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A linear stability analysis of the hydrodynamic equations with respect to thehomogeneous cooling state is carried out to identify the conditions forstability of a granular gas of rough hard spheres. The description is based onthe results for the transport coefficients derived from the Boltzmann equationfor inelastic rough hard spheres [Phys. Rev. E 90, 022205 (2014)], which takeinto account the complete nonlinear dependence of the transport coefficientsand the cooling rate on the coefficients of normal and tangential restitution.As expected, linear stability analysis shows that a doubly degeneratetransversal (shear) mode and a longitudinal ("heat") mode are unstable withrespect to long enough wavelength excitations. The instability is driven by theshear mode above a certain inelasticity threshold; at larger inelasticity,however, the instability is driven by the heat mode for aninelasticity-dependent range of medium roughness. Comparison with the case of agranular gas of inelastic smooth spheres confirms previous simulation resultsabout the dual role played by surface friction: while small and large levels ofroughness make the system less unstable than the frictionless system, theopposite happens at medium roughness. On the other hand, such an intermediatewindow of roughness values shrinks as inelasticity increases and eventuallydisappears at a certain value, beyond which the rough-sphere gas is always lessunstable than the smooth-sphere gas. A comparison with some preliminarysimulation results shows a very good agreement for conditions of practicalinterest.

Impact of roughness on the instability of a free-cooling granular gas