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Abstract

In this paper we show how, under certain restrictions, the hydrodynamic equations for the freely evolving granular fluid fit within the framework of the time dependent Landau-Ginzburg (LG) models for critical and unstable fluids (e.g. spinodal decomposition). The granular fluid, which is usually modeled as a fluid of inelastic hard spheres (IHS), exhibits two instabilities: the spontaneous formation of vortices and of high density clusters. We suppress the clustering instability by imposing constraints on the system sizes, in order to illustrate how LG-equations can be derived for the order parameter, being the rate of deformation or shear rate tensor, which controls the formation of vortex patterns. From the shape of the energy functional we obtain the stationary patterns in the flow field.Quantitative predictions of this theory for the stationary states agree well with molecular dynamics simulations of a fluid of inelastic hard disks.

Hydrodynamic instabilities in shear flows of dry cohesive granular particles

“…(7), and the homogeneous granular temperature, Eq. (20), where the dimensionless critical shear rate for the neutral stability is found to be…”

Pattern formation in the inhomogeneous cooling state of granular fluids

“…Wakou et al [12] have shown that fluctuations in these quantities obey TDGL-like equations of phase ordering dynamics with a nonconserved order parameter [10]. Typically, these equations are obtained as the overdamped limit of a Hamiltonian formulation.…”

Shear state of freely evolving granular gases

“…The existence of this free shear state has been suggested previously in the framework of a time-dependent Ginzburg-Landau model for granular gases [10], and also as the result of a nonlinear analysis of the shearing instability [16]. The theory presented in this paper differs from those studies both in the method and the results, where rather strong discrepancies occur as it will be discussed along the paper.…”

“…The latter happens to be smaller than the critical size for the instability of one of the longitudinal modes associated with the density [9,10]. Consequently, the spatial inhomogeneities of the hydrodynamic fields are expected to be small.…”