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We investigate the linear axisymmetric viscous overstability in denseplanetary rings with typical values of the dynamical optical depth $\tau\gtrsim0.5$. We develop a granular flow model which accounts for the particulatenature of a planetary ring subjected to dissipative particle collisions. Themodel captures the dynamical evolution of the disc's vertical thickness,temperature, and effects related to a finite volume filling factor of the ringfluid. We compute equilibrium states of self-gravitating andnon-self-gravitating rings, which compare well with existing results fromkinetic models and N-Body simulations. Subsequently, we conduct a linearstability analysis of our model. We briefly discuss the different lineareigenmodes of the system and compare with existing literature by applyingcorresponding limiting approximations. We then focus on the viscousoverstability, analysing the effect of temperature variations, radial andvertical self-gravity, and for the first time the effects of vertical motionson the instability. In addition, we perform local N-body simulationsincorporating radial and vertical self-gravity. Critical values for the opticaldepth and the filling factor for the onset of instability resulting from ourN-body simulations compare well with our model predictions under the neglect ofradial self-gravity. When radial self-gravity is included, agreement withN-body simulations can be achieved by adopting enhanced values of the bulkviscous stress.

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