles, one bouncing back and forth between two others, with a diverging frequency. For the same system, the inset shows with diamonds the results of a standard “double precision” run (reals on 8 bytes). In both cases, the floor of machine precision is indicated by a dashed line (approximately 10 for real⋆16 and 10 for real⋆8). Only those collisions with d∗ < 10σ have been displayed. in Figure 1. When a multi-body interaction commences, d decreases geometrically with the number of collisions, as in two dimensions [2]. Enforcing a high-precision computation allows to follow the decay over more than 26 orders of magnitude (whereas only 8 orders are accessible with a standard double precision algorithm, see the inset of Fig. 1). After a collapse has occurred, the inaccuracy of the computer disperses the collapsing cluster, before another multi-body event involving different particles occurs at a different location. Our analysis indicates that on a hypothetical infinite precision machine, the collapse would continue forever whereas roundoff errors act as an effective regularization. Throughout a collapse, the time between two successive collisions follows a geometrical decrease very similar to that displayed in Figure 1. Let us note that the seemingly low value of the packing fraction φ in Figure 1 is a misleading effect of “high” dimensionality. It turns that the reduced density n = nσ = d 2 πd/2 Γ (