Viscous overstability in dense planetary rings -- Effect of vertical motions and dense packing

We investigate the linear axisymmetric viscous overstability in dense planetary rings with typical values of the dynamical optical depth $\tau\gtrsim 0.5$. We develop a granular flow model which accounts for the particulate nature of a planetary ring subjected to dissipative particle collisions. The model captures the dynamical evolution of the disc's vertical thickness, temperature, and effects related to a finite volume filling factor of the ring fluid. We compute equilibrium states of self-gravitating and non-self-gravitating rings, which compare well with existing results from kinetic models and N-Body simulations. Subsequently, we conduct a linear stability analysis of our model. We briefly discuss the different linear eigenmodes of the system and compare with existing literature by applying corresponding limiting approximations. We then focus on the viscous overstability, analysing the effect of temperature variations, radial and vertical self-gravity, and for the first time the effects of vertical motions on the instability. In addition, we perform local N-body simulations incorporating radial and vertical self-gravity. Critical values for the optical depth and the filling factor for the onset of instability resulting from our N-body simulations compare well with our model predictions under the neglect of radial self-gravity. When radial self-gravity is included, agreement with N-body simulations can be achieved by adopting enhanced values of the bulk viscous stress.