EPJ manuscript No. (will be inserted by the editor) Consequences of using different pair-correlation functions on the stability properties of the Homogeneous Cooling State for a monodisperse system of near-elastic disks Sebasti´an Gonz´alez and Stefan Luding Multi Scale Mechanics, TS, CTW, UTwente, P.O.Box 217, 7500 AE Enschede, Netherlands, e-mail: j.s.l.gonzalezbriones@utwente.nl, s.luding@utwente.nl Abstract. We show the differences in the stability properties of the Homogeneous Cooling State (HCS) of a two-dimensional monodisperse collection of rigid and near-elastic disks, obtained by using different formulae for the pair-correlation function. For an equation of state that takes into account the crystallization and ordering of the particles (and the respective pressure drop), the critical wavelength of the heat conduction mode is considerably modified in the transition zone, involving a bifurcation and an additional mode of instability. The theoretical predictions, using the improved equation of state, are confirmed by numerical simulations. Nevertheless, some open questions remain. 1 Introduction Structure formation in a granular gas has attracted much attention during the last decades (see for example, Refs. [1–10]). Starting from a macroscopically homogeneous system, structures evolve and a dilute granular gas coexists with denser, possibly much denser and even solid, clusters – in non-equilibrium. However, the coexistence of a fluid-like granular gas with a solid- like packing also occurs in many other systems, like during avalanche flow on inclined planes or in vibrated containers, see Refs. [11,12] and references therein. In the absence of walls and external forces, the crucial phenomena in a freely cooling gran- ular gas involve the fluctuations in density, velocity and temperature, which cause position- dependent energy loss [2]. In denser areas, due to strong local dissipation, pressure and energy decay rapidly and material moves from ‘hot’ to ‘cold’ regions, there leading to even stronger dissipation and thus causing the density instability with ever growing (dense) clusters. The freely cooling granular gas will be introduced first, as an example of a case where dilute and dense granular media co-exist. Even though the need to treat walls is avoided by using periodic boundary conditions, and the initial state is macroscopically homogeneous, resembling a classical, elastic hard sphere gas, the system develops interesting dynamics and structure formation – only due to the dissipative interactions of the particles, see Fig. 1 in section 2. Hydrodynamic equations and constitutive relations are introduced in section 3 and a stability analysis of these equations is presented in section 4. This has been done several times in the literature (see for example Refs. [2,3,13,14]), but always for a given (Enskog) pair correlation function (at contact distance, g(ν ; r = d)). Our contribution is to study the consequences of an empirical pair correlation function on the instabilities of the HCS. Section 5 provides concluding comments.